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arXiv:1504.01565 · Physical Review E

Kosterlitz-Thouless transitions and phase diagrams of the interacting monomer-dimer model on a checkerboard lattice

Sazi Li, Wei Li, Ziyu Chen

Published Nov 5, 2014• 4 citations
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Abstract

Using the tensor network approach, we investigate the monomer-dimer models on a checkerboard lattice, in which there are interactions (with strength $v$) between the parallel dimers on half of the plaquettes. For the fully packed interacting dimer model, we observe a Kosterlitz-Thouless (KT) transition between the low-temperature symmetry breaking and the high-temperature critical phases; for the doped monomer-dimer case with finite chemical potential $\ensuremath{\mu}$, we also find an order-disorder phase transition which is of second order instead. We use the boundary matrix product state approach to detect the KT and second-order phase transitions and obtain the phase diagrams $v\text{\ensuremath{-}}T$ and $\ensuremath{\mu}\text{\ensuremath{-}}T$. Moreover, for the noninteracting monomer-dimer model (setting $\ensuremath{\mu}=\ensuremath{\nu}=0$), we get an extraordinarily accurate determination of the free energy per site (negative of the monomer-dimer constant ${h}_{2}$) as $f=\ensuremath{-}0.662\phantom{\rule{0.16em}{0ex}}798\phantom{\rule{0.16em}{0ex}}972\phantom{\rule{0.16em}{0ex}}833\phantom{\rule{0.16em}{0ex}}746$ with the dimer density $n=0.638\phantom{\rule{0.16em}{0ex}}123\phantom{\rule{0.16em}{0ex}}109\phantom{\rule{0.16em}{0ex}}228\phantom{\rule{0.16em}{0ex}}547$, both of 15 correct digits.

DOI
10.1103/physreve.90.052104

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Using the tensor network approach, we investigate the monomer-dimer models on a checkerboard lattice, in which there are interactions (with strength vvv) between the parallel dimers on half of the plaquettes. For the fully packed interacting dimer model, we observe a Kosterlitz-Thouless (KT) transition between the low-temperature symmetry breaking and the high-temperature critical phases; for the doped monomer-dimer case with finite chemical potential \ensuremathμ\ensuremath{\mu}\ensuremathμ, we also find an order-disorder phase transition which is of second order instead. We use the boundary matrix product state approach to detect the KT and second-order phase transitions and obtain the phase diagrams v\ensuremath-Tv\text{\ensuremath{-}}Tv\ensuremath-T and \ensuremathμ\ensuremath-T\ensuremath{\mu}\text{\ensuremath{-}}T\ensuremathμ\ensuremath-T. Moreover, for the noninteracting monomer-dimer model (setting \ensuremathμ=\ensuremathν=0\ensuremath{\mu}=\ensuremath{\nu}=0\ensuremathμ=\ensuremathν=0), we get an extraordinarily accurate determination of the free energy per site (negative of the monomer-dimer constant h2{h}_{2}h2​) as f=\ensuremath−0.662798972833746f=\ensuremath{-}0.662\phantom{\rule{0.16em}{0ex}}798\phantom{\rule{0.16em}{0ex}}972\phantom{\rule{0.16em}{0ex}}833\phantom{\rule{0.16em}{0ex}}746f=\ensuremath−0.662798972833746 with the dimer density n=0.638123109228547n=0.638\phantom{\rule{0.16em}{0ex}}123\phantom{\rule{0.16em}{0ex}}109\phantom{\rule{0.16em}{0ex}}228\phantom{\rule{0.16em}{0ex}}547n=0.638123109228547, both of 15 correct digits.
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.
We present a detailed study of a model of close-packed dimers on the square lattice with an interaction between nearest-neighbor dimers. The interaction favors parallel alignment of dimers, resulting in a low-temperature crystalline phase. With large-scale Monte Carlo and transfer matrix calculations, we show that the crystal melts through a Kosterlitz-Thouless phase transition to give rise to a high-temperature critical phase, with algebraic decays of correlations functions with exponents that vary continuously with the temperature. We give a theoretical interpretation of these results by mapping the model to a Coulomb gas, whose coupling constant and associated exponents are calculated numerically with high precision. Introducing monomers is a marginal perturbation at the Kosterlitz-Thouless transition and gives rise to another critical line. We study this line numerically, showing that it is in the Ashkin-Teller universality class, and terminates in a tricritical point at finite temperature and monomer fugacity. In the course of this work, we also derive analytic results relevant to the noninteracting case of dimer coverings, including a Bethe ansatz (at the free fermion point) analysis, a detailed discussion of the effective height model, and a free field analysis of height fluctuations.
In this work, we investigate the classical loop models doped with monomers and dimers on a square lattice, whose partition function can be expressed as a tensor network (TN). In the thermodynamic limit, we use the boundary matrix product state technique to contract the partition function TN, and determine the thermodynamic properties with high accuracy. In this monomer-dimer-loop model, we find a second-order phase transition between a trivial monomer-condensation and a loop-condensation (LC) phase, which cannot be distinguished by any local order parameter, while nevertheless the two phases have distinct topological properties. In the LC phase, we find two degenerate dominating eigenvalues in the transfer-matrix spectrum, as well as a nonvanishing (nonlocal) string order parameter, both of which identify the topological ergodicity breaking in the LC phase and can serve as the order parameter for detecting the phase transitions.
Abstract The critical behavior of a dimer model with an interaction favoring parallel dimers in each plaquette of the square lattice is studied numerically using the corner transfer matrix renormalization group algorithm. The critical exponents are known to depend on the chemical potential of vacancies, or monomers. At large average density of the latter, the phase transition becomes the first-order. We compute the scaling dimensions of both the order parameter and temperature in the second-order regime and compare them with the conjecture that the critical behavior is the same as the Ashkin–Teller model on its self-dual critical line.
We study phase diagrams of a class of doped quantum dimer models on the square lattice with ground-state wave functions whose amplitudes have the form of the Gibbs weights of a classical doped dimer model. In this dimer model, parallel neighboring dimers have attractive interactions, whereas neighboring holes either do not interact or have a repulsive interaction. We investigate the behavior of this system via analytic methods and by Monte Carlo simulations. At zero doping, we confirm the existence of a Kosterlitz-Thouless transition from a quantum critical phase to a columnar phase. At low hole densities, we find a dimer-hole liquid phase and a columnar phase, separated by a phase boundary which is a line of critical points with varying exponents. We demonstrate that this line ends at a multicritical point where the transition becomes first order and the system phase separates. The first-order transition coexistence curve is shown to become unstable with respect to more complex inhomogeneous phases in the presence of direct hole-hole interactions. We also use a variational approach to determine the spectrum of low-lying density fluctuations in the dimer-hole fluid phase.
We use Monte Carlo simulations and tensor network methods to study a classical monomer-dimer model on the square lattice with a hole (monomer) fugacity zzz, an aligning dimer-dimer interaction uuu that favors columnar order, and an attractive dimer-dimer interaction vvv between two adjacent dimers that lie on the same principal axis of the lattice. The Monte Carlo simulations of finite size systems rely on our grand-canonical generalization of the dimer worm algorithm, while the tensor network computations are based on a uniform matrix product ansatz for the eigenvector of the row-to-row transfer matrix that work directly in the thermodynamic limit. The phase diagram has nematic, columnar order and fluid phases, and a nonzero temperature multicritical point at which all three meet. For any fixed $v/u<\ensuremath{\infty}$, we argue that this multicritical point continues to be located at a nonzero hole fugacity zmc(v/u)>0{z}_{\mathrm{mc}}(v/u)>0zmc​(v/u)>0; our numerical results confirm this theoretical expectation but find that ${z}_{\mathrm{mc}}(v/u)\ensuremath{\rightarrow}0$ very rapidly as $v/u\ensuremath{\rightarrow}\ensuremath{\infty}$. Our numerical results also confirm the theoretical expectation that the corresponding multicritical behavior is in the universality class of the four-state Potts multicritical point on critical line of the two-dimensional Ashkin-Teller model.