参考论文

1. Dynamics of Bloch oscillations

Ref	Project_BlockOScillation
摘要	We study the dynamics of Bloch oscillations in a one-dimensional periodic potential plus a (relatively weak) static force. The tight-binding and single-band approximations are analysed in detail, and also in a classicalized version. A number of numerically exact results obtained from wavepacket propagation are analysed and interpreted in terms of the tight-binding and single-band model, both in co-ordinate and momentum space.

Ref

摘要

We study the dynamics of Bloch oscillations in a one-dimensional periodic potential plus a (relatively weak) static force. The tight-binding and single-band approximations are analysed in detail, and also in a classicalized version. A number of numerically exact results obtained from wavepacket propagation are analysed and interpreted in terms of the tight-binding and single-band model, both in co-ordinate and momentum space.

2. Classical and quantum regimes of the inhomogeneous Dicke model and its Ehrenfest time

Ref	Project_Dicke_Dynamics
摘要	We show that in the few-excitation regime, the classical and quantum time evolution of the inhomogeneous Dicke model for N two-level systems coupled to a single boson mode agree for \(N \gg1\). In the presence of a single excitation only, the leading term in an \(1/N\) expansion of the classical equations of motion reproduces the result of the Schrödinger equation. For a small number of excitations, the numerical solutions of the classical and quantum problems become equal for N sufficiently large. By solving the Schrödinger equation exactly for two excitations and a particular inhomogeneity, we obtain 1/N corrections which lead to a significant difference between the classical and quantum solutions at a new time scale which we identify as an Ehrenferst time, given by \(\tau_E = \sqrt{N/\langle g^2 \rangle}\), where \(\langle g^2 \rangle\) is an effective coupling strength between the two-level systems and the boson.

Ref

Project_Dicke_Dynamics

摘要

We show that in the few-excitation regime, the classical and quantum time evolution of the inhomogeneous Dicke model for N two-level systems coupled to a single boson mode agree for $N \gg1$. In the presence of a single excitation only, the leading term in an $1/N$ expansion of the classical equations of motion reproduces the result of the Schrödinger equation. For a small number of excitations, the numerical solutions of the classical and quantum problems become equal for N sufficiently large. By solving the Schrödinger equation exactly for two excitations and a particular inhomogeneity, we obtain 1/N corrections which lead to a significant difference between the classical and quantum solutions at a new time scale which we identify as an Ehrenferst time, given by $\tau_E = \sqrt{N/\langle g^2 \rangle}$, where $\langle g^2 \rangle$ is an effective coupling strength between the two-level systems and the boson.

3. Entropy and Area

Ref	Project_Entropy_Area
摘要	The ground-state density matrix for a massless free field is traced over the degrees of freedom residing inside an imaginary sphere; the resulting entropy is shown to be proportional to the area (and not the volume) of the sphere. Possible connections with the physics of black holes are discussed.

4. Ground-state cooling of a nanomechanical oscillator with N spins

Ref	Project_GS_Cooling_Nspins
摘要	It is typical of modern quantum technologies employing nanomechanical oscillators to demand few mechanical quantum excitations, for instance, to prolong coherence times of a particular task or to engineer a specific nonclassical state. For this reason, we devote the present work to exhibiting how to bring an initially thermalized nanomechanical oscillator to near its ground state. Particularly, we focus on extending the novel results of D. D. B. Rao et al. [Phys. Rev. Lett. 117, 077203 (2016)], where a mechanical object can be heated up, squeezed, or cooled down to near its ground state through conditioned single-spin measurements. In our work, we study a similar iterative spin-mechanical system when N spins interact with the mechanical oscillator. Here, we have also found that the postselection procedure acts as a discarding process; i.e., we steer the mechanics to the ground state by dynamically filtering its vibrational modes. We show that when considering symmetric collective spin postselection, the inclusion of N spins in the quantum dynamics is highly beneficial—in particular, decreasing the total number of iterations to achieve the ground state, with a success rate of probability comparable with the one obtained from the single-spin case.

Ref

Project_GS_Cooling_Nspins

摘要

It is typical of modern quantum technologies employing nanomechanical oscillators to demand few mechanical quantum excitations, for instance, to prolong coherence times of a particular task or to engineer a specific nonclassical state. For this reason, we devote the present work to exhibiting how to bring an initially thermalized nanomechanical oscillator to near its ground state. Particularly, we focus on extending the novel results of D. D. B. Rao et al. [Phys. Rev. Lett. 117, 077203 (2016)], where a mechanical object can be heated up, squeezed, or cooled down to near its ground state through conditioned single-spin measurements. In our work, we study a similar iterative spin-mechanical system when N spins interact with the mechanical oscillator. Here, we have also found that the postselection procedure acts as a discarding process; i.e., we steer the mechanics to the ground state by dynamically filtering its vibrational modes. We show that when considering symmetric collective spin postselection, the inclusion of N spins in the quantum dynamics is highly beneficial—in particular, decreasing the total number of iterations to achieve the ground state, with a success rate of probability comparable with the one obtained from the single-spin case.

5. Zurek-Kibble Causality Bounds in Time-Dependent Ginzburg-Landau Theory and Quantum Field Theory

Ref	Project_Kibble_Zurek_Dynamics
摘要	Zurek's and Kibble's causal constraints for defect production at continuous transitions are encoded in the field equations that condensed matter systems and quantum fields satisfy. In this article we highlight some of the properties of the solutions to the equations and show to what extent they support the original ideas.

6. Nonequilibrum dynamics in the strongly excited inhomogeneous Dicke model

Ref	Project_Noneq_Dickemodel
摘要	Using the exact eigenstates of the inhomogeneous Dicke model obtained by numerically solving the Bethe equations, we study the decay of bosonic excitations due to the coupling of the mode to an ensemble of two-level (spin 1/2) systems. We compare the quantum time evolution of the bosonic mode population with the mean-field description which, for a few bosons, agree up to a relatively long Ehrenfest time. We demonstrate that additional excitations lead to a dramatic shortening of the period of validity of the mean-field analysis. However, even in the limit where the number of bosons equal the number of spins, the initial instability remains adequately described by the mean-field approach leading to a finite, albeit short, Ehrenfest time. Through finite size analysis we also present indications that the mean-field approach could still provide an adequate description for thermodynamically large systems even at long times. However, for mesoscopic systems one cannot expect it to capture the behavior beyond the initial decay stage in the limit of an extremely large number of excitations.

Ref

Project_Noneq_Dickemodel

摘要

Using the exact eigenstates of the inhomogeneous Dicke model obtained by numerically solving the Bethe equations, we study the decay of bosonic excitations due to the coupling of the mode to an ensemble of two-level (spin 1/2) systems. We compare the quantum time evolution of the bosonic mode population with the mean-field description which, for a few bosons, agree up to a relatively long Ehrenfest time. We demonstrate that additional excitations lead to a dramatic shortening of the period of validity of the mean-field analysis. However, even in the limit where the number of bosons equal the number of spins, the initial instability remains adequately described by the mean-field approach leading to a finite, albeit short, Ehrenfest time. Through finite size analysis we also present indications that the mean-field approach could still provide an adequate description for thermodynamically large systems even at long times. However, for mesoscopic systems one cannot expect it to capture the behavior beyond the initial decay stage in the limit of an extremely large number of excitations.

7. Exact Spatiotemporal Dynamics of Confined Lattice Random Walks in Arbitrary Dimensions: A Century after Smoluchowski and Pólya

Ref	Project_randomwalk
摘要	A lattice random walk is a mathematical representation of movement through random steps on a lattice at discrete times. It is commonly referred to as Pólya’s walk when the steps occur in either of the nearestneighbor sites. Since Smoluchowski’s 1906 derivation of the spatiotemporal dependence of the walk occupation probability in an unbounded one-dimensional lattice, discrete random walks and their continuous counterpart, Brownian walks, have developed over the course of a century into a vast and versatile area of knowledge. Lattice random walks are now routinely employed to study stochastic processes across scales, dimensions, and disciplines, from the one-dimensional search of proteins along a DNA strand and the two-dimensional roaming of bacteria in a petri dish, to the three-dimensional motion of macromolecules inside cells and the spatial coverage of multiple robots in a disaster area. In these realistic scenarios, when the randomly moving object is constrained to remain within a finite domain, confined lattice random walks represent a powerful modeling tool. Somewhat surprisingly, and differently from Brownian walks, the spatiotemporal dependence of the confined lattice walk probability has been accessible mainly via computational techniques, and finding its analytic description has remained an open problem. Making use of a set of analytic combinatorics identities with Chebyshev polynomials, I develop a hierarchical dimensionality reduction to find the exact space and time dependence of the occupation probability for confined Pólya’s walks in arbitrary dimensions with reflective, periodic, absorbing, and mixed (reflective and absorbing) boundary conditions along each direction. The probability expressions allow one to construct the time dependence of derived quantities, explicitly in one dimension and via an integration in higher dimensions, such as the first-passage probability to a single target, return probability, average number of distinct sites visited, and absorption probability with imperfect traps. Exact mean firstpassage time formulas to a single target in arbitrary dimensions are also presented. These formulas allow one to extend the so-called discrete pseudo-Green function formalism, employed to determine analytically mean first-passage time, with reflecting and periodic boundaries, and a wealth of other related quantities, to arbitrary dimensions. For multiple targets, I introduce a procedure to construct the time dependence of the first-passage probability to one of many targets. Reduction of the occupation probability expressions to the continuous time limit, the so-called continuous time random walk, and to the space-time continuous limit is also presented.

Ref

Project_randomwalk

摘要

A lattice random walk is a mathematical representation of movement through random steps on a lattice at discrete times. It is commonly referred to as Pólya’s walk when the steps occur in either of the nearestneighbor sites. Since Smoluchowski’s 1906 derivation of the spatiotemporal dependence of the walk occupation probability in an unbounded one-dimensional lattice, discrete random walks and their continuous counterpart, Brownian walks, have developed over the course of a century into a vast and versatile area of knowledge. Lattice random walks are now routinely employed to study stochastic processes across scales, dimensions, and disciplines, from the one-dimensional search of proteins along a DNA strand and the two-dimensional roaming of bacteria in a petri dish, to the three-dimensional motion of macromolecules inside cells and the spatial coverage of multiple robots in a disaster area. In these realistic scenarios, when the randomly moving object is constrained to remain within a finite domain, confined lattice random walks represent a powerful modeling tool. Somewhat surprisingly, and differently from Brownian walks, the spatiotemporal dependence of the confined lattice walk probability has been accessible mainly via computational techniques, and finding its analytic description has remained an open problem. Making use of a set of analytic combinatorics identities with Chebyshev polynomials, I develop a hierarchical dimensionality reduction to find the exact space and time dependence of the occupation probability for confined Pólya’s walks in arbitrary dimensions with reflective, periodic, absorbing, and mixed (reflective and absorbing) boundary conditions along each direction. The probability expressions allow one to construct the time dependence of derived quantities, explicitly in one dimension and via an integration in higher dimensions, such as the first-passage probability to a single target, return probability, average number of distinct sites visited, and absorption probability with imperfect traps. Exact mean firstpassage time formulas to a single target in arbitrary dimensions are also presented. These formulas allow one to extend the so-called discrete pseudo-Green function formalism, employed to determine analytically mean first-passage time, with reflecting and periodic boundaries, and a wealth of other related quantities, to arbitrary dimensions. For multiple targets, I introduce a procedure to construct the time dependence of the first-passage probability to one of many targets. Reduction of the occupation probability expressions to the continuous time limit, the so-called continuous time random walk, and to the space-time continuous limit is also presented.

8. Optimal Coordinate as a General Method in Stochastic Dynamics.

Ref	Project_Stochastic_Phys
摘要	A general method to describe stochastic dynamics of Markov processes is suggested. The method aims to solve three related problems. The determination of an optimal coordinate for the description of stochastic dynamics. The reconstruction of time from an ensemble of stochastic trajectories. The decomposition of stationary stochastic dynamics on eigen-modes which do not decay exponentially with time. The problems are solved by introducing additive eigenvectors which are transformed by a stochastic matrix in a simple way - every component is translated on a constant distance. Such solutions have peculiar properties. For example, an optimal coordinate for stochastic dynamics with detailed balance is a multi-valued function. An optimal coordinate for a random walk on the line corresponds to the conventional eigenvector of the one dimensional Dirac equation. The equation for the optimal coordinate in a slow varying potential reduces to the Hamilton-Jacobi equation for the action function.

Ref

Project_Stochastic_Phys

摘要

A general method to describe stochastic dynamics of Markov processes is suggested. The method aims to solve three related problems. The determination of an optimal coordinate for the description of stochastic dynamics. The reconstruction of time from an ensemble of stochastic trajectories. The decomposition of stationary stochastic dynamics on eigen-modes which do not decay exponentially with time. The problems are solved by introducing additive eigenvectors which are transformed by a stochastic matrix in a simple way - every component is translated on a constant distance. Such solutions have peculiar properties. For example, an optimal coordinate for stochastic dynamics with detailed balance is a multi-valued function. An optimal coordinate for a random walk on the line corresponds to the conventional eigenvector of the one dimensional Dirac equation. The equation for the optimal coordinate in a slow varying potential reduces to the Hamilton-Jacobi equation for the action function.

9. Stable amplitude chimera states in a network of locally coupled Stuart-Landau oscillators

Ref	Project_Stuart_Landau
摘要	We investigate the occurrence of collective dynamical states such as transient amplitude chimera, stable amplitude chimera and imperfect breathing chimera states in a locally coupled network of Stuart-Landau oscillators. In an imperfect breathing chimera state, the synchronized group of oscillators exhibits oscillations with large amplitudes while the desynchronized group of oscillators oscillates with small amplitudes and this behavior of coexistence of synchronized and desynchronized oscillations fluctuates with time. Then we analyze the stability of the amplitude chimera states under various circumstances, including variations in system parameters and coupling strength, and perturbations in the initial states of the oscillators. For an increase in the value of the system parameter, namely the nonisochronicity parameter, the transient chimera state becomes a stable chimera state for a sufficiently large value of coupling strength. In addition, we also analyze the stability of these states by perturbing the initial states of the oscillators. We find that while a small perturbation allows one to perturb a large number of oscillators resulting in a stable amplitude chimera state, a large perturbation allows one to perturb a small number of oscillators to get a stable amplitude chimera state. We also find the stability of the transient and stable amplitude chimera states as well as traveling wave states for appropriate number of oscillators using Floquet theory. In addition, we also find the stability of the incoherent oscillation death states.

Ref

Project_Stuart_Landau

摘要

We investigate the occurrence of collective dynamical states such as transient amplitude chimera, stable amplitude chimera and imperfect breathing chimera states in a locally coupled network of Stuart-Landau oscillators. In an imperfect breathing chimera state, the synchronized group of oscillators exhibits oscillations with large amplitudes while the desynchronized group of oscillators oscillates with small amplitudes and this behavior of coexistence of synchronized and desynchronized oscillations fluctuates with time. Then we analyze the stability of the amplitude chimera states under various circumstances, including variations in system parameters and coupling strength, and perturbations in the initial states of the oscillators. For an increase in the value of the system parameter, namely the nonisochronicity parameter, the transient chimera state becomes a stable chimera state for a sufficiently large value of coupling strength. In addition, we also analyze the stability of these states by perturbing the initial states of the oscillators. We find that while a small perturbation allows one to perturb a large number of oscillators resulting in a stable amplitude chimera state, a large perturbation allows one to perturb a small number of oscillators to get a stable amplitude chimera state. We also find the stability of the transient and stable amplitude chimera states as well as traveling wave states for appropriate number of oscillators using Floquet theory. In addition, we also find the stability of the incoherent oscillation death states.

10. Current fluctuations in mesoscopic systems with Andreev scattering

Ref	Project_Andreev_Currents
摘要	Using a scattering theory approach we study the zero-frequency current fluctuations of the normal terminals of a phase-coherent mesoscopic structure with a superconducting region. We find that for devices where the potential of the superconducting region is externally fixed (Fig. 1), the expression for current fluctuations is a simple generalization of the corresponding expression obtained in Buttiker [Phys Rev. B 46, 12 485 (1992)] for purely normal mesoscopic systems. In contrast to purely normal mesoscopic systems, we find that the current fluctuations between two different contacts can be positive in these devices. We apply this formula to derive a simple expression for the shot noise in a normal superconducting (NS) junction and study the noise to current ratio both as a function of the applied bias and a potential barrier at the NS interface. For devices with a floating superconductor (Fig. 2), a self-consistent calculation of the current fluctuations is necessary, and here we derive an approximate formula valid in the small bias limit. We show that two similar devices with identical average currents can exhibit very different fluctuations depending on whether the superconductor is held at a fixed potential or is left floating.

Ref

Project_Andreev_Currents

摘要

Using a scattering theory approach we study the zero-frequency current fluctuations of the normal terminals of a phase-coherent mesoscopic structure with a superconducting region. We find that for devices where the potential of the superconducting region is externally fixed (Fig. 1), the expression for current fluctuations is a simple generalization of the corresponding expression obtained in Buttiker [Phys Rev. B 46, 12 485 (1992)] for purely normal mesoscopic systems. In contrast to purely normal mesoscopic systems, we find that the current fluctuations between two different contacts can be positive in these devices. We apply this formula to derive a simple expression for the shot noise in a normal superconducting (NS) junction and study the noise to current ratio both as a function of the applied bias and a potential barrier at the NS interface. For devices with a floating superconductor (Fig. 2), a self-consistent calculation of the current fluctuations is necessary, and here we derive an approximate formula valid in the small bias limit. We show that two similar devices with identical average currents can exhibit very different fluctuations depending on whether the superconductor is held at a fixed potential or is left floating.

11 Colloquium: Andreev reflection and Klein tunneling in graphene

Ref	Project_Beenakker
摘要	A colloquium-style introduction to two electronic processes in a carbon monolayer (graphene) is presented, each having an analog in relativistic quantum mechanics. Both processes couple electronlike and holelike states, through the action of either a superconducting pair potential or an electrostatic potential. The first process, Andreev reflection, is the electron-to-hole conversion at the interface with a superconductor. The second process, Klein tunneling, is the tunneling through a p−n junction. The absence of backscattering, characteristic of massless Dirac fermions, implies that both processes happen with unit efficiency at normal incidence. Away from normal incidence, retro-reflection in the first process corresponds to negative refraction in the second process. In the quantum Hall effect, both Andreev reflection and Klein tunneling induce the same dependence of the two-terminal conductance plateau on the valley isospin of the carriers. Existing and proposed experiments on Josephson junctions and bipolar junctions in graphene are discussed from a unified perspective.

Ref

Project_Beenakker

摘要

A colloquium-style introduction to two electronic processes in a carbon monolayer (graphene) is presented, each having an analog in relativistic quantum mechanics. Both processes couple electronlike and holelike states, through the action of either a superconducting pair potential or an electrostatic potential. The first process, Andreev reflection, is the electron-to-hole conversion at the interface with a superconductor. The second process, Klein tunneling, is the tunneling through a p−n junction. The absence of backscattering, characteristic of massless Dirac fermions, implies that both processes happen with unit efficiency at normal incidence. Away from normal incidence, retro-reflection in the first process corresponds to negative refraction in the second process. In the quantum Hall effect, both Andreev reflection and Klein tunneling induce the same dependence of the two-terminal conductance plateau on the valley isospin of the carriers. Existing and proposed experiments on Josephson junctions and bipolar junctions in graphene are discussed from a unified perspective.

12 Critical current of a quasi-one-dimensional superconducting wire

Ref	Project_Current_SC_wire
摘要	We solve the Bogoliubov-de Gennes equations self-consistently to obtain the critical current Ic versus Fermi energy μ for a ballistic quasi-one-dimensional superconducting wire. Instead of the `discretized' critical current I_c(μ) = N e ∆_bulk / hbar predicted for a superconducting point contact, we find I_c(μ) = [4 e ∆_wire(μ) / h] [n(μ)/n_1(μ)] for the superconducting wire. The normalized electron density n(μ)/n_1(μ)= sum_1^N (k_N/k_1) is a slowly increasing function of μ. Because we consider a uniform superconducting wire, rather than a superconducting point contact, the self-consistency condition for the superconducting order parameter ∆_wire(μ) must be solved for each value of the Fermi energy. The resulting order parameter ∆_wire(μ) follows the normal metal quasi-one-dimensional density of states N(μ) of the wire, as does the critical current I_c(μ).

Ref

Project_Current_SC_wire

摘要

We solve the Bogoliubov-de Gennes equations self-consistently to obtain the critical current Ic versus Fermi energy μ for a ballistic quasi-one-dimensional superconducting wire. Instead of the `discretized' critical current I_c(μ) = N e ∆_bulk / hbar predicted for a superconducting point contact, we find I_c(μ) = [4 e ∆_wire(μ) / h] [n(μ)/n_1(μ)] for the superconducting wire. The normalized electron density n(μ)/n_1(μ)= sum_1^N (k_N/k_1) is a slowly increasing function of μ. Because we consider a uniform superconducting wire, rather than a superconducting point contact, the self-consistency condition for the superconducting order parameter ∆_wire(μ) must be solved for each value of the Fermi energy. The resulting order parameter ∆_wire(μ) follows the normal metal quasi-one-dimensional density of states N(μ) of the wire, as does the critical current I_c(μ).

13 Specular Andreev Reflection in Graphene

Ref	Project_graphene_Andreev
摘要	By combining the Dirac equation of relativistic quantum mechanics with the Bogoliubov–de Gennes equation of superconductivity we investigate the electron-hole conversion at a normal-metal–superconductor interface in graphene. We find that the Andreev reflection of Dirac fermions has several unusual features: (1) the electron and hole occupy different valleys of the band structure; (2) at normal incidence the electron-hole conversion happens with unit efficiency in spite of the large mismatch in Fermi wavelengths at the two sides of the interface; and, most fundamentally: (3) away from normal incidence the reflection angle may be the same as the angle of incidence (retroreflection) or it may be inverted (specular reflection). Specular Andreev reflection dominates in weakly doped graphene, when the Fermi wavelength in the normal region is large compared to the superconducting coherence length.

Ref

Project_graphene_Andreev

摘要

By combining the Dirac equation of relativistic quantum mechanics with the Bogoliubov–de Gennes equation of superconductivity we investigate the electron-hole conversion at a normal-metal–superconductor interface in graphene. We find that the Andreev reflection of Dirac fermions has several unusual features: (1) the electron and hole occupy different valleys of the band structure; (2) at normal incidence the electron-hole conversion happens with unit efficiency in spite of the large mismatch in Fermi wavelengths at the two sides of the interface; and, most fundamentally: (3) away from normal incidence the reflection angle may be the same as the angle of incidence (retroreflection) or it may be inverted (specular reflection). Specular Andreev reflection dominates in weakly doped graphene, when the Fermi wavelength in the normal region is large compared to the superconducting coherence length.

14 Full-quantum simulation of hole transport and band-to-band tunneling in nanowires using the kp method

Ref	Project_hole_kp
摘要	We have developed a three-dimensional, self-consistent full-quantum transport simulator for nanowire field effect transistors based on the eight-band k⋅p method. We have constructed the mode-space Hamiltonian via a unitary transformation from the Hamiltonian discretized in the k-space, and reduced its size significantly by selecting only the modes that contribute to the transport. We have also devised an approximate but highly accurate method to solve the cross-sectional eigenvalue problems, thereby overcoming the numerical bottleneck of the mode-space approach. We have therefore been able to develop a highly efficient device simulator. We demonstrate the capability of our simulator by calculating the hole transport in a p-type Si nanowire field effect transistor and the band-to-band tunneling current in a InAs nanowire tunnel field effect transistor.

Ref

Project_hole_kp

摘要

We have developed a three-dimensional, self-consistent full-quantum transport simulator for nanowire field effect transistors based on the eight-band k⋅p method. We have constructed the mode-space Hamiltonian via a unitary transformation from the Hamiltonian discretized in the k-space, and reduced its size significantly by selecting only the modes that contribute to the transport. We have also devised an approximate but highly accurate method to solve the cross-sectional eigenvalue problems, thereby overcoming the numerical bottleneck of the mode-space approach. We have therefore been able to develop a highly efficient device simulator. We demonstrate the capability of our simulator by calculating the hole transport in a p-type Si nanowire field effect transistor and the band-to-band tunneling current in a InAs nanowire tunnel field effect transistor.

15 Correspondence between Andreev reflection and Klein tunneling in bipolar graphene

Ref	Project_Klein_Andreev_graphene
摘要	The Andreev reflection at a superconductor and the Klein tunneling through an n−p junction in graphene are two processes that couple electrons to holes—the former through the superconducting pair potential Δ and the latter through the electrostatic potential U. We derive that the energy spectra in the two systems are identical at low energies ε⪡Δ and for an antisymmetric potential profile U(−x,y)=−U(x,y). This correspondence implies that bipolar junctions in graphene may have zero density of states at the Fermi level and carry a current in equilibrium, analogous to the superconducting Josephson junctions. It also implies that nonelectronic systems with the same band structure as graphene, such as honeycomb-lattice photonic crystals, can exhibit pseudosuperconducting behavior.

Ref

Project_Klein_Andreev_graphene

摘要

The Andreev reflection at a superconductor and the Klein tunneling through an n−p junction in graphene are two processes that couple electrons to holes—the former through the superconducting pair potential Δ and the latter through the electrostatic potential U. We derive that the energy spectra in the two systems are identical at low energies ε⪡Δ and for an antisymmetric potential profile U(−x,y)=−U(x,y). This correspondence implies that bipolar junctions in graphene may have zero density of states at the Fermi level and carry a current in equilibrium, analogous to the superconducting Josephson junctions. It also implies that nonelectronic systems with the same band structure as graphene, such as honeycomb-lattice photonic crystals, can exhibit pseudosuperconducting behavior.

16 Kondo Effect in the Presence of Spin-Orbit Coupling

Ref	Project_KondoEffect
摘要	Recently, a series of noncentrosymmetric superconductors has been a subject of considerable interest since the discovery of superconductivity in CePt_3Si. In noncentrosymmetric materials, the degeneracy of bands is lifted in the presence of spin-orbit coupling. This will bring about new effects in the Kondo effect since the band degeneracy plays an important role in the scattering of electrons by localized spins. We investigate the single-impurity Kondo problem in the presence of spin-orbit coupling. We examine the effect of spin-orbit coupling on the scattering of conduction electrons, by using the Green's function method, for the s-d Hamiltonian, with employing a decoupling procedure. As a result, we obtain a closed system of equations of Green's functions, from which we can calculate physical quantities. The Kondo temperature T_K is estimated from a singularity of Green's functions. We show that T_K is reduced as the spin-orbit coupling constant \alpha is increased. When 2\alpha k_F is comparable to or greater than k_BT_K(\alpha=0), T_K shows an abrupt decrease as a result of the band splitting. This suggests a Kondo collapse accompanied with a sharp decrease of T_K. The log T-dependence of the resistivity will be concealed by the spin-orbit interaction.

Ref

Project_KondoEffect

摘要

Recently, a series of noncentrosymmetric superconductors has been a subject of considerable interest since the discovery of superconductivity in CePt_3Si. In noncentrosymmetric materials, the degeneracy of bands is lifted in the presence of spin-orbit coupling. This will bring about new effects in the Kondo effect since the band degeneracy plays an important role in the scattering of electrons by localized spins. We investigate the single-impurity Kondo problem in the presence of spin-orbit coupling. We examine the effect of spin-orbit coupling on the scattering of conduction electrons, by using the Green's function method, for the s-d Hamiltonian, with employing a decoupling procedure. As a result, we obtain a closed system of equations of Green's functions, from which we can calculate physical quantities. The Kondo temperature T_K is estimated from a singularity of Green's functions. We show that T_K is reduced as the spin-orbit coupling constant \alpha is increased. When 2\alpha k_F is comparable to or greater than k_BT_K(\alpha=0), T_K shows an abrupt decrease as a result of the band splitting. This suggests a Kondo collapse accompanied with a sharp decrease of T_K. The log T-dependence of the resistivity will be concealed by the spin-orbit interaction.

17 Andreev reflection in superconducting QCD

Ref	Project_QCD_Andreev
摘要	In this paper we discuss the phenomenon of the Andreev reflection of quarks at the interface between the 2SC and the Color-Flavor-Locked (CFL) superconductors appeared in QCD at asymptotically high densities. We also give the general introduction to the Andreev reflection in the condensed matter systems as well as the review of this subject in high density QCD.

18 Exact Spatiotemporal Dynamics of Confined Lattice Random Walks in Arbitrary Dimensions: A Century after Smoluchowski and Pólya

Ref	Project_random_walk
摘要	A lattice random walk is a mathematical representation of movement through random steps on a lattice at discrete times. It is commonly referred to as Pólya’s walk when the steps occur in either of the nearest-neighbor sites. Since Smoluchowski’s 1906 derivation of the spatiotemporal dependence of the walk occupation probability in an unbounded one-dimensional lattice, discrete random walks and their continuous counterpart, Brownian walks, have developed over the course of a century into a vast and versatile area of knowledge. Lattice random walks are now routinely employed to study stochastic processes across scales, dimensions, and disciplines, from the one-dimensional search of proteins along a DNA strand and the two-dimensional roaming of bacteria in a petri dish, to the three-dimensional motion of macromolecules inside cells and the spatial coverage of multiple robots in a disaster area. In these realistic scenarios, when the randomly moving object is constrained to remain within a finite domain, confined lattice random walks represent a powerful modeling tool. Somewhat surprisingly, and differently from Brownian walks, the spatiotemporal dependence of the confined lattice walk probability has been accessible mainly via computational techniques, and finding its analytic description has remained an open problem. Making use of a set of analytic combinatorics identities with Chebyshev polynomials, I develop a hierarchical dimensionality reduction to find the exact space and time dependence of the occupation probability for confined Pólya’s walks in arbitrary dimensions with reflective, periodic, absorbing, and mixed (reflective and absorbing) boundary conditions along each direction. The probability expressions allow one to construct the time dependence of derived quantities, explicitly in one dimension and via an integration in higher dimensions, such as the first-passage probability to a single target, return probability, average number of distinct sites visited, and absorption probability with imperfect traps. Exact mean first-passage time formulas to a single target in arbitrary dimensions are also presented. These formulas allow one to extend the so-called discrete pseudo-Green function formalism, employed to determine analytically mean first-passage time, with reflecting and periodic boundaries, and a wealth of other related quantities, to arbitrary dimensions. For multiple targets, I introduce a procedure to construct the time dependence of the first-passage probability to one of many targets. Reduction of the occupation probability expressions to the continuous time limit, the so-called continuous time random walk, and to the space-time continuous limit is also presented.

Ref

Project_random_walk

摘要

A lattice random walk is a mathematical representation of movement through random steps on a lattice at discrete times. It is commonly referred to as Pólya’s walk when the steps occur in either of the nearest-neighbor sites. Since Smoluchowski’s 1906 derivation of the spatiotemporal dependence of the walk occupation probability in an unbounded one-dimensional lattice, discrete random walks and their continuous counterpart, Brownian walks, have developed over the course of a century into a vast and versatile area of knowledge. Lattice random walks are now routinely employed to study stochastic processes across scales, dimensions, and disciplines, from the one-dimensional search of proteins along a DNA strand and the two-dimensional roaming of bacteria in a petri dish, to the three-dimensional motion of macromolecules inside cells and the spatial coverage of multiple robots in a disaster area. In these realistic scenarios, when the randomly moving object is constrained to remain within a finite domain, confined lattice random walks represent a powerful modeling tool. Somewhat surprisingly, and differently from Brownian walks, the spatiotemporal dependence of the confined lattice walk probability has been accessible mainly via computational techniques, and finding its analytic description has remained an open problem. Making use of a set of analytic combinatorics identities with Chebyshev polynomials, I develop a hierarchical dimensionality reduction to find the exact space and time dependence of the occupation probability for confined Pólya’s walks in arbitrary dimensions with reflective, periodic, absorbing, and mixed (reflective and absorbing) boundary conditions along each direction. The probability expressions allow one to construct the time dependence of derived quantities, explicitly in one dimension and via an integration in higher dimensions, such as the first-passage probability to a single target, return probability, average number of distinct sites visited, and absorption probability with imperfect traps. Exact mean first-passage time formulas to a single target in arbitrary dimensions are also presented. These formulas allow one to extend the so-called discrete pseudo-Green function formalism, employed to determine analytically mean first-passage time, with reflecting and periodic boundaries, and a wealth of other related quantities, to arbitrary dimensions. For multiple targets, I introduce a procedure to construct the time dependence of the first-passage probability to one of many targets. Reduction of the occupation probability expressions to the continuous time limit, the so-called continuous time random walk, and to the space-time continuous limit is also presented.

19 Reversing Berry phase and modulating Andreev reflection by Rashba spin-orbit coupling in graphene mono- and bilayers

Ref	Project_SOC_Andreev
摘要	Graphene with Rashba spin-orbit coupling (RSOC) has attracted much attention so far. However, no one has noticed the topologically nontrivial changes of Berry phase for RSOC from the absence to the presence. We demonstrate that the Berry phase of electronic wave functions changes from π to 2π in graphene monolayer (GML) and from 2π to π in graphene bilayer (GBL), driven by RSOC. These reversals of Berry phase result in anomalous electron-hole conversions at normal conductor-superconductor junctions. The specular Andreev reflection can be significantly reduced in GML, but obviously enhanced in GBL. Another unusual point caused by RSOC is that the spin-flipped electron reflection happens due to the spin helical structures on equal-energy surfaces. An electrically observable effect induced by RSOC is proposed such that the differential conductance at voltages below the superconducting gap decreases strongly for GML while it increases remarkably for GBL, attributed to the Berry-phase-dictated interference between incident and reflected states.

Ref

Project_SOC_Andreev

摘要

Graphene with Rashba spin-orbit coupling (RSOC) has attracted much attention so far. However, no one has noticed the topologically nontrivial changes of Berry phase for RSOC from the absence to the presence. We demonstrate that the Berry phase of electronic wave functions changes from π to 2π in graphene monolayer (GML) and from 2π to π in graphene bilayer (GBL), driven by RSOC. These reversals of Berry phase result in anomalous electron-hole conversions at normal conductor-superconductor junctions. The specular Andreev reflection can be significantly reduced in GML, but obviously enhanced in GBL. Another unusual point caused by RSOC is that the spin-flipped electron reflection happens due to the spin helical structures on equal-energy surfaces. An electrically observable effect induced by RSOC is proposed such that the differential conductance at voltages below the superconducting gap decreases strongly for GML while it increases remarkably for GBL, attributed to the Berry-phase-dictated interference between incident and reflected states.

20 含顺磁杂质超导体中的束缚态

Ref	Project_含顺磁杂质超导体中的束缚态
摘要	本文利用广义正则变换和自洽场方法,讨论了单个杂质对超导体的影响。证明在磁性杂质附近,可能形成一个束缚态的元激发,其能量位于能隙之中。求出了能级和波函数的解析表达式,并计算了束缚能级所引起的附加电磁吸收。讨论了与此有关的隧道和高频吸收实验。此外,还讨论了非磁性杂质对连续谱元激发的影响及杂质附近能隙的变化。

Ref

Project_含顺磁杂质超导体中的束缚态

摘要

本文利用广义正则变换和自洽场方法,讨论了单个杂质对超导体的影响。证明在磁性杂质附近,可能形成一个束缚态的元激发,其能量位于能隙之中。求出了能级和波函数的解析表达式,并计算了束缚能级所引起的附加电磁吸收。讨论了与此有关的隧道和高频吸收实验。此外,还讨论了非磁性杂质对连续谱元激发的影响及杂质附近能隙的变化。

21 布朗运动理论一百年

Ref	Project_布朗运动理论一百年
摘要	文章基于作者在2005 年纪念爱因斯坦奇迹年的香山会议上的综述报告, 扼要叙述了从布朗运动到统计涨落场论的发展历程, 特别提及了与中国物理学家有关的贡献。

22 Brownian yet Non-Gaussian Diffusion: From Superstatistics to Subordination of Diffusing Diffusivities

Ref	Project_non_Gaussian_Brownian
摘要	A growing number of biological, soft, and active matter systems are observed to exhibit normal diffusive dynamics with a linear growth of the mean-squared displacement, yet with a non-Gaussian distribution of increments. Based on the Chubinsky-Slater idea of a diffusing diffusivity, we here establish and analyze a minimal model framework of diffusion processes with fluctuating diffusivity. In particular, we demonstrate the equivalence of the diffusing diffusivity process with a superstatistical approach with a distribution of diffusivities, at times shorter than the diffusivity correlation time. At longer times, a crossover to a Gaussian distribution with an effective diffusivity emerges. Specifically, we establish a subordination picture of Brownian but non-Gaussian diffusion processes, which can be used for a wide class of diffusivity fluctuation statistics. Our results are shown to be in excellent agreement with simulations and numerical evaluations.

Ref

Project_non_Gaussian_Brownian

摘要

A growing number of biological, soft, and active matter systems are observed to exhibit normal diffusive dynamics with a linear growth of the mean-squared displacement, yet with a non-Gaussian distribution of increments. Based on the Chubinsky-Slater idea of a diffusing diffusivity, we here establish and analyze a minimal model framework of diffusion processes with fluctuating diffusivity. In particular, we demonstrate the equivalence of the diffusing diffusivity process with a superstatistical approach with a distribution of diffusivities, at times shorter than the diffusivity correlation time. At longer times, a crossover to a Gaussian distribution with an effective diffusivity emerges. Specifically, we establish a subordination picture of Brownian but non-Gaussian diffusion processes, which can be used for a wide class of diffusivity fluctuation statistics. Our results are shown to be in excellent agreement with simulations and numerical evaluations.

23 Ratchet, pawl and spring Brownian motor

Ref	Project_ratchet_pawl_spring_Brown
摘要	We present a model for a thermal Brownian motor based on Feynman’s famous ratchet and pawl device. Its main feature is that the ratchet and the pawl are in different thermal baths and connected by a harmonic spring. We simulate its dynamics, explore its main features and also derive an approximate analytical solution for the mean velocity as a function of the external torque applied and the temperatures of the baths. Such theoretical predictions and results from numerical simulations agree within the ranges of the approximations performed.

Ref

Project_ratchet_pawl_spring_Brown

摘要

We present a model for a thermal Brownian motor based on Feynman’s famous ratchet and pawl device. Its main feature is that the ratchet and the pawl are in different thermal baths and connected by a harmonic spring. We simulate its dynamics, explore its main features and also derive an approximate analytical solution for the mean velocity as a function of the external torque applied and the temperatures of the baths. Such theoretical predictions and results from numerical simulations agree within the ranges of the approximations performed.

24 Quantum Langevin model for nonequilibrium condensation

Ref	Project_quantum_Langevin_nonequilibrium
摘要	We develop a quantum model for nonequilibrium Bose-Einstein condensation of photons and polaritons in planar microcavity devices. The model builds on laser theory and includes the spatial dynamics of the cavity field, a saturation mechanism, and some frequency dependence of the gain: quantum Langevin equations are written for a cavity field coupled to a continuous distribution of externally pumped two-level emitters with a well-defined frequency. As an example of application, the method is used to study the linearized quantum fluctuations around a steady-state condensed state. In the good-cavity regime, an effective equation for the cavity field only is proposed in terms of a stochastic Gross-Pitaevskii equation. Perspectives in view of a full quantum simulation of the nonequilibrium condensation process are finally sketched.

Ref

Project_quantum_Langevin_nonequilibrium

摘要

We develop a quantum model for nonequilibrium Bose-Einstein condensation of photons and polaritons in planar microcavity devices. The model builds on laser theory and includes the spatial dynamics of the cavity field, a saturation mechanism, and some frequency dependence of the gain: quantum Langevin equations are written for a cavity field coupled to a continuous distribution of externally pumped two-level emitters with a well-defined frequency. As an example of application, the method is used to study the linearized quantum fluctuations around a steady-state condensed state. In the good-cavity regime, an effective equation for the cavity field only is proposed in terms of a stochastic Gross-Pitaevskii equation. Perspectives in view of a full quantum simulation of the nonequilibrium condensation process are finally sketched.

25 Quantum dynamics of a four-well Bose-Hubbard model with two different tunneling rates

Ref	Project_dynamics_Bose_Hubbard
摘要	We consider a theoretical model of a four-mode Bose-Hubbard model consisting of two pairs of wells coupled via two processes with two different rates. The model is naturally divided into two subsystems with strong intrasystem coupling and much weaker coupling between the two subsystems and has previously been introduced as a model for Josephson heat oscillations by Strzys and Anglin [Phys. Rev. A 81, 043616 (2010)]. We examine the quantum dynamics of this model for a range of different initial conditions, in terms of both the number distribution among the wells and the quantum statistics. We find that the time evolution is different to that predicted by a mean-field model and that this system exhibits a wide range of interesting behaviours. We find that the system equilibrates to a maximum entropy state and is thus a useful model for quantum thermalisation. As our model may be realized to a good approximation in the laboratory, it becomes a candidate for experimental investigation.

Ref

Project_dynamics_Bose_Hubbard

摘要

We consider a theoretical model of a four-mode Bose-Hubbard model consisting of two pairs of wells coupled via two processes with two different rates. The model is naturally divided into two subsystems with strong intrasystem coupling and much weaker coupling between the two subsystems and has previously been introduced as a model for Josephson heat oscillations by Strzys and Anglin [Phys. Rev. A 81, 043616 (2010)]. We examine the quantum dynamics of this model for a range of different initial conditions, in terms of both the number distribution among the wells and the quantum statistics. We find that the time evolution is different to that predicted by a mean-field model and that this system exhibits a wide range of interesting behaviours. We find that the system equilibrates to a maximum entropy state and is thus a useful model for quantum thermalisation. As our model may be realized to a good approximation in the laboratory, it becomes a candidate for experimental investigation.

26 Quench-induced dynamical phase transitions and π-synchronization in the Bose-Hubbard model

Ref	Project_quench_Bose_Hubbard
摘要	We investigate the nonequilibrium behavior of a fully connected (or all-to-all coupled) Bose-Hubbard model after a Mott to superfluid quench, in the limit of large boson densities and for an arbitrary number V of lattice sites, with potential relevance in experiments ranging from cold atoms to superconducting qubits. By means of the truncated Wigner approximation, we predict that crossing a critical quench strength the system undergoes a dynamical phase transition between two regimes that are characterized at long times either by an inhomogeneous population of the lattice (i.e., macroscopical self-trapping) or by the tendency of the mean-field bosonic variables to split into two groups with phase difference π, that we refer to as π-synchronization. We show the latter process to be intimately connected to the presence, only for V≥4, of a manifold of infinitely many fixed points of the dynamical equations. Finally, we show that no fine-tuning of the model parameters is needed for the emergence of such π-synchronization, that is in fact found to vanish smoothly in presence of an increasing site-dependent disorder, in what we call a synchronization crossover.

Ref

Project_quench_Bose_Hubbard

摘要

We investigate the nonequilibrium behavior of a fully connected (or all-to-all coupled) Bose-Hubbard model after a Mott to superfluid quench, in the limit of large boson densities and for an arbitrary number V of lattice sites, with potential relevance in experiments ranging from cold atoms to superconducting qubits. By means of the truncated Wigner approximation, we predict that crossing a critical quench strength the system undergoes a dynamical phase transition between two regimes that are characterized at long times either by an inhomogeneous population of the lattice (i.e., macroscopical self-trapping) or by the tendency of the mean-field bosonic variables to split into two groups with phase difference π, that we refer to as π-synchronization. We show the latter process to be intimately connected to the presence, only for V≥4, of a manifold of infinitely many fixed points of the dynamical equations. Finally, we show that no fine-tuning of the model parameters is needed for the emergence of such π-synchronization, that is in fact found to vanish smoothly in presence of an increasing site-dependent disorder, in what we call a synchronization crossover.

27 Time-Dependent Statistics of the Ising Model

Ref	Project_Ising_model
摘要	The individual spins of the Ising model are assumed to interact with an external agency (e.g., a heat reservoir) which causes them to change their states randomly with time. Coupling between the spins is introduced through the assumption that the transition probabilities for any one spin depend on the values of the neighboring spins. This dependence is determined, in part, by the detailed balancing condition obeyed by the equilibrium state of the model. The Markoff process which describes the spin functions is analyzed in detail for the case of a closed N‐member chain. The expectation values of the individual spins and of the products of pairs of spins, each of the pair evaluated at a different time, are found explicitly. The influence of a uniform, time‐varying magnetic field upon the model is discussed, and the frequency‐dependent magnetic susceptibility is found in the weak‐field limit. Some fluctuation‐dissipation theorems are derived which relate the susceptibility to the Fourier transform of the time‐dependent correlation function of the magnetization at equilibrium.

Ref

Project_Ising_model

摘要

The individual spins of the Ising model are assumed to interact with an external agency (e.g., a heat reservoir) which causes them to change their states randomly with time. Coupling between the spins is introduced through the assumption that the transition probabilities for any one spin depend on the values of the neighboring spins. This dependence is determined, in part, by the detailed balancing condition obeyed by the equilibrium state of the model. The Markoff process which describes the spin functions is analyzed in detail for the case of a closed N‐member chain. The expectation values of the individual spins and of the products of pairs of spins, each of the pair evaluated at a different time, are found explicitly. The influence of a uniform, time‐varying magnetic field upon the model is discussed, and the frequency‐dependent magnetic susceptibility is found in the weak‐field limit. Some fluctuation‐dissipation theorems are derived which relate the susceptibility to the Fourier transform of the time‐dependent correlation function of the magnetization at equilibrium.

28 Many-fermion generalization of the Caldeira-Leggett model

Ref	Project_fermion_Caldeira_Leggett
摘要	We analyze a model system of fermions in a harmonic oscillator potential under the influence of a dissipative environment: The fermions are subject to a fluctuating force deriving from a bath of harmonic oscillators. This represents an extension of the well-known Caldeira-Leggett model to the case of many fermions. Using the method of bosonization, we calculate one- and two-particle Green’s functions of the fermions. We discuss the relaxation of a single extra particle added above the Fermi sea, considering also dephasing of a particle added in a coherent superposition of states. The consequences of the separation of center-of-mass and relative motion, the Pauli principle, and the bath-induced effective interaction are discussed. Finally, we extend our analysis to a more generic coupling between system and bath, which results in complete thermalization of the system.

Ref

Project_fermion_Caldeira_Leggett

摘要

We analyze a model system of fermions in a harmonic oscillator potential under the influence of a dissipative environment: The fermions are subject to a fluctuating force deriving from a bath of harmonic oscillators. This represents an extension of the well-known Caldeira-Leggett model to the case of many fermions. Using the method of bosonization, we calculate one- and two-particle Green’s functions of the fermions. We discuss the relaxation of a single extra particle added above the Fermi sea, considering also dephasing of a particle added in a coherent superposition of states. The consequences of the separation of center-of-mass and relative motion, the Pauli principle, and the bath-induced effective interaction are discussed. Finally, we extend our analysis to a more generic coupling between system and bath, which results in complete thermalization of the system.

29 Many-body dynamical localisation of coupled quantum kicked rotors

Ref	Project_quantum_kicked_rotors
摘要	The quantum motion of N coupled kicked rotors is mapped to an interacting N-particle Anderson-Aubry-André tight-binding problem supporting many-body localised (MBL) phases. Interactions in configuration space are known to be insufficient for destroying Anderson localisation in a system in the MBL phase. The mapping we establish here predicts that a similar effect takes place in momentum space and determines the quantum dynamics of the coupled kicked rotors. Due to the boundedness of the Floquet quasi-energy spectrum there exists limitations on the interacting lattice models that can be mapped to quantum kicked rotors; in particular, no extensive observable can be mapped in the thermodynamic limit.

Ref

Project_quantum_kicked_rotors

摘要

The quantum motion of N coupled kicked rotors is mapped to an interacting N-particle Anderson-Aubry-André tight-binding problem supporting many-body localised (MBL) phases. Interactions in configuration space are known to be insufficient for destroying Anderson localisation in a system in the MBL phase. The mapping we establish here predicts that a similar effect takes place in momentum space and determines the quantum dynamics of the coupled kicked rotors. Due to the boundedness of the Floquet quasi-energy spectrum there exists limitations on the interacting lattice models that can be mapped to quantum kicked rotors; in particular, no extensive observable can be mapped in the thermodynamic limit.

30 Mobility edges in one-dimensional bichromatic incommensurate potentials

Ref	project_A_A_model
摘要	We theoretically study a one-dimensional (1D) mutually incommensurate bichromatic lattice system, which has been implemented in ultracold atoms to study quantum localization. It has been universally believed that the tight-binding version of this bichromatic incommensurate system is represented by the well-known Aubry-Andre model capturing all the essential localization physics in the experimental cold atom optical lattice system. Here we establish that this belief is incorrect and that the Aubry-Andre model description, which applies only in the extreme tight-binding limit of a very deep primary lattice potential, generically breaks down near the localization transition due to the unavoidable appearance of single-particle mobility edges (SPME). In fact, we show that the 1D bichromatic incommensurate potential system manifests generic mobility edges, which disappear in the tight-binding limit, leading to the well-studied Aubry-Andre physics. We carry out an extensive study of the localization properties of the 1D incommensurate optical lattice without making any tight-binding approximation. We find that, for the full lattice system, an intermediate phase between completely localized and completely delocalized regions appears due to the existence of the SPME, making the system qualitatively distinct from the Aubry-Andre prediction. Using the Wegner flow approach, we show that the SPME in the real lattice system can be attributed to significant corrections of higher-order harmonics in the lattice potential, which are absent in the strict tight-binding limit. We calculate the dynamical consequences of the intermediate phase in detail to guide future experimental investigations for the observation of 1D SPME and the associated intermediate (i.e., neither purely localized nor purely delocalized) phase. We consider effects of interaction numerically, and conjecture the stability of SPME to weak interaction effects, thus leading to the exciting possibility of an experimentally viable nonergodic extended phase in interacting 1D optical lattices. Our work provides precise quantitative protocols for future optical lattice based experiments searching for mobility edges in one-dimensional bichromatic incommensurate lattices, both in noninteracting and interacting systems.

Ref

project_A_A_model

摘要

We theoretically study a one-dimensional (1D) mutually incommensurate bichromatic lattice system, which has been implemented in ultracold atoms to study quantum localization. It has been universally believed that the tight-binding version of this bichromatic incommensurate system is represented by the well-known Aubry-Andre model capturing all the essential localization physics in the experimental cold atom optical lattice system. Here we establish that this belief is incorrect and that the Aubry-Andre model description, which applies only in the extreme tight-binding limit of a very deep primary lattice potential, generically breaks down near the localization transition due to the unavoidable appearance of single-particle mobility edges (SPME). In fact, we show that the 1D bichromatic incommensurate potential system manifests generic mobility edges, which disappear in the tight-binding limit, leading to the well-studied Aubry-Andre physics. We carry out an extensive study of the localization properties of the 1D incommensurate optical lattice without making any tight-binding approximation. We find that, for the full lattice system, an intermediate phase between completely localized and completely delocalized regions appears due to the existence of the SPME, making the system qualitatively distinct from the Aubry-Andre prediction. Using the Wegner flow approach, we show that the SPME in the real lattice system can be attributed to significant corrections of higher-order harmonics in the lattice potential, which are absent in the strict tight-binding limit. We calculate the dynamical consequences of the intermediate phase in detail to guide future experimental investigations for the observation of 1D SPME and the associated intermediate (i.e., neither purely localized nor purely delocalized) phase. We consider effects of interaction numerically, and conjecture the stability of SPME to weak interaction effects, thus leading to the exciting possibility of an experimentally viable nonergodic extended phase in interacting 1D optical lattices. Our work provides precise quantitative protocols for future optical lattice based experiments searching for mobility edges in one-dimensional bichromatic incommensurate lattices, both in noninteracting and interacting systems.

31 Bloch Electrons in Magnetic Fields: Classical Chaos and Hofstadter's ButterHy

Ref	project_bloch_electrons_in_magnetic_fields
摘要	The single-band description of Bloch electrons in magnetic fields (e.g., by Harper's equation) leads to classically integrable Hamiltonians and thus fails for lateral surface superlattices (LSSLs) where chaotic trajectories prevail near the classical limit. We derive a new model [Eq. (18)], which is reminescent of Harper’s equation, but is exact under the most general conditions and exhibits chaos in the classical limit. A matrix continued fraction expansion allows us to study the influence of classical chaos on the fractal spectrum known as Hofstadter's butterfly and to make predictions on its observability in LSSLs.

Ref

project_bloch_electrons_in_magnetic_fields

摘要

The single-band description of Bloch electrons in magnetic fields (e.g., by Harper's equation) leads to classically integrable Hamiltonians and thus fails for lateral surface superlattices (LSSLs) where chaotic trajectories prevail near the classical limit. We derive a new model [Eq. (18)], which is reminescent of Harper’s equation, but is exact under the most general conditions and exhibits chaos in the classical limit. A matrix continued fraction expansion allows us to study the influence of classical chaos on the fractal spectrum known as Hofstadter's butterfly and to make predictions on its observability in LSSLs.

32 Theory of nonlinear Landau-Zener tunneling

Ref	project_Landau_Zener_tunneling
摘要	We present a comprehensive analysis of the nonlinear Landau-Zener tunneling. We find characteristic scaling or power laws for the critical behavior that occurs as the nonlinear parameter equals to the gap of avoided crossing energy levels. For the nonlinear parameter larger than the energy gap, a closed-form solution is derived for the nonlinear tunneling probability, which is shown to be a good approximation to the exact solution for a wide range of the parameters. Finally, we discuss the experimental realization of the nonlinear model and possible observation of the scaling or power laws using a Bose-Einstein condensate in an accelerating optical lattice.

Ref

project_Landau_Zener_tunneling

摘要

We present a comprehensive analysis of the nonlinear Landau-Zener tunneling. We find characteristic scaling or power laws for the critical behavior that occurs as the nonlinear parameter equals to the gap of avoided crossing energy levels. For the nonlinear parameter larger than the energy gap, a closed-form solution is derived for the nonlinear tunneling probability, which is shown to be a good approximation to the exact solution for a wide range of the parameters. Finally, we discuss the experimental realization of the nonlinear model and possible observation of the scaling or power laws using a Bose-Einstein condensate in an accelerating optical lattice.

33 Many-body localization edge in the random-field Heisenberg chain

Ref	project_MBL_random_field_Heisenberg
摘要	We present a large-scale exact diagonalization study of the one-dimensional spin-1/2 Heisenberg model in a random magnetic field. In order to access properties at varying energy densities across the entire spectrum for system sizes up to L = 22 spins, we use a spectral transformation which can be applied in a massively parallel fashion. Our results allow for an energy-resolved interpretation of the many-body localization transition including the existence of an extensive many-body mobility edge. The ergodic phase is well characterized by Gaussian orthogonal ensemble statistics, volume-law entanglement, and a full delocalization in the Hilbert space. Conversely, the localized regime displays Poisson statistics, area-law entanglement, and nonergodicity in the Hilbert space where a true localization never occurs. We perform finite-size scaling to extract the critical edge and exponent of the localization length divergence.

Ref

project_MBL_random_field_Heisenberg

摘要

We present a large-scale exact diagonalization study of the one-dimensional spin-1/2 Heisenberg model in a random magnetic field. In order to access properties at varying energy densities across the entire spectrum for system sizes up to L = 22 spins, we use a spectral transformation which can be applied in a massively parallel fashion. Our results allow for an energy-resolved interpretation of the many-body localization transition including the existence of an extensive many-body mobility edge. The ergodic phase is well characterized by Gaussian orthogonal ensemble statistics, volume-law entanglement, and a full delocalization in the Hilbert space. Conversely, the localized regime displays Poisson statistics, area-law entanglement, and nonergodicity in the Hilbert space where a true localization never occurs. We perform finite-size scaling to extract the critical edge and exponent of the localization length divergence.

34 Spectral statistics across the many-body localization transition

Ref	project_MBL_spectral_statistics
摘要	The many-body localization transition (MBLT) between ergodic and many-body localized phases in disordered interacting systems is a subject of much recent interest. The statistics of eigenenergies is known to be a powerful probe of crossovers between ergodic and integrable systems in simpler examples of quantum chaos. We consider the evolution of the spectral statistics across the MBLT, starting with mapping to a Brownian motion process that analytically relates the spectral properties to the statistics of matrix elements. We demonstrate that the flow from Wigner-Dyson to Poisson statistics is a two-stage process. First, a fractal enhancement of matrix elements upon approaching the MBLT from the delocalized side produces an effective power-law interaction between energy levels, and leads to a plasma model for level statistics. At the second stage, the gas of eigenvalues has local interactions and the level statistics belongs to a semi-Poisson universality class. We verify our findings numerically on the XXZ spin chain. We provide a microscopic understanding of the level statistics across the MBLT and discuss implications for the transition that are strong constraints on possible theories.

Ref

project_MBL_spectral_statistics

摘要

The many-body localization transition (MBLT) between ergodic and many-body localized phases in disordered interacting systems is a subject of much recent interest. The statistics of eigenenergies is known to be a powerful probe of crossovers between ergodic and integrable systems in simpler examples of quantum chaos. We consider the evolution of the spectral statistics across the MBLT, starting with mapping to a Brownian motion process that analytically relates the spectral properties to the statistics of matrix elements. We demonstrate that the flow from Wigner-Dyson to Poisson statistics is a two-stage process. First, a fractal enhancement of matrix elements upon approaching the MBLT from the delocalized side produces an effective power-law interaction between energy levels, and leads to a plasma model for level statistics. At the second stage, the gas of eigenvalues has local interactions and the level statistics belongs to a semi-Poisson universality class. We verify our findings numerically on the XXZ spin chain. We provide a microscopic understanding of the level statistics across the MBLT and discuss implications for the transition that are strong constraints on possible theories.

35 Monte Carlo study of the two-dimensional Bose-Hubbard model

Ref	project_MC_Bose_Hubbard_model
摘要	One of the most promising applications of ultracold gases in optical lattices is the possibility to use them as quantum emulators of more complex condensed matter systems. We provide benchmark calculations, based on exact quantum Monte Carlo simulations, for the emulator to be tested against. We report results for the ground state phase diagram of the two-dimensional Bose-Hubbard model at unity filling factor. We precisely trace out the critical behavior of the system and resolve the region of small insulating gaps, Δ⪡J. The critical point is found to be (J/U)c=0.05974(3), in perfect agreement with the high-order strong-coupling expansion method of Elstner and Monien [Phys. Rev. B 59, 12184 (1999)]. In addition, we present data for the effective mass of particle and hole excitations inside the insulating phase and obtain the critical temperature for the superfluid-normal transition at unity filling factor.

Ref

project_MC_Bose_Hubbard_model

摘要

One of the most promising applications of ultracold gases in optical lattices is the possibility to use them as quantum emulators of more complex condensed matter systems. We provide benchmark calculations, based on exact quantum Monte Carlo simulations, for the emulator to be tested against. We report results for the ground state phase diagram of the two-dimensional Bose-Hubbard model at unity filling factor. We precisely trace out the critical behavior of the system and resolve the region of small insulating gaps, Δ⪡J. The critical point is found to be (J/U)c=0.05974(3), in perfect agreement with the high-order strong-coupling expansion method of Elstner and Monien [Phys. Rev. B 59, 12184 (1999)]. In addition, we present data for the effective mass of particle and hole excitations inside the insulating phase and obtain the critical temperature for the superfluid-normal transition at unity filling factor.

36 Quantum random walks

Ref	project_quantum_random_walks
摘要	We introduce the concept of quantum rundom walk, and show that due to quantum interference effects the average path length can be much larger than the maximum allowed path in the corresponding classical random walk. A quantum-optics application is described.

37 Quantum Trajectories for Brownian Motion

Ref	project_quantum_trajectories_for_Brownian_Motion
摘要	We present the stochastic Schrödinger equation for the dynamics of a quantum particle coupled to a high temperature environment and apply it to the dynamics of a driven, damped, nonlinear quantum oscillator. Apart from an initial slip on the environmental memory time scale, in the mean, our result recovers the solution of the known non-Lindblad quantum Brownian motion master equation. A remarkable feature of our powerful stochastic approach is its localization property: individual quantum trajectories remain localized wave packets for all times, even for the classically chaotic system considered here, the localization being stronger as $\hbar \rightarrow 0$

Ref

project_quantum_trajectories_for_Brownian_Motion

摘要

We present the stochastic Schrödinger equation for the dynamics of a quantum particle coupled to a high temperature environment and apply it to the dynamics of a driven, damped, nonlinear quantum oscillator. Apart from an initial slip on the environmental memory time scale, in the mean, our result recovers the solution of the known non-Lindblad quantum Brownian motion master equation. A remarkable feature of our powerful stochastic approach is its localization property: individual quantum trajectories remain localized wave packets for all times, even for the classically chaotic system considered here, the localization being stronger as $\hbar \rightarrow 0$

38 Dynamics of dissipative Landau-Zener transitions

Ref	project_dynamics_Landau-Zener
摘要	A nonperturbative treatment, the Dirac-Frenkel time-dependent variation is employed to examine dynamics of the Landau-Zener model with both diagonal and off-diagonal qubit-bath coupling using the multiple Davydov trial states. It is shown that steady-state transition probabilities agree with analytical predictions at long times. Landau-Zener dynamics at intermediate times is little affected by diagonal coupling, and is found to be determined by off-diagonal coupling and tunneling between two diabatic states. We investigate effects of bath spectral densities, coupling strengths, and interaction angles on Laudau-Zener dynamics. Thanks to the multiple Davydov trial states, detailed boson dynamics can also be analyzed in Landau-Zener transitions. The results presented here may help provide guiding principles to manipulate the Laudau-Zener transitions in circuit QED architectures by tuning off-diagonal coupling and tunneling strength.

Ref

project_dynamics_Landau-Zener

摘要

A nonperturbative treatment, the Dirac-Frenkel time-dependent variation is employed to examine dynamics of the Landau-Zener model with both diagonal and off-diagonal qubit-bath coupling using the multiple Davydov trial states. It is shown that steady-state transition probabilities agree with analytical predictions at long times. Landau-Zener dynamics at intermediate times is little affected by diagonal coupling, and is found to be determined by off-diagonal coupling and tunneling between two diabatic states. We investigate effects of bath spectral densities, coupling strengths, and interaction angles on Laudau-Zener dynamics. Thanks to the multiple Davydov trial states, detailed boson dynamics can also be analyzed in Landau-Zener transitions. The results presented here may help provide guiding principles to manipulate the Laudau-Zener transitions in circuit QED architectures by tuning off-diagonal coupling and tunneling strength.

39 Numerical experiments on quantum chaotic billiards

Ref	project_quantum_chaotic_billiards
摘要	A recently proposed numerical technique for generation of high-quality unstructured meshes is combined with a finite-element method to solve the Helmholtz equation that describes the quantum mechanics of a particle confined in two-dimensional cavities. Different shapes are treated on equal footing, including Sinai, stadium, annular, threefold symmetric, mushroom, cardioid, triangle, and coupled billiards. The results are shown to be in excellent agreement with available measurements in flat microwave resonator counterparts with nonintegrable geometries.

Ref

project_quantum_chaotic_billiards

摘要

A recently proposed numerical technique for generation of high-quality unstructured meshes is combined with a finite-element method to solve the Helmholtz equation that describes the quantum mechanics of a particle confined in two-dimensional cavities. Different shapes are treated on equal footing, including Sinai, stadium, annular, threefold symmetric, mushroom, cardioid, triangle, and coupled billiards. The results are shown to be in excellent agreement with available measurements in flat microwave resonator counterparts with nonintegrable geometries.

40 Obtaining Highly Excited Eigenstates of Many-Body Localized Hamiltonians by the Density Matrix Renormalization Group Approach

Ref	project_Highly_Excited_Eigenstates_MBL_DMRG
摘要	The eigenstates of many-body localized (MBL) Hamiltonians exhibit low entanglement. We adapt the highly successful density-matrix renormalization group method, which is usually used to find modestly entangled ground states of local Hamiltonians, to find individual highly excited eigenstates of MBL Hamiltonians. The adaptation builds on the distinctive spatial structure of such eigenstates. We benchmark our method against the well-studied random field Heisenberg model in one dimension. At moderate to large disorder, the method successfully obtains excited eigenstates with high accuracy, thereby enabling a study of MBL systems at much larger system sizes than those accessible to exact-diagonalization methods.

Ref

project_Highly_Excited_Eigenstates_MBL_DMRG

摘要

The eigenstates of many-body localized (MBL) Hamiltonians exhibit low entanglement. We adapt the highly successful density-matrix renormalization group method, which is usually used to find modestly entangled ground states of local Hamiltonians, to find individual highly excited eigenstates of MBL Hamiltonians. The adaptation builds on the distinctive spatial structure of such eigenstates. We benchmark our method against the well-studied random field Heisenberg model in one dimension. At moderate to large disorder, the method successfully obtains excited eigenstates with high accuracy, thereby enabling a study of MBL systems at much larger system sizes than those accessible to exact-diagonalization methods.

41 Many-Body Localization in Periodically Driven Systems

Ref	project_MBL_Periodically_Driven_Systems
摘要	We consider disordered many-body systems with periodic time-dependent Hamiltonians in one spatial dimension. By studying the properties of the Floquet eigenstates, we identify two distinct phases: (i) a many-body localized (MBL) phase, in which almost all eigenstates have area-law entanglement entropy, and the eigenstate thermalization hypothesis (ETH) is violated, and (ii) a delocalized phase, in which eigenstates have volume-law entanglement and obey the ETH. The MBL phase exhibits logarithmic in time growth of entanglement entropy when the system is initially prepared in a product state, which distinguishes it from the delocalized phase. We propose an effective model of the MBL phase in terms of an extensive number of emergent local integrals of motion, which naturally explains the spectral and dynamical properties of this phase. Numerical data, obtained by exact diagonalization and time-evolving block decimation methods, suggest a direct transition between the two phases.

Ref

project_MBL_Periodically_Driven_Systems

摘要

We consider disordered many-body systems with periodic time-dependent Hamiltonians in one spatial dimension. By studying the properties of the Floquet eigenstates, we identify two distinct phases: (i) a many-body localized (MBL) phase, in which almost all eigenstates have area-law entanglement entropy, and the eigenstate thermalization hypothesis (ETH) is violated, and (ii) a delocalized phase, in which eigenstates have volume-law entanglement and obey the ETH. The MBL phase exhibits logarithmic in time growth of entanglement entropy when the system is initially prepared in a product state, which distinguishes it from the delocalized phase. We propose an effective model of the MBL phase in terms of an extensive number of emergent local integrals of motion, which naturally explains the spectral and dynamical properties of this phase. Numerical data, obtained by exact diagonalization and time-evolving block decimation methods, suggest a direct transition between the two phases.

42 Destruction of Anderson Localization by a Weak Nonlinearity

Ref	project_Anderson_Localization_Weak_Nonlinearity
摘要	We study numerically the spreading of an initially localized wave packet in a one-dimensional discrete nonlinear Schrödinger lattice with disorder. We demonstrate that above a certain critical strength of nonlinearity the Anderson localization is destroyed and an unlimited subdiffusive spreading of the field along the lattice occurs. The second moment grows with time ∝tα, with the exponent α being in the range 0.3–0.4. For small nonlinearities the distribution remains localized in a way similar to the linear case.

Ref

project_Anderson_Localization_Weak_Nonlinearity

摘要

We study numerically the spreading of an initially localized wave packet in a one-dimensional discrete nonlinear Schrödinger lattice with disorder. We demonstrate that above a certain critical strength of nonlinearity the Anderson localization is destroyed and an unlimited subdiffusive spreading of the field along the lattice occurs. The second moment grows with time ∝tα, with the exponent α being in the range 0.3–0.4. For small nonlinearities the distribution remains localized in a way similar to the linear case.

43 Numerical Chaos, Roundofl' Errors, and Homoclinic Manifolds

Ref	project_Numerical_Chaos_Roundofl_Errors_Homoclinic_Manifolds
摘要	The focusing nonlinear Schrodinger equation is numerically integrated over moderate to long time intervals. In certain parameter regimes small errors on the order of roundoA' grow rapidly and saturate at values comparable to the main wave. Although the constants of motion are nearly preserved, a serious phase instability (chaos) develops in the numerical solutions. The instability is found to be associated with homoclinic structures and the underlying mechanisms apply equally well to many Hamiltonian wave systems.

Ref

project_Numerical_Chaos_Roundofl_Errors_Homoclinic_Manifolds

摘要

The focusing nonlinear Schrodinger equation is numerically integrated over moderate to long time intervals. In certain parameter regimes small errors on the order of roundoA' grow rapidly and saturate at values comparable to the main wave. Although the constants of motion are nearly preserved, a serious phase instability (chaos) develops in the numerical solutions. The instability is found to be associated with homoclinic structures and the underlying mechanisms apply equally well to many Hamiltonian wave systems.

44 Many-body localization phase transition: A simplified strong-randomness approximate renormalization group

Ref	project_MBL_strong_randomness_approx_RG
摘要	We present a simplified strong-randomness renormalization group (RG) that captures some aspects of the many-body localization (MBL) phase transition in generic disordered one-dimensional systems. This RG can be formulated analytically and is mathematically equivalent to a domain coarsening model that has been previously solved. The critical fixed-point distribution and critical exponents (that satisfy the Chayes inequality) are thus obtained analytically or to numerical precision. This reproduces some, but not all, of the qualitative features of the MBL phase transition that are indicated by previous numerical work and approximate RG studies: our RG might serve as a “zeroth-order” approximation for future RG studies. One interesting feature that we highlight is that the rare Griffiths regions are fractal. For thermal Griffiths regions within the MBL phase, this feature might be qualitatively correctly captured by our RG. If this is correct beyond our approximations, then these Griffiths effects are stronger than has been previously assumed.

Ref

project_MBL_strong_randomness_approx_RG

摘要

We present a simplified strong-randomness renormalization group (RG) that captures some aspects of the many-body localization (MBL) phase transition in generic disordered one-dimensional systems. This RG can be formulated analytically and is mathematically equivalent to a domain coarsening model that has been previously solved. The critical fixed-point distribution and critical exponents (that satisfy the Chayes inequality) are thus obtained analytically or to numerical precision. This reproduces some, but not all, of the qualitative features of the MBL phase transition that are indicated by previous numerical work and approximate RG studies: our RG might serve as a “zeroth-order” approximation for future RG studies. One interesting feature that we highlight is that the rare Griffiths regions are fractal. For thermal Griffiths regions within the MBL phase, this feature might be qualitatively correctly captured by our RG. If this is correct beyond our approximations, then these Griffiths effects are stronger than has been previously assumed.

45 Many-body localization and thermalization: Insights from the entanglement spectrum

Ref	project_MBL_thermalization_entanglement_spectrum
摘要	We study the entanglement spectrum in the many-body localizing and thermalizing phases of one- and two-dimensional Hamiltonian systems and periodically driven “Floquet” systems. We focus on the level statistics of the entanglement spectrum as obtained through numerical diagonalization, finding structure beyond that revealed by more limited measures such as entanglement entropy. In the thermalizing phase the entanglement spectrum obeys level statistics governed by an appropriate random matrix ensemble. For Hamiltonian systems this can be viewed as evidence in favor of a strong version of the eigenstate thermalization hypothesis (ETH). Similar results are also obtained for Floquet systems, where they constitute a result “beyond ETH” and show that the corrections to ETH governing the Floquet entanglement spectrum have statistical properties governed by a random matrix ensemble. The particular random matrix ensemble governing the Floquet entanglement spectrum depends on the symmetries of the Floquet drive and therefore can depend on the choice of origin of time. In the many-body localized phase the entanglement spectrum is also found to show level repulsion, following a semi-Poisson distribution (in contrast to the energy spectrum, which follows a Poisson distribution). This semi-Poisson distribution is found to come mainly from states at high entanglement energies. The observed level repulsion occurs only for interacting localized phases. We also demonstrate that equivalent results can be obtained by calculating with a single typical eigenstate or by averaging over a microcanonical energy window, a surprising result in the localized phase. This discovery of new structure in the pattern of entanglement of localized and thermalizing phases may open up new lines of attack on many-body localization, thermalization, and the localization transition.

Ref

project_MBL_thermalization_entanglement_spectrum

摘要

We study the entanglement spectrum in the many-body localizing and thermalizing phases of one- and two-dimensional Hamiltonian systems and periodically driven “Floquet” systems. We focus on the level statistics of the entanglement spectrum as obtained through numerical diagonalization, finding structure beyond that revealed by more limited measures such as entanglement entropy. In the thermalizing phase the entanglement spectrum obeys level statistics governed by an appropriate random matrix ensemble. For Hamiltonian systems this can be viewed as evidence in favor of a strong version of the eigenstate thermalization hypothesis (ETH). Similar results are also obtained for Floquet systems, where they constitute a result “beyond ETH” and show that the corrections to ETH governing the Floquet entanglement spectrum have statistical properties governed by a random matrix ensemble. The particular random matrix ensemble governing the Floquet entanglement spectrum depends on the symmetries of the Floquet drive and therefore can depend on the choice of origin of time. In the many-body localized phase the entanglement spectrum is also found to show level repulsion, following a semi-Poisson distribution (in contrast to the energy spectrum, which follows a Poisson distribution). This semi-Poisson distribution is found to come mainly from states at high entanglement energies. The observed level repulsion occurs only for interacting localized phases. We also demonstrate that equivalent results can be obtained by calculating with a single typical eigenstate or by averaging over a microcanonical energy window, a surprising result in the localized phase. This discovery of new structure in the pattern of entanglement of localized and thermalizing phases may open up new lines of attack on many-body localization, thermalization, and the localization transition.

46 A numerical analysis of chaos in the double pendulum

Ref	project_numerical_chaos_double_pendulum
摘要	We analyse the double pendulum system numerically, using a modified mid-point integrator. Poincare´ sections and bifurcation diagrams are constructed for certain, characteristic values of energy. The largest Lyapunov characteristic exponents are also calculated. All three methods confirm the passing of the system from the regular low-energy limit into chaos as energy is increased

47 Band structure and topological properties of twisted double bilayer graphene

Ref	project_band_topo_TDBG
摘要	We study the electronic band structure and the topological properties of the twisted double bilayer graphene, or a pair of AB-stacked bilayer graphenes rotationally stacked on top of each other. We consider two different arrangements, AB-AB and AB-BA, which differ in the relative orientation. For each system, we calculate the energy band and the valley Chern number using the continuum Hamiltonian. We show that the AB-AB and the AB-BA have similar band structures, while the Chern numbers associated with the corresponding bands are completely different. In the absence of the perpendicular electric field, in particular, the AB-AB system is a trivial insulator when the Fermi energy is in a gap, while the AB-BA is a valley Hall insulator. Also, the lowest electron and hole bands of the AB-AB are entangled by the symmetry protected band touching points, while they are separated in the AB-BA. In both cases, the perpendicular electric field immediately opens an energy gap at the charge neutral point, where the electron branch becomes much narrower than the hole branch, due to the significant electron-hole asymmetry.

Ref

project_band_topo_TDBG

摘要

We study the electronic band structure and the topological properties of the twisted double bilayer graphene, or a pair of AB-stacked bilayer graphenes rotationally stacked on top of each other. We consider two different arrangements, AB-AB and AB-BA, which differ in the relative orientation. For each system, we calculate the energy band and the valley Chern number using the continuum Hamiltonian. We show that the AB-AB and the AB-BA have similar band structures, while the Chern numbers associated with the corresponding bands are completely different. In the absence of the perpendicular electric field, in particular, the AB-AB system is a trivial insulator when the Fermi energy is in a gap, while the AB-BA is a valley Hall insulator. Also, the lowest electron and hole bands of the AB-AB are entangled by the symmetry protected band touching points, while they are separated in the AB-BA. In both cases, the perpendicular electric field immediately opens an energy gap at the charge neutral point, where the electron branch becomes much narrower than the hole branch, due to the significant electron-hole asymmetry.

48 Energy spectrum and quantum Hall effect in twisted bilayer graphene

Ref	project_band_QHE_magnetic_TBG
摘要	We investigate the electronic structure and the quantum Hall effect in twisted bilayer graphenes with various rotation angles in the presence of magnetic field. Using a low-energy approximation, which incorporates the rigorous interlayer interaction, we computed the energy spectrum and the quantized Hall conductivity in a wide range of magnetic field from the semiclassical regime to the fractal spectrum regime. In weak magnetic fields, the low-energy conduction band is quantized into electronlike and holelike Landau levels at energies below and above the van Hove singularity, respectively, and the Hall conductivity sharply drops from positive to negative when the Fermi energy goes through the transition point. In increasing magnetic field, the spectrum gradually evolves into a fractal band structure called Hofstadter's butterfly, where the Hall conductivity exhibits a nonmonotonic behavior as a function of Fermi energy. The typical electron density and magnetic field amplitude characterizing the spectrum monotonically decrease as the rotation angle is reduced, indicating that the rich electronic structure may be observed in a moderate condition.

Ref

project_band_QHE_magnetic_TBG

摘要

We investigate the electronic structure and the quantum Hall effect in twisted bilayer graphenes with various rotation angles in the presence of magnetic field. Using a low-energy approximation, which incorporates the rigorous interlayer interaction, we computed the energy spectrum and the quantized Hall conductivity in a wide range of magnetic field from the semiclassical regime to the fractal spectrum regime. In weak magnetic fields, the low-energy conduction band is quantized into electronlike and holelike Landau levels at energies below and above the van Hove singularity, respectively, and the Hall conductivity sharply drops from positive to negative when the Fermi energy goes through the transition point. In increasing magnetic field, the spectrum gradually evolves into a fractal band structure called Hofstadter's butterfly, where the Hall conductivity exhibits a nonmonotonic behavior as a function of Fermi energy. The typical electron density and magnetic field amplitude characterizing the spectrum monotonically decrease as the rotation angle is reduced, indicating that the rich electronic structure may be observed in a moderate condition.

49 Screening and collective modes in gapped bilayer graphene

Ref	DOI: 10.1103/PhysRevB.86.161408
摘要	We study the static and dynamic screening of gapped AB-stacked bilayer graphene. Unlike previous works we use the full 4-band model instead of the simplified 2-band model. We find that there are important qualitative differences between the dielectric screening function obtained using the simplified 2-band model and the 4-band model. In particular, within the 4-band model, in the presence of a band gap, the static screening exhibits Kohn anomalies that are absent within the simplified 2-band model. Moreover, using the 4-band model, we examine the effect of trigonal warping on the screening properties of bilayer graphene. We also find that the plasmon modes have a qualitatively different character in the 4-band model compared to the ones obtained using the simplified 2-band model.

Ref

DOI: 10.1103/PhysRevB.86.161408

摘要

We study the static and dynamic screening of gapped AB-stacked bilayer graphene. Unlike previous works we use the full 4-band model instead of the simplified 2-band model. We find that there are important qualitative differences between the dielectric screening function obtained using the simplified 2-band model and the 4-band model. In particular, within the 4-band model, in the presence of a band gap, the static screening exhibits Kohn anomalies that are absent within the simplified 2-band model. Moreover, using the 4-band model, we examine the effect of trigonal warping on the screening properties of bilayer graphene. We also find that the plasmon modes have a qualitatively different character in the 4-band model compared to the ones obtained using the simplified 2-band model.

50 Dielectric function, screening, and plasmons in two-dimensional graphene

Ref	DOI: 10.1103/PhysRevB.75.205418
摘要	The dynamical dielectric function of two-dimensional graphene at arbitrary wave vector q and frequency ω, ϵ(q,ω), is calculated in the self-consistent-field approximation. The results are used to find the dispersion of the plasmon mode and the electrostatic screening of the Coulomb interaction in two-dimensional (2D) graphene layer within the random-phase approximation. At long wavelengths (q→0), the plasmon dispersion shows the local classical behavior ωcl=ω0√q, but the density dependence of the plasma frequency (ω0∝n1/4) is different from the usual 2D electron system (ω0∝n1/2). The wave-vector-dependent plasmon dispersion and the static screening function show very different behavior than the usual 2D case. We show that the intrinsic interband contributions to static graphene screening can be effectively absorbed in a background dielectric constant.

Ref

DOI: 10.1103/PhysRevB.75.205418

摘要

The dynamical dielectric function of two-dimensional graphene at arbitrary wave vector q and frequency ω, ϵ(q,ω), is calculated in the self-consistent-field approximation. The results are used to find the dispersion of the plasmon mode and the electrostatic screening of the Coulomb interaction in two-dimensional (2D) graphene layer within the random-phase approximation. At long wavelengths (q→0), the plasmon dispersion shows the local classical behavior ωcl=ω0√q, but the density dependence of the plasma frequency (ω0∝n1/4) is different from the usual 2D electron system (ω0∝n1/2). The wave-vector-dependent plasmon dispersion and the static screening function show very different behavior than the usual 2D case. We show that the intrinsic interband contributions to static graphene screening can be effectively absorbed in a background dielectric constant.

51 Transport between twisted graphene layers

Ref

DOI: 10.1103/PhysRevB.81.245412

摘要

Commensurate-incommensurate transitions are ubiquitous in physics and are often accompanied by intriguing phenomena. In few-layer graphene (FLG) systems, commensurability between honeycomb lattices on adjacent layers is regulated by their relative orientation angle θ, which is in turn dependent on sample preparation procedures. Because incommensurability suppresses interlayer hybridization, it is often claimed that graphene layers can be electrically isolated by a relative twist, even though they are vertically separated by a fraction of a nanometer. We present a theory of interlayer transport in FLG systems which reveals a richer picture in which the specific conductance depends sensitively on θ, single-layer Bloch-state lifetime, in-plane magnetic field, and bias voltage. We find that linear and differential conductances are generally large and negative near commensurate values of θ, and small and positive otherwise. We show that accounting for interlayer coupling may be essential for describing transport in FLG despite its physically insignificant effect on the band structure of the system.

52 Exact continuum model for low-energy electronic states of twisted bilayer graphene (kp methed)

Ref

DOI: 10.1103/PhysRevResearch.1.013001

摘要

We introduce a complete physical model for the single-particle electronic structure of twisted bilayer graphene (TBLG), which incorporates the crucial role of lattice relaxation. Our model, based on k \dot p perturbation theory and openly available, combines the accuracy of density functional theory calculations through effective tight-binding Hamiltonians with the computational efficiency and complete control of the twist angle offered by continuum models. The inclusion of relaxation significantly changes the band structure at the first magic-angle twist corresponding to flat bands near the Fermi level (the “low-energy” states), and eliminates the appearance of a second magic-angle twist. We show that minimal models for the low-energy states of TBLG can be easily modified to capture the changes in electronic states as a function of twist angle.