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arXiv:cond-mat/0010242 · Physical review. B, Condensed matter

Microscopic models of two-dimensional magnets with fractionalized excitations

Chetan Nayak, Kirill Shtengel

Published Jul 24, 2001• 41 citations
arXivHTMLPDF

Abstract

We demonstrate that spin-charge separation can occur in two dimensions and note its confluence with superconductivity, topology, gauge theory, and fault-tolerant quantum computation. We construct a microscopic Ising-like model and, at a special coupling constant value, find its exact ground state as well as neutral spin-$\frac{1}{2}$ (spinon), spinless charge e (holon), and ${Z}_{2}$ vortex (vison) states and energies. The fractionalized excitations reflect the topological order of the ground state which is evinced by its fourfold degeneracy on the torus---a degeneracy which is unrelated to translational or rotational symmetry---and is described by a ${Z}_{2}$ gauge theory. A magnetic moment coexists with the topological order. Our model is a member of a family of topologically ordered models, one of which is integrable and realizes the toric quantum error correction code but does not conserve any component of the spin. We relate our model to a dimer model which could be a spin $\mathrm{SU}(2)$ symmetric realization of topological order and its concomitant quantum number fractionalization.

DOI
10.1103/physrevb.64.064422

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We demonstrate that spin-charge separation can occur in two dimensions and note its confluence with superconductivity, topology, gauge theory, and fault-tolerant quantum computation. We construct a microscopic Ising-like model and, at a special coupling constant value, find its exact ground state as well as neutral spin-12\frac{1}{2}21​ (spinon), spinless charge e (holon), and Z2{Z}_{2}Z2​ vortex (vison) states and energies. The fractionalized excitations reflect the topological order of the ground state which is evinced by its fourfold degeneracy on the torus---a degeneracy which is unrelated to translational or rotational symmetry---and is described by a Z2{Z}_{2}Z2​ gauge theory. A magnetic moment coexists with the topological order. Our model is a member of a family of topologically ordered models, one of which is integrable and realizes the toric quantum error correction code but does not conserve any component of the spin. We relate our model to a dimer model which could be a spin SU(2)\mathrm{SU}(2)SU(2) symmetric realization of topological order and its concomitant quantum number fractionalization.
We construct a family of exactly solvable spin models that illustrate a mechanism for fractionalization in topologically ordered phases, dubbed the string flux mechanism. The essential idea is that an anyon of a topological phase can be endowed with fractional quantum numbers when the string attached to it slides over a background pattern of flux in the ground state. The string flux models that illustrate this mechanism are Zn{\mathbb{Z}}_{n}Zn​ quantum double models defined on specially constructed ddd-dimensional lattices, and possess Zn{\mathbb{Z}}_{n}Zn​ topological order for $d\ensuremath{\ge}2$. The models have a unitary, internal symmetry GGG, where GGG is an arbitrary finite group. The simplest string flux model is a Z2{\mathbb{Z}}_{2}Z2​ toric code defined on a bilayer square lattice, where G=Z2G={\mathbb{Z}}_{2}G=Z2​ is layer-exchange symmetry. In general, by varying the pattern of Zn{\mathbb{Z}}_{n}Zn​ flux in the ground state, any desired fractionalization class [element of H2(G,Zn)]{H}^{2}(G,{\mathbb{Z}}_{n})]H2(G,Zn​)] can be realized for the Zn{\mathbb{Z}}_{n}Zn​ charge excitations. While the string flux models are not gauge theories, they map to Zn{\mathbb{Z}}_{n}Zn​ gauge theories in a certain limit, where they follow a magnetic route for the emergence of low-energy gauge structure. The models are analyzed by studying the action of GGG symmetry on Zn{\mathbb{Z}}_{n}Zn​ charge excitations, and by gauging the GGG symmetry. The latter analysis confirms that distinct fractionalization classes give rise to distinct quantum phases, except that classes $[\ensuremath{\omega}],{[\ensuremath{\omega}]}^{\ensuremath{-}1}\ensuremath{\in}{H}^{2}(G,{\mathbb{Z}}_{n})$ give rise to the same phase. We conclude with a discussion of open issues and future directions.
We develop a new theoretical framework for describing and analyzing exotic phases of strongly correlated electrons which support excitations with fractional quantum numbers. Starting with a class of microscopic models believed to capture much of the essential physics of the cuprate superconductors, we derive a new gauge theory---based upon a discrete Ising or Z2{Z}_{2}Z2​ symmetry---which interpolates naturally between an antiferromagnetic Mott insulator and a conventional d-wave superconductor. We explore the intervening regime, and demonstrate the possible existence of an exotic fractionalized insulator, the nodal liquid, as well as various more conventional insulating phases exhibiting broken lattice symmetries. A crucial role is played by vortex configurations in the Z2{Z}_{2}Z2​ gauge field. Fractionalization is obtained if they are uncondensed. Within the insulating phases, the dynamics of these Z2{Z}_{2}Z2​ vortices in two dimensions is described, after a duality transformation, by an Ising model in a transverse field, the Ising spins representing the Z2{Z}_{2}Z2​ vortices. The presence of an unusual Berry's phase term in the gauge theory leads to a doping-dependent ``frustration'' in the dual Ising model, being fully frustrated at half filling. The Z2{Z}_{2}Z2​ gauge theory is readily generalized to a variety of different situations, in particular, it can also describe three-dimensional insulators with fractional quantum numbers. We point out that the mechanism of fractionalization for d>1d>1d>1 is distinct from the well-known one-dimensional spin--charge separation. Other interesting results include a description of an exotic fractionalized superconductor in two or higher dimensions.
Following an earlier construction of exactly soluble lattice models for Abelian fractional topological insulators in two and three dimensions, we construct here an exactly soluble lattice model for a non-Abelian $\ensuremath{\nu}=1$ quantum Hall state and a non-Abelian topological insulator in two dimensions. We show that both models are topologically ordered, exhibiting fractionalized charge, ground-state degeneracy on the torus, and protected edge modes. The models feature non-Abelian vortices which carry fractional electric charge in the quantum Hall case and spin in the topological insulator case. We analyze the statistical properties of the excitations in detail and discuss the possibility of extending this construction to three-dimensional non-Abelian topological insulators.
The topological defects of Spin(n+1n+1n+1) nematics in two spatial dimensions, known as disclinations, are characterized by the ${\ensuremath{\pi}}_{1}(\mathbb{R}{P}^{n})={\text{Z}}_{2}$ homotopy group for $n\ensuremath{\ge}2$. We argue that incompressible quantum liquids of disclinations can exist as stable low-temperature phases and host composite quasiparticles which combine a fractional amount of fundamental Z2{\mathrm{Z}}_{2}Z2​ charge with a unit of topological charge. The fourfold topological ground state degeneracy on a torus admits a fermionic or semionic quasiparticle exchange statistics. The topological nontriviality of these states is visible in the existence of protected gapless edge modes. While the fermionic nematic and gapped Z2{\mathrm{Z}}_{2}Z2​ spin liquids have equivalent topological orders, they are still thermodynamically distinct due to having different edge modes, in analogy to the topologically nontrivial and trivial states of quantum spin-Hall systems. The analysis proceeds by recasting the Z2{\mathrm{Z}}_{2}Z2​ gauge theory of spin nematics as a continuum limit theory with a larger gauge structure. Nematic fractionalization parallels that of a quantum Hall liquid, but the large gauge symmetry restores the time-reversal symmetry and restricts the quasiparticle fusion rules and statistics. The conclusions from field theory analysis are complemented with the construction of plausible host microscopic models.
We study the electron spectral function of various zero-temperature spin-charge separated phases in two dimensions. In these phases, the electron is not a fundamental excitation of the system, but rather ``decays'' into a spin-1/2 chargeless fermion (the spinon) and a spinless charge e boson (the chargon). Using low-energy effective theories for the spinons (d-wave pairing plus possible N\'eel order) and the chargons (condensed or quantum-disordered bosons), we explore three phases of possible relevance to the cuprate superconductors: (1) AF∗,{\mathrm{AF}}^{*},AF∗, a fractionalized antiferromagnet where the spinons are paired into a state with long-ranged N\'eel order and the chargons are 1/2-filled and (Mott) insulating; (2) the nodal liquid, a fractionalized insulator where the spinons are d-wave paired and the chargons are uncondensed; and (3) the d-wave superconductor, where the chargons are condensed and the spinons retain a d-wave gap. Working within the Z2{Z}_{2}Z2​ gauge theory of such fractionalized phases, our results should be valid at scales below the energy gap of the vison---the basic vortex excitation in the theory. However, on a phenomenological level, our results should apply to any spin-charge separated system where the excitations have these low-energy effective forms. Comparison with angle-resolved photoemission spectroscopy data in the undoped, pseudogapped, and superconducting regions is made.