Zheng-Cheng Gu

Classification of gapped symmetric phases in one-dimensional spin systems

Quantum many-body systems divide into a variety of phases with very different physical properties. The questions of what kinds of phases exist and how to identify them seem hard, especially for strongly interacting systems. Here we make an attempt to answer these questions for gapped interacting quantum spin systems whose ground states are short-range correlated. Based on the local unitary equivalence relation between short-range-correlated states in the same phase, we classify possible quantum phases for one-dimensional (1D) matrix product states, which represent well the class of 1D gapped ground states. We find that in the absence of any symmetry all states are equivalent to trivial product states, which means that there is no topological order in 1D. However, if a certain symmetry is required, many phases exist with different symmetry-protected topological orders. The symmetric local unitary equivalence relation also allows us to obtain some simple results for quantum phases in higher dimensions when some symmetries are present.

Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order

Two gapped quantum ground states in the same phase are connected by an adiabatic evolution which gives rise to a local unitary transformation that maps between the states. On the other hand, gapped ground states remain within the same phase under local unitary transformations. Therefore, local unitary transformations define an equivalence relation and the equivalence classes are the universality classes that define the different phases for gapped quantum systems. Since local unitary transformations can remove local entanglement, the above equivalence/universality classes correspond to pattern of long range entanglement, which is the essence of topological order. The local unitary transformation also allows us to define a wave function renormalization scheme, under which a wave function can flow to a simpler one within the same equivalence/universality class. Using such a setup, we find conditions on the possible fixed-point wave functions where the local unitary transformations have emph{finite} dimensions. The solutions of the conditions allow us to classify this type of topological orders, which generalize the string-net classification of topological orders. We also describe an algorithm of wave function renormalization induced by local unitary transformations. The algorithm allows us to calculate the flow of tensor-product wave functions which are not at the fixed points. This will allow us to calculate topological orders as well as symmetry breaking orders in a generic tensor-product state.

Tensor-entanglement-filtering renormalization approach and symmetry-protected topological order

We study the renormalization group flow of the Lagrangian for statistical and quantum systems by representing their path integral in terms of a tensor network. Using a tensor-entanglement-filtering renormalization approach that removes local entanglement and produces a coarse-grained lattice, we show that the resulting renormalization flow of the tensors in the tensor network has a nice fixed-point structure. The isolated fixedpoint tensors Tinv plus the symmetry group Gsym of the tensors i.e., the symmetry group of the Lagrangian characterize various phases of the system. Such a characterization can describe both the symmetry breaking phases and topological phases, as illustrated by two-dimensional 2D statistical Ising model, 2D statistical loop-gas model, and 1+1D quantum spin-1/2 and spin-1 models. In particular, using such a Gsym,Tinv characterization, we show that the Haldane phase for a spin-1 chain is a phase protected by the time-reversal, parity, and translation symmetries. Thus the Haldane phase is a symmetry-protected topological phase. The Gsym,Tinv characterization is more general than the characterizations based on the boundary spins and string order parameters. The tensor renormalization approach also allows us to study continuous phase transitions between symmetry breaking phases and/or topological phases. The scaling dimensions and the central charges for the critical points that describe those continuous phase transitions can be calculated from the fixed-point tensors at those critical points.

Symmetry-Protected Topological Orders in Interacting Bosonic Systems

Symmetry Semantics Topological insulators (TIs) are characterized by boundary states that are protected by time-reversal symmetry. A systematic study of this, and other symmetry-protected states, is possible in noninteracting systems, but complications arise when interactions are present. Chen et al. (p. 1604; see the Perspective by Qi) used group cohomology theory to predict symmetry-protected phases of interacting bosons. The analysis enabled the generalization of a known result in one dimension by using a path-integral formulation and suggests the existence of three counterparts of TIs in three dimensions, and one in two dimensions, as well as phases protected by other symmetries. The formalism is applicable to any symmetry group and dimension and is valid for interactions of arbitrary strength. Counterparts of topological insulators are predicted to exist in interacting bosonic systems. Symmetry-protected topological (SPT) phases are bulk-gapped quantum phases with symmetries, which have gapless or degenerate boundary states as long as the symmetries are not broken. The SPT phases in free fermion systems, such as topological insulators, can be classified; however, it is not known what SPT phases exist in general interacting systems. We present a systematic way to construct SPT phases in interacting bosonic systems. Just as group theory allows us to construct 230 crystal structures in three-dimensional space, we use group cohomology theory to systematically construct different interacting bosonic SPT phases in any dimension and with any symmetry, leading to the discovery of bosonic topological insulators and superconductors.

Tensor-product representations for string-net condensed states

We show that general string-net condensed states have a natural representation in terms of tensor product states (TPSs). These TPSs are built from local tensors. They can describe both states with short-range entanglement (such as the symmetry-breaking states) and states with long-range entanglement (such as string-net condensed states with topological/quantum order). The tensor product representation provides a kind of ``mean-field description for topologically ordered states and could be a powerful way to study quantum phase transitions between such states. As an attempt in this direction, we show that the constructed TPSs are fixed points under a certain wave-function renormalization-group transformation for quantum states.

Emergence of helicity ±2 modes (gravitons) from qubit models

Abstract We construct two quantum qubit models (or quantum spin models) on three-dimensional lattice in space, L-type model and N-type model. We show that, under a controlled approximation, all the low energy excitations of the L-type model are described by one set of helicity ±2 modes with ω ∝ k 3 dispersion. We also argue that all the low energy excitations of the N-type model are described by one set of helicity ±2 modes with ω ∝ k dispersion. In both model, the low energy helicity ±2 modes can be described by a symmetric tensor field h μ ν in continuum limit, and the gaplessness of the helicity ±2 modes is protected by an emergent linearized diffeomorphism gauge symmetry h μ ν → h μ ν + ∂ μ f ν + ∂ ν f μ at low energies. Thus the linearized quantum gravity emerge from our lattice models . It turns out that the low energy effective Lagrangian density of the L-type model is invariant under the linearized diffeomorphism gauge transformation. Such a property protects the gapless ω ∝ k 3 helicity ±2 modes. In contrast, the low energy effective Lagrangian of the N-type model changes by a boundary term under the linearized diffeomorphism gauge transformation. Such a property protects the gapless ω ∝ k helicity ±2 modes. From many-body physics point of view, the ground states of the our two qubit model represent new states of quantum matter, whose low energy excitations are all described by one set of gapless helicity ±2 modes.

Constructing a Gapless Spin-Liquid State for the Spin-1/2 J_1-J_2 Heisenberg Model on a Square Lattice

We construct a class of projected entangled pair states which is exactly the resonating valence bond wave functions endowed with both short range and long range valence bonds. With an energetically preferred resonating valence bond pattern, the wave function is simplified to live in a one-parameter variational space. We tune this variational parameter to minimize the energy for the frustrated spin-1/2 J(1)-J(2) antiferromagnetic Heisenberg model on the square lattice. Taking a cylindrical geometry, we are able to construct four topological sectors with an even or odd number of fluxes penetrating the cylinder and an even or odd number of spinons on the boundary. The energy splitting in different topological sectors is exponentially small with the cylinder perimeter. We find a power law decay of the dimer correlation function on a torus, and a lnL correction to the entanglement entropy, indicating a gapless spin-liquid phase at the optimum parameter.

Gapped Two-Body Hamiltonian Whose Unique Ground State Is Universal for One-Way Quantum Computation

Many-body entangled quantum states studied in condensed matter physics can be primary resources for quantum information, allowing any quantum computation to be realized using measurements alone, on the state. Such a universal state would be remarkably valuable, if only it were thermodynamically stable and experimentally accessible, by virtue of being the unique ground state of a physically reasonable Hamiltonian made of two-body, nearest-neighbor interactions. We introduce such a state, composed of six-state particles on a hexagonal lattice, and describe a general method for analyzing its properties based on its projected entangled pair state representation.

Tensor-product state approach to spin-1/2 square J_1−J_2 antiferromagnetic Heisenberg model: Evidence for deconfined quantum criticality

The ground state phase of a spin-

A lattice bosonic model as a quantum theory of gravity

frac{1}{2} {J}_{1}text{ensuremath{-}}{J}_{2}