We study the robustness of topological ground state degeneracy to long-range interactions in quantum many-body systems. We focus on slowly decaying two-body interactions that scale like a power law 1/r(alpha) where alpha is smaller than the spatial dimension; such interactions are beyond the reach of known stability theorems which only apply to short-range or rapidly decaying long-range perturbations. Our main result is a computation of the ground state splitting of several toy models, which are variants of the one-dimensional Ising model H =-Sigma(i) sigma(z)(i)sigma(z) (i+1) +lambda Sigma(ij)|i-j |(-alpha) sigma(x)(i)sigma(x)(j)with lambda>0 and alpha<1. In one variant, the power-law interactions are replaced by all-to-all interactions lambda/4L(alpha)Sigma(ij)sigma(x) (i)sigma(x )(j) , where L is the system size, while the other variant has true power-law interactions but is built out of quantum rotors rather than Ising spins. These models are also closely connected to the Kitaev p-wave wire model with power-law density-density interactions. In these examples, we find that the splitting delta scales like a stretched exponential delta similar to exp(-CL 1+alpha/2 ). Our computations are based on path integral techniques similar to the instanton method introduced by Coleman. We also study another toy model with long-range interactions that can be analyzed without path integral techniques and that shows similar behavior.

Although symmetry-protected topological phases (SPTs) can be distinguished by their entanglement properties, it has been unclear how to extract this information directly from expectation values beyond the 1D case. Here, we close this gap and propose a pair of nonlocal order parameters that can detect and distinguish all bosonic SPTs in 2D protected by internal discrete, Abelian unitary symmetries. The desired topological invariants are extracted by these quantities by effectively simulating the SPT path integral on topologically non-trivial spacetime manifolds. Our order parameters are defined in terms of expectation values of partial symmetry and permutation operations acting on fixed numbers of replicas of the system in finite spatial regions. These expectation values correspond to symmetry-twisted versions of multipartite entanglement quantities known as multi-entropies. We show explicitly that our two order parameters detect symmetry-protected four-party and six-party entanglement, respectively, and we constrain possible "spurious" contributions. We analytically test our proposal in fixed-point lattice models. Our results suggest multipartite entanglement to be a defining feature of SPTs; indeed, we expect our methods to generalize to fermionic and higher-dimensional systems.

We present counterexamples to the lore that symmetries that cannot be gauged or made on site are necessarily anomalous. Specifically, we construct unitary, internal symmetries of two-dimensional lattice models that cannot be consistently coupled to background or dynamical gauge fields or disentangled to a tensor product of on-site operators. These symmetries are nevertheless anomaly-free in the sense that they admit symmetric, gapped Hamiltonians with unique, invertible ground states. We show that symmetries of this kind are characterized by an index [ω]∈H^{2}(G,Q_{+}), where Q_{+} is the multiplicative group of positive rational numbers labeling one-dimensional quantum cellular automata.

Recent developments in the study of topological defects highlight the importance of understanding the multi-dimensional structure of bulk excitations inside a quantum system. When the bulk ground state is trivial, i.e. a product state, excitations on top of it are decoupled from each other and correspond to lower-dimensional phases and their defects within. In this paper, we expand the discussion to invertible phases and study the bulk excitations in, for example, SPT phases, Majorana chains, p + ip superconductors etc. We find that there is a one-to-one correspondence between bulk excitations inside a nontrivial invertible phase and those in a product state. For SPT phases, this can be shown using the symmetric Quantum Cellular Automaton that maps from the product state to the SPT state. More generally, for invertible phases realizable using the Topological Holography construction, we demonstrate the correspondence using the fact that certain gapped boundary conditions of a topological bulk state have only relative distinctions but no absolute ones.

The study of topological superconductivity is largely based on the analysis of simple mean-field models that do not conserve particle number. A major open question in the field is whether the remarkable properties of these mean-field models persist in more realistic models with a conserved total particle number and long-range interactions. For applications to quantum computation, two key properties that one would like to verify in more realistic models are (i) the existence of a set of low-energy states (the qubit states) that are separated from the rest of the spectrum by a finite energy gap, and (ii) an exponentially small (in system size) bound on the splitting of the energies of the qubit states. It is well known that these properties hold for mean-field models, but so far only property (i) has been verified in a number-conserving model. In this work we fill this gap by rigorously establishing both properties (i) and (ii) for a number-conserving toy model of two topological superconducting wires coupled to a single bulk superconductor. Our result holds in a broad region of the parameter space of this model, suggesting that properties (i) and (ii) are robust properties of number-conserving models and not just artifacts of the mean-field approximation.

Recently it was shown that the topological entanglement entropy (TEE) of a two-dimensional gapped ground state obeys the universal inequality gamma >= log D, where gamma is the TEE and D is the total quantum dimension of all anyon excitations, D = root Sigma(da2)(a). Here we present an alternative, more direct proof of this inequality. Our proof uses only the strong subadditivity property of the von Neumann entropy together with a few physical assumptions about the ground-state density operator. Our derivation naturally generalizes to a variety of systems, including spatially inhomogeneous systems with defects and boundaries, higher-dimensional systems, and mixed states.

Recently, it was proposed that the chiral central charge of a gapped, two-dimensional quantum many-body system is proportional to a bulk ground state entanglement measure known as the modular commutator. While there is significant evidence to support this relation, we show in this Letter that it is not universal. We give examples of lattice systems that have vanishing chiral central charge, which nevertheless give nonzero "spurious" values for the modular commutator for arbitrarily large system sizes, in both one and two dimensions. Our examples are based on cluster states and utilize the fact that they can generate nonlocal modular Hamiltonians.

Entanglement entropies of two-dimensional gapped ground states are expected to satisfy an area law, with a constant correction term known as the topological entanglement entropy (TEE). In many models, the TEE takes a universal value that characterizes the underlying topological phase. However, the TEE is not truly universal: it can differ even for two states related by constant-depth circuits, which are necessarily in the same phase. The difference between the TEE and the value predicted by the anyon theory is often called the "spurious" topological entanglement entropy. We show that this spurious contribution is always non-negative, thus the value predicted by the anyon theory provides a universal lower bound. This observation also leads to a definition of TEE that is invariant under constant-depth quantum circuits.

We construct an exactly solvable lattice model for a deconfined quantum critical point (DQCP) in (1+1) dimensions. This DQCP occurs in an unusual setting, namely, at the edge of a (2+1) dimensional bosonic symmetry protected topological (SPT) phase with Z_{2}×Z_{2} symmetry. The DQCP describes a transition between two gapped edges that break different Z_{2} subgroups of the full Z_{2}×Z_{2} symmetry. Our construction is based on an exact mapping between the SPT edge theory and a Z_{4} spin chain. This mapping reveals that DQCPs in this system are directly related to ordinary Z_{4} symmetry breaking critical points.

We show that some gapped quantum many-body systems have a ground state degeneracy that is stable to long-range (e.g., power-law) perturbations, in the sense that any ground state energy splitting induced by such perturbations is exponentially small in the system size. More specifically, we consider an Ising symmetry-breaking Hamiltonian with several exactly degenerate ground states and an energy gap, and we then perturb the system with Ising symmetric long-range interactions. For these models we prove (1) the stability of the gap, and (2) that the residual splitting of the low-energy states below the gap is exponentially small in the system size. Our proof relies on a convergent polymer expansion that is adapted to handle the long-range interactions in our model. We also discuss applications of our result to several models of physical interest, including the Kitaev p-wave wire model perturbed by power-law density-density interactions with an exponent greater than 1.

We present a method for deriving bulk and edge invariants for interacting, many-body localized Floquet systems in two spatial dimensions. This method is based on a general mathematical object which we call a flow. As an application of our method, we derive bulk invariants for Floquet systems without symmetry, as well as for systems with $U(1)$ symmetry. We also derive new formulations of previously known single-particle and many-body invariants. For bosonic systems without symmetry, our invariant gives a bulk counterpart of the rational-valued GNVW index $\frac{p}{q}$ quantifying transport of quantum information along the edge.

Fracton order features point excitations that either cannot move at all or are only allowed to move in a lowerdimensional submanifold of the whole system. In this paper, we generalize the (2 + 1)-dimensional [(2 + 1)D] U(1) Chern-Simons (CS) theory, a powerful tool in the study of (2 + 1)D topological orders, to include infinite gauge field components and find that they can describe interesting types of (3 + 1)-dimensional fracton order beyond what is known from exactly solvable models and tensor gauge theories. On the one hand, they can describe foliated fractonic systems for which increasing the system size requires insertion of nontrivial (2 + 1)D topological states. The CS formulation provides an easier approach to study the phase relation among foliated models. More interestingly, we find simple examples that lie beyond the foliation framework, characterized by 2D excitations of infinite order and irrational braiding statistics. This finding extends our realm of understanding of possible fracton phenomena.

We consider the process of flux insertion for ground states of almost local commuting projector Hamiltoniansin two spatial dimensions. In the case of finite dimensional local Hilbert spaces, we prove that this process cannot pump any charge and we conclude that the Hall conductance must vanish

We construct a new class of edge theories for a family of fermionic Abelian topological phases with $K$-matrices of the form $K = \begin{pmatrix} k_1 & 0 \\ 0 & - k_2 \end{pmatrix}$, where $k_1, k_2 > 0$ are odd integers. Our edge theories are notable for two reasons: (i) they have finite dimensional Hilbert spaces (for finite sized systems) and (ii) depending on the values of $k_1, k_2$, some of the edge theories describe boundaries that cannot be gapped by any local interaction. The simplest example of such an ungappable boundary occurs for $(k_1, k_2) = (1, 3)$, which is realized by the $\nu = 2/3$ FQH state. We derive our edge theories by starting with the standard chiral boson edge theory, consisting of two counterpropagating chiral boson modes, and then introducing an array of pointlike impurity scatterers. We solve this impurity model exactly in the limit of infinite impurity scattering, and we show that the energy spectrum consists of a gapped phonon spectrum together with a ground state degeneracy that scales exponentially with the number of impurities. This ground state subspace forms the Hilbert space for our edge theory. We believe that similar edge theories can be constructed for any Abelian topological phase with vanishing thermal Hall coefficient, $\kappa_H = 0$.

We investigate whether there could exist topological invariants of gapped 2D materials related to dissipationless thermoelectric transport at low temperatures. We give both macroscopic and microscopic arguments showing that thermoelectric transport coefficients vanish in the limit of zero temperature and thus topological invariants arise only from the electric Hall conductance and the thermal Hall conductance. Our arguments apply to systems with arbitrarily strong interactions. We also show that there is no analog of the Thouless pump for entropy.

We describe a systematic procedure for determining the identity of a 2D bosonic symmetry protected topological (SPT) phase from the properties of its edge excitations. Our approach applies to general bosonic SPT phases with either unitary or antiunitary symmetries, and with either continuous or discrete symmetry groups, with the only restriction being that the symmetries must be on-site. Concretely, our procedure takes a bosonic SPT edge theory as input, and produces an element $\omega$ of the cohomology group $H^3(G, U_T(1))$. This element $\omega \in H^3(G, U_T(1))$ can be interpreted as either a label for the bulk 2D SPT phase or a label for the anomaly carried by the SPT edge theory. The basic idea behind our approach is to compute the $F$-symbol associated with domain walls in a symmetry broken edge theory; this domain wall $F$-symbol is precisely the anomaly we wish to compute. We demonstrate our approach with several SPT edge theories including both lattice models and continuum field theories.

Previous articleNext article No AccessBook ReviewsChurch and State in Spanish Italy: Rituals and Legitimacy in the Kingdom of Naples. By Céline Dauverd. Cambridge: Cambridge University Press, 2020. Pp. x+300. $99.99 (cloth); $80.00 (Adobe eBook Reader).Michael J. LevinMichael J. LevinUniversity of Akron Search for more articles by this author PDFPDF PLUSFull Text Add to favoritesDownload CitationTrack CitationsPermissionsReprints Share onFacebookTwitterLinkedInRedditEmail SectionsMoreDetailsFiguresReferencesCited by The Journal of Modern History Volume 93, Number 4December 2021 Article DOIhttps://doi.org/10.1086/716788 Views: 30Total views on this site For permission to reuse, please contact [email protected]PDF download Crossref reports no articles citing this article.

We derive a complete classification of Floquet phases of interacting bosons and fermions with $U(1)$ symmetry in two spatial dimensions. According to our classification, there is a one-to-one correspondence between these Floquet phases and rational functions $\ensuremath{\pi}(z)=a(z)/b(z)$, where $a(z)$ and $b(z)$ are polynomials obeying certain conditions and $z$ is a formal parameter. The physical meaning of $\ensuremath{\pi}(z)$ involves the stroboscopic edge dynamics of the corresponding Floquet system: in the case of bosonic systems, $\ensuremath{\pi}(z)=\frac{p}{q}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\pi}}(z)$, where $\frac{p}{q}$ is a rational number that characterizes the flow of quantum information at the edge during each driving period and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\pi}}(z)$ is a rational function which characterizes the flow of $U(1)$ charge at the edge. A similar decomposition exists in the fermionic case. We also show that $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\pi}}(z)$ is directly related to the time-averaged $U(1)$ current that flows in a particular geometry. This $U(1)$ current is a generalization of the quantized current and quantized magnetization density found in previous studies of noninteracting fermionic Floquet phases.