Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1802.09576 · Physical review. B./Physical review. B

Phases of quantum dimers from ensembles of classical stochastic trajectories

Tom Oakes, Stephen Powell, Claudio Castelnovo, Austen Lamacraft, Juan P. Garrahan

Published Aug 8, 2018• 26 citations
arXivHTMLPDF

Abstract

We study the connection between the phase behavior of quantum dimers and the dynamics of classical stochastic dimers. At the so-called Rokhsar-Kivelson (RK) point a quantum dimer Hamiltonian is equivalent to the Markov generator of the dynamics of classical dimers. A less well understood fact is that away from the RK point the quantum-classical connection persists: in this case the Hamiltonian corresponds to a nonstochastic ``tilted'' operator that encodes the statistics of time-integrated observables of the classical stochastic problem. This implies a direct relation between the phase behavior of quantum dimers and properties of ensembles of stochastic trajectories of classical dimers. We make these ideas concrete by studying fully packed dimers on the square lattice. Using transition path sampling---supplemented by trajectory umbrella sampling---we obtain the large deviation statistics of dynamical activity in the classical problem, and show the correspondence between the phase behavior of the classical and quantum systems. The transition at the RK point between quantum phases of distinct order corresponds, in the classical case, to a trajectory phase transition between active and inactive dynamical phases. Furthermore, from the structure of stochastic trajectories in the active dynamical phase we infer that the ground state of quantum dimers has columnar order to one side of the RK point. We discuss how these results relate to those from quantum Monte Carlo, and how our approach may generalize to other problems.

DOI
10.1103/physrevb.98.064302

Related papers

Dynamical fluctuations of classical and quantum square dimer models

Tom Oakes

The work in this thesis is split into two main parts; the dynamical study of the fully packed classical dimer model, and the development of advanced large deviation techniques applied to the classical dimer model, highlighting the connection to the quantum dimer model. The classical dimer model on the square lattice is a paradigmatic example of a system subject to strong local constraints. We study its behavior under local stochastic dynamics, by means of Monte Carlo simulations and theoretical arguments. We observe clear signatures of correlated dynamics in both global and local observables and over a broad range of time scales, indicating a breakdown of the simple continuum description that approximates well the statics. We show that this collective dynamics can be understood in terms of one-dimensional “strings” of high mobility. We introduce a coarse-grained description of the strings, which leads to exact results in the limit of low string density and provides a detailed qualitative understanding of the dynamics in all flux sectors. We then go on, still using the fully packed dimers on the square lattice as a paradigmatic system, to study the connection between the phase behaviour of the ground state of quantum dimers and the dynamics of classical stochastic dimers. At the so-called Rokhsar-Kivelson (RK) point, a quantum dimer Hamiltonian is equivalent to the Markov generator of the dynamics of classical dimers. A less well understood fact is that away from the RK point the quantum-classical connection persists: in this case the quantum Hamiltonian maps to a non-stochastic "tilted" master operator of the classical stochastic problem. This implies a direct relation between the phase behaviour of quantum dimers and properties of ensembles of stochastic trajectories of classical dimers. Using transition path sampling (TPS) -- supplemented by trajectory umbrella sampling -- we obtain the large deviation statistics of dynamical activity in the classical problem, and show the correspondence between the phase behaviour of the classical and quantum systems. Finally, we take these ideas further by developing a feedback-based modified dynamics used for umbrella sampling alongside TPS. This feedback approach was first used in the cloning (or population dynamics) method. We compare different modified dynamics within TPS and show how much more efficient the feedback-modified dynamics can be than the original dynamics, for the purpose of rare event sampling. We also discuss how our approach may generalise to other problems with slow complex dynamics.

•Jul 17, 2019•0 citations

RANDOM WALKS NEAR ROKHSAR–KIVELSON POINTS

Olav F. Syljuåsen

There is a class of quantum Hamiltonians known as Rokhsar–Kivelson (RK)–Hamiltonians for which static ground state properties can be obtained by evaluating thermal expectation values for classical models. The ground state of an RK–Hamiltonian is known explicitly, and its dynamical properties can be obtained by performing a classical Monte Carlo simulation. We discuss the details of a Diffusion Monte Carlo method that is a good tool for studying statics and dynamics of perturbed RK–Hamiltonians without time discretization errors. As a general result we point out that the relation between the quantum dynamics and classical Monte Carlo simulations for RK–Hamiltonians follows from the known fact that the imaginary-time evolution operator describing optimal importance sampling, where the exact ground state is used as guiding function, is Markovian. Thus quantum dynamics can be studied by classical Monte Carlo for any Hamiltonian provided its ground state is known explicitly and that there is no sign problem.

arXiv:cond-mat/0504195•International Journal of Modern Physics B•May 10, 2005•11 citations

From classical to quantum dynamics at Rokhsar-Kivelson points

Christopher L. Henley

Abstract. For any classical statistical-mechanics model with a discrete state space, and endowed with a dynamics satisfying detailed balance, it is possible to generalize the Rokhsar-Kivelson point for the quantum dimer model. That is, a quantum Hamiltonian can be constructed (on the same state space), such that the ground state wavefunction coincides with the classical equilibrium distribution. Furthermore the excited eigenstates correspond to classical relaxation modes, which (in cases with a symmetry or conserved quantity) permits extraction of the dispersion law of long-wavelength excitations. The mapping is natural mainly when the states have equal weight, as is typical of a highly frustrated model. Quantum and classical correlation functions are related by analytic continuation to the imaginary time axis. The quantum dimer model [1] of Rokhsar and Kivelson (RK) was inspired by the resonating-valence bond state of a quantum antiferromagnet, which had just been proposed as a starting point for the explanation of high-temperature superconductivity. That was not to be, but models of this type are of interest among

•Jul 26, 2013•52 citations

Dynamical dimer correlations at bipartite and non-bipartite Rokhsar–Kivelson points

Andreas Martin Läuchli, Sylvain Capponi, Fakher F. Assaad

We determine the dynamical dimer correlation functions of quantum dimer models at the Rokhsar–Kivelson point on the bipartite square and cubic lattices and the non-bipartite triangular lattice. On the basis of an algorithmic idea by Henley, we simulate a stochastic process of classical dimer configurations in continuous time and perform a stochastic analytical continuation to obtain the dynamical correlations in momentum space and the frequency domain. This approach allows us to observe directly the dispersion relations and the evolution of the spectral intensity within the Brillouin zone beyond the single-mode approximation. On the square lattice, we confirm analytical predictions related to soft modes close to the wavevectors (π,π) and (π,0) and further reveal the existence of shadow bands close to the wavevector (0,0). On the cubic lattice the spectrum is also gapless but here only a single soft mode at (π,π,π) is found, as predicted by the single-mode approximation. The soft mode has a quadratic dispersion at very long wavelength, but crosses over to a linear behavior very rapidly. We believe this to be the remnant of the linearly dispersing 'photon' of the Coulomb phase. Finally the triangular lattice is in a fully gapped liquid phase where the bottom of the dimer spectrum exhibits a rich structure. At the M point the gap is minimal and the spectral response is dominated by a sharp quasiparticle peak. On the other hand, at the X point the spectral function is much broader. We sketch a possible explanation based on the crossing of the coherent dimer excitations into the two-vison continuum.

arXiv:0711.0752•Journal of Statistical Mechanics Theory and Experiment•Jan 9, 2008•20 citations

Interacting Classical Dimers on the Square Lattice

Fabien Alet, Jesper Lykke Jacobsen, Grégoire Misguich +3 more

We study a model of close-packed dimers on the square lattice with a nearest neighbor interaction between parallel dimers. This model corresponds to the classical limit of quantum dimer models [D. S. Rokhsar and S. A. Kivelson, Phys. Rev. Lett. 61, 2376 (1988)]. By means of Monte Carlo and transfer matrix calculations, we show that this system undergoes a Kosterlitz-Thouless transition separating a low temperature ordered phase where dimers are aligned in columns from a high temperature critical phase with continuously varying exponents. This is understood by constructing the corresponding Coulomb gas, whose coupling constant is computed numerically. We also discuss doped models and implications on the finite-temperature phase diagram of quantum dimer models.

arXiv:cond-mat/0501241•Physical Review Letters•Jun 15, 2005•116 citations

From classical to quantum dynamics at Rokhsar–Kivelson points

Christopher L. Henley

For any classical statistical-mechanics model with a discrete state space, and endowed with a dynamics satisfying detailed balance, it is possible to generalize the Rokhsar–Kivelson point for the quantum dimer model. That is, a quantum Hamiltonian can be constructed (on the same state space) such that the ground state wavefunction coincides with the classical equilibrium distribution. Furthermore the excited eigenstates correspond to classical relaxation modes, which (in cases with a symmetry or conserved quantity) permits extraction of the dispersion law of long-wavelength excitations. The mapping is natural mainly when the states have equal weight, as is typical of a highly frustrated model. Quantum and classical correlation functions are related by analytic continuation to the imaginary time axis.

arXiv:cond-mat/0311345•Journal of Physics Condensed Matter•Mar 4, 2004•115 citations
We study the connection between the phase behavior of quantum dimers and the dynamics of classical stochastic dimers. At the so-called Rokhsar-Kivelson (RK) point a quantum dimer Hamiltonian is equivalent to the Markov generator of the dynamics of classical dimers. A less well understood fact is that away from the RK point the quantum-classical connection persists: in this case the Hamiltonian corresponds to a nonstochastic ``tilted'' operator that encodes the statistics of time-integrated observables of the classical stochastic problem. This implies a direct relation between the phase behavior of quantum dimers and properties of ensembles of stochastic trajectories of classical dimers. We make these ideas concrete by studying fully packed dimers on the square lattice. Using transition path sampling---supplemented by trajectory umbrella sampling---we obtain the large deviation statistics of dynamical activity in the classical problem, and show the correspondence between the phase behavior of the classical and quantum systems. The transition at the RK point between quantum phases of distinct order corresponds, in the classical case, to a trajectory phase transition between active and inactive dynamical phases. Furthermore, from the structure of stochastic trajectories in the active dynamical phase we infer that the ground state of quantum dimers has columnar order to one side of the RK point. We discuss how these results relate to those from quantum Monte Carlo, and how our approach may generalize to other problems.
The work in this thesis is split into two main parts; the dynamical study of the fully packed classical dimer model, and the development of advanced large deviation techniques applied to the classical dimer model, highlighting the connection to the quantum dimer model. The classical dimer model on the square lattice is a paradigmatic example of a system subject to strong local constraints. We study its behavior under local stochastic dynamics, by means of Monte Carlo simulations and theoretical arguments. We observe clear signatures of correlated dynamics in both global and local observables and over a broad range of time scales, indicating a breakdown of the simple continuum description that approximates well the statics. We show that this collective dynamics can be understood in terms of one-dimensional “strings” of high mobility. We introduce a coarse-grained description of the strings, which leads to exact results in the limit of low string density and provides a detailed qualitative understanding of the dynamics in all flux sectors. We then go on, still using the fully packed dimers on the square lattice as a paradigmatic system, to study the connection between the phase behaviour of the ground state of quantum dimers and the dynamics of classical stochastic dimers. At the so-called Rokhsar-Kivelson (RK) point, a quantum dimer Hamiltonian is equivalent to the Markov generator of the dynamics of classical dimers. A less well understood fact is that away from the RK point the quantum-classical connection persists: in this case the quantum Hamiltonian maps to a non-stochastic "tilted" master operator of the classical stochastic problem. This implies a direct relation between the phase behaviour of quantum dimers and properties of ensembles of stochastic trajectories of classical dimers. Using transition path sampling (TPS) -- supplemented by trajectory umbrella sampling -- we obtain the large deviation statistics of dynamical activity in the classical problem, and show the correspondence between the phase behaviour of the classical and quantum systems. Finally, we take these ideas further by developing a feedback-based modified dynamics used for umbrella sampling alongside TPS. This feedback approach was first used in the cloning (or population dynamics) method. We compare different modified dynamics within TPS and show how much more efficient the feedback-modified dynamics can be than the original dynamics, for the purpose of rare event sampling. We also discuss how our approach may generalise to other problems with slow complex dynamics.
There is a class of quantum Hamiltonians known as Rokhsar–Kivelson (RK)–Hamiltonians for which static ground state properties can be obtained by evaluating thermal expectation values for classical models. The ground state of an RK–Hamiltonian is known explicitly, and its dynamical properties can be obtained by performing a classical Monte Carlo simulation. We discuss the details of a Diffusion Monte Carlo method that is a good tool for studying statics and dynamics of perturbed RK–Hamiltonians without time discretization errors. As a general result we point out that the relation between the quantum dynamics and classical Monte Carlo simulations for RK–Hamiltonians follows from the known fact that the imaginary-time evolution operator describing optimal importance sampling, where the exact ground state is used as guiding function, is Markovian. Thus quantum dynamics can be studied by classical Monte Carlo for any Hamiltonian provided its ground state is known explicitly and that there is no sign problem.
Abstract. For any classical statistical-mechanics model with a discrete state space, and endowed with a dynamics satisfying detailed balance, it is possible to generalize the Rokhsar-Kivelson point for the quantum dimer model. That is, a quantum Hamiltonian can be constructed (on the same state space), such that the ground state wavefunction coincides with the classical equilibrium distribution. Furthermore the excited eigenstates correspond to classical relaxation modes, which (in cases with a symmetry or conserved quantity) permits extraction of the dispersion law of long-wavelength excitations. The mapping is natural mainly when the states have equal weight, as is typical of a highly frustrated model. Quantum and classical correlation functions are related by analytic continuation to the imaginary time axis. The quantum dimer model [1] of Rokhsar and Kivelson (RK) was inspired by the resonating-valence bond state of a quantum antiferromagnet, which had just been proposed as a starting point for the explanation of high-temperature superconductivity. That was not to be, but models of this type are of interest among
We determine the dynamical dimer correlation functions of quantum dimer models at the Rokhsar–Kivelson point on the bipartite square and cubic lattices and the non-bipartite triangular lattice. On the basis of an algorithmic idea by Henley, we simulate a stochastic process of classical dimer configurations in continuous time and perform a stochastic analytical continuation to obtain the dynamical correlations in momentum space and the frequency domain. This approach allows us to observe directly the dispersion relations and the evolution of the spectral intensity within the Brillouin zone beyond the single-mode approximation. On the square lattice, we confirm analytical predictions related to soft modes close to the wavevectors (π,π) and (π,0) and further reveal the existence of shadow bands close to the wavevector (0,0). On the cubic lattice the spectrum is also gapless but here only a single soft mode at (π,π,π) is found, as predicted by the single-mode approximation. The soft mode has a quadratic dispersion at very long wavelength, but crosses over to a linear behavior very rapidly. We believe this to be the remnant of the linearly dispersing 'photon' of the Coulomb phase. Finally the triangular lattice is in a fully gapped liquid phase where the bottom of the dimer spectrum exhibits a rich structure. At the M point the gap is minimal and the spectral response is dominated by a sharp quasiparticle peak. On the other hand, at the X point the spectral function is much broader. We sketch a possible explanation based on the crossing of the coherent dimer excitations into the two-vison continuum.
We study a model of close-packed dimers on the square lattice with a nearest neighbor interaction between parallel dimers. This model corresponds to the classical limit of quantum dimer models [D. S. Rokhsar and S. A. Kivelson, Phys. Rev. Lett. 61, 2376 (1988)]. By means of Monte Carlo and transfer matrix calculations, we show that this system undergoes a Kosterlitz-Thouless transition separating a low temperature ordered phase where dimers are aligned in columns from a high temperature critical phase with continuously varying exponents. This is understood by constructing the corresponding Coulomb gas, whose coupling constant is computed numerically. We also discuss doped models and implications on the finite-temperature phase diagram of quantum dimer models.
For any classical statistical-mechanics model with a discrete state space, and endowed with a dynamics satisfying detailed balance, it is possible to generalize the Rokhsar–Kivelson point for the quantum dimer model. That is, a quantum Hamiltonian can be constructed (on the same state space) such that the ground state wavefunction coincides with the classical equilibrium distribution. Furthermore the excited eigenstates correspond to classical relaxation modes, which (in cases with a symmetry or conserved quantity) permits extraction of the dispersion law of long-wavelength excitations. The mapping is natural mainly when the states have equal weight, as is typical of a highly frustrated model. Quantum and classical correlation functions are related by analytic continuation to the imaginary time axis.