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arXiv:2011.04506 · Physical review. E

Bilayer Coulomb phase of two-dimensional dimer models: Absence of power-law columnar order

Nisheeta Desai, Sumiran Pujari, Kedar Damle

Published Apr 26, 2021• 12 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreve.103.042136

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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.
We study phases and transitions of the square-lattice double dimer model, consisting of two coupled replicas of the classical dimer model. As on the cubic lattice, we find a thermal phase transition from the Coulomb phase, a disordered but correlated dimer liquid, to a phase where fluctuations of the two replicas are closely synchronized with one another. Surprisingly, and in contrast to the cubic case, the phase boundary includes the noninteracting point, as we establish using a symmetry-based analysis of an effective height theory, indicating that infinitesimal coupling is sufficient to synchronize the double dimer model. In addition, we observe a novel antisynchronized phase when the coupling between replicas is repulsive, and use Monte Carlo simulations to establish the full phase diagram, including (anti)synchronized columnar and staggered phases, with interactions between parallel dimers in each replica.
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 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.
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×LL\times LL×L, where L≤8L\leq 8L≤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/JV/JV/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 LLL. It exhibits columnar dimer order, but has disorder {\it within} the dimer columns. This state persists up to V/J<1V/J<1V/J<1, and the system changes to a dimer liquid state only at V/J=1V/J=1V/J=1.
We consider an efficient scheme to simulate fermionic Hubbard models with nonlocal density-density interactions in two dimensions, based on bond-centered auxiliary-field quantum Monte Carlo. The simulations are shown to be sign-problem-free within a finite, restricted parameter range. Using this approach, we first study the Hubbard model on the half-filled square lattice bilayer, including an interlayer repulsion term in addition to the local repulsion, and present the ground-state phase diagram within the accessible parameter region. Starting from the antiferromagnetically ordered state in the absence of interlayer repulsion, the interlayer interactions are found to destabilize the antiferromagnetic order, leading to a band insulator state. Moreover, for sufficiently strong interlayer tunneling, we also observe the emergence of a direct dimer product state of mixed D-Mott and S-Mott character along the equal coupling line. We discuss the stability range of this state within strong-coupling perturbation theory. Furthermore, we consider the Hubbard model on the honeycomb lattice with next-nearest-neighbor interactions. Such an interaction is found to enhance both charge density and spin-current correlations within the semimetallic region. However, inside the accessible parameter region, they do not stabilize long-ranged charge density wave order nor a quantum spin Hall state, and the only insulating state that we observe exhibits long-range antiferromagnetism.