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arXiv:1511.01539 · Physical review. B./Physical review. B

Exactly solvable models of spin liquids with spinons, and of three-dimensional topological paramagnets

Daniel Ben-Zion, Diptarka Das, John McGreevy

Published Apr 25, 2016• 14 citations
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Abstract

We develop a scheme to make exactly solvable gauge theories whose electric flux lines host (1+1)-dimensional topological phases. We use this exact ``decorated-string-net'' framework to construct several classes of interesting models. In particular, we construct an exactly solvable model of a quantum spin liquid whose (gapped) elementary excitations form doublets under an internal symmetry, and hence may be regarded as spin-carrying spinons. The model may be formulated, and is solvable, in any number of dimensions on any bipartite graph. Another example, in any dimension, has ${\mathbb{Z}}_{2}$ topological order and anyons which are Kramers' doublets of time-reversal symmetry. Further, we make exactly solvable models of three-dimensional topological paramagnets.

DOI
10.1103/physrevb.93.155147

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We develop a scheme to make exactly solvable gauge theories whose electric flux lines host (1+1)-dimensional topological phases. We use this exact ``decorated-string-net'' framework to construct several classes of interesting models. In particular, we construct an exactly solvable model of a quantum spin liquid whose (gapped) elementary excitations form doublets under an internal symmetry, and hence may be regarded as spin-carrying spinons. The model may be formulated, and is solvable, in any number of dimensions on any bipartite graph. Another example, in any dimension, has Z2{\mathbb{Z}}_{2}Z2​ topological order and anyons which are Kramers' doublets of time-reversal symmetry. Further, we make exactly solvable models of three-dimensional topological paramagnets.
We construct exactly solvable models for a wide class of symmetry-enriched topological (SET) phases. Our construction applies to two-dimensional (2D) bosonic SET phases with finite unitary on-site symmetry group GGG and we conjecture that our models realize every phase in this class that can be described by a commuting projector Hamiltonian. Our models are designed so that they have a special property: If we couple them to a dynamical lattice gauge field with gauge group GGG, the resulting gauge theories are equivalent to string-net models. This property is what allows us to analyze our models in generality. As an example, we present a model for a phase with the same anyon excitations as the toric code and with a Z2{\mathbb{Z}}_{2}Z2​ symmetry which exchanges the eee and mmm type anyons. We further illustrate our construction with a number of additional examples.
We propose a generic construction of exactly soluble local bosonic models that realize various topological orders with gappable boundaries. In particular, we construct an exactly soluble bosonic model that realizes a (3+1)-dimensional [(3+1)D] Z2{Z}_{2}Z2​-gauge theory with emergent fermionic Kramers doublet. We show that the emergence of such a fermion will cause the nucleation of certain topological excitations in space-time without pin+{\mathrm{pin}}^{+}pin+ structure. The exactly soluble model also leads to a statistical transmutation in (3+1)D. In addition, we construct exactly soluble bosonic models that realize 2 types of time-reversal symmetry-enriched Z2{Z}_{2}Z2​ topological orders in 2+1 dimensions, and 20 types of simplest time-reversal symmetry-enriched topological (SET) orders which have only one nontrivial pointlike and stringlike topological excitation. Many physical properties of those topological states are calculated using the exactly soluble models. We find that some time-reversal SET orders have pointlike excitations that carry Kramers doublet, a fractionalized time-reversal symmetry. We also find that some Z2{Z}_{2}Z2​ SET orders have stringlike excitations that carry anomalous (nononsite) Z2{Z}_{2}Z2​ symmetry, which can be viewed as a fractionalization of Z2{Z}_{2}Z2​ symmetry on strings. Our construction is based on cochains and cocycles in algebraic topology, which is very versatile. In principle, it can also realize emergent topological field theory beyond the twisted gauge theory.
Topological order is a new paradigm for quantum phases of matter developed to explain phase transitions which do not fit the symmetry breaking scheme for classifying phases of matter. They are characterized by patterns of entanglement that lead to topologically depended ground state degeneracy and anyonic excitations. One common approach for studying such phases in two-dimensional systems is through exactly solvable lattice Hamiltonian models such as quantum double models and String-Net models. The former can be understood as the Hamiltonian formulation of lattice gauge theories and, as such, it is defined by a finite gauge group. However, not much is known about topological phases in tridimensional systems. Motivated by this we develop a new class of three-dimensional exactly solvable models which go beyond quantum double models by using finite crossed modules instead of gauge groups. This approach relies on a lattice implementation of 2-gauge theory to obtain models with a richer topological structure. We construct the Hamiltonian model explicitly and provide a rigorous proof that the ground state degeneracy is a topological invariant and that the ground states can only be characterized with nonlocal order parameters.
The formation of quantum spin liquids in frustrated magnets represents an exciting possibility due to the rather exotic features they harbor, including fractionalized excitations and emergent gauge fields. \nUnfortunately, they are notoriously difficult to study as there are often no good analytical methods available and quantum Monte Carlo simulations are hindered by the negative sign problem. \nThe Kitaev honeycomb model is a notable exception of a frustrated quantum model which is exactly solvable and which hosts a number of distinct quantum spin liquid ground states. \nAs such, it allows for a rare opportunity to study the physics of spin liquids with full analytical control. \n \nIn this thesis, we study the fractionalization of spin-1/2 moments into Majorana fermions and an emergent Z2 gauge field in a generalization of the Kitaev honeycomb model to a number of three-dimensional lattices. \nWhile the excitations of the gauge field are always gapped, the fermionic quasiparticles may exhibit a gapless dispersion, forming fully two-dimensional Fermi surfaces, symmetry protected nodal lines, or topological Weyl nodes. \nWe show that one can deduce rather general constraints on the possible gapless excitations by making use of an object called the projective symmetry group. \nIn doing so we provide a scheme for classifying the various gapless Kitaev spin liquids. \nA thorough analysis is carried out for a number of these spin liquids, primarily investigating the stability of the gapless modes and the novel features resulting from their sometimes non-trivial topology, as well as their effects on certain equal-time correlation functions.
We construct spin- 3 2 and spin- 7 2 models on the square-octagon and checkerboard lattices that are exactly solvable with Majorana representations. They give rise to spin-liquid phases with full spin-rotation and lattice-translational symmetries but broken time-reversal symmetry. Although nonzero on elementary plaquettes, the net orbital magnetic moment is guaranteed to vanish as a result of point symmetries; due to the analogy to long-range ordered altermagnets, these types of phases were dubbed altermagnetic spin liquids in [Sobral ., ]. For the spin- 3 2 model, we find that a g -wave altermagnetic spin liquid emerges as the unique ground state. In contrast, the spin- 7 2 model exhibits a significantly richer phase diagram, involving different types of chiral spin liquids competing with a d -wave altermagnetic spin liquid. Finally, we identify and characterize the topological and nontopological excitations, illustrating the rich physics of altermagnetic spin liquids resulting from the interplay of nontrivial topological and symmetry aspects of this novel phase of matter.
Topological spin liquids can be described by topological gauge theories with global symmetry. Due to the presence of both nontrivial bulk deconfined gauge fluxes and global symmetry, topological spin liquids are examples of the so-called ``symmetry enriched topological phases'' (SETs). In this paper, we find that, in some twisted versions of topological gauge theories (with discrete Abelian gauge group Gg{G}_{g}Gg​), implementing a global symmetry (denoted by Gs{G}_{s}Gs​) is anomalous although symmetry charge carried by topological pointlike excitations is normally fractionalized and classified by the second cohomology group. To demonstrate the anomaly, we fully gauge the global symmetry, rendering a new gauge theory that is not gauge invariant. Therefore, the SET order of the ground state is anomalous, which cannot exist in the three-dimensional system alone. Such an anomalous state construction generalizes the ``2D surface topological order'' to three dimensions. A concrete example with ${G}_{g}={\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{4}$ and Gs=Z2{G}_{s}={\mathbb{Z}}_{2}Gs​=Z2​ is calculated.