Qing-Rui Wang

Bio: Qing-Rui Wang is an academic researcher from The Chinese University of Hong Kong. The author has contributed to research in topics: Wave function & Physics. The author has an hindex of 7, co-authored 17 publications receiving 278 citations. Previous affiliations of Qing-Rui Wang include Tsinghua University & Yale University.

Towards a Complete Classification of Symmetry-Protected Topological Phases for Interacting Fermions in Three Dimensions and a General Group Supercohomology Theory

Abstract: A new analysis presents a complete classification scheme for symmetry-protected topological phases in three-dimensional systems of interacting fermions, extending previous work in classifying such phases in bosonic matter.

Towards a complete classification of fermionic symmetry protected topological phases in 3D and a general group supercohomology theory

Abstract: Classification and construction of symmetry protected topological (SPT) phases in interacting boson and fermion systems have become a fascinating theoretical direction in recent years. It has been shown that the (generalized) group cohomology theory or cobordism theory can give rise to a complete classification of SPT phases in interacting boson/spin systems. Nevertheless, the construction and classification of SPT phases in interacting fermion systems are much more complicated, especially in 3D. In this work, we revisit this problem based on the equivalent class of fermionic symmetric local unitary (FSLU) transformations. We construct very general fixed point SPT wavefunctions for interacting fermion systems. We naturally reproduce the partial classifications given by special group super-cohomology theory, and we show that with an additional $\tilde{B}H^2(G_b, \mathbb Z_2)$ (the so-called obstruction free subgroup of $H^2(G_b, \mathbb Z_2)$) structure, a complete classification of SPT phases for three-dimensional interacting fermion systems with a total symmetry group $G_f=G_b\times \mathbb Z_2^f$ can be obtained for unitary symmetry group $G_b$. We also discuss the procedure of deriving a general group super-cohomology theory in arbitrary dimensions.

Construction and classification of symmetry protected topological phases in interacting fermion systems

Abstract: A classification of symmetry-protected topological phases in systems of interacting fermions provides a road map for discovering new types of topological phases in strongly correlated electron systems.

Topological quantum field theory for Abelian topological phases and loop braiding statistics in (3 +1 ) -dimensions

Abstract: Topological quantum field theory (TQFT) is a very powerful theoretical tool to study topological phases and phase transitions. In $2+1d$, it is well known that the Chern-Simons theory captures all the universal topological data of topological phases, e.g., quasiparticle braiding statistics, chiral central charge, and even provides us a deep insight for the nature of topological phase transitions. Recently, topological phases of quantum matter are also intensively studied in $3+1d$ and it has been shown that looplike excitation obeys the so-called three-loop-braiding statistics. In this paper, we will try to establish a TQFT framework to understand the quantum statistics of particle and looplike excitation in $3+1d$. We will focus on Abelian topological phases for simplicity, however, the general framework developed here is not limited to Abelian topological phases.

Single-hole wave function in two dimensions: A case study of the doped Mott insulator

Abstract: We study a ground-state ansatz for the single-hole-doped $t\text{\ensuremath{-}}J$ model in two dimensions via a variational Monte Carlo method. Such a single-hole wave function possesses finite angular momenta generated by hidden spin currents, which give rise to a novel ground-state degeneracy in agreement with recent exact diagonalization (ED) and density matrix renormalization group (DMRG) results. We further show that the wave function can be decomposed into a quasiparticle component and an incoherent momentum distribution in excellent agreement with the DMRG results up to an $8\ifmmode\times\else\texttimes\fi{}8$ lattice. Such a two-component structure indicates the breakdown of Landau's one-to-one correspondence principle, and in particular, the quasiparticle spectral weight vanishes by a power law in the large sample size limit. By contrast, turning off the phase string induced by the hole hopping in the so-called $\ensuremath{\sigma}\ifmmode\cdot\else\textperiodcentered\fi{}t\text{\ensuremath{-}}J$ model, a conventional Bloch-wave wave function with a finite quasiparticle spectral weight can be recovered, also in agreement with the ED and DMRG results. The present study shows that a singular effect already takes place in the single-hole-doped Mott insulator, by which the bare hole is turned into a non-Landau quasiparticle with translational-symmetry breaking. Generalizations to pairing and finite doping are briefly discussed.

Quantum criticality: beyond the Landau-Ginzburg-Wilson paradigm

Abstract: We present the critical theory of a number of zero-temperature phase transitions of quantum antiferromagnets and interacting boson systems in two dimensions. The most important example is the transition of the $S=1∕2$ square lattice antiferromagnet between the N\'eel state (which breaks spin rotation invariance) and the paramagnetic valence bond solid (which preserves spin rotation invariance but breaks lattice symmetries). We show that these two states are separated by a second-order quantum phase transition. This conflicts with Landau-Ginzburg-Wilson theory, which predicts that such states with distinct broken symmetries are generically separated either by a first-order transition, or by a phase with co-existing orders. The critical theory of the second-order transition is not expressed in terms of the order parameters characterizing either state, but involves fractionalized degrees of freedom and an emergent, topological, global conservation law. A closely related theory describes the superfluid-insulator transition of bosons at half filling on a square lattice, in which the insulator has a bond density wave order. Similar considerations are shown to apply to transitions of antiferromagnets between the valence bond solid and the ${Z}_{2}$ spin liquid: the critical theory has deconfined excitations interacting with an emergent $\mathrm{U}(1)$ gauge force. We comment on the broader implications of our results for the study of quantum criticality in correlated electron systems.

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

Abstract: 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.

Braiding statistics approach to symmetry-protected topological phases

Abstract: We construct a 2D quantum spin model that realizes an Ising paramagnet with gapless edge modes protected by Ising symmetry. This model provides an example of a \"symmetry-protected topological phase.\" We describe a simple physical construction that distinguishes this system from a conventional paramagnet: we couple the system to a Z_2 gauge field and then show that the \\pi-flux excitations have different braiding statistics from that of a usual paramagnet. In addition, we show that these braiding statistics directly imply the existence of protected edge modes. Finally, we analyze a particular microscopic model for the edge and derive a field theoretic description of the low energy excitations. We believe that the braiding statistics approach outlined in this paper can be generalized to a large class of symmetry-protected topological phases.

On the Cobordism Classification of Symmetry Protected Topological Phases

Abstract: In the framework of Atiyah’s axioms of topological quantum field theory with unitarity, we give a direct proof of the fact that symmetry protected topological phases without Hall effects are classified by cobordism invariants. We first show that the partition functions of those theories are cobordism invariants after a tuning of the Euler term. Conversely, for a given cobordism invariant, we construct a unitary topological field theory whose partition function is given by the cobordism invariant, assuming that a certain bordism group is finitely generated. Two theories having the same cobordism invariant partition functions are isomorphic.

Symmetry protected topological phases and generalized cohomology

Abstract: We discuss the classification of SPT phases in condensed matter systems. We review Kitaev’s argument that SPT phases are classified by a generalized cohomology theory, valued in the spectrum of gapped physical systems [20, 23]. We propose a concrete description of that spectrum and of the corresponding cohomology theory. We compare our proposal to pre-existing constructions in the literature.