Ising nematic fluid phase of hard-core dimers on the square lattice

We present a model of classical hard-core dimers on the square lattice that contains an Ising nematic phase in its phase diagram. We consider a model with an attractive interaction for parallel dimers on a given plaquette of the square lattice and an attractive interaction for neighboring parallel dimers on the same row (viz column) of the lattice. By extensive Monte Carlo simulations we find that with a finite density of holes the phase diagram has, with rising temperatures, a columnar crystalline phase, an Ising nematic liquid phase and a disordered fluid phase, separated by Ising continuous phase transitions. We present strong evidence for the Ising universality class of both transitions. The Ising nematic phase can be interpreted as either an intermediate classical thermodynamic phase (possibly of a strongly correlated antiferromagnet) or as a phase of a two-dimensional quantum dimer model using the Rokhsar-Kivelson construction of exactly solvable quantum Hamiltonians.

Figure 1

A typical configuration of an Ising nematic phase for a hard-core dimer model with a finite density of holes (see text).

Figure 2

Evolution of phase diagram for different hole densities $\delta$, and for different attractive dimer plaquette interactions $u$ and attractive dimer-aligning interactions $v$ as a function of temperature $T$. (a) When only the plaquette interactions are present there is a single phase transition present between a featureless dimer-hole liquid and the columnar solid [4]. At zero hole density this transition is in the KT universality class [8]. At finite hole density there is a line of varying critical exponents lead to a rather unconventional multicritical point at ${\delta }_m=0.11,$ $T_m=0.4$ (endpoint of solid line). At lower temperatures $T<T_m$ the transition is discontinuous, and when in a fixed-density ensemble there is phase coexistence. The dotted lines represent the range of observed coexistence, derived from simulations [4]. (b) As soon as the dimer-aligning interaction $v$ is finite, a new phase emerges in between the dimer-hole liquid and the columnar solid, the Ising nematic dimer liquid. The phase transition from the columnar solid to the nematic liquid appears to resemble quantitatively (almost overlapping) the columnar solid to dimer-hole liquid transition line at zero nematicity. The transition line is in the Ising universality class. At higher hole densities there is a subsequent Ising transition line separating the Ising nematic phase from the isotropic dimer-hole liquid. (c) When only the dimer-aligning interaction $v$ is present, the columnar solid phase is absent. Only a single Ising transition line separates the nematic liquid from the dimer-hole liquid. In the close-packed limit $\delta =0$ and for $u=0$, the nematic liquid is infinitesimally unstable to a columnar solid, possibly at finite tilt [9, 10], due to a subextensive entropy of aligned dimer lines. The solid line for $T>1$ signifies the presence of the finite tilt columnar solid. Due to the presence of this subextensive entropy, the nature of the transition, expected to be in the KT universality class, is unclear from the simulations.

Figure 3

Columnar order parameter as a function of temperature for multiple system sizes in a close-packed dimer model without a plaquette interaction, $u=0$, and zero hole density, $\delta =0$. The columnar order vanishes quickly in the thermodynamic limit, with no signs of an instability.

Figure 4

Rotational order parameter as a function of temperature for multiple system sizes in a close-packed dimer model without a plaquette interaction, $u=0$, and zero hole density, $\delta =0$. Rotational order displays a clear transition.

Figure 5

Specific heat for different system sizes in a close-packed dimer model without a plaquette interaction, $u=0$, and zero hole density, $\delta =0$. The specific heat clearly displays a cusp or possibly a logarithmic divergence, consistent with both an Ising and a KT transition.

Figure 6

Rotational order parameter susceptibility displaying a divergence with an anomalous exponent $\eta =0.25$ consistent with an Ising transition, in a close-packed dimer model without a plaquette interaction, $u=0$, and zero hole density, $\delta =0$. The position of the peak appears to be very close to the position of the specific heat peak. For a KT transition, there would be expected a discrepancy of the two peaks.

Figure 7

Columnar (top) and nematic (bottom) order parameters in a close-packed dimer model with dimer-aligning and plaquette interactions at zero hole density. The plaquette interaction has $u=0.1$ and the dimer-aligning interactions have $v=1.0$. In the thermodynamic limit, both order parameters appear to display a strong transition, signifying the emergence of a columnar solid phase. The same happens at any $u$ we studied $(0.2,0.5)$ (not shown).

Figure 8

Columnar susceptibility for different sizes for a close-packed dimer model with dimer-aligning and plaquette interactions and zero hole density. The dimer-aligning interaction has $v=1.0$ and the plaquette interaction $u=0.1$. Slowly, the susceptibility displays a divergence, but the anomalous exponent we uncovered is very low $0.1$, compared to the KT expectation of $0.25$. We believe that this is still a complex crossover due to the competition of the columnar solid with the nematic liquid in finite systems, which has a finite entropy, albeit subextensive.

Figure 9

Nematic susceptibility in a close-packed dimer model with dimer-aligning and plaquette interactions and zero hole density for $u=0.1$ and $v=1.0$. Rotational susceptibility for different system sizes. This susceptibility displays a stronger behavior, and an apparent anomalous exponent of $0.3$, close but different from the Ising value. It appears that this exponent also is the outcome of a strong crossover towards the KT universality class, albeit impossible to resolve in simulations for systems up to ${512}^2$.

Figure 10

Specific heat for different system sizes displaying a divergence with an exponent $\alpha /\nu \simeq 0.5$ for a close-packed dimer model with dimer-aligning, and plaquette interactions at zero hole density. Here $v=1.0$ and $u=0.1$.

Figure 11

Binder cumulants for the columnar (top) and nematic (bottom) order parameters in a close-packed dimer model with dimer-aligning interactions and plaquette interactions, at zero hole density. Here, $u=0.1$ and $v=1.0$. The columnar order displays no crossing whatsoever, signature of a nontrivial crossover, while the nematic order displays a crossing at a value that is not consistent with the Ising expectation, but nevertheless shows the emergence of a correlation length critical exponent $\nu \simeq 1$ as expected for Ising transitions. These discrepancies suggest that there is a strong crossover even at the largest sizes we studied.

Figure 12

Nematic (top) and columnar (bottom) order parameters in a dimer model with only dimer-aligning interactions. The plaquette interaction is $u=0$ and the dimer-aligning interactions $v=1$. The hole density is $\delta =0.03125=1/32$. (Top) The nematic order parameter as a function of temperature for increasing system size, displaying a strong transition. (Bottom) The columnar order parameter, clearly vanishing in the thermodynamic limit.

Figure 13

Nematic susceptibility of the dimer model with only dimer-aligning interactions, $v=1$. The plaquette interaction is not included and has $u=0$. The hole density is $\delta =0.03125=1/32$. The susceptibility of the nematic order parameter displays a strong divergence with the system size (up to a lattice of $2^{20}$ sites) and in the inset scaling of the peak giving anomalous exponent $\eta \simeq 0.25$, consistent with an Ising transition.

Figure 14

Temperature dependence of the specific heat of a dimer model with only dimer-aligning interactions, with $v=1$. The plaquette interaction is absent, $u=0$. The hole density is $\delta =0.03125=1/32$. The temperature dependence of the specific heat displays a clear phase transition with corresponding exponent from the peak scaling $\alpha /\nu \simeq 0.1\pm 0.1$, consistent with the expected logarithmic singularity ($\alpha =0$) at a 2D Ising transition.

Figure 15

Binder ratio crossings for the nematic order parameter of a dimer model with only dimer-aligning interactions, with $v=1$, at finite hole density. The plaquette interaction is not included, $u=0$. The hole density is $\delta =0.03125=1/32$. The observed crossing point is consistent with the expected 2D Ising transition and its derivative at the crossing gives clearly $\nu \simeq 1$.

Figure 16

Finite-size dependence of the correlations for columnar order in the nematic phase of a dimer model at finite hole density with only dimer-aligning interactions. Here, $u$=0, $T=0.5$. The dilution is $1/8$. The evolution of the longitudinal correlation profile is shown as a function of distance. Clearly, a saturation of the profile is seen and the infinite-distance value is almost twice the fully disordered value ($T\to \infty$, same density), consistent with the phase structure, since most dimers in the nematic liquid are oriented in the same direction.

Figure 17

Columnar order correlations at different hole densities in the nematic phase for a dimer model with only dimer-aligning interactions. Here $L=128$ and $T=0.5$, deep in the nematic phase when at low hole densities. The crystal correlation at close-packed dimers transforms to liquidlike at large distances immediately as dilution becomes nonzero.

Figure 18

Evolution of the columnar and the nematic order parameters as a function of temperature in the dimer model at finite hole density with both dimer-aligning and plaquette interactions. Here $u=0.1$ and $v=1$. The hole density is $\delta =0.0625=1/16$. Two Ising phase transitions are observed as temperature decreases. The columnar order parameter shows a distinctly different transition temperature from the nematic order. The isotropic-nematic transition takes place at a higher temperature due to the presence of the attractive columnar interaction.

Figure 19

Specific heat for a dimer model with both dimer-aligning and plaquette interactions at finite hole density with both dimer-aligning and plaquette interactions. The plaquette coupling is $u=0.1$ and $v=1$. The hole density is $\delta =0.0625=1/16$. The specific heat shows a signature for both transitions, but the lower temperature transition's peak is much weaker, between the columnar solid and the Ising nematic liquid, converging much slower as the system size increases.

Figure 20

Columnar order susceptibility of the dimer model with both dimer-aligning and plaquette interactions at finite hole density. The plaquette coupling is $u=0.1$ and the two (and column) coupling are $v=1$. The hole density is $\delta =0.0625=1/16$. The susceptibility of the columnar order parameter displays a strong divergent peak with Ising exponents at the lower transition temperature.

Figure 21

Susceptibility for the nematic order parameter in a dimer model with both dimer-aligning and plaquette interactions at finite hole density. The plaquette coupling is $u=0.1$ and $v=1$. The hole density is $\delta =0.0625=1/16$. The susceptibility of the nematic order parameter displays a strong divergent peak with the Ising exponents at the higher transition temperature.

Figure 22

Binder cumulant for the columnar order parameter for a dimer model with both dimer-aligning and plaquette interactions at finite hole density. The plaquette coupling is $u=0.1$ and the dimer-aligning couplings are $v=1$. The hole density is $\delta =0.0625=1/16$. The Binder cumulant for the columnar order parameter displays crossings at both critical temperatures, since the columnar order parameter, by construction, is sensitive to both transitions in opposite ways.

Figure 23

Binder cumulant for the nematic order parameter of a dimer model with both dimer-aligning and plaquette interactions at finite hole density. The plaquette coupling is $u=0.1$ and the dimer-aligning couplings are $v=1$. The hole density is $\delta =0.0625=1/16$. The Binder cumulant for the nematic order parameter displays a crossing only at the highest critical temperature with slightly lower than Ising, but consistent, critical exponent. The same overall behavior is displayed in all the hole densities studied.

Figure 24

Mean-field approach to the phase diagram in the $u=0,v=1$ case. The thick dashed line describes a mean-field estimate from an associated mean-field theory discussed in the text. The points signify the transition line, as estimated from numerical simulations.