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arXiv:1608.00594 · Physical review. B./Physical review. B

Constructing topological models by symmetrization: A projected entangled pair states study

Carlos Fernández-González, Roger S. K. Mong, Olivier Landon-Cardinal, David Pérez-Garcı́a, Norbert Schuch

Published Oct 5, 2016• 14 citations
arXivHTMLPDF

Abstract

Symmetrization of topologically ordered wave functions is a powerful method for constructing new topological models. Here we study wave functions obtained by symmetrizing quantum double models of a group $G$ in the projected entangled pair states (PEPS) formalism. We show that symmetrization naturally gives rise to a larger symmetry group $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{G}$ which is always non-Abelian. We prove that by symmetrizing on sufficiently large blocks, one can always construct wave functions in the same phase as the double model of $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{G}$. In order to understand the effect of symmetrization on smaller patches, we carry out numerical studies for the toric code model, where we find strong evidence that symmetrizing on individual spins gives rise to a critical model which is at the phase transitions of two inequivalent toric codes, obtained by anyon condensation from the double model of $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{G}$.

DOI
10.1103/physrevb.94.155106

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Symmetrization of topologically ordered wave functions is a powerful method for constructing new topological models. Here we study wave functions obtained by symmetrizing quantum double models of a group GGG in the projected entangled pair states (PEPS) formalism. We show that symmetrization naturally gives rise to a larger symmetry group G\ifmmode~\else˜\fi\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{G}G\ifmmode~\else˜\fi​ which is always non-Abelian. We prove that by symmetrizing on sufficiently large blocks, one can always construct wave functions in the same phase as the double model of G\ifmmode~\else˜\fi\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{G}G\ifmmode~\else˜\fi​. In order to understand the effect of symmetrization on smaller patches, we carry out numerical studies for the toric code model, where we find strong evidence that symmetrizing on individual spins gives rise to a critical model which is at the phase transitions of two inequivalent toric codes, obtained by anyon condensation from the double model of G\ifmmode~\else˜\fi\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{G}G\ifmmode~\else˜\fi​.
We use projected entangled pair states (PEPS) to study topological quantum phase transitions. The local description of topological order in the PEPS formalism allows us to set up order parameters which measure condensation and deconfinement of anyons and serve as substitutes for conventional order parameters. We apply these order parameters, together with anyon-anyon correlation functions and some further probes, to characterize topological phases and phase transitions within a family of models based on a Z4{\mathbb{Z}}_{4}Z4​ symmetry, which contains Z4{\mathbb{Z}}_{4}Z4​ quantum double, toric code, double semion, and trivial phases. We find a diverse phase diagram which exhibits a variety of different phase transitions of both first and second order which we comprehensively characterize, including direct transitions between the toric code and the double semion phase.
We develop and investigate several quantum many-body spin models of use for topological quantum information processing and storage. These models fall into two categories: those that are designed to be more realistic than alternative models with similar phenomenology, and those that are designed to have richer phenomenology than related models. In the first category, we present a procedure to obtain the Hamiltonians of the toric code and Kitaev quantum double models as the perturbative low-energy limits of entirely two-body Hamiltonians. This construction reproduces the target models' behavior using only couplings which are natural in terms of the original Hamiltonians. As an extension of this work, we construct parent Hamiltonians involving only local 2-body interactions for a broad class of Projected Entangled Pair States (PEPS). We define a perturbative Hamiltonian with a finite order low energy effective Hamiltonian that is a gapped, frustration-free parent Hamiltonian for an encoded version of a desired PEPS. For topologically ordered PEPS, the ground space of the low energy effective Hamiltonian is shown to be in the same phase as the desired state to all orders of perturbation theory. We then move on to define models that generalize the phenomenology of several well-known systems. We first define generalized cluster states based on finite group algebras, and investigate properties of these states including their PEPS representations, global symmetries, relationship to the Kitaev quantum double models, and possible applications. Finally, we propose a generalization of the color codes based on finite groups. For non-Abelian groups, the resulting model supports non-Abelian anyonic quasiparticles and topological order. We examine the properties of these models such as their relationship to Kitaev quantum double models, quasiparticle spectrum, and boundary structure.
This thesis contributes to the understanding of symmetry-enriched topological phases focusing on their descriptions in terms of tensor network states. The Projected Entangled Pair State (PEPS) formalism allows us to locally encode the main properties of the models (like topological order, symmetries and their interplay) in the tensors. We have used that encoding to classify, construct and detect some classes of symmetry-enriched topological phases in 2D PEPS. For that purpose, we have studied what the allowed freedom in the tensors generating the same tensor network state is. These results are the so-called 'fundamental theorems' and they give the necessary knowledge to properly study symmetries (actions that leave the states invariant). We focus our study on the family of PEPS describing quantum double models of GGG, the so-called GGG-injective PEPS, and on global on-site symmetries coming from a finite group QQQ: as a result of the classification we have a finite number of phases closely related to the theory of group extensions. We also provide the maps that appear in group extension theory with physical meaning and we characterize their actions on the ground subspace and on the anyons. We construct a representative of each class of our classification. Moreover, we connect our construction with an interesting physical phenomenon, the so-called anyon condensation, which describes topological phase transitions. We have also proposed a family of gauge invariant quantities and their corresponding order parameters (in the form of expectation values of a local operator in 2D) to detect the corresponding quantum phases, in particular their symmetry fractionalization patterns. We conclude this thesis by touching on some mathematical open problems in PEPS.
We study projected entangled pair states (PEPS) with continuous virtual symmetries, i.e., symmetries in the virtual degrees of freedom, through an elementary class of models with SU(2)\mathrm{SU}(2)SU(2) symmetry. Discrete symmetries of that kind have previously allowed for a comprehensive explanation of topological order in the PEPS formalism. We construct local parent Hamiltonians whose ground space with open boundaries is exactly parametrized by the PEPS wave function, and show how the ground state can be made unique by a suitable choice of boundary conditions. We also find that these models exhibit a logarithmic correction to the entanglement entropy and an extensive ground space degeneracy on systems with periodic boundaries, which suggests that they do not describe conventional gapped topological phases, but either critical models or some other exotic phase.
Topological phases of matter describe systems beyond the paradigm of spontaneous symmetry breaking, giving rise to unconventional phenomena with possible applications in topological quantum computing. Almost 40 years after their discovery, topological phases remain among the most active research fields of condensed matter physics. As evidenced by the recent discovery of higher order topological insulators (HOTIs), novel physics can emerge even in non-interacting systems. Theoretical investigations of strongly correlated systems are extremely difficult due to the exponential growth of the quantum-mechanical Hilbert space and frequently require the application of approximative methods such as model wave functions. Tensor network states (TNS) are a class of variational wave functions which allow an efficient encoding of relevant quantum many body states and which form the basis for very successful numerical algorithms. Projected entangled pair states (PEPS), a class of TNS in two and higher dimensions, permit the exact representation of many interacting topological phases with time reversal symmetry. However, the description of experimentally relevant chiral topological phases like the integer and fractional quantum Hall effects is much more subtle using the TNS framework. The first part of this thesis focuses on the description of chiral topological phases using PEPS. Firstly, we study a previously proposed chiral PEPS that was conjectured to possess anyonic excitations. By a careful analysis of its symmetries we are able to align important entanglement observables more closely with the expected universal behavior. This highlights that efficient PEPS can possess characteristic properties of chiral topological phases. Secondly, we focus on the issue that all known chiral PEPS have algebraic bulk correlation functions and therefore cannot be the ground states of gapped local Hamiltonians. Since this problem arises already for non-interacting chiral topological phases, we focus on two examples of such systems and show that their ground states can be represented exactly by efficient PEPS in a hybrid lattice with one momentum-space direction. After an inverse Fourier transform, the PEPS with only real-space coordinates requires an exponentially growing number of parameters for an exact representation. This provides a concrete illustration of the impossibility to encode gapped chiral phases exactly with efficient PEPS. Finally, we analyze a model wave function for a three-dimensional (3D) HOTI with strong intrinsic correlations using large-scale variational Monte Carlo simulations. This wave function is obtained by projection of two copies of a non-interacting HOTI with chiral hinge states. We characterize the gapless hinge states of the interacting system and show that they are of the same nature as the edge states of the 1/21/21/2 Laughlin state. Surprisingly, the gapped surfaces host a two-dimensional (2D) phase whose topological entanglement entropy is half of that of the 1/21/21/2 Laughlin state. Such a value cannot be obtained by any of the known 2D topological orders, showing a clear departure not only from the Laughlin 1/21/21/2 physics but from conventional 2D topological order. This demonstrates that 3D topological phases can host rich phenomena that are not yet fully understood.
Topological phase transition of deformed Z3{\mathbb Z}_3Z3​ toric code
We investigate topological phase transitions in a family of deformed Z3\mathbb Z_3Z3​ toric-code wavefunctions prepared from a cluster state by local deformations and projective measurements. Their norms map to the Q=3Q=3Q=3 Potts model for single-parameter deformations and to a three-state Ashkin--Teller-like (AT3_33​) construction with two independent four-spin couplings in the general case. Projected entangled-pair-state (PEPS) and variational uniform matrix-product-state (VUMPS) calculations identify the toric-code (TC) phase and phases in which electric (eee) anyons are confined or condensed. These phases are separated by critical structures with central charges c=4/5c=4/5c=4/5, 8/58/58/5, and isolated c=1c=1c=1 antiferromagnetic (AFM) endpoints. A normalized finite-distance eee-anyon pair-state norm provides a Fredenhagen--Marcu-type check of the confinement boundary, while the topological data of the quantum double D(Z3)D(\mathbb Z_3)D(Z3​) imply a topological entanglement entropy γ=log⁡3γ=\log3γ=log3 throughout the gapped toric-code phase. Relative to the Z2\mathbb Z_2Z2​ case, the absence of sign-change folding leaves the AFM endpoints unfolded, and the extreme deformation reaches square ice with an emergent U(1)U(1)U(1) one-form symmetry, Hilbert-space fragmentation, and exact scar configurations.