We introduce a many-body topological invariant, called the topological disorder parameter (TDP), to characterize gapped quantum phases with global internal symmetry in $2+1$ dimensions. TDP is defined as the constant correction that appears in the ground-state expectation value of a partial symmetry transformation applied to a connected spatial region $M$ , the absolute value of which scales generically as $exp(\ensuremath{-}\ensuremath{\alpha}l+\ensuremath{\gamma})$ where $l$ is the perimeter of $M$ and $\ensuremath{\gamma}$ is the TDP. Motivated by a topological quantum field theory interpretation of the operator, we show that ${e}^{\ensuremath{\gamma}}$ can be related to the quantum dimension of the symmetry defect, and provide a general formula for $\ensuremath{\gamma}$ when the entanglement Hamiltonian of the topological phase can be described by a ( $1+1$ )-dimensional conformal field theory (CFT). A special case of TDP is equivalent to the topological R\'enyi entanglement entropy when the symmetry is the cyclic permutation of the replica of the gapped phase. We then investigate several examples of lattice models of topological phases, both analytically and numerically, in particular when the assumption of having a CFT edge theory is not satisfied. We also consider an example of partial translation symmetry in Wen's plaquette model and show that the result can be understood using the edge CFT. Our results establish an alternative tool to detect quantum topological order.We introduce a many-body topological invariant, called the topological disorder parameter (TDP), to characterize gapped quantum phases with global internal symmetry in $2+1$ dimensions. TDP is defined as the constant correction that appears in the ground-state expectation value of a partial symmetry transformation applied to a connected spatial region $M$, the absolute value of which scales generically as $exp(\ensuremath{-}\ensuremath{\alpha}l+\ensuremath{\gamma})$ where $l$ is the perimeter of $M$ and $\ensuremath{\gamma}$ is the TDP. Motivated by a topological quantum field theory interpretation of the operator, we show that ${e}^{\ensuremath{\gamma}}$ can be related to the quantum dimension of the symmetry defect, and provide a general formula for $\ensuremath{\gamma}$ when the entanglement Hamiltonian of the topological phase can be described by a ($1+1$)-dimensional conformal field theory (CFT). A special case of TDP is equivalent to the topological R\'enyi entanglement entropy when the symmetry is the cyclic permutation of the replica of the gapped phase. We then investigate several examples of lattice models of topological phases, both analytically and numerically, in particular when the assumption of having a CFT edge theory is not satisfied. We also consider an example of partial translation symmetry in Wen's plaquette model and show that the result can be understood using the edge CFT. Our results establish an alternative tool to detect quantum topological order.

arXiv:2203.08847Physical review. B./Physical review. BSep 12, 202231 citations

Disordered topological crystalline phases

Ian Mondragon-Shem, Taylor L. Hughes

APSJan 1, 20160 citations

Topological phases and delocalization of quantum walks in random environments

Hideaki Obuse, Norio Kawakami

We investigate one-dimensional (1D) discrete-time quantum walks (QWs) with spatially or temporally random defects as a consequence of interactions with random environments. We focus on the QWs with chiral symmetry in a topological phase, and reveal that chiral symmetry together with the bipartite nature of the QWs brings about intriguing behaviors such as coexistence of topologically protected edge states at zero energy and Anderson transitions in the 1D chiral class at nonzero energy in their dynamics. In contrast to results of previous studies, therefore, the spatially disordered QWs can avoid complete localization due to the Anderson transition. It is further confirmed that the edge states are robust to spatial disorder but not to temporal disorder.We investigate one-dimensional (1D) discrete-time quantum walks (QWs) with spatially or temporally random defects as a consequence of interactions with random environments. We focus on the QWs with chiral symmetry in a topological phase, and reveal that chiral symmetry together with the bipartite nature of the QWs brings about intriguing behaviors such as coexistence of topologically protected edge states at zero energy and Anderson transitions in the 1D chiral class at nonzero energy in their dynamics. In contrast to results of previous studies, therefore, the spatially disordered QWs can avoid complete localization due to the Anderson transition. It is further confirmed that the edge states are robust to spatial disorder but not to temporal disorder.

arXiv:1103.5545Physical Review BNov 28, 2011111 citations

Topological Phases with Average Symmetries: The Decohered, the Disordered, and the Intrinsic

Ruochen Ma, Jian-Hao Zhang, Zhen Bi +2 more

Global symmetries greatly enrich the landscape of topological quantum phases, playing an essential role from topological insulators to fractional quantum Hall effect. Topological phases in mixed quantum states, originating from decoherence in open quantum systems or disorders in imperfect crystalline solids, have recently garnered significant interest. Unlike pure states, mixed quantum states can exhibit average symmetries—symmetries that keep the total ensemble invariant but not on each individual state. In this work, we present a systematic classification and characterization of average symmetry-protected topological (ASPT) phases applicable to generic symmetry groups, encompassing both average and exact symmetries, for bosonic and fermionic systems. Moreover, we formulate the theory of average symmetry-enriched topological (ASET) orders in disordered bosonic systems. Our systematic approach helps clarify nuanced issues in previous literature and uncovers compelling new physics. Notably, we discover that (1) the definition and classification of ASPT phases in decohered and disordered systems exhibit subtle differences, (2) despite these differences, ASPT phases in both settings can be classified and characterized under a unified framework of defect decoration and spectral sequence, (3) this systematic classification uncovers a plethora of ASPT phases that are intrinsically mixed, implying they can exclusively manifest in decohered or disordered systems where part of the symmetry is average, and (4) similarly for ASET, we find intrinsically disordered phases exhibiting exotic anyon behaviors—the ground states of such phases necessarily contain localized anyons, with gapless (yet still localized) excitation blue spectra.Global symmetries greatly enrich the landscape of topological quantum phases, playing an essential role from topological insulators to fractional quantum Hall effect. Topological phases in mixed quantum states, originating from decoherence in open quantum systems or disorders in imperfect crystalline solids, have recently garnered significant interest. Unlike pure states, mixed quantum states can exhibit average symmetries—symmetries that keep the total ensemble invariant but not on each individual state. In this work, we present a systematic classification and characterization of average symmetry-protected topological (ASPT) phases applicable to generic symmetry groups, encompassing both average and exact symmetries, for bosonic and fermionic systems. Moreover, we formulate the theory of average symmetry-enriched topological (ASET) orders in disordered bosonic systems. Our systematic approach helps clarify nuanced issues in previous literature and uncovers compelling new physics. Notably, we discover that (1) the definition and classification of ASPT phases in decohered and disordered systems exhibit subtle differences, (2) despite these differences, ASPT phases in both settings can be classified and characterized under a unified framework of defect decoration and spectral sequence, (3) this systematic classification uncovers a plethora of ASPT phases that are intrinsically mixed, implying they can exclusively manifest in decohered or disordered systems where part of the symmetry is average, and (4) similarly for ASET, we find intrinsically disordered phases exhibiting exotic anyon behaviors—the ground states of such phases necessarily contain localized anyons, with gapless (yet still localized) excitation blue spectra.

arXiv:2305.16399Physical Review XMay 21, 202548 citations

Topological Phases and Phase Transitions in Clean and Disordered Systems

Jakub Mildner

Nov 1, 20240 citations

We introduce a many-body topological invariant, called the topological disorder parameter (TDP), to characterize gapped quantum phases with global internal symmetry in

2+1

dimensions. TDP is defined as the constant correction that appears in the ground-state expectation value of a partial symmetry transformation applied to a connected spatial region

M

, the absolute value of which scales generically as $exp(\ensuremath{-}\ensuremath{\alpha}l+\ensuremath{\gamma})$ where

l

is the perimeter of

M

and $\ensuremath{\gamma}$ is the TDP. Motivated by a topological quantum field theory interpretation of the operator, we show that ${e}^{\ensuremath{\gamma}}$ can be related to the quantum dimension of the symmetry defect, and provide a general formula for $\ensuremath{\gamma}$ when the entanglement Hamiltonian of the topological phase can be described by a (

1+1

)-dimensional conformal field theory (CFT). A special case of TDP is equivalent to the topological R\'enyi entanglement entropy when the symmetry is the cyclic permutation of the replica of the gapped phase. We then investigate several examples of lattice models of topological phases, both analytically and numerically, in particular when the assumption of having a CFT edge theory is not satisfied. We also consider an example of partial translation symmetry in Wen's plaquette model and show that the result can be understood using the edge CFT. Our results establish an alternative tool to detect quantum topological order.