Gapless Kitaev Spin Liquid to Classical String Gas through Tensor Networks

Abstract

We provide a framework for understanding the gapless Kitaev spin liquid (KSL) in the language of tensor network (TN). Without introducing Majorana fermion, most of the features of the KSL including the symmetries, gauge structure, criticality and vortex-freeness are explained in a compact TN representation. Our construction reveals a hidden string gas structure of the KSL. With only two variational parameters to adjust, we obtain an accurate KSL ansatz with the bond dimension D = 8 in a compact form, where the energy is about 0.007% higher than the exact one. arXiv:1901.05786v3 [cond-mat.str-el]

Tensor network wave function of S=1 Kitaev spin liquids

“…Note that the Q tensor for the spin-1/2 model obeys similar relations, but with a non-Hermitian unitary matrix v, defined in Ref. [28], rather than σ x in the above relations. That is because U γ obeys U α U β = U γ with (α, β, γ ) being an arbitrary permutation of (x, y, z), while the Pauli matrices obey σ α σ β = iε αβγ σ γ with ε αβγ being the Levi-Civita symbol.…”

“…In a similar fashion to the spin-1/2 model [28], one can define the LG operator Q LG for the spin-1 model in a bond dimension D = 2 TN representation with a local tensor Q λμν = τ λμν (U x ) 1−λ (U y ) 1−μ (U z ) 1−ν , where λ, μ, ν = 0, 1 are the virtual indices. Here nonzero elements of the tensor τ are only τ λμν = 1 for λ + μ + ν = even.…”

“…One can examine the physical symmetries of Q LG at the local tensor level as done in Ref. [28] and derive the Z 2 gauge symmetry: λ ,μ ,ν σ z λλ σ z μμ σ z νν Q λ μ ν = Q λμν . Furthermore, it is straightforward to show that the local Q tensor obeys the following equations:…”

“…In this paper, we provide the tensor network (TN) representation of the ground-state wave function for the S = 1 Kitaev model, with both ferromagnetic and antiferromagnetic interactions between the S = 1 local moments. Our approach is inspired by the recent construction of the TN ground-state wave function of the S = 1/2 Kitaev model [28], where the wave function is written in terms of physical spin variables, instead of the Majorana fermion operators in the exact solution. We start from the bond dimension D = 2 TN representation of the wave function, which is an explicit eigenstate of the flux operator via the projector Q LG , or the loop gas (LG) operator.…”

“…Using the mapping between the norm of the wave function and the partition function of the classical loop gas, it is shown that the wave function is in a gapped state, which is corroborated by the direct computation of the correlation length from the transfer matrix. This wave function is further improved by applying an additional dimer gas (DG) operator [28] and imaginary time evolution (ITE), resulting in excellent variational energy.…”