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

Dynamical topology of quantum quenches in two dimensions

Haiping Hu, Erhai Zhao

Published Nov 6, 2019• 0 citations
OpenAlex

Abstract

Recent experiments began to explore the topological properties of quench dynamics, i.e. the time evolution following a sudden change in the Hamiltonian, via tomography of quantum gases in optical lattices. In contrast to the well established theory for static band insulators or periodically driven systems, at present it is not clear whether, and how, topological invariants can be defined for a general quench of band insulators. Previous work solved a special case of this problem beautifully using Hopf mapping of two-band Hamiltonians in two dimensions. But it only works for topologically trivial initial state and is hard to generalize to multiband systems or other dimensions. Here we introduce the concept of loop unitary constructed from the unitary time-evolution operator, and show its homotopy invariant fully characterizes the dynamical topology. For two-band systems in two dimensions, we prove that the invariant is precisely equal to the change in the Chern number across the quench regardless of the initial state. We further show that the nontrivial dynamical topology manifests as hedgehog defects in the loop unitary, and also as winding and linking of its eigenvectors along a curve where dynamical quantum phase transition occurs. This opens up a systematic route to classify and characterize quantum quench dynamics.

Related papers

Topological Invariants for Quantum Quench Dynamics from Unitary Evolution

Haiping Hu, Erhai Zhao

Recent experiments began to explore the topological properties of quench dynamics, i.e., the time evolution following a sudden change in the Hamiltonian, via tomography of quantum gases in optical lattices. In contrast to the well-established theory for static band insulators or periodically driven systems, at present it is not clear whether, and how, topological invariants can be defined for a general quench of band insulators. Previous work solved a special case of this problem beautifully using Hopf mapping of two-band Hamiltonians in two dimensions. However, it only works for a topologically trivial initial state and is hard to generalize to multiband systems or other dimensions. Here we introduce the concept of loop unitary constructed from the unitary time-evolution operator and show its homotopy invariant fully characterizes the dynamical topology. For two-band systems in two dimensions, we prove that the invariant is precisely equal to the change in the Chern number across the quench, regardless of the initial state. We further show that the nontrivial dynamical topology manifests as hedgehog defects in the loop unitary and also as winding and linking of its eigenvectors along a curve where dynamical quantum phase transition occurs. This opens up a systematic route to classify and characterize quantum quench dynamics.

arXiv:1911.02211•Physical Review Letters•Apr 21, 2020•61 citations

Quench dynamics of Hopf insulators

Haiping Hu, Chao Yang, Erhai Zhao

Hopf insulators are exotic topological states of matter outside the standard tenfold-way classification based on discrete symmetries. Its topology is captured by an integer invariant that describes the linking structures of the Hamiltonian in the three-dimensional momentum space. In this paper, we investigate the quantum dynamics of Hopf insulators across a sudden quench and show that the quench dynamics is characterized by a ${\mathbb{Z}}_{2}$ invariant $\ensuremath{\nu}$ which reveals a rich interplay between quantum quench and static band topology. We construct the ${\mathbb{Z}}_{2}$ topological invariant using the loop unitary operator and prove that $\ensuremath{\nu}$ relates the pre- and postquench Hopf invariants through $\ensuremath{\nu}=(\mathcal{L}\ensuremath{-}{\mathcal{L}}_{0})\phantom{\rule{0.28em}{0ex}}mod\phantom{\rule{0.28em}{0ex}}2$. The ${\mathbb{Z}}_{2}$ nature of the dynamical invariant is in sharp contrast to the $\mathbb{Z}$ invariant for the quench dynamics of Chern insulators in two dimensions. The nontrivial dynamical topology is further attributed to the emergence of $\ensuremath{\pi}$ defects in the phase band of the loop unitary. These $\ensuremath{\pi}$ defects are generally closed curves in the momentum-time space, for example, as nodal rings carrying Hopf charge.

arXiv:1912.03436•Physical review. B./Physical review. B•Apr 24, 2020•9 citations

Topological quantum quench dynamics carrying arbitrary Hopf and second Chern numbers

Motohiko Ezawa

A quantum quench is a nonequilibrium dynamics governed by the unitary evolution. We propose a two-band model whose quench dynamics is characterized by an arbitrary Hopf number belonging to the homotopy group ${\ensuremath{\pi}}_{3}({S}^{2})=\mathbb{Z}$. When we quench a system from an insulator with the Chern number ${C}_{i}\ensuremath{\in}{\ensuremath{\pi}}_{2}({S}^{2})=\mathbb{Z}$ to another insulator with the Chern number ${C}_{f}$, the preimage of the Hamiltonian vector forms links having the Hopf number ${C}_{f}\ensuremath{-}{C}_{i}$. We also investigate a quantum-quench dynamics for a four-band model carrying an arbitrary second Chern number $N\ensuremath{\in}{\ensuremath{\pi}}_{4}({S}^{4})=\mathbb{Z}$, which can be realized by quenching a three-dimensional topological insulator having the three-dimensional winding number $N\ensuremath{\in}{\ensuremath{\pi}}_{3}({S}^{3})=\mathbb{Z}$.

arXiv:1808.08069•Physical review. B./Physical review. B•Nov 9, 2018•22 citations

Topology and entanglement in quench dynamics

Po-Yao Chang

We classify the topology of the quench dynamics by homotopy groups. A relation between the topological invariant of a postquench order parameter and the topological invariant of a static Hamiltonian is shown in $d+1$ dimensions ($d=1,2,3$). The midgap states in the entanglement spectrum of the postquench states reveal their topological nature. When a trivial quantum state is under a sudden quench to a Chern insulator, the midgap states in the entanglement spectrum form rings. These rings are analogous to the boundary Fermi rings in the Hopf insulators. Finally, we show a postquench order parameter in 3+1 dimensions can be characterized by the second Chern number. The number of Dirac cones in the entanglement spectrum is equal to the second Chern number.

arXiv:1804.10625•Physical review. B./Physical review. B•Jun 11, 2018•32 citations

Linking invariant for the quench dynamics of a two-dimensional two-band Chern insulator

Xin Chen, Ce Wang, Jinlong Yu

We discuss the topological invariant in the ($2+1$)-dimensional quench dynamics of a two-dimensional two-band Chern insulator starting from a topological initial state (i.e., with a nonzero Chern number ${c}_{i}$), evolved by a postquench Hamiltonian (with Chern number ${c}_{f}$). This process is classified by the torus homotopy group ${\ensuremath{\tau}}_{3}({S}^{2})$. In contrast to the process with ${c}_{i}=0$ studied in previous works, this process cannot be characterized by the Hopf invariant that is described by the sphere homotopy group ${\ensuremath{\pi}}_{3}({S}^{2})=\mathbb{Z}$. It is possible, however, to calculate a variant of the Chern-Simons integral with a complementary part to cancel the Chern number of the initial spin configuration, which at the same time does not affect the ($2+1$)-dimensional topology. We show that the modified Chern-Simons integral gives rise to a topological invariant of this quench process, i.e., the linking invariant in the ${\mathbb{Z}}_{2{c}_{i}}$ class: $\ensuremath{\nu}=({c}_{f}\ensuremath{-}{c}_{i})\phantom{\rule{3.33333pt}{0ex}}mod\phantom{\rule{0.28em}{0ex}}(2{c}_{i})$. We give concrete examples to illustrate this result and also show the detailed deduction to get this linking invariant.

arXiv:1904.12552•Physical Review A•Mar 10, 2020•17 citations

Emergent topology and dynamical quantum phase transitions in two-dimensional closed quantum systems

Utso Bhattacharya, Amit Dutta

Dynamical quantum phase transitions (DQPTs) manifested in the nonanalyticities in the temporal evolution of a closed quantum system generated by the time-independent final Hamiltonian, following a quench (or ramping) of a parameter of the Hamiltonian, is an emerging frontier of nonequilibrium quantum dynamics. We, here, introduce the notion of a dynamical topological order parameter (DTOP) that characterizes these DQPTs occurring in quenched (or ramped) two-dimensional closed quantum systems; this is quite a nontrivial generalization of the notion of DTOP introduced in Budich and Heyl [Phys. Rev. B 93, 085416 (2016)] for one-dimensional situations. This DTOP is obtained from the ``gauge-invariant'' Pancharatnam phase extracted from the Loschmidt overlap, i.e., the modulus of the overlap between the initially prepared state and its time-evolved counterpart reached following a temporal evolution generated by the time-independent final Hamiltonian. This generic proposal is illustrated considering DQPTs occurring in the subsequent temporal evolution following a sudden quench of the staggered mass of the topological Haldane model on a hexagonal lattice where it stays fixed to zero or unity and makes a discontinuous jump between these two values at critical times at which DQPTs occur. What is remarkable is that while the topology of the equilibrium model is characterized by the Chern number, the emergent topology associated with the DQPTs is characterized by a generalized winding number.

arXiv:1701.03911•Physical review. B./Physical review. B•Jul 11, 2017•75 citations
Recent experiments began to explore the topological properties of quench dynamics, i.e. the time evolution following a sudden change in the Hamiltonian, via tomography of quantum gases in optical lattices. In contrast to the well established theory for static band insulators or periodically driven systems, at present it is not clear whether, and how, topological invariants can be defined for a general quench of band insulators. Previous work solved a special case of this problem beautifully using Hopf mapping of two-band Hamiltonians in two dimensions. But it only works for topologically trivial initial state and is hard to generalize to multiband systems or other dimensions. Here we introduce the concept of loop unitary constructed from the unitary time-evolution operator, and show its homotopy invariant fully characterizes the dynamical topology. For two-band systems in two dimensions, we prove that the invariant is precisely equal to the change in the Chern number across the quench regardless of the initial state. We further show that the nontrivial dynamical topology manifests as hedgehog defects in the loop unitary, and also as winding and linking of its eigenvectors along a curve where dynamical quantum phase transition occurs. This opens up a systematic route to classify and characterize quantum quench dynamics.
Recent experiments began to explore the topological properties of quench dynamics, i.e., the time evolution following a sudden change in the Hamiltonian, via tomography of quantum gases in optical lattices. In contrast to the well-established theory for static band insulators or periodically driven systems, at present it is not clear whether, and how, topological invariants can be defined for a general quench of band insulators. Previous work solved a special case of this problem beautifully using Hopf mapping of two-band Hamiltonians in two dimensions. However, it only works for a topologically trivial initial state and is hard to generalize to multiband systems or other dimensions. Here we introduce the concept of loop unitary constructed from the unitary time-evolution operator and show its homotopy invariant fully characterizes the dynamical topology. For two-band systems in two dimensions, we prove that the invariant is precisely equal to the change in the Chern number across the quench, regardless of the initial state. We further show that the nontrivial dynamical topology manifests as hedgehog defects in the loop unitary and also as winding and linking of its eigenvectors along a curve where dynamical quantum phase transition occurs. This opens up a systematic route to classify and characterize quantum quench dynamics.
Hopf insulators are exotic topological states of matter outside the standard tenfold-way classification based on discrete symmetries. Its topology is captured by an integer invariant that describes the linking structures of the Hamiltonian in the three-dimensional momentum space. In this paper, we investigate the quantum dynamics of Hopf insulators across a sudden quench and show that the quench dynamics is characterized by a Z2{\mathbb{Z}}_{2}Z2​ invariant $\ensuremath{\nu}$ which reveals a rich interplay between quantum quench and static band topology. We construct the Z2{\mathbb{Z}}_{2}Z2​ topological invariant using the loop unitary operator and prove that $\ensuremath{\nu}$ relates the pre- and postquench Hopf invariants through $\ensuremath{\nu}=(\mathcal{L}\ensuremath{-}{\mathcal{L}}_{0})\phantom{\rule{0.28em}{0ex}}mod\phantom{\rule{0.28em}{0ex}}2$. The Z2{\mathbb{Z}}_{2}Z2​ nature of the dynamical invariant is in sharp contrast to the Z\mathbb{Z}Z invariant for the quench dynamics of Chern insulators in two dimensions. The nontrivial dynamical topology is further attributed to the emergence of $\ensuremath{\pi}$ defects in the phase band of the loop unitary. These $\ensuremath{\pi}$ defects are generally closed curves in the momentum-time space, for example, as nodal rings carrying Hopf charge.
A quantum quench is a nonequilibrium dynamics governed by the unitary evolution. We propose a two-band model whose quench dynamics is characterized by an arbitrary Hopf number belonging to the homotopy group ${\ensuremath{\pi}}_{3}({S}^{2})=\mathbb{Z}$. When we quench a system from an insulator with the Chern number ${C}_{i}\ensuremath{\in}{\ensuremath{\pi}}_{2}({S}^{2})=\mathbb{Z}$ to another insulator with the Chern number Cf{C}_{f}Cf​, the preimage of the Hamiltonian vector forms links having the Hopf number ${C}_{f}\ensuremath{-}{C}_{i}$. We also investigate a quantum-quench dynamics for a four-band model carrying an arbitrary second Chern number $N\ensuremath{\in}{\ensuremath{\pi}}_{4}({S}^{4})=\mathbb{Z}$, which can be realized by quenching a three-dimensional topological insulator having the three-dimensional winding number $N\ensuremath{\in}{\ensuremath{\pi}}_{3}({S}^{3})=\mathbb{Z}$.
We classify the topology of the quench dynamics by homotopy groups. A relation between the topological invariant of a postquench order parameter and the topological invariant of a static Hamiltonian is shown in d+1d+1d+1 dimensions (d=1,2,3d=1,2,3d=1,2,3). The midgap states in the entanglement spectrum of the postquench states reveal their topological nature. When a trivial quantum state is under a sudden quench to a Chern insulator, the midgap states in the entanglement spectrum form rings. These rings are analogous to the boundary Fermi rings in the Hopf insulators. Finally, we show a postquench order parameter in 3+1 dimensions can be characterized by the second Chern number. The number of Dirac cones in the entanglement spectrum is equal to the second Chern number.
We discuss the topological invariant in the (2+12+12+1)-dimensional quench dynamics of a two-dimensional two-band Chern insulator starting from a topological initial state (i.e., with a nonzero Chern number ci{c}_{i}ci​), evolved by a postquench Hamiltonian (with Chern number cf{c}_{f}cf​). This process is classified by the torus homotopy group ${\ensuremath{\tau}}_{3}({S}^{2})$. In contrast to the process with ci=0{c}_{i}=0ci​=0 studied in previous works, this process cannot be characterized by the Hopf invariant that is described by the sphere homotopy group ${\ensuremath{\pi}}_{3}({S}^{2})=\mathbb{Z}$. It is possible, however, to calculate a variant of the Chern-Simons integral with a complementary part to cancel the Chern number of the initial spin configuration, which at the same time does not affect the (2+12+12+1)-dimensional topology. We show that the modified Chern-Simons integral gives rise to a topological invariant of this quench process, i.e., the linking invariant in the Z2ci{\mathbb{Z}}_{2{c}_{i}}Z2ci​​ class: $\ensuremath{\nu}=({c}_{f}\ensuremath{-}{c}_{i})\phantom{\rule{3.33333pt}{0ex}}mod\phantom{\rule{0.28em}{0ex}}(2{c}_{i})$. We give concrete examples to illustrate this result and also show the detailed deduction to get this linking invariant.
Dynamical quantum phase transitions (DQPTs) manifested in the nonanalyticities in the temporal evolution of a closed quantum system generated by the time-independent final Hamiltonian, following a quench (or ramping) of a parameter of the Hamiltonian, is an emerging frontier of nonequilibrium quantum dynamics. We, here, introduce the notion of a dynamical topological order parameter (DTOP) that characterizes these DQPTs occurring in quenched (or ramped) two-dimensional closed quantum systems; this is quite a nontrivial generalization of the notion of DTOP introduced in Budich and Heyl [Phys. Rev. B 93, 085416 (2016)] for one-dimensional situations. This DTOP is obtained from the ``gauge-invariant'' Pancharatnam phase extracted from the Loschmidt overlap, i.e., the modulus of the overlap between the initially prepared state and its time-evolved counterpart reached following a temporal evolution generated by the time-independent final Hamiltonian. This generic proposal is illustrated considering DQPTs occurring in the subsequent temporal evolution following a sudden quench of the staggered mass of the topological Haldane model on a hexagonal lattice where it stays fixed to zero or unity and makes a discontinuous jump between these two values at critical times at which DQPTs occur. What is remarkable is that while the topology of the equilibrium model is characterized by the Chern number, the emergent topology associated with the DQPTs is characterized by a generalized winding number.