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arXiv:2012.15346 · Physical review. B./Physical review. B

On the stability of topological order in tensor network states

Dominic J. Williamson, Clement Delcamp, Frank Verstraete, Norbert Schuch

Published Dec 27, 2021• 16 citations
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Abstract

We construct a tensor network representation of the three-dimensional (3D) toric code ground state that is stable to a generating set of uniform local tensor perturbations, including those that do not map to local operators on the physical Hilbert space. The stability is established by mapping the phase diagram of the perturbed tensor network to that of the 3D Ising gauge theory, which has a nonzero finite temperature transition. More generally, we find that the stability of a topological tensor network state is determined by the form of its virtual symmetries and the topological excitations created by virtual operators that break those symmetries. In particular, a dual representation of the 3D toric code ground state, as well as representations of the $X$-cube and cubic code ground states, for which pointlike excitations are created by such operators, are found to be unstable.

DOI
10.1103/physrevb.104.235151

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We construct a tensor network representation of the three-dimensional (3D) toric code ground state that is stable to a generating set of uniform local tensor perturbations, including those that do not map to local operators on the physical Hilbert space. The stability is established by mapping the phase diagram of the perturbed tensor network to that of the 3D Ising gauge theory, which has a nonzero finite temperature transition. More generally, we find that the stability of a topological tensor network state is determined by the form of its virtual symmetries and the topological excitations created by virtual operators that break those symmetries. In particular, a dual representation of the 3D toric code ground state, as well as representations of the XXX-cube and cubic code ground states, for which pointlike excitations are created by such operators, are found to be unstable.
We define two dual tensor network representations of the (3+1)d toric code ground state subspace. These two representations, which are obtained by initially imposing either family of stabilizer constraints, are characterized by different virtual symmetries generated by string-like and membrane-like operators, respectively. We discuss the topological properties of the model from the point of view of these virtual symmetries, emphasizing the differences between both representations. In particular, we argue that, depending on the representation, the phase diagram of boundary entanglement degrees of freedom is naturally associated with that of a (2+1)d Hamiltonian displaying either a global or a gauge Z2 -symmetry.
The tensor network representation of many-body quantum states, given by local tensors, provides a promising numerical tool for the study of strongly correlated topological phases in two dimensions. However, the representation may be vulnerable to instabilities caused by small variations in the local tensors. For example, the topological order in the tensor network representations of the toric code ground state has been shown in Chen, Zeng, Gu, Chuang, and Wen, Phys. Rev. B 82, 165119 (2010) to be unstable if the variations break certain Z2{Z}_{2}Z2​ symmetry of the tensor. In this work, we ask whether other types of topological orders suffer from similar kinds of instability and if so, what is the underlying physical mechanism and whether we can protect the order by enforcing certain symmetries on the tensor. We answer these questions by showing that the tensor network representations of all string-net models are indeed unstable, but the matrix product operator (MPO) symmetries of the tensors identified in \ifmmode \mbox{\c{S}}\else \c{S}\fi{}ahino\ifmmode \breve{g}\else \u{g}\fi{}lu, Williamson, Bultinck, Mari\"en, Haegeman, Schuch, and Verstraete, arXiv:1409.2150 can help to protect the order. In particular, we show that a subset of variations that break the MPO symmetries lead to instability by inducing the condensation of bosonic quasiparticles, which destroys the topological order in the wave function. Therefore such variations must be forbidden for the encoded topological order to be reliably extracted from the local tensors. On the other hand, if a tensor network based variational algorithm is used to simulate the phase transition due to boson condensation, such variation directions may prove important to access the continuous transition correctly.
We identify a three-dimensional system that exhibits long-range entanglement at sufficiently small but nonzero temperature-it therefore constitutes a quantum topological order at finite temperature. The model of interest is known as the fermionic toric code, a variant of the usual 3D toric code, which admits emergent fermionic pointlike excitations. The fermionic toric code, importantly, possesses an anomalous two-form symmetry, associated with the spacelike Wilson loops of the fermionic excitations. We argue that it is this symmetry that imbues low-temperature thermal states with a novel topological order and long-range entanglement. Based on the current classification of three-dimensional topological orders, we expect that the low-temperature thermal states of the fermionic toric code belong to a phase of matter that, in the context of equilibrium phases of matter, only exists at nonzero temperatures. We conjecture that further examples of topological orders at nonzero temperatures are given by discrete gauge theories with anomalous two-form symmetries. Our Letter therefore opens the door to studying quantum topological order at nonzero temperature in physically realistic dimensions.
Due to the absence of local order parameters, it is a challenging task to characterize the quantum topological phase transitions between topologically ordered phases in two dimensions. In this paper, we construct a topologically ordered tensor network wave function with one parameter $\ensuremath{\lambda}$, describing both the toric code state $(\ensuremath{\lambda}=1)$ and double semion state $(\ensuremath{\lambda}=\ensuremath{-}1)$. Via calculating the correlation length defined from the one-dimensional quantum transfer operator of the wave function norm, we can map out the complete phase diagram in terms of the parameter $\ensuremath{\lambda}$, and three different quantum critical points (QCPs) at $\ensuremath{\lambda}=0,\ifmmode\pm\else\textpm\fi{}1.73$ are identified. The first one separates the toric code phase and double semion phase, while latter two describe the topological phase transitions from the toric code phase or double semion phase to the symmetry-breaking phase, respectively. When mapping the quantum tensor network wave function to the exactly solved statistical model, the norm of the wave function is identified as the partition function of the classical eight-vertex model, and both QCPs at $\ensuremath{\lambda}=\ifmmode\pm\else\textpm\fi{}1.73$ correspond to the eight-vertex model at the critical point $\ensuremath{\lambda}=\sqrt{3}$, while the QCP at $\ensuremath{\lambda}=0$ corresponds to the critical six-vertex model. Actually such a quantum-classical mapping cannot yield the complete low-energy excitations at these three QCPs. We further demonstrate that the full eigenvalue spectra of the transfer operators without/with the flux insertions can give rise to the complete quantum criticalities, which are described by the (2+0)-dimensional free boson conformal field theories (CFTs) compactified on a circle with the radius R=6R=\sqrt{6}R=6​ at $\ensuremath{\lambda}=\ifmmode\pm\else\textpm\fi{}\sqrt{3}$ and R=8/3R=\sqrt{8/3}R=8/3​ at $\ensuremath{\lambda}=0$. From the complete transfer operator spectra, the finite-size spectra of the CFTs for the critical eight-vertex model are obtained, and the topological sectors of anyonic excitations are yielded as well. Furthermore, for the QCP at $\ensuremath{\lambda}=0$, no anyon condensation occurs, but the emerged symmetries of the matrix product operators significantly enrich the topological sectors of the CFT spectra. Finally, we provide our understanding of the (2+0)-dimensional conformal quantum criticalities and their possible connection with the generic (2+1)-dimensional CFTs for quantum topological phase transitions.
Tensor networks are an efficient platform for representing topological states of matter as well as computing both physical observables and information-theoretic quantities. We present a general protocol to construct tensor network representations for intrinsically mixed-state topological phases. Such phases naturally emerge in noisy long-range entangled quantum systems, and exhibit nontrivial topological phenomena without pure-state counterparts. The method exploits the isomorphism between anyon decoherence in the physical Hilbert space and anyon condensation in the corresponding Choi space. The protocol is applicable to a broad class of systems arising from decoherence of pure-state topological phases, where the incoherent noise creates excitations in the Choi space that have trivial monodromy with all excitations. Representative examples at renormalization group fixed points include maebm^a e^bmaeb decoherence in the ZN\Z_NZN​ toric code, decohered non-Abelian quantum double of S3S_3S3​ as well as pure ZZZ/XXX decoherence of arbitrary CSS codes. Another notable example is that of the mixed-state chiral semion, whose tensor network representation has no pure-state analogue. We then generalize the formalism beyond RG fixed points and, as an example, examine the decoherence transition in a Z3\mathbb{Z}_3Z3​ toric code channel that suffers from sign problem when mapped to a statistical mechanical model. Together, these results establish tensor networks as a natural platform for analyzing noisy intrinsically mixed-state topological matter.
Understanding many-body quantum systems is one of the most challenging problems in contemporary condensed-matter physics. Tensor network representation of quantum states and operators are taking central stage in this pursuit and beyond. They prove to be a powerful numerical and conceptual tool, and indeed a new language altogether. This thesis investigates various aspects of these representations by focusing on two specific problems: the first half of the thesis is devoted to examining how 'stable' a tensor network representation is for two-dimensional quantum states with topological order, and the second half explores the representability of various unitary loop operators with tensor networks. In the numerical usage of the tensor networks, the tensor is varied as to find the representation of the ground states of the given Hamiltonian. In chapter two and three of this thesis we show that such a numerical program for topological phases can be 'ill-posed'. We show that tensor network can be an unstable representation for a topological phase: even an infinitesimal variation in the representation results in the loss of topological order, completely or partially. We diagnose this problem by identifying the exact causes of this instability, and find that it is only tensor variations in certain directions that result in instability, because they result in the condensation of bosonic quasi-particles of the phase. Such unstable variations are characterized by two properties: (1) they can replace a tensor in the tensor network without making the network collapse, and (2) their presence in the network represents the presence of a non-trivial topological charge. We prove that the general tensor representation of all string-net models suffer with such instabilities. We propose an exact mathematical operator to project out all such unstable variations and show its efficacy for a few models by direct calculations. Such an operator can be useful in numerical programs involving such tensor representations. We also point out that such variations play a crucial role in simulating topological phase transitions and their presence can be vital in an accurate simulation. In chapter four and five of this thesis we focus on the representability of unitary loop operators by tensor networks. Such operators not only provide an important tool in the study of dynamical process in one-dimensional systems, but also in understanding and classification of symmetry protected topological phases in two dimensions. To characterize all such operators, we find a necessary and sufficient condition for any loop tensor network operator of a given length to represent a unitary operator. In particular, it is shown that all unitary operators that map local operators to local operators (locality-preserving) can always be represented by a tensor network. Locality-preserving unitary loop operators are classified by a rational index called the GNVW index defined in Ref. [1] which measures how much information 'flows' along the loop. We define Rank-Ratio index for tensor network operators and show that it is completely equivalent to the GNVW index. Therefore, GNVW index of a unitary operator can be easily extracted from its tensor network representation. We find that, other than representing locality-preserving unitary maps, tensor networks can also represent unitary operators that map local operators to global (non-local) operators. These tensor network operators are found to have a long-ranged order similar to tensors that represent topological tensor network states in two dimensions.