Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review Letters

Critical Points for Dimer Models with 32 -Order Transitions

John F. Nagle

Published May 5, 1975• 41 citations
Publisher

Abstract

A new extensive variable $A$ and its conjugate field II are introduced for the Kasteleyn dimer model on the honeycomb lattice via an isomorphic membrane-polymer chain model. The resultant phase diagram has a first-order phase line, thereby revealing a non-local order parameter in this model for the first time. The first-order line terminates in a critical point with critical exponents $\ensuremath{\alpha}={\ensuremath{\alpha}}^{\ensuremath{'}}=0$, $\ensuremath{\beta}=1=\ensuremath{\gamma}={\ensuremath{\gamma}}^{\ensuremath{'}}$, and $\ensuremath{\delta}=2$. The $\frac{3}{2}$-order exponents appear on the line $\ensuremath{\Pi}={\ensuremath{\Pi}}_{\mathrm{critical}}$.

DOI
10.1103/physrevlett.34.1150

Related papers

Interacting dimers on the honeycomb lattice: an exact solution of the five-vertex model

H. Y. Huang, Fa Yueh Wu, Kunz, H. +1 more

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.

•Jan 1, 1996•18 citations

Exact solution of the 4-8/4/1 dimer model with OK multicritical point

Carlos S. O. Yokoi, John F. Nagle

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.

Journal of Physics A Mathematical and General•Feb 7, 1992•2 citations

Quantum phase transition, O(3) universality class, and phase diagram of the spin- 12 Heisenberg antiferromagnet on a distorted honeycomb lattice: A tensor renormalization-group study

Wei Li, Shou-Shu Gong, Yang Zhao +1 more

The spin-$\frac{1}{2}$ Heisenberg antiferromagnet on the distorted honeycomb (DHC) lattice is studied by means of the tensor renormalization-group method. It is unveiled that the system has a quantum phase transition of second order between the gapped quantum dimer phase and a collinear N\'eel phase at the critical point of coupling ratio ${\ensuremath{\alpha}}_{c}\ensuremath{\simeq}0.54$, where the quantum critical exponents $\ensuremath{\nu}\ensuremath{\simeq}0.69(2)$ and $\ensuremath{\gamma}\ensuremath{\simeq}1.363(8)$ are obtained. The quantum criticality is found to fall into the $O(3)$ universality class. A ground-state phase diagram in the field-coupling ratio plane is proposed, where the phases such as the dimer, semiclassical N\'eel, and polarized phases are identified. A link between the present spin system to the boson Hubbard model on the DHC lattice is also discussed.

arXiv:1005.0932•Physical Review B•May 21, 2010•34 citations

CTMRG study of the critical behavior of an interacting-dimer model

Christophe Chatelain

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.

arXiv:2401.14830•Journal of Statistical Mechanics Theory and Experiment•Aug 2, 2024•1 citation

FIRST AND SECOND ORDER TRANSITIONS IN DILUTE O (n) MODELS

Wenan Guo, Henk W. J. Blöte, Bernard Nienhuis

We explore the phase diagram of an O (n) model on the honeycomb lattice with vacancies, using finite-size scaling and transfer-matrix methods. We make use of the loop representation of the O (n) model, so that n is not restricted to positive integers. For low activities of the vacancies, we observe critical points of the known universality class. At high activities the transition becomes first order. For n=0 the model includes an exactly known theta point, used to describe a collapsing polymer in two dimensions. When we vary n from 0 to 1, we observe a tricritical point which interpolates between the universality classes of the theta point and the Ising tricritical point.

arXiv:cond-mat/9812270•International Journal of Modern Physics C•Feb 1, 1999•5 citations

Renormalization-group study of the critical end point in 4−ε dimensions

Timothy A. L. Ziman, D J Amit, G. Grinstein +1 more

The phase diagrams of a variety of physical systems, among them certain antiferromagnets and ternary fluid mixtures, may include a critical line intersecting a line of first-order transitions at a critical end point rather than at the familiar tricritical point. At the critical end point the order parameter has critical fluctuations in the paramagnetic phase and a finite jump. Within Landau theory this phenomenon appears with the introduction of a negative ${\ensuremath{\varphi}}^{6}$ coupling; here we extend that theory to include fluctuations by a momentum-shell renormalization-group analysis close to four dimensions. In previous real-space analyses the critical end point was associated with a new fixed point. In contrast, the Wilson-Fisher fixed point which characterizes the entire critical line describes the critical end point here. We derive the $O(\ensuremath{\epsilon})$ corrections to the mean-field exponents which, in Landau theory, describe the approach to the critical end point from high temperatures. At the same time, the discontinuity in $〈\ensuremath{\varphi}〉$ at the critical end point predicted by Landau theory is essentially unaltered by fluctuations. It emerges from a renormalization-group calculation of the Gibbs free energy for the Ginzburg-Landau-Wilson model with a negative ${\ensuremath{\varphi}}^{6}$ term and a positive ${\ensuremath{\varphi}}^{8}$ term.

Physical review. B, Condensed matter•Jan 1, 1982•33 citations
A new extensive variable AAA and its conjugate field II are introduced for the Kasteleyn dimer model on the honeycomb lattice via an isomorphic membrane-polymer chain model. The resultant phase diagram has a first-order phase line, thereby revealing a non-local order parameter in this model for the first time. The first-order line terminates in a critical point with critical exponents \ensuremathα=\ensuremathα\ensuremath′=0\ensuremath{\alpha}={\ensuremath{\alpha}}^{\ensuremath{'}}=0\ensuremathα=\ensuremathα\ensuremath′=0, \ensuremathβ=1=\ensuremathγ=\ensuremathγ\ensuremath′\ensuremath{\beta}=1=\ensuremath{\gamma}={\ensuremath{\gamma}}^{\ensuremath{'}}\ensuremathβ=1=\ensuremathγ=\ensuremathγ\ensuremath′, and \ensuremathδ=2\ensuremath{\delta}=2\ensuremathδ=2. The 32\frac{3}{2}23​-order exponents appear on the line \ensuremathΠ=\ensuremathΠcritical\ensuremath{\Pi}={\ensuremath{\Pi}}_{\mathrm{critical}}\ensuremathΠ=\ensuremathΠcritical​.
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.
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.
The spin-12\frac{1}{2}21​ Heisenberg antiferromagnet on the distorted honeycomb (DHC) lattice is studied by means of the tensor renormalization-group method. It is unveiled that the system has a quantum phase transition of second order between the gapped quantum dimer phase and a collinear N\'eel phase at the critical point of coupling ratio ${\ensuremath{\alpha}}_{c}\ensuremath{\simeq}0.54$, where the quantum critical exponents $\ensuremath{\nu}\ensuremath{\simeq}0.69(2)$ and $\ensuremath{\gamma}\ensuremath{\simeq}1.363(8)$ are obtained. The quantum criticality is found to fall into the O(3)O(3)O(3) universality class. A ground-state phase diagram in the field-coupling ratio plane is proposed, where the phases such as the dimer, semiclassical N\'eel, and polarized phases are identified. A link between the present spin system to the boson Hubbard model on the DHC lattice is also discussed.
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 explore the phase diagram of an O (n) model on the honeycomb lattice with vacancies, using finite-size scaling and transfer-matrix methods. We make use of the loop representation of the O (n) model, so that n is not restricted to positive integers. For low activities of the vacancies, we observe critical points of the known universality class. At high activities the transition becomes first order. For n=0 the model includes an exactly known theta point, used to describe a collapsing polymer in two dimensions. When we vary n from 0 to 1, we observe a tricritical point which interpolates between the universality classes of the theta point and the Ising tricritical point.
The phase diagrams of a variety of physical systems, among them certain antiferromagnets and ternary fluid mixtures, may include a critical line intersecting a line of first-order transitions at a critical end point rather than at the familiar tricritical point. At the critical end point the order parameter has critical fluctuations in the paramagnetic phase and a finite jump. Within Landau theory this phenomenon appears with the introduction of a negative ${\ensuremath{\varphi}}^{6}$ coupling; here we extend that theory to include fluctuations by a momentum-shell renormalization-group analysis close to four dimensions. In previous real-space analyses the critical end point was associated with a new fixed point. In contrast, the Wilson-Fisher fixed point which characterizes the entire critical line describes the critical end point here. We derive the $O(\ensuremath{\epsilon})$ corrections to the mean-field exponents which, in Landau theory, describe the approach to the critical end point from high temperatures. At the same time, the discontinuity in $〈\ensuremath{\varphi}〉$ at the critical end point predicted by Landau theory is essentially unaltered by fluctuations. It emerges from a renormalization-group calculation of the Gibbs free energy for the Ginzburg-Landau-Wilson model with a negative ${\ensuremath{\varphi}}^{6}$ term and a positive ${\ensuremath{\varphi}}^{8}$ term.