arXiv:0707.1986 · Physical Review B

Crystallization of the resonating valence bond liquid as vortex condensation

Arnaud Ralko, Michel Ferrero, Federico Becca, Dmitri A. Ivanov, Frédéric Mila

Published Oct 15, 2007• 25 citations

Abstract

We show that the liquid-to-crystal quantum phase transition in the Rokhsar-Kivelson dimer model on the two-dimensional triangular lattice occurs as a condensation of vortexlike excitations called ``visons.'' This conclusion is drawn from numerical studies of the vison spectrum in the liquid phase by using the Green's function Monte Carlo method. We find that visons remain the lowest excitation throughout the liquid phase and that their gap decreases continuously to zero at the phase transition. The nature of the crystal phase and the second order of the phase transition are in agreement with an earlier prediction of Moessner and Sondhi [Phys. Rev. B 63, 224401 (2001)].

DOI: 10.1103/physrevb.76.140404

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arXiv:0707.1986 · Physical Review B

Crystallization of the resonating valence bond liquid as vortex condensation

Arnaud Ralko, Michel Ferrero, Federico Becca, Dmitri A. Ivanov, Frédéric Mila

Published Oct 15, 2007• 25 citations

arXiv HTML PDF

Abstract

DOI: 10.1103/physrevb.76.140404

Related papers

Quantum dimer model on the triangular lattice: Semiclassical and variational approaches to vison dispersion and condensation

Grégoire Misguich, Frédéric Mila

After reviewing the concept of vison excitations in ${\mathbb{Z}}_{2}$ dimer liquids, we study the liquid-crystal transition of the quantum dimer model on the triangular lattice by means of a semiclassical spin-wave approximation to the dispersion of visons in the context of a ``soft-dimer'' version of the model. This approach captures some important qualitative features of the transition: continuous nature of the transition, linear dispersion at the critical point, and $\sqrt{12}\ifmmode\times\else\texttimes\fi{}\sqrt{12}$ symmetry-breaking pattern. In a second part, we present a variational calculation of the vison dispersion relation at the Rokhsar-Kivelson (RK) point, which reproduces the qualitative shape of the dispersion relation and the order of magnitude of the gap. This approach provides a simple but reliable approximation of the vison wave functions at the RK point.After reviewing the concept of vison excitations in ${\mathbb{Z}}_{2}$ dimer liquids, we study the liquid-crystal transition of the quantum dimer model on the triangular lattice by means of a semiclassical spin-wave approximation to the dispersion of visons in the context of a ``soft-dimer'' version of the model. This approach captures some important qualitative features of the transition: continuous nature of the transition, linear dispersion at the critical point, and $\sqrt{12}\ifmmode\times\else\texttimes\fi{}\sqrt{12}$ symmetry-breaking pattern. In a second part, we present a variational calculation of the vison dispersion relation at the Rokhsar-Kivelson (RK) point, which reproduces the qualitative shape of the dispersion relation and the order of magnitude of the gap. This approach provides a simple but reliable approximation of the vison wave functions at the RK point.

arXiv:0712.1102Physical Review BApr 9, 200840 citations

Vison excitations in near-critical quantum dimer models

Grégory Strübi, Dmitri A. Ivanov

We study vison excitations in a quantum dimer model interpolating between the Rokhsar-Kivelson models on the square and triangular lattices. In the square-lattice case, the model is known to be critical and characterized by U (1) topological quantum numbers. Introducing diagonal dimers brings the model to a Z 2 resonating-valence-bond phase. We study variationally the emergence of vison excitations at low concentration of diagonal dimers, close to the critical point. We find that, in this regime, vison excitations are large in size and their structure resembles vortices in type-II superconductors.We study vison excitations in a quantum dimer model interpolating between the Rokhsar-Kivelson models on the square and triangular lattices. In the square-lattice case, the model is known to be critical and characterized by U (1) topological quantum numbers. Introducing diagonal dimers brings the model to a Z 2 resonating-valence-bond phase. We study variationally the emergence of vison excitations at low concentration of diagonal dimers, close to the critical point. We find that, in this regime, vison excitations are large in size and their structure resembles vortices in type-II superconductors.

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Three-dimensional resonating-valence-bond liquids and their excitations

Roderich Moessner, Shivaji L. Sondhi

We show that there are two types of resonating-valence-bond (RVB) liquid phases present in three-dimensional quantum dimer models, corresponding to the deconfining phases of $U(1)$ and ${Z}_{2}$ gauge theories in $d=3+1.$ The former is found on the bipartite cubic lattice and is the generalization of the critical point in the square lattice quantum dimer model found originally by Rokhsar and Kivelson. The latter exists on the nonbipartite face-centered cubic lattice and generalizes the RVB phase found earlier by us on the triangular lattice. We discuss the excitation spectrum and the nature of the ordering in both cases. Both phases exhibit gapped spinons. In the $U(1)$ case we find a collective, linearly dispersing, transverse excitation, which is the photon of the low-energy Maxwell Lagrangian and we identify the ordering as quantum order in Wen's sense. In the ${Z}_{2}$ case all collective excitations are gapped and, as in $d=2,$ the low-energy description of this topologically ordered state is the purely topological BF action. As a byproduct of this analysis, we unearth a further gapless excitation, the pi0n, in the square lattice quantum dimer model at its critical point.We show that there are two types of resonating-valence-bond (RVB) liquid phases present in three-dimensional quantum dimer models, corresponding to the deconfining phases of $U(1)$ and ${Z}_{2}$ gauge theories in $d=3+1.$ The former is found on the bipartite cubic lattice and is the generalization of the critical point in the square lattice quantum dimer model found originally by Rokhsar and Kivelson. The latter exists on the nonbipartite face-centered cubic lattice and generalizes the RVB phase found earlier by us on the triangular lattice. We discuss the excitation spectrum and the nature of the ordering in both cases. Both phases exhibit gapped spinons. In the $U(1)$ case we find a collective, linearly dispersing, transverse excitation, which is the photon of the low-energy Maxwell Lagrangian and we identify the ordering as quantum order in Wen's sense. In the ${Z}_{2}$ case all collective excitations are gapped and, as in $d=2,$ the low-energy description of this topologically ordered state is the purely topological BF action. As a byproduct of this analysis, we unearth a further gapless excitation, the pi0n, in the square lattice quantum dimer model at its critical point.

arXiv:cond-mat/0307592Physical review. B, Condensed matterNov 18, 2003170 citations

Columnar dimer and plaquette resonating valence bond states of the quantum dimer model

P. W. Leung, Chiu, K. C., Karl J. Runge

We study the nature of the ground state of the quantum dimer model proposed by Rokhsar and Kivelson by diagonalizing the Hamiltonian of the model on square lattices of size $L\times L$ , where $L\leq 8$ , with periodic boundary conditions. Finite-size scaling studies of the columnar order parameter and the low lying excitation spectrum show no evidence of a dimer liquid state in any finite region of the zero temperature phase diagram. In addition, we find evidence of a transition from the columnar dimer state to an intermediate state at a negative value of $V/J$ . This state is identified to be the plaquette resonating valence bond (RVB) state. The energy gap of this state vanishes as a power law of $L$ . It exhibits columnar dimer order, but has disorder {\it within} the dimer columns. This state persists up to $V/J<1$ , and the system changes to a dimer liquid state only at $V/J=1$ .We study the nature of the ground state of the quantum dimer model proposed by Rokhsar and Kivelson by diagonalizing the Hamiltonian of the model on square lattices of size $L\times L$, where $L\leq 8$, with periodic boundary conditions. Finite-size scaling studies of the columnar order parameter and the low lying excitation spectrum show no evidence of a dimer liquid state in any finite region of the zero temperature phase diagram. In addition, we find evidence of a transition from the columnar dimer state to an intermediate state at a negative value of $V/J$. This state is identified to be the plaquette resonating valence bond (RVB) state. The energy gap of this state vanishes as a power law of $L$. It exhibits columnar dimer order, but has disorder {\it within} the dimer columns. This state persists up to $V/J<1$, and the system changes to a dimer liquid state only at $V/J=1$.

arXiv:cond-mat/9605179May 29, 19960 citations

Columnar dimer and plaquette resonating-valence-bond orders in the quantum dimer model

P. W. Leung, K. C. Chiu, Karl J. Runge

We study the nature of the ground state of the quantum dimer model proposed by Rokhsar and Kivelson by diagonalizing the Hamiltonian of the model on square lattices of size L\ifmmode\times\else\texttimes\fi{}L, where L\ensuremath{\leqslant}8, with periodic boundary conditions. Finite-size scaling studies of the columnar order parameter and the low lying excitation spectrum show no evidence of a dimer liquid state in any finite region of the zero temperature phase diagram. In addition, we find evidence of a transition from the columnar dimer state to an intermediate state at a negative value of V/J. This state is consistent with the plaquette resonating-valence-bond state. The energy gap of this state vanishes as a power law of L. It exhibits columnar dimer order, but has disorder within the dimer columns. This state persists up to V/J1, and the system changes to a dimer liquid state only at V/J=1. \textcopyright{} 1996 The American Physical Society.We study the nature of the ground state of the quantum dimer model proposed by Rokhsar and Kivelson by diagonalizing the Hamiltonian of the model on square lattices of size L\ifmmode\times\else\texttimes\fi{}L, where L\ensuremath{\leqslant}8, with periodic boundary conditions. Finite-size scaling studies of the columnar order parameter and the low lying excitation spectrum show no evidence of a dimer liquid state in any finite region of the zero temperature phase diagram. In addition, we find evidence of a transition from the columnar dimer state to an intermediate state at a negative value of V/J. This state is consistent with the plaquette resonating-valence-bond state. The energy gap of this state vanishes as a power law of L. It exhibits columnar dimer order, but has disorder within the dimer columns. This state persists up to V/J1, and the system changes to a dimer liquid state only at V/J=1. \textcopyright{} 1996 The American Physical Society.

Physical review. B, Condensed matterNov 1, 199672 citations

Single hole and vortex excitations in the doped Rokhsar-Kivelson quantum dimer model on the triangular lattice

Hugo Ribeiro, Samuel Bieri, Dmitri A. Ivanov

We consider the doped Rokhsar-Kivelson quantum dimer model on the triangular lattice with one mobile hole (monomer) at the Rokhsar-Kivelson point. The motion of the hole is described by two branches of excitations: the hole may either move with or without a trapped ${\mathrm{Z}}_{2}$ vortex (vison). We perform a study of the hole dispersion in the limit where the hole-hopping amplitude is much smaller than the interdimer interaction. In this limit, the hole without vison moves freely and has a tight-binding spectrum. On the other hand, the hole with a trapped vison is strongly constrained due to interference effects and can only move via higher-order virtual processes.We consider the doped Rokhsar-Kivelson quantum dimer model on the triangular lattice with one mobile hole (monomer) at the Rokhsar-Kivelson point. The motion of the hole is described by two branches of excitations: the hole may either move with or without a trapped ${\mathrm{Z}}_{2}$ vortex (vison). We perform a study of the hole dispersion in the limit where the hole-hopping amplitude is much smaller than the interdimer interaction. In this limit, the hole without vison moves freely and has a tight-binding spectrum. On the other hand, the hole with a trapped vison is strongly constrained due to interference effects and can only move via higher-order virtual processes.

arXiv:0708.0508Physical Review BNov 30, 20077 citations

After reviewing the concept of vison excitations in

{\mathbb{Z}}_{2}

dimer liquids, we study the liquid-crystal transition of the quantum dimer model on the triangular lattice by means of a semiclassical spin-wave approximation to the dispersion of visons in the context of a ``soft-dimer'' version of the model. This approach captures some important qualitative features of the transition: continuous nature of the transition, linear dispersion at the critical point, and $\sqrt{12}\ifmmode\times\else\texttimes\fi{}\sqrt{12}$ symmetry-breaking pattern. In a second part, we present a variational calculation of the vison dispersion relation at the Rokhsar-Kivelson (RK) point, which reproduces the qualitative shape of the dispersion relation and the order of magnitude of the gap. This approach provides a simple but reliable approximation of the vison wave functions at the RK point.

We show that there are two types of resonating-valence-bond (RVB) liquid phases present in three-dimensional quantum dimer models, corresponding to the deconfining phases of

U(1)

and

{Z}_{2}

gauge theories in

d=3+1.

The former is found on the bipartite cubic lattice and is the generalization of the critical point in the square lattice quantum dimer model found originally by Rokhsar and Kivelson. The latter exists on the nonbipartite face-centered cubic lattice and generalizes the RVB phase found earlier by us on the triangular lattice. We discuss the excitation spectrum and the nature of the ordering in both cases. Both phases exhibit gapped spinons. In the

U(1)

case we find a collective, linearly dispersing, transverse excitation, which is the photon of the low-energy Maxwell Lagrangian and we identify the ordering as quantum order in Wen's sense. In the

{Z}_{2}

case all collective excitations are gapped and, as in

d=2,

the low-energy description of this topologically ordered state is the purely topological BF action. As a byproduct of this analysis, we unearth a further gapless excitation, the pi0n, in the square lattice quantum dimer model at its critical point.

We study the nature of the ground state of the quantum dimer model proposed by Rokhsar and Kivelson by diagonalizing the Hamiltonian of the model on square lattices of size

L\times L

, where

L\leq 8

, with periodic boundary conditions. Finite-size scaling studies of the columnar order parameter and the low lying excitation spectrum show no evidence of a dimer liquid state in any finite region of the zero temperature phase diagram. In addition, we find evidence of a transition from the columnar dimer state to an intermediate state at a negative value of

V/J

. This state is identified to be the plaquette resonating valence bond (RVB) state. The energy gap of this state vanishes as a power law of

L

. It exhibits columnar dimer order, but has disorder {\it within} the dimer columns. This state persists up to

V/J<1

, and the system changes to a dimer liquid state only at

V/J=1

{\mathrm{Z}}_{2}

vortex (vison). We perform a study of the hole dispersion in the limit where the hole-hopping amplitude is much smaller than the interdimer interaction. In this limit, the hole without vison moves freely and has a tight-binding spectrum. On the other hand, the hole with a trapped vison is strongly constrained due to interference effects and can only move via higher-order virtual processes.