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arXiv:cond-mat/0408393

Critical gaps of first-order phase transition in infinitely long Ising cylinders with antiperiodically joined circumference

Tsong‐Ming Liaw, Ming-Chang Huang, S. C. Lin, Yu-Pin Luo

Published Aug 18, 2004• 0 citations
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Abstract

Based on the analytic expression of free energy for infinitely long Ising strip with finite width joined antiperiodically on a variety of planar lattices, we show the existence of first-order phase transition at the critical point of Ising transition. The critical gaps of the transition are also calculated analytically by measuring the discontinuities in the internal energy and the specific heat.

DOI
10.48550/arxiv.cond-mat/0408393

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Based on the analytic expression of free energy for infinitely long Ising strip with finite width joined antiperiodically on a variety of planar lattices, we show the existence of first-order phase transition at the critical point of Ising transition. The critical gaps of the transition are also calculated analytically by measuring the discontinuities in the internal energy and the specific heat.
The critical properties of an infinitely long Ising strip with finite width L joined periodically or antiperiodically are investigated by analyzing the distribution of partition function zeros. For periodic boundary condition, the the leading finite-size scaling of partition function zeros and its corrections are given. For antiperiodic boundary condition, the critical point of 2D Ising transition is one of the loci of the zeros, and the associated non-analyticity is identified as a first-order phase transition. The exact amount of the latent heat released by the transition is 4/L.
To gain a better understanding of the interplay between frustrated long-range interactions and zero-temperature quantum fluctuations, we investigate the ground-state phase diagram of the transverse-field Ising model with algebraically decaying long-range Ising interactions on quasi-one-dimensional infinite-cylinder triangular lattices. Technically, we apply various approaches including low- and high-field series expansions. For the classical long-range Ising model, we investigate cylinders with an arbitrary even circumference. We show the occurrence of gapped stripe-ordered phases emerging out of the infinitely degenerate nearest-neighbor Ising ground-state space on the two-dimensional triangular lattice. Further, while cylinders with circumferences 6, 10, 14, etc., are always in the same stripe phase for any decay exponent of the long-range Ising interaction, the family of cylinders with circumferences 4, 8, 12, etc., displays a phase transition between two different types of stripe structures. For the full long-range transverse-field Ising model, we concentrate on cylinders with circumference four and six. The ground-state phase diagram consists of several quantum phases in both cases including an xxx-polarized phase, stripe-ordered phases, and clock-ordered phases which emerge from an order-by-disorder scenario already present in the nearest-neighbor model. In addition, the generic presence of a potential intermediate gapless phase with algebraic correlations and associated Kosterlitz-Thouless transitions is discussed for both cylinders.
First order phase transitions are rounded in solids of finite size. It is shown here that the above rounding is monitored by the correlation length ξL of the finite system, or equivalently by the so-called mass gap for the quantum Hamiltonian version of the model. Scaling with size is studied as a function of variable boundary conditions for the cylinder geometry (infinite strips with finite width), and a striking crossover is found in the mass gap behavior when the coupling g along the boundary becomes anti-periodic. For g>0, the rounding is exponential with size and an accurate determination of the spontaneous magnetization (order parameter) of the infinite system is obtained from numerical extrapolations.
The critical Ising model in two dimensions with a specific defect line is analyzed to deliver the first exact solution with twisted boundary conditions. We derive exact expressions for the eigenvalues of the transfer matrix and obtain analytically the partition function and the asymptotic expansions of the free energy and inverse correlation lengths for an infinitely long cylinder of circumference L x . We find that finite-size corrections to scaling are of the form ak/Lx2k−1a_k/L^{2k-1}_xak​/Lx2k−1​ for the free energy f and bk(p)/Lx2k−1b_k(p)/L_x^{2k-1}bk​(p)/Lx2k−1​ and ck(p)/Lx2k−1c_k(p)/L_x^{2k-1}ck​(p)/Lx2k−1​ for inverse correlation lengths xip−1\\xi^{-1}_pxip−1​ and xiL−p−1\\xi^{-1}_{L-p}xiL−p−1​ , respectively, with integer values of k. By exact evaluation we find that the amplitude ratios bk(p)/akb_k(p)/a_kbk​(p)/ak​ and ck(p)/akc_k(p)/a_kck​(p)/ak​ are universal and verify this universal behavior using a perturbative conformal approach.
The critical Ising model in two dimensions with a specific defect line is analyzed to deliver the first exact solution with twisted boundary conditions. We derive exact expressions for the eigenvalues of the transfer matrix and obtain analytically the partition function and the asymptotic expansions of the free energy and inverse correlation lengths for an infinitely long cylinder of circumference L x . We find that finite-size corrections to scaling are of the form a k / L x 2 k − 1 for the free energy f and b k ( p ) / L x 2 k − 1 and c k ( p ) / L x 2 k − 1 for inverse correlation lengths ξ p − 1 and ξ L − p − 1 , respectively, with integer values of k . By exact evaluation we find that the amplitude ratios b k ( p ) / a k and c k ( p ) / a k are universal and verify this universal behavior using a perturbative conformal approach.