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arXiv:1201.5312 · Physical Review B

Parent Hamiltonian for the non-Abelian chiral spin liquid

Martin Greiter, Darrell F. Schroeter, Ronny Thomale

Published Apr 21, 2014• 83 citations
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Abstract

We construct a parent Hamiltonian for the family of non-Abelian chiral spin liquids proposed recently by two of us [Phys. Rev. Lett. 102, 207203 (2009)] which includes the Abelian chiral spin liquid proposed by Kalmeyer and Laughlin as the special case $s=\frac{1}{2}$. As we use a circular disk geometry with an open boundary, both the annihilation operators we identify and the Hamiltonians we construct from these are exact only in the thermodynamic limit.

DOI
10.1103/physrevb.89.165125

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We construct a parent Hamiltonian for the family of non-Abelian chiral spin liquids proposed recently by two of us [Phys. Rev. Lett. 102, 207203 (2009)] which includes the Abelian chiral spin liquid proposed by Kalmeyer and Laughlin as the special case s=12s=\frac{1}{2}s=21​. As we use a circular disk geometry with an open boundary, both the annihilation operators we identify and the Hamiltonians we construct from these are exact only in the thermodynamic limit.
Abstract We construct a family of 1D and 2D long-range SU (2) spin models as parent Hamiltonians associated with infinite dimensional matrix product states that arise from simple current correlation functions in SU (2) k WZW models. Our results provide a conformal field theory foundation for recent proposals by Greiter and coauthors regarding the realization of non-abelian chiral spin liquids. We explain, in particular, how the symmetrization procedure for the amplitudes of the ground state wave function suggested by Greiter M and Thomale R (2009 Phys. Rev. Lett. 102 207203) originates from the conformal field theory description.
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 signalling 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.
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.
We search for short-range Hamiltonians of finite spin-1 kagome systems, maximizing the overlaps with lattice Moore-Read states. Our starting point is an exact, long-range parent Hamiltonian for such a state on a finite plane, obtained from conformal field theory. A truncation procedure is applied to it, which retains only short-range terms and makes it easy to define the Hamiltonian on a torus. Finally, the remaining coefficients are optimized, to yield maximized overlaps between exact diagonalization results and model ground states. In the best cases, the squared overlaps exceed 0.9 and 0.8 for the three lowest states of 12- and 18-site systems, respectively, suggesting that the obtained Hamiltonians are good parent Hamiltonians for a non-Abelian topological order.
We describe a general method to identify exact, local parent Hamiltonians for trial states such as quantum Hall or spin-liquid states, which we have used extensively during the past decade. It can be used to identify exact parent Hamiltonians, either directly or via the construction of simpler annihilation operators from which a parent Hamiltonian respecting all the required symmetries can be constructed. Most remarkably, however, the method provides approximate parent Hamiltonians whenever an exact solution is not available within the space of presumed interaction terms.