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

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

Souvik Kumar Bandyopadhyay, Amit Dutta

Published May 4, 2020• 0 citations
OpenAlex

Abstract

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.

Related papers

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

Souvik Kumar Bandyopadhyay, Amit Dutta

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.

arXiv:2005.01455•Physical review. B./Physical review. B•Sep 2, 2020•8 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

Dynamical bulk-boundary correspondence and dynamical quantum phase transitions in higher-order topological insulators

Tomasz Masłowski, Nicholas Sedlmayr

Dynamical quantum phase transitions occur in dynamically evolving quantum systems when nonanalyticities occur at critical times in the return rate, a dynamical analog of the free energy. This extension of the concept of phase transitions can be brought into contact with another, namely, that of topological phase transitions in which the phase transition is marked by a change in a topological invariant. Following a quantum quench dynamical quantum phase transitions can happen in topological matter, a fact which has already been explored in one-dimensional topological insulators and in two-dimensional Chern insulators. Additionally, in one-dimensional systems a dynamical bulk boundary correspondence has been seen, related to the periodic appearance of zero modes of the Loschmidt echo itself. Here we extend both of these concepts to two-dimensional higher-order topological matter in which the topologically protected boundary modes are corner modes. We consider a minimal model which encompasses all possible forms of higher-order topology in two-dimensional topological band structures. We find that dynamical quantum phase transitions can still occur, and can occur for quenches which cross both bulk and boundary gap closings. Furthermore, a dynamical bulk boundary correspondence is also found, which takes a different form to that in one dimension.

arXiv:2305.06241•Physical review. B./Physical review. B•Sep 21, 2023•19 citations

Non-equilibrium aspects of topological Floquet quantum systems

Lorenzo Privitera

The manifestations of topology are ubiquitous in condensed matter physics. One of the most striking examples is the integer quantum Hall effect, which served as a theoretical basis for developing many classes of new materials in the last years such as topological insulators. It has been realized that topology can manifest itself also in time-dependent quantum systems. The question that we try to address in this thesis is how much of these topological structures can be harnessed in the realistic dynamics of a periodically driven closed quantum system. We investigate two kinds of topological periodically driven systems Thouless pumping and Floquet Chern insulators. Our analysis shows that the nonequilibrium occupations of the eigenvalues of the Floquet Hamiltonian, the so called quasienergies, play a crucial role. In the case of Thouless pumping we analysed the pumped charge averaged over many cycles and discovered that the corrections to perfect quantization of the pumped charge are actually quadratic in the frequency of the driving for a suddenly switched-on driving. In the case of Floquet Chern insulators, we studied fermions on the honeycomb lattice subjected to a circularly polarised field. We found that while the quasienergy bulk states can be populated in a controlled way, edge states get unavoidable excitations due to exponentially small gaps preventing adiabaticity. These defects reflect in nonequilibrium currents flowing at the edges, that can be manipulated by tailoring the parameters of the periodic driving. Nevertheless, the nonequilibrium state undergoes a generalized form of topological transition, thanks to the controlled population of bulk states. We describe it with a generalized version of the Chern marker of Bianco and Resta

•Oct 16, 2017•1 citation

Dynamics of edge currents in a linearly quenched Haldane model

Sougata Mardanya, Utso Bhattacharya, Amit Agarwal +1 more

In a finite-time quantum quench of the Haldane model, the Chern number determining the topology of the bulk remains invariant, as long as the dynamics is unitary. Nonetheless, the corresponding boundary attribute, the edge current, displays interesting dynamics. For the case of sudden and adiabatic quenches the postquench edge current is solely determined by the initial and the final Hamiltonians, respectively. However for a finite-time ($\ensuremath{\tau}$) linear quench in a Haldane nanoribbon, we show that the evolution of the edge current from the sudden to the adiabatic limit is not monotonic in $\ensuremath{\tau}$ and has a turning point at a characteristic time scale $\ensuremath{\tau}={\ensuremath{\tau}}_{0}$. For small $\ensuremath{\tau}$, the excited states lead to a huge unidirectional surge in the edge current of both edges. On the other hand, in the limit of large $\ensuremath{\tau}$, the edge current saturates to its expected equilibrium ground-state value. This competition between the two limits lead to the observed nonmonotonic behavior. Interestingly, ${\ensuremath{\tau}}_{0}$ seems to depend only on the Semenoff mass and the Haldane flux. A similar dynamics for the edge current is also expected in other systems with topological phases.

arXiv:1801.07015•Physical review. B./Physical review. B•Mar 26, 2018•17 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
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.
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.
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.
Dynamical quantum phase transitions occur in dynamically evolving quantum systems when nonanalyticities occur at critical times in the return rate, a dynamical analog of the free energy. This extension of the concept of phase transitions can be brought into contact with another, namely, that of topological phase transitions in which the phase transition is marked by a change in a topological invariant. Following a quantum quench dynamical quantum phase transitions can happen in topological matter, a fact which has already been explored in one-dimensional topological insulators and in two-dimensional Chern insulators. Additionally, in one-dimensional systems a dynamical bulk boundary correspondence has been seen, related to the periodic appearance of zero modes of the Loschmidt echo itself. Here we extend both of these concepts to two-dimensional higher-order topological matter in which the topologically protected boundary modes are corner modes. We consider a minimal model which encompasses all possible forms of higher-order topology in two-dimensional topological band structures. We find that dynamical quantum phase transitions can still occur, and can occur for quenches which cross both bulk and boundary gap closings. Furthermore, a dynamical bulk boundary correspondence is also found, which takes a different form to that in one dimension.
The manifestations of topology are ubiquitous in condensed matter physics. One of the most striking examples is the integer quantum Hall effect, which served as a theoretical basis for developing many classes of new materials in the last years such as topological insulators. It has been realized that topology can manifest itself also in time-dependent quantum systems. The question that we try to address in this thesis is how much of these topological structures can be harnessed in the realistic dynamics of a periodically driven closed quantum system. We investigate two kinds of topological periodically driven systems Thouless pumping and Floquet Chern insulators. Our analysis shows that the nonequilibrium occupations of the eigenvalues of the Floquet Hamiltonian, the so called quasienergies, play a crucial role. In the case of Thouless pumping we analysed the pumped charge averaged over many cycles and discovered that the corrections to perfect quantization of the pumped charge are actually quadratic in the frequency of the driving for a suddenly switched-on driving. In the case of Floquet Chern insulators, we studied fermions on the honeycomb lattice subjected to a circularly polarised field. We found that while the quasienergy bulk states can be populated in a controlled way, edge states get unavoidable excitations due to exponentially small gaps preventing adiabaticity. These defects reflect in nonequilibrium currents flowing at the edges, that can be manipulated by tailoring the parameters of the periodic driving. Nevertheless, the nonequilibrium state undergoes a generalized form of topological transition, thanks to the controlled population of bulk states. We describe it with a generalized version of the Chern marker of Bianco and Resta
In a finite-time quantum quench of the Haldane model, the Chern number determining the topology of the bulk remains invariant, as long as the dynamics is unitary. Nonetheless, the corresponding boundary attribute, the edge current, displays interesting dynamics. For the case of sudden and adiabatic quenches the postquench edge current is solely determined by the initial and the final Hamiltonians, respectively. However for a finite-time ($\ensuremath{\tau}$) linear quench in a Haldane nanoribbon, we show that the evolution of the edge current from the sudden to the adiabatic limit is not monotonic in $\ensuremath{\tau}$ and has a turning point at a characteristic time scale $\ensuremath{\tau}={\ensuremath{\tau}}_{0}$. For small $\ensuremath{\tau}$, the excited states lead to a huge unidirectional surge in the edge current of both edges. On the other hand, in the limit of large $\ensuremath{\tau}$, the edge current saturates to its expected equilibrium ground-state value. This competition between the two limits lead to the observed nonmonotonic behavior. Interestingly, ${\ensuremath{\tau}}_{0}$ seems to depend only on the Semenoff mass and the Haldane flux. A similar dynamics for the edge current is also expected in other systems with topological phases.
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.