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Physical review. B, Condensed matter

Properties of the chiral-spin-liquid state

R. B. Laughlin, Z. Zou

Published Jan 1, 1990• 100 citations
Publisher

Abstract

It is shown that one of the class of variational spin-liquid wave functions recently derived by Wen, Wilczek, and Zee [Phys. Rev. B 39, 11413 (1989)] is identical to the fractional-quantum-Hall analog state proposed by Kalmeyer and Laughlin [Phys. Rev. Lett. 59, 2095 (1987); Phys. Rev. B 39, 11 879 (1989)] and Laughlin [Ann. Phys. (N.Y.) 191, 163 (1989)], and that the neutral spin-1/2 excitations of the two states are also the same. The 1/2 fractional statistics obeyed by these particles is demonstrated explicitly. The spin-spin correlation function and quasiparticle profile for the Wen-Wilczek-Zee states are calculated both numerically and by a hypernetted-chain procedure. Both are consistent with the idea that spontaneous breaking of time-reversal symmetry is an essential feature of any spin state lacking magnetic order. A version of this state for a three-dimensional spin system is reported and shown to have similar properties. A case is made that the neutral spin-1/2 excitations of the three-dimensional spin liquid behave like anisotropic monopoles.

DOI
10.1103/physrevb.41.664

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It is shown that one of the class of variational spin-liquid wave functions recently derived by Wen, Wilczek, and Zee [Phys. Rev. B 39, 11413 (1989)] is identical to the fractional-quantum-Hall analog state proposed by Kalmeyer and Laughlin [Phys. Rev. Lett. 59, 2095 (1987); Phys. Rev. B 39, 11 879 (1989)] and Laughlin [Ann. Phys. (N.Y.) 191, 163 (1989)], and that the neutral spin-1/2 excitations of the two states are also the same. The 1/2 fractional statistics obeyed by these particles is demonstrated explicitly. The spin-spin correlation function and quasiparticle profile for the Wen-Wilczek-Zee states are calculated both numerically and by a hypernetted-chain procedure. Both are consistent with the idea that spontaneous breaking of time-reversal symmetry is an essential feature of any spin state lacking magnetic order. A version of this state for a three-dimensional spin system is reported and shown to have similar properties. A case is made that the neutral spin-1/2 excitations of the three-dimensional spin liquid behave like anisotropic monopoles.
Recent years have seen the discovery of a chiral spin liquid state - a bosonic analogue of a fractional Quantum Hall state first put forward by Kalmeyer and Laughlin in 1987 - in several deformations of the Heisenberg model on the Kagome lattice. Here, we apply state-of-the-art numerical techniques to one such model, where breaking of the time-reversal symmetry drives the system into the chiral phase. Our methods allow us to obtain explicit matrix-product state representations of the low-lying excitations of the chiral spin liquid state, including the topologically non-trivial semionic excitation. We characterize these excitations and study their energetics as the model is tuned towards a topological phase transition.
The relation between the chiral-spin-liquid state and the correlators of the Wess-Zumino-Witten model is shown. I argue that in the Kalmeyer-Laughlin state there is a hidden symmetry, associated with the conformal invariance in the continuum limit. The fractional statistics of the spinons naturally follows from the conformal weight for the primary fields. The relation between the Kalmeyer-Laughlin and the spin-singlet quantum-Hall-effect (QHE) wave functions is shown and leads to the possibility of the spinon excitations in the spin-singlet QHE.
We propose that the disordered state of a two-dimensional spin-1/2 Heisenberg antiferromagnet is physically equivalent to the incompressible liquid state of the fractional quantum Hall system. The fractional quantum Hall state for bosons is shown to be an exact spin singlet and to possess a low variational energy for the near-neighbor Heisenberg model on a triangular lattice. Variational wave functions for neutral spin-1/2 excitations are constructed and shown to form an exact spin doublet. Variational energies of these states are calculated, and their spin density profiles are determined. We find that a localized spin-1/2 quaisparticle has a size comparable to a lattice bond length and an excitation energy \ensuremath{\Delta}=1.3J. The energy-momentum dispersion of quasiparticles and spin-1 collective modes, obtained variationally, supports the hypothesis that the spin liquid state has a finite energy gap. The 1/2 fractional statistics exhibited by the quasiparticle excitations is explicitly demonstrated.
The Kalmeyer-Laughlin chiral spin liquids (CSL) and the Z2 spin liquids are two of the simplest topologically ordered states. Here I develop a theory of a direct quantum phase transition between them. Each CSL is characterized by an integer n and is topologically equivalent to the 1/2n Laughlin fractional quantum Hall (FQH) state. Depending on the parity of n, the transition from the CSL is either to a "twisted" version of the Z2 spin liquid, the "doubled semion" model, or the conventional Z_2 spin liquid ("toric code"). In the presence of SU(2) spin symmetry, the triplet gap remains open through the transition and only singlet operators acquire algebraic correlations. An essential observation is that the CSL/Laughlin FQH states can be understood in terms of bosonic integer quantum Hall (BIQH) states of Schwinger bosons or vortices, respectively. I propose several novel many-body wave functions that can interpolate through the transition.
The properties of a class of wave functions describing charged spinless excitations of the Kalmeyer-Laughlin spin-liquid state are studied within the context of the t-J model and the case made that they are legitimate elementary excitations of such a state. Using these wave functions and those previously proposed for the neutral spin-1/2 excitations, a variational calculation of the hole propagator is performed. The spectral density at a given momentum is found to be a broad continuum with resonant structures that move as the momentum is varied, and to be consistent with exact-diagonalization studies.
We describe a coupled-chain construction for chiral spin liquids in two-dimensional spin systems. Starting from a one-dimensional zigzag spin chain and imposing SU(2) symmetry in the framework of non-Abelian bosonization, we first show that our approach faithfully describes the low-energy physics of an exactly solvable model with a three-spin interaction. Generalizing the construction to the two-dimensional case, we obtain a theory that incorporates the universal properties of the chiral spin liquid predicted by Kalmeyer and Laughlin: charge-neutral edge states, gapped spin-1/2 bulk excitations, and ground-state degeneracy on the torus signaling the topological order of this quantum state. In addition, we show that the chiral spin liquid phase is more easily stabilized in frustrated lattices containing corner-sharing triangles, such as the extended kagome lattice, than in the triangular lattice. Our field-theoretical approach invites generalizations to more exotic chiral spin liquids and may be used to assess the existence of the chiral spin liquid as the ground state of specific lattice systems.