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Entanglement Entropy in Quantum Dimer Models
Abstract Book of the XXIII IUPAP International Conference on Statistical Physics, Genova, Italy, (9-13 July 2007)
Аннотация
A universal characterization of topological order in terms of entanglement in the ground state
was proposed recently by Kitaev and Preskill (KP) 1 and by Levin and Wen (LW) 2.
It was argued that, if $Ømega$ is a disk with a smooth boundary of length $L$,
the entanglement entropy (measure of entanglement between $Ømega$ and the rest of the system)
scales as $S_Ømega = L - + \cdots$
in the limit $L\toınfty$.
While the coefficient $\alpha$ depends on the microscopic details of the system,
$\gamma$ is a universal constant characterizing the underlying gauge theory for the topological order
and is dubbed the topological entanglement entropy.
In general, it is difficult to separate the topological term $-\gamma$ from the boundary term in the above scaling
because of the generic ambiguity in defining the boundary length $L$ in a lattice system.
However, KP and LW formulated ways to define $\gamma$ unambiguously
by forming a linear combination of the entanglement entropies on several areas sharing the boundaries,
and canceling the boundary terms out to leave the topological term.
Here we consider the quantum dimer model on the triangular lattice
and examine the effectiveness of the proposal in numerical calculations of finite-size systems 3.
This model is known to exhibit a liquid phase with $Z_2$ topological order 4.
Unlike a solvable model studied in 2,
this model shows a finite correlation length and thus can be used to probe finite-size effects.
We mainly consider the Rokhsar-Kivelson point with exact ground states,
for which the reduced density matrices can be calculated by
counting the number of dimer coverings of the lattice
satisfying some particular constraints.
We also investigate other points by performing Lanczos diagonalization
of the Hamiltonian for small systems.
We examine the two original constructions to measure the topological entropy
by combining entropies on plural areas,
and we observe that in the large-area limit they both approach the value expected for $Z_2$
topological order. We also consider the
entanglement entropy on a topologically non-trivial ``zigzag'' area
and propose to use it as another way to measure the topological
entanglement entropy.
1) A. Kitaev and J. Preskill, Phys. Rev. Lett. 96 (2006) 110404.\\
2) M. Levin and X.-G. Wen, Phys. Rev. Lett. 96 (2006) 110405. \\
3) S. Furukawa and G. Misguich, arXiv:cond-mat/0612227.\\
4) R. Moessner and S. L. Sondhi, Phys. Rev. Lett. 86 (2001) 1881.
