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arXiv:2310.17612 · Physical review. B./Physical review. B

Steady-state topological order

X. W. Dai, Zijian Wang, He-Ran Wang, Zhong Wang

Published Mar 19, 2025• 8 citations
arXivHTMLPDF

Abstract

We investigate a generalization of topological order from closed systems to open systems, for which the steady states take the place of ground states. We construct concrete lattice models with steady-state topological order, and characterize them by complementary approaches based on topological degeneracy of steady states, topological entropy, dissipative gauge theory, and relaxation dynamics. We take the topological degeneracy of steady states as the defining property of steady-state topological order, and study its stability using degenerate perturbation theory. We show that while the topological degeneracy is fragile in two dimensions, it is robust against perturbations in three and higher dimensions. We unveil how and why nontrivial steady-state topological order is always endowed with unusual relaxation dynamics. Particularly, the relaxation time is algebraically divergent in the thermodynamic limit, which is indicative of a gapless Liouvillian spectrum. It is shown that steady-state topological order remains definable in the presence of (Liouvillian) gapless modes: Whereas the (Liouvillian) level splitting between topologically degenerate steady states is exponentially small with respect to the system size, the Liouvillian gap between the steady states and the rest of the spectrum decays algebraically as the system size grows. We also investigate the topological phase transition to the trivial phase under additional noise, and find that the lifting of topological degeneracy is accompanied by gapping out the gapless modes. The topological phase transition is also diagnosed from the perspective of dissipative gauge theory, where the topologically ordered phase is identified as the deconfined phase and the trivial phase as the confined phase. Our paper offers a toolbox for investigating open-system topology of steady states.

DOI
10.1103/physrevb.111.115142

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We investigate a generalization of topological order from closed systems to open systems, for which the steady states take the place of ground states. We construct concrete lattice models with steady-state topological order, and characterize them by complementary approaches based on topological degeneracy of steady states, topological entropy, dissipative gauge theory, and relaxation dynamics. We take the topological degeneracy of steady states as the defining property of steady-state topological order, and study its stability using degenerate perturbation theory. We show that while the topological degeneracy is fragile in two dimensions, it is robust against perturbations in three and higher dimensions. We unveil how and why nontrivial steady-state topological order is always endowed with unusual relaxation dynamics. Particularly, the relaxation time is algebraically divergent in the thermodynamic limit, which is indicative of a gapless Liouvillian spectrum. It is shown that steady-state topological order remains definable in the presence of (Liouvillian) gapless modes: Whereas the (Liouvillian) level splitting between topologically degenerate steady states is exponentially small with respect to the system size, the Liouvillian gap between the steady states and the rest of the spectrum decays algebraically as the system size grows. We also investigate the topological phase transition to the trivial phase under additional noise, and find that the lifting of topological degeneracy is accompanied by gapping out the gapless modes. The topological phase transition is also diagnosed from the perspective of dissipative gauge theory, where the topologically ordered phase is identified as the deconfined phase and the trivial phase as the confined phase. Our paper offers a toolbox for investigating open-system topology of steady states.
The interplay between dissipation and correlation can lead to novel emergent phenomena in open systems. Here we investigate “steady-state topological order” defined by the robust topological degeneracy of steady states, which is a generalization of the ground-state topological degeneracy of closed systems. Specifically, we construct two representative Liouvillians using engineered dissipation, and exactly solve the steady states with topological degeneracy. We find that while the steady-state topological degeneracy is fragile under noise in two dimensions, it is stable in three dimensions, where a genuine many-body phase with topological degeneracy is realized. We identify universal features of steady-state topological physics such as the deconfined emergent gauge field and slow relaxation dynamics of topological defects. The transition from a topologically ordered phase to a trivial phase is also investigated via numerical simulation. Our work highlights the essential difference between ground-state topological order in closed systems and steady-state topological order in open systems.
We study a new kind of ordering — topological order — in rigid states (the states with no local gapless excitations). We concentrate on characterization of the different topological orders. As an example we discuss in detail chiral spin states of 2 + 1 dimensional spin systems. Chiral spin states are described by the topological Chern-Simons theories in the continuum limit. We show that the topological orders can be characterized by a non-Abelian gauge structure over the moduli space which parametrizes a family of the model Hamiltonians supporting topologically ordered ground states. In 2 + 1 dimensions, the non-Abelian gauge structure determines possible fractional statistics of the quasi-particle excitations over the topologically ordered ground states. The dynamics of the low lying global excitations is shown to be independent of random spatial dependent perturbations. The ground state degeneracy and the non-Abelian gauge structures discussed in this paper are very robust, even against those perturbations that break translation symmetry. We also discuss the symmetry properties of the degenerate ground states of chiral spin states. We find that some degenerate ground states of chiral spin states on torus carry non-trivial quantum numbers of the 90° rotation.
Topological order represents a profound state of matter characterized by exotic properties robust against local perturbations, such as fractionalized excitations and ground state degeneracy dependent on system topology. While extensively studied in closed quantum systems, the detection and characterization of topological order in open quantum systems, which inevitably interact with their environment, remain a significant challenge. Environmental coupling typically introduces decoherence and dissipation, potentially obscuring or even destroying fragile quantum correlations. This paper investigates the concept of "entropic fingerprints" as a robust diagnostic tool for identifying topological order in systems subjected to environmental influences. We explore how various entanglement measures, particularly the topological entanglement entropy and related information-theoretic quantities, can be adapted and reinterpreted in the context of non-equilibrium steady states or dynamically evolving open systems. Through a theoretical framework employing Lindblad master equations and tensor network methods, we demonstrate that universal signatures of topological order, though potentially modified, can persist in the steady-state entanglement properties of open systems. Our findings suggest that specific patterns in the bipartite entanglement entropy, conditional entropies, and mutual information, robust against a finite range of environmental parameters, can serve as reliable indicators of underlying topological phases. This research not only offers a pathway for experimentally probing topological materials in realistic, noisy environments but also deepens our fundamental understanding of how quantum information is preserved or degraded under open system dynamics.
The growing field of topological orders has been extensively studied both form the communities of condensed matter and quantum simulation. However, very little is known about the fate of topological order in the presence of disturbing effects such as external noise or dissipation. In the first part of this thesis, we start by studying how the edge states of a topological insulator become unstable when interacting with thermal baths. Motivated by these results, we generalise the notion of Chern insulators from the well-known Hamiltonian case to Liouvillian dynamics. We achieve this goal by defining a new topological witness that is still related to the quantum Hall conductivity at finite temperature. The mixed character of edge states is also well captured by our formalism, and explicit models for topological insulators and dissipative channels are considered. Additionally, we find new topological phases that remain quantised at finite temperature. The construction is based on the Uhlmann phase, a geometric quantum phase defined for general density matrices. Using this new tool, we are able to characterise topological insulators and superconductors at finite temperature both in one and two spatial dimensions. From the experimental side, we propose a state-independent protocol to measure the topological Uhlmann phase in the context of quantum simulation. Symmetry-protected topological orders have traditionally emerged from shortrange interactions. It remains very much unknown what the role played by longrange interactions is, within the physics of these topological systems. In the second part of this thesis, we analyse how topological superconducting phases are affected by the inclusion of long-range couplings. Remarkably, we unveil new topological quasi-particles due to long-range interactions, that were absent in short-range models. We also study how topological invariants are modified by the presence of long-range effects. In the appendix section of the thesis, we explore new numerical methods for driven-dissipative phase transitions. We consider quantum systems with a dissipative term driving the system into a non-equilibrium steady state. The inclusion of short-range fluctuations out-of-equilibrium deeply modifies the shape of the phase-diagram, something never observed in equilibrium thermodynamics.
Topological order is a new paradigm for quantum phases of matter developed to explain phase transitions which do not fit the symmetry breaking scheme for classifying phases of matter. They are characterized by patterns of entanglement that lead to topologically depended ground state degeneracy and anyonic excitations. One common approach for studying such phases in two-dimensional systems is through exactly solvable lattice Hamiltonian models such as quantum double models and String-Net models. The former can be understood as the Hamiltonian formulation of lattice gauge theories and, as such, it is defined by a finite gauge group. However, not much is known about topological phases in tridimensional systems. Motivated by this we develop a new class of three-dimensional exactly solvable models which go beyond quantum double models by using finite crossed modules instead of gauge groups. This approach relies on a lattice implementation of 2-gauge theory to obtain models with a richer topological structure. We construct the Hamiltonian model explicitly and provide a rigorous proof that the ground state degeneracy is a topological invariant and that the ground states can only be characterized with nonlocal order parameters.
This study delves into the concept of quantum phases in open quantum systems, examining the shortcomings of existing approaches that focus on steady states of Lindbladians and highlighting their limitations in capturing key phase transitions. In contrast to these methods, we introduce the concept of imaginary-time Lindbladian evolution as an alternative framework. This new approach defines gapped quantum phases in open systems through the spectrum properties of the imaginary-Liouville superoperator. We find that, in addition to all pure gapped ground states, the Gibbs state of a stabilizer Hamiltonian at any finite temperature can also be characterized by our scheme, demonstrated through explicit construction. Moreover, the closing of the imaginary Liouville gap is associated with the divergence of the Markov length, which has recently been proposed as an indicator of phase transitions in open quantum systems. To illustrate the effectiveness of this framework, we apply it to investigate the phase diagram for open systems withZ2σ×Z2τsymmetry, including cases with nontrivial average symmetry protected topological order or spontaneous symmetry breaking order. Our findings demonstrate universal properties at quantum criticality, such as nonanalytic behaviors of steady-state observables, divergence of correlation lengths, and closing of the imaginary-Liouville gap. These results advance our understanding of quantum phase transitions in open quantum systems. In contrast, we find that the steady states of real-time Lindbladians do not provide an effective framework for characterizing phase transitions in open systems.