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arXiv:2306.02578 · Physical Review Research

Ashkin-Teller phase transition and multicritical behavior in a classical monomer-dimer model

Satoshi Morita, Hyun‐Yong Lee, Kedar Damle, Naoki Kawashima

Published Oct 19, 2023• 4 citations
arXivHTMLPDF

Abstract

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 $z$, an aligning dimer-dimer interaction $u$ that favors columnar order, and an attractive dimer-dimer interaction $v$ 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 ${z}_{\mathrm{mc}}(v/u)>0$; 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.

DOI
10.1103/physrevresearch.5.043061

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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∞v/u<\ensuremath{\infty}v/u<\ensuremath∞, 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 zmc(v/u)\ensuremath→0{z}_{\mathrm{mc}}(v/u)\ensuremath{\rightarrow}0zmc​(v/u)\ensuremath→0 very rapidly as v/u\ensuremath→\ensuremath∞v/u\ensuremath{\rightarrow}\ensuremath{\infty}v/u\ensuremath→\ensuremath∞. 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.
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.
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.
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.
Using renormalization group (RG) analyses and Monte Carlo (MC) simulations, we study the fully packed dimer model on the bilayer square lattice with fugacity equal to z (1) for interlayer (intralayer) dimers, and intralayer interaction V between neighboring parallel dimers on any elementary plaquette in either layer. For a range of not-too-large z>0 and repulsive interactions 0<V<V_{s} (with V_{s}≈2.1), we demonstrate the existence of a bilayer Coulomb phase with purely dipolar two-point functions, i.e., without the power-law columnar order that characterizes the usual Coulomb phase of square and honeycomb lattice dimer models. The transition line z_{c}(V) separating this bilayer Coulomb phase from a large-z disordered phase is argued to be in the inverted Kosterlitz-Thouless universality class. Additionally, we argue for the possibility of a tricritical point at which the bilayer Coulomb phase, the large-z disordered phase and the large-V staggered phase meet in the large-z, large-V part of the phase diagram. In contrast, for the attractive case with V_{cb}<V≤0 (V_{cb}≈-1.2), we argue that any z>0 destroys the power-law correlations of the z=0 decoupled layers, and leads immediately to a short-range correlated state, albeit with a slow crossover for small |V|. For V_{c}<V<V_{cb} (V_{c}≈-1.55), we predict that any small nonzero z immediately gives rise to long-range bilayer columnar order although the z=0 decoupled layers remain power-law correlated in this regime; this implies a nonmonotonic z dependence of the columnar order parameter for fixed V in this regime. Further, our RG arguments predict that this bilayer columnar ordered state is separated from the large-z disordered state by a line of Ashkin-Teller transitions z_{AT}(V). Finally, for V<V_{c}, the z=0 decoupled layers are already characterized by long-range columnar order, and a small nonzero z leads immediately to a locking of the order parameters of the two layers, giving rise to the same bilayer columnar ordered state for small nonzero z.
Two nonperturbative methods by means of the transfer-matrix finite-size-scaling and Monte Carlo simulations are used to investigate the spin-1 Ashkin-Teller model (ATM). We have obtained rich phase diagrams with first- and second-order phase transitions with several multicritical points of higher order. Also this model exhibits a partially ordered phase PO2{\mathrm{PO}}_{2}PO2​ which does not exist in the spin-1/2 Ashkin-Teller model (ATM). Finally, the critical behavior of this model is discussed.
The results of the large-scale Monte Carlo simulations of the continuous phase transitions in the three-dimensional Ashkin-Teller model on a cubic lattice are presented in the two-parameter space. The invariance of the ratio of the square of the second moment of the order parameter to its fourth moment in the critical region and the Metropolis algorithm are exploited. Using the universality hypothesis and the finite-size-scaling analysis, the accurate positions of the continuous phase-transition lines are calculated and their Ising character is confirmed. Approximate locations of the tricritical points are also estimated.