Correlation lengths in quasi-one-dimensional systems via transfer matrices

Abstract

Using transfer matrices up to next-nearest-neighbour (NNN) interactions, we examine the structural correlations of quasi-one-dimensional systems of hard disks confined by two parallel lines and hard spheres confined in cylinders. Simulations have shown that the non-monotonic and non-smooth growth of the correlation length in these systems accompanies structural crossovers (Fu et al., Soft Matter, 2017, 13, 3296). Here, we identify the theoretical basis for these behaviour. In particular, we associate kinks in the growth of correlation lengths with eigenvalue crossing and splitting. Understanding the origin of such structural crossovers answers questions raised by earlier studies, and thus bridges the gap between theory and simulations for these reference models.arXiv:1804.00693v1 [cond-mat.soft]

Kosterlitz-Thouless-type caging-uncaging transition in a quasi-one-dimensional hard disk system

“…The interest in a quasi-one-dimensional (q1D) hard-core fluid has both basic [1][2][3][4][5][6] and applied [7][8][9][10] aspects. The fundamental interest is because this system allows for a more detailed [1] and more advanced analytical approach [11][12][13][14] which can facilitate studies in higher dimensions.…”

Inherent Structure Landscape of Hard Spheres Confined to Narrow Cylindrical Channels

“…However, if the interactions between defects become sufficiently long-ranged, phase transitions can occur [59,60]. Hu et al [61] recently obtained numerically exact results using the transfer matrix method that shows a transition does not occur in this system, suggesting the configurational entropy, which is related to the number of defects, still dominates at high density, despite the defect-defect attraction driven by the vibrational entropy. The fact that the PMF for the odd-sized helical sections only exhibits a single minimum at large n, and that the PMF for the even-sized sections has a large barrier at n = 2, reducing defect-defect annihilation, may work to counter the attractive interaction and help prevent a transition.…”

Numerical transfer matrix study of frustrated next-nearest-neighbor Ising models on square lattices

“…2(a), inset]. This phenomenon, which generically accompanies a structural crossover [27], here bespeaks a stepwise change in the modulation period [14]. For L = 24, for example, two distinct steps can be identified, both involving ξ 1 = ξ 2 (associated with doubly degenerate eigenvalues) crossing the subleading ξ 3 and ξ 4 .…”

“…Exponential improvement to computational hardware over the years (Moore's Law) coupled with more efficient eigensolvers [23] offer hope that the situation might have since improved. For instance, TM now provide definitive solutions of even fairly complex (quasi-)one-dimensional continuum-space systems [24][25][26][27][28].…”