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arXiv:1401.5557 · Physical Review B

Modular matrices as topological order parameter by a gauge-symmetry-preserved tensor renormalization approach

Huan He, Heidar Moradi, Xiao-Gang Wen

Published Nov 10, 2014• 55 citations
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Abstract

Topological order has been proposed to go beyond Landau symmetry breaking theory for more than 20 years. But it is still a challenging problem to generally detect it in a generic many-body state. In this paper, we will introduce a systematic numerical method based on tensor network to calculate modular matrices in two-dimensional systems, which can fully identify topological order with gapped edge. Moreover, it is shown numerically that modular matrices, including $S$ and $T$ matrices, are robust characterization to describe phase transitions between topologically ordered states and trivial states, which can work as topological order parameters. This method only requires local information of one ground state in the form of a tensor network, and directly provides the universal data ($S$ and $T$ matrices), without any nonuniversal contributions. Furthermore, it is generalizable to higher dimensions. Unlike calculating topological entanglement entropy by extrapolating, in which numerical complexity is exponentially high, this method extracts a much more complete set of topological data (modular matrices) with much lower numerical cost.

DOI
10.1103/physrevb.90.205114

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Topological order has been proposed to go beyond Landau symmetry breaking theory for more than 20 years. But it is still a challenging problem to generally detect it in a generic many-body state. In this paper, we will introduce a systematic numerical method based on tensor network to calculate modular matrices in two-dimensional systems, which can fully identify topological order with gapped edge. Moreover, it is shown numerically that modular matrices, including SSS and TTT matrices, are robust characterization to describe phase transitions between topologically ordered states and trivial states, which can work as topological order parameters. This method only requires local information of one ground state in the form of a tensor network, and directly provides the universal data (SSS and TTT matrices), without any nonuniversal contributions. Furthermore, it is generalizable to higher dimensions. Unlike calculating topological entanglement entropy by extrapolating, in which numerical complexity is exponentially high, this method extracts a much more complete set of topological data (modular matrices) with much lower numerical cost.
Symmetry-protected topological (SPT) phases exhibit nontrivial order if symmetry is respected but are adiabatically connected to the trivial product phase if symmetry is not respected. However, unlike the symmetry-breaking phase, there is no local order parameter for SPT phases. Here we employ a tensor-network method to compute the topological invariants characterized by the simulated modular SSS and TTT matrices to study transitions in a few families of two-dimensional (2D) wave functions which are ZN{\mathbb{Z}}_{N}ZN​ (N=2and3)(N=2\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}3)(N=2and3) symmetric. We find that in addition to the topologically ordered phases, the modular matrices can be used to identify nontrivial SPT phases and detect transitions between different SPT phases as well as between symmetric and symmetry-breaking phases. Therefore modular matrices can be used to characterize various types of gapped phases in a unifying way.
The combination of quantum effects and interactions in quantum many-body systems can result in exotic phases with fundamentally entangled ground state wavefunctions -- topological phases. Topological phases come in two types, both of which will be studied in this thesis. In topologically ordered phases, the pattern of entanglement in the ground state wavefunction encodes the statistics of exotic emergent excitations, a universal indicator of a phase that is robust to all types of perturbations. In symmetry protected topological phases, the entanglement instead encodes a universal response of the system to symmetry defects, an indicator that is robust only to perturbations respecting the protecting symmetry.Finding and creating these phases in physical systems is a motivating challenge that tests all aspects - analytical, numerical, and experimental - of our understanding of the quantum many-body problem. Nearly three decades ago, the creation of simple ansatz wavefunctions - such as the Laughlin fractional quantum hall state, the AKLT state, and the resonating valence bond state - spurred analytical understanding of both the role of entanglement in topological physics and physical mechanisms by which it can arise. However, quantitative understanding of the relevant phase diagrams is still challenging. For this purpose, tensor networks provide a toolbox for systematically improving wavefunction ansatz while still capturing the relevant entanglement properties.In this thesis, we use the tools of entanglement and tensor networks to analyze ansatz states for several proposed new phases. In the first part, we study a featureless phase of bosons on the honeycomb lattice and argue that this phase can be topologically protected under any one of several distinct subsets of the crystalline lattice symmetries. We discuss methods of detecting such phases with entanglement and without.In the second part, we consider the problem of constructing fixed-point wavefunctions for intrinsically fermionic topological phases, i.e. topological phases contructed out of fermions with a nontrivial response to fermion parity defects. A zero correlation length wavefunction and a commuting projector Hamiltonian that realizes this wavefunction as its ground state are constructed. Using an appropriate generalization of the minimally entangled states method for extraction of topological order from the ground states on a torus to the intrinsically fermionic case, we fully characterize the corresponding topological order as Ising x (p - i p). We argue that this phase can be captured using fermionic tensor networks, expanding the applicability of tensor network methods.
The modern conception of phases of matter has undergone tremendous developments since the first observation of topologically ordered state in fractional quantum Hall systems in the 1980s. Topological orders are exotic states of matter characterized by patterns of long-range entanglement that lie beyond Landau's symmetry breaking paradigm. In 2+1 dimensions, these phases support signature anyonic excitations with fractional statistics or even non-Abelian braiding effects. One important theoretical question is how we should characterize these phases. In 2+1 dimensions, plenty of evidence suggests that the \emph{modular matrices}, S and T, suffice to characterize the order completely. It is thus of equal importance to find out if these quantities are experimental observables. Here we demonstrate how such measurements can indeed be achieved using only the given Hamiltonian, allowing one to identify the corresponding topological order using a nuclear magnetic resonance simulator. This is a first experimental realization of these fundamental signatures of a topological order; it opens up new future avenues toward identifying more generic topological orders based purely on experimental measurements.
The most general description of a quantum many-body system is given by a wave- function that lives in a Hilbert space with dimension exponential in the number of particles. This makes it extremely hard to study strongly correlated phenomena like the fractional quantum Hall effect and high-temperature superconductivity. Whenever interactions are sufficiently local and temperature is low, the system does not explore the full Hilbert space, but its ground state resides in the small corner of Hilbert space described by the area law. Containing little entanglement, the states can then be expressed as tensor networks, a family of wavefunctions with a polynomial number of parameters. On the one hand, tensor networks can be used as a variational manifold in nu- merical computations. On the other hand, they allow building model wavefunctions much like locality allows writing down physically realistic Hamiltonians. Besides allowing for an analytical treatment, these models grant access both to the physical and the entanglement degrees of freedom. This is particularly useful in classifying phases of matter. A large number of phases can be explained in terms of Landau’s symmetry-breaking paradigm. This framework, however, is not complete, as exemplified by the existence of phases with intrinsic topological order in two dimensions. It was a major conceptual advance when tensor networks could explain (non-chiral) topological phases as those where the symmetry resides in the entanglement degrees of freedom. The symmetries corresponding to those topological phases act as discrete, finite groups on the virtual degrees of freedom. The purpose of this Thesis is to generalize this program to include other symmetries. We investigate a class of tensor networks with continuous symmetries and find that they cannot describe gapped physics with a unique ground state. The abelian case is found to describe a non-Lorentz invariant phase transition point into a topologically ordered phase. The physics of the non- abelian case is that of a plaquette state that spontaneously breaks the translation symmetry of the lattice. The non-abelian PEPS arises as the ground state of a local parent Hamiltonian whose ground state manifold is completely characterized by the tensor network. In both cases, we find two types of corrections to the entanglement entropy: first there is a correction that is logarithmic in the size of the boundary and independent of the shape. A further correction depends only on the shape of the partition, imposing further restrictions on regions that are suffciently thin. Finally, we investigate symmetries that mix the virtual with the physical degrees of freedom and are furthermore anisotropic. Their physics is described by subsystem symmetry protected topological order. In particular, we focus on the entanglement entropy in the cluster phase and show that there is a universal constant correction to the entropy throughout the phase. This is important in the program of establishing the entanglement entropy as a detection mechanism for topologically ordered phases. We put forward a numerical algorithm to compute the correction and use it to discover a novel phase of matter in which the cluster phase is embedded.
Abstract One of the most striking features of gapped quantum phases that exhibit topological order is the presence of long-range entanglement that cannot be detected by any local order parameter. The formalism of projected entangled-pair states is a natural framework for the parameterization of gapped ground state wavefunctions which allows one to characterize topological order in terms of the virtual symmetries of the local tensors that encode the wavefunction. In their most general form, these symmetries are represented by matrix product operators acting on the virtual level, which leads to a set of algebraic rules characterizing states with topological quantum order. This construction generalizes the concepts of G{\mathsf {G}}G G - and twisted injectivity; the corresponding matrix product operators encode all topological features of the theory and provide a complete picture of the ground state manifold on the torus. We show how the string-net models of Levin and Wen fit within this formalism and in doing so provide a particularly intuitive interpretation of the pentagon equation for F-symbols as the pulling of matrix product operators through the string-net tensor network. Our approach paves the way to finding novel topological phases beyond string nets and elucidates the description of topological phases in terms of entanglement Hamiltonians and edge theories.
Tensor network states are capable of describing many-body systems with complex quantum entanglement, including systems with non-trivial topological order. In this paper, we study methods to calculate the topological properties of a tensor network state from the tensors that form the state. Motivated by the concepts of gauge group and projective symmetry group in the slave-particle/projective construction, and by the low-dimensional gauge-like symmetries of some exactly solvable Hamiltonians, we study the ddd-dimensional gauge structure and the ddd-dimensional symmetry structure of a tensor network state, where d≤dspaced\leq d_{space}d≤dspace​ with dspaced_{space}dspace​ the dimension of space. The ddd-dimensional gauge structure and ddd-dimensional symmetry structure allow us to calculate the string operators and ddd-brane operators of the tensor network state. This in turn allows us to calculate many topological properties of the tensor network state, such as ground state degeneracy and quasiparticle statistics.