Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2005.01455 · Physical review. B./Physical review. B

Unitary preparation of many-body Chern insulators: Adiabatic bulk-boundary correspondence

Souvik Kumar Bandyopadhyay, Amit Dutta

Published Sep 2, 2020• 8 citations
arXivHTMLPDF

Abstract

We approach the long-standing problem of preparing an out-of-equilibrium many-body Chern insulator (CI) and associated bulk-boundary correspondence unitarily. Herein, this is addressed by constructing a dynamical many-body Chern invariant exploiting the property of the bulk macroscopic electric polarization (Resta polarization) of the CI. This Chern invariant defined from observable correlations is also established to topologically classify many-body Chern states in equilibrium. The nonequilibrium behavior of the invariant is probed by ramping the paradigmatic Haldane model of graphene from its trivial to the topological phase. We show that a nonlinear ramp may work more efficiently in approaching the topological state, thereby establishing the existence of an optimal topological state preparation. Furthermore, to ensure the near adiabatic dynamics across the quantum critical point, we propose a counterdiabatic scheme. The topological nature of the prepared state is firmly established by observing an emerging $U(1)$ topological charge. We also compute the edge current in the time-evolved state of the system under a semiperiodic boundary condition, and we clearly establish an adiabatic bulk-boundary correspondence that firmly ensconces the validity of the many-body invariant.

DOI
10.1103/physrevb.102.094301

Related papers

Generalised many body topological invariant and the unitary preparation of Chern insulators: An emergent bulk-boundary correspondence

Souvik Kumar Bandyopadhyay, Amit Dutta

Building on the non-uniqueness of the macroscopic electric polarisation in the topological phase of a Chern insulator, we construct a many body Chern invariant. Under a smooth unitary temporal evolution, this quantity evolves with time and successfully reflects the topology of the out-of-equilibrium state of the system by assuming an integer-quantised value in the adiabatic limit. Considering a linear ramping of the staggered Semenoff mass of the paradigmatic Haldane model of graphene, we illustrate that starting from a trivial state, it is indeed possible to obtain a topologically non-trivial state even by an adiabatic unitary driving protocol. We further propose a counter-diabatic protocol that enables us to suppress the otherwise inevitable diabatic excitations generated while crossing the gapless quantum critical point. Notably, the corresponding emergence of an edge current in the final evolved state of the model under a semi-periodic boundary condition necessitates the application of a novel counter-diabatic field manifested as anisotropic hopping in the real space Hamiltonian. The generalised topology of this dynamical Chern invariant takes into account the dynamical occupation of instantaneous single-particle bands, and consequently ensconces an emergent bulk-boundary correspondence that is expected to manifest in experiments.

•May 4, 2020•0 citations

Dissipative preparation of many-body Floquet Chern insulators

Souvik Kumar Bandyopadhyay, Amit Dutta

Considering coupling to a microstructured bath as a relaxation mechanism in a periodically driven dissipative Haldane model, we establish that the system may be tuned to a stroboscopic topological steady state at all finite temperatures. The amplitude and frequency of the periodic drive is so chosen that the Floquet Hamiltonian describing the Haldane model at stroboscopic instants of time in the unitary situation is topologically nontrivial. We establish that in the stroboscopic steady state, the system reaches a thermal state of the Floquet Hamiltonian at a controlled temperature. Further, it is observed that even with a coupling to a bath, remarkably a Chern insulator can indeed be prepared in a Chern nontrivial pure steady state which is expected to exhibit a stroboscopic bulk-boundary correspondence. Using the nonuniqueness of the macroscopic bulk electric polarization of a Chern insulator in its topological phase, we propose a generalized Chern invariant that reflects the topology of out-of-equilibrium many-body stroboscopic states of the Haldane model even in a dissipative ambience. The generalized topology of dynamical Chern insulators being dependent on single-particle correlations is expected to manifest in experiments probing many-body quantum observables.

arXiv:2005.09972•Physical review. B./Physical review. B•Nov 2, 2020•19 citations

Hall response and edge current dynamics in Chern insulators out of equilibrium

M. D. Caio, Nigel R. Cooper, M. J. Bhaseen

We investigate the transport properties of Chern insulators following a quantum quench between topological and nontopological phases. Recent works have shown that this yields an excited state for which the Chern number is preserved under unitary evolution. However, this does not imply the preservation of other physical observables, as we stressed in our previous work. Here we provide an analysis of the Hall response following a quantum quench in an isolated system, with explicit results for the Haldane model. We show that the Hall conductance is no longer related to the Chern number in the postquench state, in agreement with previous work. We also examine the dynamics of the edge currents in finite-size systems with open boundary conditions along one direction. We show that the late-time behavior is captured by a Generalized Gibbs Ensemble, after multiple traversals of the sample. We discuss the effects of generic open boundary conditions and confinement potentials.

arXiv:1604.04216•Physical review. B./Physical review. B•Oct 4, 2016•54 citations

Quantum cluster approach to the topological invariants in correlated Chern insulators

Zhao-Long Gu, Kai Li, Jian‐Xin Li

Abstract We detect the topological properties of Chern insulators with strong Coulomb interactions by use of cluster perturbation theory and variational cluster approach. The common scheme in previous studies only involves the calculation of the interacting bulk Chern number within the natural-unit cell by means of the so-called topological Hamiltonian. With close investigations on a prototype model, the half-filled Haldane Hubbard model, which is subject to both periodic and open boundary conditions, we uncover the unexpected failure of this scheme due to the explicit breaking of the translation symmetry. Instead, we assert that the faithful interacting bulk Chern number in the framework of quantum cluster approaches can be computed in the enlarged unit cell, which is free of the fault caused by the explicit translation symmetry breaking and consistent with the interacting bulk-edge correspondence.

arXiv:1512.05118•New Journal of Physics•Jun 18, 2019•6 citations

The Two Dimensional Dynamical Bulk Boundary Correspondence: Beyond Two Band Models

Tomasz Masłowski, Nicholas Sedlmayr

A dynamical equivalent of the bulk-boundary correspondence has been suggested to occur in one and two dimensional topological models following sudden quenches. Depending on the topological invariant of the time evolving and initial phases involved large boundary contributions to a dynamical free energy occur. Moreover they occur periodically between the critical times at which this dynamical free energy becomes non-analytic, \emph{i.e.}~at dynamical quantum phase transitions. At these critical times the eigenvalue spectrum of the Loschmidt matrix which underlies the dynamical free energy closes its gap. The boundary contributions are understood to be due to zero-modes or in-gap bands of this matrix, forming a close analogy with equilibrium topological models and their edge modes. The exact cause of this phenomena and its generality remain unknown. In this article we test the dynamical bulk-boundary correspondence for a complicated two dimensional topological superconductor with a rich phase diagram, allowing quenches between many different Chern numbers. We show that there is no straightforward correspondence between the equilibrium phases quenched between and the dynamical bulk boundary correspondence. Furthermore the correspondence can depend on the orientation of the edges, suggesting a possible weak topological variant.

arXiv:2606.10731•Jun 9, 2026•0 citations

Electric polarization and discrete shift from boundary and corner charge in crystalline Chern insulators

Yuxuan Zhang, Maissam Barkeshli

Recently, it has been shown how topological phases of matter with crystalline symmetry and $U(1)$ charge conservation can be partially characterized by a set of many-body invariants, the discrete shift ${\mathcal{S}}_{\mathrm{o}}$ and electric polarization ${\stackrel{P\vec}{\mathcal{P}}}_{\mathrm{o}}$, where $\mathrm{o}$ labels a high-symmetry point. Crucially, these can be defined even with nonzero Chern number and/or magnetic field. One manifestation of these invariants is through quantized fractional contributions to the charge in the vicinity of a lattice disclination or dislocation. In this paper, we show that these invariants can also be extracted from the length and corner dependence of the total charge (mod 1) on the boundary of the system. We provide a general formula in terms of ${\mathcal{S}}_{\mathrm{o}}$ and ${\stackrel{P\vec}{\mathcal{P}}}_{\mathrm{o}}$ for the total charge of any subregion of the system which can include full boundaries or bulk lattice defects, unifying boundary, corner, disclination, and dislocation charge responses into a single general theory. These results hold for Chern insulators, despite their gapless chiral edge modes, and for which an unambiguous definition of an intrinsically two-dimensional electric polarization has been unclear until recently. We also discuss how our theory can fully characterize the topological response of quadrupole insulators.

arXiv:2410.03821•Physical review. B./Physical review. B•Feb 27, 2025•4 citations
We approach the long-standing problem of preparing an out-of-equilibrium many-body Chern insulator (CI) and associated bulk-boundary correspondence unitarily. Herein, this is addressed by constructing a dynamical many-body Chern invariant exploiting the property of the bulk macroscopic electric polarization (Resta polarization) of the CI. This Chern invariant defined from observable correlations is also established to topologically classify many-body Chern states in equilibrium. The nonequilibrium behavior of the invariant is probed by ramping the paradigmatic Haldane model of graphene from its trivial to the topological phase. We show that a nonlinear ramp may work more efficiently in approaching the topological state, thereby establishing the existence of an optimal topological state preparation. Furthermore, to ensure the near adiabatic dynamics across the quantum critical point, we propose a counterdiabatic scheme. The topological nature of the prepared state is firmly established by observing an emerging U(1)U(1)U(1) topological charge. We also compute the edge current in the time-evolved state of the system under a semiperiodic boundary condition, and we clearly establish an adiabatic bulk-boundary correspondence that firmly ensconces the validity of the many-body invariant.
Building on the non-uniqueness of the macroscopic electric polarisation in the topological phase of a Chern insulator, we construct a many body Chern invariant. Under a smooth unitary temporal evolution, this quantity evolves with time and successfully reflects the topology of the out-of-equilibrium state of the system by assuming an integer-quantised value in the adiabatic limit. Considering a linear ramping of the staggered Semenoff mass of the paradigmatic Haldane model of graphene, we illustrate that starting from a trivial state, it is indeed possible to obtain a topologically non-trivial state even by an adiabatic unitary driving protocol. We further propose a counter-diabatic protocol that enables us to suppress the otherwise inevitable diabatic excitations generated while crossing the gapless quantum critical point. Notably, the corresponding emergence of an edge current in the final evolved state of the model under a semi-periodic boundary condition necessitates the application of a novel counter-diabatic field manifested as anisotropic hopping in the real space Hamiltonian. The generalised topology of this dynamical Chern invariant takes into account the dynamical occupation of instantaneous single-particle bands, and consequently ensconces an emergent bulk-boundary correspondence that is expected to manifest in experiments.
Considering coupling to a microstructured bath as a relaxation mechanism in a periodically driven dissipative Haldane model, we establish that the system may be tuned to a stroboscopic topological steady state at all finite temperatures. The amplitude and frequency of the periodic drive is so chosen that the Floquet Hamiltonian describing the Haldane model at stroboscopic instants of time in the unitary situation is topologically nontrivial. We establish that in the stroboscopic steady state, the system reaches a thermal state of the Floquet Hamiltonian at a controlled temperature. Further, it is observed that even with a coupling to a bath, remarkably a Chern insulator can indeed be prepared in a Chern nontrivial pure steady state which is expected to exhibit a stroboscopic bulk-boundary correspondence. Using the nonuniqueness of the macroscopic bulk electric polarization of a Chern insulator in its topological phase, we propose a generalized Chern invariant that reflects the topology of out-of-equilibrium many-body stroboscopic states of the Haldane model even in a dissipative ambience. The generalized topology of dynamical Chern insulators being dependent on single-particle correlations is expected to manifest in experiments probing many-body quantum observables.
We investigate the transport properties of Chern insulators following a quantum quench between topological and nontopological phases. Recent works have shown that this yields an excited state for which the Chern number is preserved under unitary evolution. However, this does not imply the preservation of other physical observables, as we stressed in our previous work. Here we provide an analysis of the Hall response following a quantum quench in an isolated system, with explicit results for the Haldane model. We show that the Hall conductance is no longer related to the Chern number in the postquench state, in agreement with previous work. We also examine the dynamics of the edge currents in finite-size systems with open boundary conditions along one direction. We show that the late-time behavior is captured by a Generalized Gibbs Ensemble, after multiple traversals of the sample. We discuss the effects of generic open boundary conditions and confinement potentials.
Abstract We detect the topological properties of Chern insulators with strong Coulomb interactions by use of cluster perturbation theory and variational cluster approach. The common scheme in previous studies only involves the calculation of the interacting bulk Chern number within the natural-unit cell by means of the so-called topological Hamiltonian. With close investigations on a prototype model, the half-filled Haldane Hubbard model, which is subject to both periodic and open boundary conditions, we uncover the unexpected failure of this scheme due to the explicit breaking of the translation symmetry. Instead, we assert that the faithful interacting bulk Chern number in the framework of quantum cluster approaches can be computed in the enlarged unit cell, which is free of the fault caused by the explicit translation symmetry breaking and consistent with the interacting bulk-edge correspondence.
A dynamical equivalent of the bulk-boundary correspondence has been suggested to occur in one and two dimensional topological models following sudden quenches. Depending on the topological invariant of the time evolving and initial phases involved large boundary contributions to a dynamical free energy occur. Moreover they occur periodically between the critical times at which this dynamical free energy becomes non-analytic, \emph{i.e.}~at dynamical quantum phase transitions. At these critical times the eigenvalue spectrum of the Loschmidt matrix which underlies the dynamical free energy closes its gap. The boundary contributions are understood to be due to zero-modes or in-gap bands of this matrix, forming a close analogy with equilibrium topological models and their edge modes. The exact cause of this phenomena and its generality remain unknown. In this article we test the dynamical bulk-boundary correspondence for a complicated two dimensional topological superconductor with a rich phase diagram, allowing quenches between many different Chern numbers. We show that there is no straightforward correspondence between the equilibrium phases quenched between and the dynamical bulk boundary correspondence. Furthermore the correspondence can depend on the orientation of the edges, suggesting a possible weak topological variant.
Recently, it has been shown how topological phases of matter with crystalline symmetry and U(1)U(1)U(1) charge conservation can be partially characterized by a set of many-body invariants, the discrete shift So{\mathcal{S}}_{\mathrm{o}}So​ and electric polarization ${\stackrel{P\vec}{\mathcal{P}}}_{\mathrm{o}}$, where o\mathrm{o}o labels a high-symmetry point. Crucially, these can be defined even with nonzero Chern number and/or magnetic field. One manifestation of these invariants is through quantized fractional contributions to the charge in the vicinity of a lattice disclination or dislocation. In this paper, we show that these invariants can also be extracted from the length and corner dependence of the total charge (mod 1) on the boundary of the system. We provide a general formula in terms of So{\mathcal{S}}_{\mathrm{o}}So​ and ${\stackrel{P\vec}{\mathcal{P}}}_{\mathrm{o}}$ for the total charge of any subregion of the system which can include full boundaries or bulk lattice defects, unifying boundary, corner, disclination, and dislocation charge responses into a single general theory. These results hold for Chern insulators, despite their gapless chiral edge modes, and for which an unambiguous definition of an intrinsically two-dimensional electric polarization has been unclear until recently. We also discuss how our theory can fully characterize the topological response of quadrupole insulators.