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arXiv:cond-mat/0410693 · Physical Review B

Quantum dimer model with Z2 liquid ground state: Interpolation between cylinder and disk topologies and toy model for a topological quantum bit

Grégoire Misguich, Vincent Pasquier, Frédéric Mila, C. Lhuillier

Published May 27, 2005• 10 citations
arXivHTMLPDF

Abstract

We consider a quantum dimer model on the kagome lattice, which was introduced recently [Phys. Rev. Lett. 89, 137202 (2002)]. It realizes a ${\mathbb{Z}}_{2}$ liquid phase and its spectrum was obtained exactly. It displays a topological degeneracy when the lattice has a nontrivial geometry (cylinder, torus, etc). We discuss and solve two extensions of the model where perturbations along lines are introduced: first a potential-energy term repelling (or attracting) the dimers along a line and second, a perturbation allowing to create, move, or destroy monomers. For each of these perturbations we show that there exists a critical value above which, in the thermodynamic limit, the degeneracy of the ground state is lifted from 2 (on a cylinder) to 1. In both cases the exact value of the gap between the first two levels is obtained by a mapping to an Ising chain in a transverse field. This model provides an example of a solvable Hamiltonian for a topological quantum bit where the two perturbations act as diagonal and transverse operators in the two-dimensional subspace. We discuss how crossing the transitions may be used in the manipulation of the quantum bit to simultaneously optimize the frequency of operation and the losses due to decoherence.

DOI
10.1103/physrevb.71.184424

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We consider a quantum dimer model on the kagome lattice, which was introduced recently [Phys. Rev. Lett. 89, 137202 (2002)]. It realizes a Z2{\mathbb{Z}}_{2}Z2​ liquid phase and its spectrum was obtained exactly. It displays a topological degeneracy when the lattice has a nontrivial geometry (cylinder, torus, etc). We discuss and solve two extensions of the model where perturbations along lines are introduced: first a potential-energy term repelling (or attracting) the dimers along a line and second, a perturbation allowing to create, move, or destroy monomers. For each of these perturbations we show that there exists a critical value above which, in the thermodynamic limit, the degeneracy of the ground state is lifted from 2 (on a cylinder) to 1. In both cases the exact value of the gap between the first two levels is obtained by a mapping to an Ising chain in a transverse field. This model provides an example of a solvable Hamiltonian for a topological quantum bit where the two perturbations act as diagonal and transverse operators in the two-dimensional subspace. We discuss how crossing the transitions may be used in the manipulation of the quantum bit to simultaneously optimize the frequency of operation and the losses due to decoherence.
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Extend of the Z2\mathbb{Z}_2Z2​-spin liquid phase on the Kagomé-lattice
The Z2\mathbb{Z}_2Z2​ topological phase in the quantum dimer model on the Kagomé-lattice is a candidate for the description of the low-energy physics of the anti-ferromagnetic Heisenberg model on the same lattice. We study the extend of the topological phase by interpolating between the exactly solvable parent Hamiltonian of the topological phase and an effective low-energy description of the Heisenberg model in terms of a quantum-dimer Hamiltonian. Therefore, we perform a perturbative treatment of the low-energy excitations in the topological phase including free and interacting quasi-particles. We find a phase transition out of the topological phase far from the Heisenberg point. The resulting phase is characterized by a spontaneously broken rotational symmetry and a unit cell involving six sites.
Thanks to Pfaffian techniques, we study the Rényi entanglement entropies and the entanglement spectrum of large subsystems for two-dimensional Rokhsar–Kivelson wavefunctions constructed from a dimer model on the triangular lattice. By including a fugacity t on some suitable bonds, one interpolates between the triangular lattice ( t = 1) and the square lattice ( t = 0). The wavefunction is known to be a massive topological liquid for t > 0 whereas it is a gapless critical state at t = 0. We mainly consider two geometries for the subsystem: that of a semi-infinite cylinder and the disc-like set-up proposed by Kitaev and Preskill (2006 Phys. Rev. Lett. 96 110404). In the cylinder case, the entropies contain an extensive term—proportional to the length of the boundary—and a universal subleading constant s n ( t ). Fitting these cylinder data (up to a perimeter of L = 32 sites) provides s n with a very high numerical accuracy (10 − 9 at t = 1 and 10 − 6 at t = 0.5). In the topological liquid phase we find s n ( t > 0) = − ln2, independent of the fugacity t and the Rényi parameter n . At t = 0 we recover a previously known result, for n < 1 and s n ( t = 0) = − ln(2)/( n − 1) for n > 1. In the disc-like geometry—designed to get rid of the boundary contributions—we find an entropy s n KP ( t 0) = − ln2 in the whole massive phase whatever n > 0, in agreement with the result of Flammia et al (2009 Phys. Rev. Lett. 103 261601). Some results for the gapless limit are discussed.
We study a generalized quantum hard-core dimer model on the square and honeycomb lattices, allowing for first and second neighbor dimers. At generalized Rokhsar-Kivelson points, the exact ground states can be constructed, and ground-state correlation functions can be equated to those of interacting (1+1)-dimensional Grassmann fields. When the concentration of second neighbor dimers is small, the ground-state correlations are shown to be short ranged corresponding to a (gaped) spin liquid phase. On a 2-torus, the ground states exhibit fourfold topological degeneracy. On a finite cylinder we have found a dramatic even-odd effect depending on the circumference and propose that this can be used as a numerical diagnostic of gapped spin-liquid phases, more generally.
Quantum spin liquids are exotic many-body states featured with long-range entanglement and fractional anyon quasiparticles. Quantum phase transitions of spin liquids are particularly interesting problems related with novel phenomena of anyon condensation and anyon confinement. Here we study a quantum dimer model, which implements a transition between a Z 2 spin liquid ( Z 2 SL ) and a valence bond solid (VBS) on the kagome lattice. The transition is driven by the condensation of vison excitation of the Z 2 spin liquid, which impacts on other anyon excitations especially leading to the confinement of spinon excitations. By numerical exact diagonalization of the dimer model, we directly measure the vison condensation using vison string operators, and explicitly check a confining potential acting on spinon excitations in the VBS state. It is observed that topological degeneracy of the spin-liquid state is lifted concomitantly with the vison condensation. The dimer ordering pattern of the VBS state is identified by investigating dimer structure factor. Furthermore, we find an interesting state that exhibits features of spin liquid and VBS simultaneously. We discuss the origin of the mixed behaviors and possible scenarios expected in thermodynamic limit. This work complements the previous analytical studies on the dimer model [ and ] by providing numerical evidences on the vison condensation and the spinon confinement in the Z 2 SL -to-VBS transition. Published by the American Physical Society 2024
Quantum dimer models typically arise in various low energy theories like those of frustrated antiferromagnets. We introduce a quantum dimer model on the kagome lattice which stabilizes an alternative Z2{\mathbb{Z}}_{2}Z2​ topological order, namely the so-called ``double semion'' order. For a particular set of parameters, the model is exactly solvable, allowing us to access the ground state as well as the excited states. We show that the double semion phase is stable over a wide range of parameters using numerical exact diagonalization. Furthermore, we propose a simple microscopic spin Hamiltonian for which the low-energy physics is described by the derived quantum dimer model.