Hard Hexagon Research Articles

New congruences involving statistical mechanics partition functions in regime III and IV of Baxter’s solution of the hard-hexagon model

New congruences involving statistical mechanics partition functions in regime III and IV of Baxter’s solution of the hard-hexagon model

On the free energy of solvable lattice models

We conjecture the inversion relations for thermalized solvable interaction round the face (IRF) two dimensional lattice models. We base ourselves on an ansatz for the Baxterization described in the 90's. We solve these inversion relations in the four main regimes of the models, to give the free energy of the models, in these regimes. We use the method of Baxter in the calculation of the free energy of the hard hexagon model. We believe these results to be quite general, shared by most of the known IRF models. Our results apply equally well to solvable vertex models. Using the expression for the free energy we calculate the critical exponent α, and from it the dimension of the perturbing (thermal) operator in the fixed point conformal field theory (CFT). We show that it matches either the coset O/G or G/O, where O is the original CFT used to define the model and G is some unknown CFT, depending on the regime. This agrees with known examples of such models by Huse and Jimbo et al.

Q-Series congruences involving statistical mechanics partition functions in regime III and IV of Baxter’s solution of the hard-hexagon model

For each $$s\in \{2,4\}$$ , the generating function of $$R_s(n)$$ , the number of partitions of n into odd parts or congruent to 0, $$\pm \, s\pmod {10}$$ , arises naturally in regime III of Rodney Baxter’s solution of the hard-hexagon model of statistical mechanics. For each $$s\in \{1,3\}$$ , the generating function of $$R^*_s(n)$$ , the number of partitions of n into parts not congruent to 0, $$\pm \, s\pmod {10}$$ and $$10-2s \pmod {20}$$ , arises naturally in regime IV of Rodney Baxter’s solution of the hard-hexagon model of statistical mechanics. In this paper, we investigate the parity of $$R_s(n)$$ and $$R^*_s(n)$$ , providing new parity results involving sums of partition numbers p(n) and squares in arithmetic progressions.

Galois Conjugated Tensor Fusion Categories and Nonunitary Conformal Field Theory.

We provide a generalization of the matrix product operator formalism for string-net projected entangled pair states (PEPS) to include nonunitary solutions of the pentagon equation. These states provide the explicit lattice realization of the Galois conjugated counterparts of (2+1)-dimensional topological quantum field theories, based on tensor fusion categories. Although the parent Hamiltonians of these renormalization group fixed point states are gapless, these states can still be the topological ground states of a gapped non-Hermitian Hamiltonian. We show by example that the topological sectors of the Yang-Lee theory (the nonunitary counterpart of the Fibonacci fusion category) can be constructed, even in the absence of closure under Hermitian conjugation of the basis elements of the Ocneanu tube algebra. The topological sector construction is demonstrated by applying the concept of strange correlators to the Yang-Lee model, giving rise to a nonunitary version of the classical hard hexagon model in the Yang-Lee universality class and obtaining all generalized twisted boundary conditions on a finite cylinder of the Yang-Lee edge singularity. Finally, we construct the PEPS transfer matrix and show that taking the Hermitian conjugate changes the topological phase for these nonunitary string-net models.

Frustrated honeycomb-lattice bilayer quantum antiferromagnet in a magnetic field: Unconventional phase transitions in a two-dimensional isotropic Heisenberg model

We consider the spin-1/2 antiferromagnetic Heisenberg model on a bilayer honeycomb lattice including interlayer frustration in the presence of an external magnetic field. In the vicinity of the saturation field, we map the low-energy states of this quantum system onto the spatial configurations of hard hexagons on a honeycomb lattice. As a result, we can construct effective classical models (lattice-gas as well as Ising models) on the honeycomb lattice to calculate the properties of the frustrated quantum Heisenberg spin system in the low-temperature regime. We perform classical Monte Carlo simulations for a hard-hexagon model and adopt known results for an Ising model to discuss the finite-temperature order-disorder phase transition that is driven by a magnetic field at low temperatures. We also discuss an effective-model description around the ideal frustration case and find indications for a spin-flop-like transition in the considered isotropic spin model.

Hard hexagons, hard squares and hard mathematics

Hard hexagons, hard squares and hard mathematics

The Rogers-Ramanujan Identities

In 1894, Rogers found the two identities for the first time. In 1913, Ramanujan found the two identities later and then the two identities are known as The Rogers-Ramanujan Identities. In 1982, Baxter used the two identities in solving the Hard Hexagon Model in Statistical Mechanics. In 1829 Jacobi proved his triple product identity; it is used in proving The Rogers-Ramanujan Identities. In 1921, Ramanujan used Jacobi's triple product identity in proving his famous partition congruences. This paper shows how to generate the generating function for ( ) Cn ' , ( ) 1 Cn ' , ( ) Cn '' and ( ) 1 Cn '' , and shows how to prove the Corollaries 1 and 2 with the help of Jacobi's triple product identity. This paper shows how to prove the Remark 3 with the help of various auxiliary functions and shows how to prove The Rogers-Ramanujan Identities with help of Ramanujan's device of the introduction of a second parameter a.

Integrability versus non-integrability: hard hexagons and hard squares compared

In this paper we compare the integrable hard hexagon model with the non-integrable hard squares model by means of partition function roots and transfer matrix eigenvalues. We consider partition functions for toroidal, cylindrical, and free–free boundary conditions up to sizes 40 × 40 and transfer matrices up to 30 sites. For all boundary conditions the hard squares roots are seen to lie in a bounded area of the complex fugacity plane along with the universal hard core line segment on the negative real fugacity axis. The density of roots on this line segment matches the derivative of the phase difference between the eigenvalues of largest (and equal) moduli and exhibits much greater structure than the corresponding density of hard hexagons. We also study the special point of hard squares, where all eigenvalues have unit modulus, and we give several conjectures for the value at of the partition functions.

On the dual nature of partial theta functions and Appell–Lerch sums

In recent work, Hickerson and the author demonstrated that it is useful to think of Appell–Lerch sums as partial theta functions. This notion can be used to relate identities involving partial theta functions with identities involving Appell–Lerch sums. In this sense, Appell–Lerch sums and partial theta functions appear to be dual to each other. This duality theory is not unlike that found by Andrews between various sets of identities of Rogers–Ramanujan type with respect to Baxter's solution to the hard hexagon model of statistical mechanics. As an application we construct bilateral q-series with mixed mock modular behaviour. In subsequent work we see that our bilateral series are well-suited for computing radial limits of Ramanujan's mock theta functions.

The hard hexagon partition function for complex fugacity

We study the analyticity of the partition function of the hard hexagon model in the complex fugacity plane by computing zeros and transfer matrix eigenvalues for large finite size systems. We find that the partition function per site computed by Baxter in the thermodynamic limit for positive real values of the fugacity is not sufficient to describe the analyticity in the full complex fugacity plane. We also obtain a new algebraic equation for the low density partition function per site.

The role of available sites in the activity of lattice gases with geometric constraints

The activity in lattice-gas systems with geometric constraints is shown to be the ratio of the number of particles to the number of available sites. The key role of sites available for occupation is emphasized. Available sites may be different for different species and are not necessarily just unoccupied sites. Location-specific or non-local constraints are allowed. An analytical expression for the number of available sites is given for the hard-hexagon model. The utility of an expression for available sites is illustrated for the non-trivial case of a mixed Langmuir/hard-hexagon adsorption system, where the influence of the Langmuir adsorbates on the hard-hexagon phase transition is investigated. The dependence on available sites indicates how to extend these results to the kinetic regime and simulations of kinetic voltammograms for the hard-hexagon model are given as an example.

Fundamental-measure density functional for the fluid of aligned hard hexagons: Further insights in fundamental measure theory

In this article we obtain a fundamental measure functional for the model of aligned hard hexagons in the plane. Our aim is not just to provide a functional for an admittedly academic model, but to investigate the structure of fundamental measure theory. A model of aligned hard hexagons has similarities with the hard disk model. Both share "lost cases," i.e. admit configurations of three particles in which there is pairwise overlap but not triple overlap. These configurations are known to be problematic for fundamental measure functionals, which are not able to capture their contribution correctly. This failure lies in the inability of these functionals to yield a correct low density limit of the third order direct correlation function. Here we derive the functional by projecting aligned hard cubes on the plane x+y+z=0 . The correct dimensional crossover behavior of these functionals permits us to follow this strategy. The functional of aligned hard cubes, however, does not have lost cases, so neither had the resulting functional for aligned hard hexagons. The latter exhibits, in fact, a peculiar structure as compared to the one for hard disks. It depends on a uniparametric family of weighted densities through an additional term not appearing in the functional for hard disks. Apart from studying the freezing of this system, we discuss the implications of the functional structure for further developments of fundamental measure theory.

High Field Properties of Geometrically Frustrated Magnets

Above the saturation field, geometrically frustrated quantum antiferromagnets have dispersionless low-energy branches of excitations corresponding to localized spin-flip modes. Transition into a partially magnetized state occurs via condensation of an infinite number of degrees of freedom. The ground state below the phase transition is a magnon crystal, which breaks only translational symmetry and preserves spin-rotations about the field direction. We give a detailed review of recent works on physics of such phase transitions and present further theoretical developments. Specifically, the low-energy degrees of freedom of a spin-1/2 kagom\'e antiferromagnet are mapped to a hard hexagon gas on a triangular lattice. Such a mapping allows to obtain a quantitative description of the magnetothermodynamics of a quantum kagom\'e antiferromagnet from the exact solution for a hard hexagon gas. In particular, we find the exact critical behavior at the transition into a magnon crystal state, the universal value of the entropy at the saturation field, and the position of peaks in temperature- and field-dependence of the specific heat. Analogous mapping is presented for the sawtooth chain, which is mapped onto a model of classical hard dimers on a chain. The finite macroscopic entropies of geometrically frustrated magnets at the saturation field lead to a large magnetocaloric effect.

Triangular Ising model with nearest- and next-nearest-neighbor couplings in a field

The authors study the Ising model on the triangular lattice with nearest-neighbor couplings K(nn) , next-nearest-neighbor couplings K(nnn) >0 , and a magnetic field H . This work is done by means of finite-size scaling of numerical results of transfer matrix calculations, and Monte Carlo simulations. We determine the phase diagram and confirm the character of the critical manifolds. The emphasis of this work is on the antiferromagnetic case K(nn) <0 , but we also explore the ferromagnetic regime K(nn) >/=0 for H=0 . For K(nn) <0 and H=0 we locate a critical phase presumably covering the whole range -infinity< K(nn) <0 . For K(nn) <0 , H not equal 0 we locate a plane of phase transitions containing a line of tricritical three-state Potts transitions. In the limit H-->infinity this line leads to a tricritical model of hard hexagons with an attractive next-nearest-neighbor potential.

Exact low-temperature behavior of a kagomé antiferromagnet at high fields

Low-energy degrees of freedom of a spin-$1∕2$ kagom\'e antiferromagnet in the vicinity of the saturation field are mapped to a hard-hexagon model on a triangular lattice. The latter model is exactly solvable. The presented mapping allows us to obtain the quantitative description of the magnetothermodynamics of a quantum kagom\'e antiferromagnet up to exponentially small corrections as well as predict the critical behavior for the transition into a magnon crystal state. Analogous mapping is presented for the sawtooth chain, which is mapped onto a model of classical hard dimers on a chain.

A thermal desorption study of iodine on Pt( [formula omitted

Thermal desorption spectra of iodine atoms adsorbed on Pt(1 1 1) are reported for coverages θ up to 0.43 monolayers (ML), the saturation coverage at 300 K. Analysis of the spectra as a function of heating rate and initial coverage shows that for θ<0.28 ML, desorption is first order with a coverage-dependent desorption energy. These data are fitted quantitatively to a Bragg–Williams statistical–mechanical transition-state model that includes the effect of vibrations of the adsorbed iodine layer according to the Debye theory. Measurements of the temperature variation of low-energy electron diffraction beam intensities gave surface Debye temperatures of 105±15 K for the (√3×√3) R30° phase and 155±15 K for the (√7×√7) R19.1° phase. Lateral repulsions between iodine atoms are 10 kJ mol −1 for θ<0.28 ML. A feature at θ=0.28 ML is interpreted in the hard-hexagon model as a phase transition between 2-D gas and 2-D solid phases. Another such phase transition occurs at θ>0.38 ML, and explains the zero-order desorption peak observed at these coverages. Thermodynamic quantities were estimated from the data, including Δ des H o=219±20 kJ mol −1 for (√3×√3) R30°-I→I(g) and Δ des H o=201±20 kJ mol −1 for (√7×√7) R19.1°-I→I(g).

Ordering of adsorbed proteins

Adapting a hard hexagon model to describing protein adsorption, we show by using Monte Carlo simulations that the ordering of adsorbed proteins may strongly depend on the relative location of the hydrophilic and hydrophobic patches on their surface. Specifically, proteins may form dimers, trimers, small rings, and zig-zag or straight chains or islands consisting of such fragments.

Complete non-diagonal reflection matrices of RSOS/SOS and hard hexagon models

In this paper we compute the most general nondiagonal reflection matrices of the RSOS/SOS models and hard hexagon model using the boundary Yang-Baxter equations. We find new one-parameter family of reflection matrices for the RSOS model in addition to the previous result without any parameter. We also find three classes of reflection matrices for the SOS model, which has one or two parameters. For the hard hexagon model which can be mapped to RSOS(5) model by folding four RSOS heights into two, the solutions can be obtained similarly with a main difference in the boundary unitarity conditions. Due to this, the reflection matrices can have two free parameters. We show that these extra terms can be identified with the `decorated' solutions. We also generalize the hard hexagon model by `folding' the RSOS heights of the general RSOS(p) model and show that they satisfy the integrability conditions such as the Yang- Baxter and boundary Yang-Baxter equations. These models can be solved using the results for the RSOS models.

Thermodynamic Bethe ansatz for the subleading magnetic perturbation of the tricritical Ising model

We give further support to Smirnov's conjecture on the exact kink S-matrix for the massive quantum field theory describing the integrable perturbation of the c = 0.7 minimal conformal field theory (known to describe the tricritical Ising model) by the operator φ 2,1 This operator has conformal dimensions ( 7 16 , 7 16 ) and is identified with the subleading magnetic operator of the tricritical Ising model. In this paper we apply the thermodynamic Bethe ansatz (TBA) approach to the kink scattering theory by explicitly utilising its relationship with the solvable lattice hard hexagon model. Analytically examining the ultraviolet scaling limit we recover the expected central charge c = 0.7 of the tricritical Ising model. We also compare numerical values for the ground state energy of the finite-size system obtained from the TBA equations with the results obtained by the truncated conformal space approach and conformal perturbation theory.

Auxiliary Vertices Method for Kagomé-Lattice Eight-Vertex Model

We investigate a class of eight-vertex models on a Kagome lattice. With the help of auxiliary vertices, the Kagome-lattice eight-vertex model (KEVM) is related to an inhomogeneous system which leads to a one-parameter family of commuting transfer matrices. Using an equation for commuting transfer matrices, we determine their eigenvalues. From calculated eigenvalues the correlation length of the KEVM is derived with its full anisotropy. There are two cases: In the first case the anisotropic correlation length (ACL) is the same as that of the triangular/honeycomb-lattice Ising model. By the use of an algebraic curve, it is shown that the Kagome-lattice Ising model, the diced-lattice Ising model, and the hard-hexagon model also have (essentially) the same ACL as the KEVM. In the second case we find that the ACL displays 12fold rotational symmetry.