Identifying topological order by entanglement entropy

Abstract

Topological phases are unique states of matter incorporating long-range quantum entanglement, hosting exotic excitations with fractional quantum statistics. We report a practical method to identify topological phases in arbitrary realistic models by accurately calculating the Topological Entanglement Entropy (TEE) using the Density Matrix Renormalization Group (DMRG). We argue that the DMRG algorithm naturally produces a minimally entangled state, from amongst the quasi-degenerate ground states in a topological phase.This proposal both explains the success of this method, and the absence of ground state degeneracy found in prior DMRG sightings of topological phases.We demonstrate the effectiveness of the calculational procedure by obtaining the TEE for several microscopic models, with an accuracy of order 10 −3 when the circumference of the cylinder is around ten times the correlation length.As an example, we definitively show the ground state of the quantum S = 1/2 antiferromagnet on the kagomé lattice is a topological spin liquid, and strongly constrain the full identification of this phase of matter. 1 arXiv:1205.4289v1 [cond-mat.str-el]

Phase diagram of theJ1−J2Heisenberg model on the kagome lattice

“…Correspondingly, the triplet gap decreases, suggesting a vanishing gap in the thermodynamic limit. Within the system sizes accessible to the simulations, our results are consistent with the presence of a magnetically disordered phase for −0.1 J 2 0.2, which is compatible with spinliquid behavior [31]. In this region, the spin gap shows a weaker dependence on the cylinder size.…”

“…On the other hand, for antiferromagnetic J 2 , signatures of the q = 0 state already appear at J 2 0.2. In the narrow region at −0.1 J 2 0.2, although strong finite-size effects are present, our data are compatible with an extended disordered region, suggestive of a spin-liquid behavior [31]. Finally, we analyze the structure of the entanglement spectrum (ES) [51] in the q = 0 ordered phase at large J 2 0.2.…”

“…However, recent density-matrix renormalization-group (DMRG) simulations [26][27][28] provided convincing evidence that the ground state of the KHA is a gapped spin liquid with topological entanglement entropy γ = log(2) [29][30][31]. This is compatible with both a spin liquid of the toric-code or double-semion [32,33] type.…”

“…Moreover, it has been suggested that the spin-liquid behavior survives upon introducing a small antiferromagnetic nextnearest-neighbor interaction [31], i.e., in the J 1 -J 2 Heisenberg model (J 1 -J 2 KHA). This is in contrast to the T = 0 phase diagram of the classical version of the model.…”

Entanglement spectroscopy of SU(2)-broken phases in two dimensions

“…54,55 Interestingly, there is also evidence that the spin liquid behavior might survive at small positive J 2 with the formation of an extended spin liquid region. 56 c…”