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arXiv:1606.06673 · Physical review. B./Physical review. B

Locating topological phase transitions using nonequilibrium signatures in local bulk observables

Sthitadhi Roy, Roderich Moessner, Arnab Das

Published Jan 19, 2017• 26 citations
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Abstract

Topological quantum phases cannot be characterized by local order parameters in the bulk. In this work, however, we show that nonanalytic signatures of a topological quantum critical point do remain in local observables in the bulk, and manifest themselves as nonanalyticities in their expectation values taken over a family of nonequilibrium states generated using a quantum quench protocol. The signature can be used for precisely locating the critical points in parameter space. A large class of initial states can be chosen for the quench, including finite temperature states. We demonstrate these results in tractable models of noninteracting fermions exhibiting topological phase transitions in one and two spatial dimensions. We also show that the nonanalyticities can be absent if the gap closing is nontopological, i.e., when it corresponds to no phase transition.

DOI
10.1103/physrevb.95.041105

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Topological quantum phases cannot be characterized by local order parameters in the bulk. In this work, however, we show that nonanalytic signatures of a topological quantum critical point do remain in local observables in the bulk, and manifest themselves as nonanalyticities in their expectation values taken over a family of nonequilibrium states generated using a quantum quench protocol. The signature can be used for precisely locating the critical points in parameter space. A large class of initial states can be chosen for the quench, including finite temperature states. We demonstrate these results in tractable models of noninteracting fermions exhibiting topological phase transitions in one and two spatial dimensions. We also show that the nonanalyticities can be absent if the gap closing is nontopological, i.e., when it corresponds to no phase transition.
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Quantum phase transitions are central to our understanding of why matter at very low temperatures can exhibit starkly different properties upon small changes of microscopic parameters.Accurately locating those transitions is challenging experimentally and theoretically.Here, we show that the antithetic strategy of forcing systems out of equilibrium via sudden quenches provides a route to locate quantum phase transitions.Specifically, we show that such transitions imprint distinctive features in the intermediate-time dynamics, and results after equilibration, of local observables in quantum chaotic spin chains.Furthermore, we show that the effective temperature in the expected thermal-like states after equilibration can exhibit minima in the vicinity of the quantum critical points.We discuss how to test our results in experiments with Rydberg atoms and explore nonequilibrium signatures of quantum critical points in models with topological transitions.
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.
We demonstrate the existence of topological phase transitions in interacting, symmetry-protected quantum matter at finite temperatures. Using a combined numerical and analytical approach, we study a one-dimensional Su-Schrieffer-Heeger model with added Hubbard interactions, where no thermodynamic phase transition occurs at finite temperatures. The transition is signalled by a quantized, nonlocal bulk topological order parameter. It is driven by defects, which are enabled by the combination of interaction and thermal activation, with no counterpart in the noninteracting limit. The defects localize topological zero modes, which, when sufficiently abundant, cause the order parameter to vanish. This phenomenon, interpreted via bulk-boundary correspondence, reflects the loss of a topological edge mode at a well-defined critical temperature in the thermodynamic limit. Unlike zero-temperature topological transitions, these finite-temperature transitions lack thermodynamic signatures but remain observable in controlled quantum systems, such as ultracold fermionic atoms in optical lattices.
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.
The presence of nonanalyticity in observables is a manifestation of phase transitions. Through the study of two paradigmatic topological models in one and two dimensions, in this work we show that the circuit complexity based on our specific quantification can reveal the occurrence of topological phase transitions, both in and out of equilibrium, by the presence of nonanalyticity. By quenching the system out of equilibrium, we find that the circuit complexity grows linearly or quadratically in the short-time regime if the quench is finished instantaneously or in a finite time, respectively. Notably, we find that for both the sudden quench and the finite-time quench, a topological phase transition in the pre-quench Hamiltonian will be manifested by the presence of nonanalyticity in the first-order or second-order derivative of circuit complexity with respect to time in the short-time regime, and a topological phase transition in the post-quench Hamiltonian will be manifested by the presence of nonanalyticity in the steady value of circuit complexity in the long-time regime. We also show that the increase of dimension does not remove, but only weakens the nonanalyticity of circuit complexity. Our findings can be tested in quantum simulators and cold-atom systems.