Gapless Symmetry-Protected Topological Order

Scaffidi, Thomas; Parker, Daniel E.; Vasseur, Romain

doi:10.1103/physrevx.7.041048

Cited by 116 publications

(131 citation statements)

References 138 publications

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“…We now observe that, due to (16), the vertex operator e iπΦ(x) introduces a π kink in the phase Θ and hence from (17) it creates π kinks in the phases of all the flavors, θ a . We therefore conclude that switching the parity of all N chains simultaneously results in inserting a current N .…”

Section: Low-energy Description Of the Topological Phasementioning

confidence: 73%

From one-dimensional charge conserving superconductors to the gapless Haldane phase

2018

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We develop a framework to analyze one-dimensional topological superconductors with charge conservation. In particular, we consider models with N flavors of fermions and (Z2) N symmetry, associated with the conservation of the fermionic parity of each flavor. For a single flavor, we recover the result that a distinct topological phase with exponentially localized zero modes does not exist due to absence of a gap to single particles in the bulk. For N > 1, however, we show that the ends of the system can host low-energy, exponentially-localized modes. The analysis can readily be generalized to systems in other symmetry classes. To illustrate these ideas, we focus on lattice models with SO (N ) symmetric interactions, and study the phase transition between the trivial and the topological gapless phases using bosonization and a weak-coupling renormalization group analysis. As a concrete example, we study in detail the case of N = 3. We show that in this case, the topologically non-trivial superconducting phase corresponds to a gapless analogue of the Haldane phase in spin-1 chains. In this phase, although the bulk is gapless to single particle excitations, the ends host spin-1/2 degrees of freedom which are exponentially localized and protected by the spin gap in the bulk. We obtain the full phase diagram of the model numerically, using density matrix renormalization group calculations. Within this model, we identify the self-dual line studied by Andrei and Destri [Nucl. Phys. B, 231(3), 445-480 (1984)], as a first-order transition line between the gapless Haldane phase and a trivial gapless phase. This allows us to identify the propagating spin-1/2 kinks in the Andrei-Destri model as the topological end-modes present at the domain walls between the two phases.

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Section: Low-energy Description Of the Topological Phasementioning

confidence: 73%

From one-dimensional charge conserving superconductors to the gapless Haldane phase

2018

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“…Another interesting extension to the interacting case was touched upon in Section 3.1.2. This suggested that the Z classification of gapped phases should be stable under interactions if we allow 24 We are grateful to the participants of the AIM workshop 'Fisher-Harwig asymptotics, Szegő expansions and statistical physics' for discussions on this point. 25 However, numerical simulations indicated the stability away from the non-interacting limit.…”

Section: Resultsmentioning

confidence: 99%

Asymptotic Correlations in Gapped and Critical Topological Phases of 1D Quantum Systems

Jones

Verresen

2019

J Stat Phys

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Topological phases protected by symmetry can occur in gapped and-surprisingly-in critical systems. We consider the class of non-interacting fermions in one dimension with spinless timereversal symmetry. It is known that the phases in this class are classified by a topological invariant ω and a central charge c. Here we investigate the correlations of string operators in order to gain insight into the interplay between topology and criticality. In the gapped phases, these non-local operators are the string order parameters that allow us to extract ω. More remarkable is that the correlation lengths of these operators show universal features, depending only on ω. In the critical phases, the scaling dimensions of these operators serve as an order parameter, encoding both ω and c. More generally, we derive the exact long-distance asymptotics of these correlation functions using the theory of Toeplitz determinants. We include physical discussion in light of the mathematical results. This includes an expansion of the lattice operators in terms of the operator content of the relevant conformal field theory. Moreover, we discuss the spin chains which are dual to these fermionic systems. Contents

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“…79, in the sense that operators charged under this symmetry in the bulk necessarily create gapped excitations (in this case, defects). The coexistence of gapless bulk with these additional gapped degrees of freedom ensures the twofold degeneracy of the ground state, up to an exponentially small finite size splitting [78,79]. In this particular model, due to the boundary SLIOMs, this degeneracy is exact (and present throughout the spectrum).…”

Section: Largest Sectors and Spt Ordermentioning

confidence: 99%

Statistical localization: From strong fragmentation to strong edge modes

et al. 2020

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Certain disorder-free Hamiltonians can be non-ergodic due to a strong fragmentation of the Hilbert space into disconnected sectors. Here, we characterize such systems by introducing the notion of 'statistically localized integrals of motion' (SLIOM), whose eigenvalues label the connected components of the Hilbert space. SLIOMs are not spatially localized in the operator sense, but appear localized to sub-extensive regions when their expectation value is taken in typical states with a finite density of particles. We illustrate this general concept on several Hamiltonians, both with and without dipole conservation. Furthermore, we demonstrate that there exist perturbations which destroy these integrals of motion in the bulk of the system, while keeping them on the boundary. This results in statistically localized strong zero modes, leading to infinitely long-lived edge magnetizations along with a thermalizing bulk, constituting the first example of such strong edge modes in a non-integrable model. We also show that in a particular example, these edge modes lead to the appearance of topological string order in a certain subset of highly excited eigenstates. Some of our suggested models can be realized in Rydberg quantum simulators.CONTENTS * These authors contributed equally to this work.

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Gapless Symmetry-Protected Topological Order

Cited by 116 publications

References 138 publications

From one-dimensional charge conserving superconductors to the gapless Haldane phase

From one-dimensional charge conserving superconductors to the gapless Haldane phase

Asymptotic Correlations in Gapped and Critical Topological Phases of 1D Quantum Systems

Statistical localization: From strong fragmentation to strong edge modes

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