We analyze the ground state of a strongly interacting fermion chain with a supersymmetry. We conjecture a number of exact results, such as a hidden duality between weak and strong couplings. By exploiting a scale-free property of the perturbative expansions, we find exact expressions for the order parameters, yielding the critical exponents. We show that the ground state of this fermion chain and another model in the same universality class, the XYZ chain along a line of couplings, are both written in terms of the same polynomials. We demonstrate this explicitly for up to N = 24 sites and provide consistency checks for large N . These polynomials satisfy a recursion relation related to the Painlevé VI differential equation and, using a scale-free property of these polynomials, we derive a simple and exact formula for their limit.
For the Ising model, the calculation of the spontaneous magnetization leads to the problem of evaluating a determinant. Yang did this by calculating the eigenvalues in the large-lattice limit. Montroll, Potts and Ward expressed it as a Toeplitz determinant and used Szegő's theorem: this is almost certainly the route originally travelled by Onsager. For the corresponding problem in the superintegrable chiral Potts model, neither approach appears to work: here we show that the determinant D PQ can be expressed as that of a product of two Cauchy-like matrices. One can then use the elementary exact formula for the Cauchy determinant. One of course regains the known result, originally conjectured in 1989.
We study the free energy of the classical square-lattice Ising model in a uniform real magnetic field
h, for which no exact solution is known. We first conjecture a closed-form expression built from the one-dimensional Ising chain and Onsager's hypergeometric solution. Comparing with high-precision tensor renormalization group (TRG) data, we find that it deviates by up to
2.6%, has no spontaneous magnetization, and gives an effective critical-isotherm exponent
δ→1. However, using the Lee--Yang circle theorem, we show that the exact free energy is a superposition of one-dimensional Ising chains in the same field, weighted by a temperature-dependent (normalized, but not always positive) distribution of squared Lee--Yang gaps, of which our first expression is a single-chain approximation. Exact results (Onsager, Yang, the strong-field expansion and the Lee--Yang solution at
βh=iπ/2) become sum rules for this distribution. We propose a simple form for it, fixed by Onsager's free energy and the exact strong-field limit without any fit to numerical data. This \emph{improved} conjecture agrees with TRG to
0.15% for all
β and
h, reproduces Yang's spontaneous magnetization exactly, has a positive specific heat, and yields the critical isotherm
M∝h1/δ with
δ=15.2 and amplitude
1.055, compared with the exact
δ=15 and
1.059. A refined form, which builds in additional input from Ising field theory (including the exact critical isotherm,
δ=15) and a few constants adjusted to transfer-matrix and TRG data, matches the TRG free energy to
4×10−5.
We adapt our previous results for the “partition function ” of the superintegrable chiral Potts model with open boundaries to obtain the corresponding matrix elements of e −αH, where H is the associated hamiltonian. The spontaneous magnetization Mr can be expressed in terms of particular matrix elements of e −αH S r 1 e−βH, where S1 is a diagonal matrix. We present a conjecture for these matrix elements as an m by m determinant, where m is proportional to the width of the lattice. The author has previously derived the spontaneous magnetization of the chiral Potts model by analytic means, but hopes that this work will facilitate a more algebraic derivation, similar to that of Yang for the Ising model. KEY WORDS: Statistical mechanics, lattice models, transfer matrices. 1
We study a model of strongly correlated fermions in one dimension with extended N = 2 supersymmetry. The model is related to the spin S = 1/2 XXZ Heisenberg chain at anisotropy Delta = -1/2 with a real magnetic field on the boundary. We exploit the combinatorial properties of the ground state to determine its exact wave function on finite lattices with up to 30 sites. We compute several correlation functions of the fermionic and spin fields. We discuss the continuum limit by constructing lattice observables with well defined finite-size scaling behavior. For the fermionic model with periodic boundary conditions we give the emptiness formation probability in closed form.