A dimer model with competing interactions and ground state frustration is defined on the 4-8 lattice and the exact solution is obtained for the specific heat and densities of dimers. The model contains an energy parameter r, so phase transitions appear as lines in the (T,r) phase diagram. Two distinct kinds of striped incommensurate phase appear, separated from two commensurate and two disordered phases by a K-type transition line, and separated from each other by an O-type transition line. The phase diagram contains a novel OK multicritical point where a K-type transition line crosses an O-type transition line. The asymptotic scaling form at the OK multicritical point is the first member of a new class of scaling forms distinct from the two classes that encompassed all previous exact solutions of dimer models. The 4-8/4/1 model corresponds to the F-model at one temperature in a new kind of field.
A dimer model with competing interactions and ground state frustration is defined on the 4-8 lattice and the exact solution is obtained for the specific heat and densities of dimers. The model contains an energy parameter r, so phase transitions appear as lines in the (T,r) phase diagram. Two distinct kinds of striped incommensurate phase appear, separated from two commensurate and two disordered phases by a K-type transition line, and separated from each other by an O-type transition line. The phase diagram contains a novel OK multicritical point where a K-type transition line crosses an O-type transition line. The asymptotic scaling form at the OK multicritical point is the first member of a new class of scaling forms distinct from the two classes that encompassed all previous exact solutions of dimer models. The 4-8/4/1 model corresponds to the F-model at one temperature in a new kind of field.
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.
The problem of close-packed dimers on the honeycomb lattice was solved by Kasteleyn in 1963. Here we extend the solution to include interactions between neighboring dimers in two spatial lattice directions. The solution is obtained by using the method of Bethe ansatz and by converting the dimer problem into a five-vertex problem. The complete phase diagram is obtained and it is found that a new frozen phase, in which the attracting dimers prevail, arises when the interaction is attractive. For repulsive dimer interactions a new first-order line separating two frozen phases occurs. The transitions are continuous and the critical behavior in the disorder regime is found to be the same as in the case of noninteracting dimers characterized by a specific heat exponent α = 12.
The antiferromagnetic spin-1 chain has a venerable history and has been thought to be well understood. Here, we show that inclusion of both next-nearest-neighbor $(\ensuremath{\alpha})$ and biquadratic $(\ensuremath{\beta})$ interactions results in a rich phase diagram with a multicritical point that has not been observed before. We study the problem using a combination of the density matrix renormalization group (DMRG), an analytic variational matrix product state wave function, and conformal field theory. For negative $\ensuremath{\beta}<{\ensuremath{\beta}}^{*}$, we establish the existence of a spontaneously dimerized phase, separated from the Haldane phase by the critical line ${\ensuremath{\alpha}}_{c}(\ensuremath{\beta})$ of second-order phase transitions. In the opposite regime, $\ensuremath{\beta}>{\ensuremath{\beta}}^{*}$, the transition from the Haldane phase becomes first order into the next-nearest-neighbor (NNN) AKLT phase. Based on the field theoretical arguments and DMRG calculations, we find that these two regimes are separated by a multicritical point $({\ensuremath{\beta}}^{*},{\ensuremath{\alpha}}^{*})$ of a different universality class, described by the level-4 SU(2) Wess-Zumino-Witten conformal theory. From the DMRG calculations, we estimate this multicritical point to lie in the range $\ensuremath{-}0.2<{\ensuremath{\beta}}^{*}<\ensuremath{-}0.15$ and $0.47<{\ensuremath{\alpha}}^{*}<0.53$. We further find that the dimerized and NNN-AKLT phases are separated from each other by a line of first-order phase transitions that terminates at the multicritical point. We establish that transitions out of the Haldane phase into the dimer or NNN-AKLT phases are topological in nature and occur either with or without closing of the bulk gap, respectively. We also study short-range incommensurate-to-commensurate transitions in the resulting phase diagram. Inside the Haldane phase, we show the existence of two incommensurate crossovers: the Lifshitz transition and the disorder transition of the first kind, marking incommensurate correlations in momentum and real space, respectively. Notably, these crossover lines stretch across the entire $(\ensuremath{\beta},\ensuremath{\alpha})$ phase diagram, merging into a single incommensurate-to-commensurate transition line for negative $\ensuremath{\beta}\ensuremath{\lesssim}{\ensuremath{\beta}}^{*}$ inside the dimer phase. This behavior is qualitatively similar to that seen in classical frustrated two-dimensional spin models, by way of the quantum (1+1)D to classical 2D correspondence.
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 zmc(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.
Abstract We use scale invariant scattering theory to exactly determine the lines of renormalization group fixed points for O(N )-symmetric models with quenched disorder in two dimensions. Random fixed points are characterized by two disorder parameters: a modulus that vanishes when approaching the pure case, and a phase angle. The critical lines fall into three classes depending on the values of the disorder modulus. Besides the class corresponding to the pure case, a second class has maximal value of the disorder modulus and includes Nishimori-like multicritical points as well as zero temperature fixed points. The third class contains critical lines that interpolate, as N varies, between the first two classes. For positive N , it contains a single line of infrared fixed points spanning the values of N from 2−1 2 − 1 to 1. The symmetry sector of the energy density operator is superuniversal (i.e. N -independent) along this line. For N = 2 a line of fixed points exists only in the pure case, but accounts also for the Berezinskii-Kosterlitz-Thouless phase observed in presence of disorder.
We consider the thermal phase transition from a paramagnetic to a stripe-antiferromagnetic phase in the frustrated two-dimensional square-lattice Ising model with competing interactions J1<0 (nearest neighbor, ferromagnetic) and J2>0 (second neighbor, antiferromagnetic). The striped phase breaks a Z4 symmetry and is stabilized at low temperatures for g=J2/∣J1∣>1/2. Despite the simplicity of the model, it has proved difficult to precisely determine the order and the universality class of the phase transitions. This was done convincingly only recently by Jin et al. [Phys. Rev. Lett. 108, 045702 (2012)]. Here, we further elucidate the nature of these transitions and their anomalies by employing a combination of cluster mean-field theory, Monte Carlo simulations, and transfer-matrix calculations. The J1-J2 model has a line of very weak first-order phase transitions in the whole region 1/2<g<g∗, where ${g}^{*}=0.67\ifmmode\pm\else\textpm\fi{}0.01$. Thereafter, the transitions from g=g∗ to $g\ensuremath{\rightarrow}\ensuremath{\infty}$ are continuous and can be fully mapped, using universality arguments, to the critical line of the well-known Ashkin-Teller model from its four-state Potts point to the decoupled Ising limit. We also comment on the pseudo-first-order behavior at the Potts point and its neighborhood in the Ashkin-Teller model on finite lattices, which in turn leads to the appearance of similar effects in the vicinity of the multicritical point g∗ in the J1-J2 model. The continuous transitions near g∗ can therefore be mistaken for first-order transitions, and this realization was the key to understanding the paramagnetic-striped transition for the full range of g>1/2. Most of our conclusions are based on Monte Carlo calculations, while the cluster mean-field and transfer-matrix results provide useful methodological benchmarks for weakly first-order behaviors and Ashkin-Teller criticality.