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arXiv:cond-mat/9710144 · Physical review. B, Condensed matter

Density profiles, Casimir amplitudes, and critical exponents in the two-dimensional Potts model: A density-matrix renormalization study

Enrico Carlon, Ferenc Iglói

Published Apr 1, 1998• 28 citations
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Abstract

We use the density matrix renormalization group (DMRG) to perform a detailed study of the critical properties of the two-dimensional $Q$ state Potts model, including the magnetization and energy-density profiles, bulk and surface critical exponents, and the Casimir amplitudes. We apply symmetry breaking boundary conditions to a $L\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$ strip and diagonalize the corresponding transfer matrix for a series of moderately large systems $(L<~64)$ by the DMRG method. The numerically very accurate finite lattice results are then extrapolated by efficient sequence extrapolation techniques. The density profiles and the Casimir amplitudes at the critical point are found to follow precisely the conformal predictions for $Q=2$ and $3$. Similarly, the bulk and surface critical exponents of the models are in very good agreement with the conformal and exact values: their accuracy has reached or even exceeded the accuracy of other available numerical methods. For the $Q=4$ model both the profiles and the critical exponents show strong logarithmic corrections, which are also studied.

DOI
10.1103/physrevb.57.7877

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We use the density matrix renormalization group (DMRG) to perform a detailed study of the critical properties of the two-dimensional QQQ state Potts model, including the magnetization and energy-density profiles, bulk and surface critical exponents, and the Casimir amplitudes. We apply symmetry breaking boundary conditions to a L\ifmmode×\else\texttimes\fi\ensuremath∞L\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}L\ifmmode×\else\texttimes\fi\ensuremath∞ strip and diagonalize the corresponding transfer matrix for a series of moderately large systems (L< 64)(L<~64)(L< 64) by the DMRG method. The numerically very accurate finite lattice results are then extrapolated by efficient sequence extrapolation techniques. The density profiles and the Casimir amplitudes at the critical point are found to follow precisely the conformal predictions for Q=2Q=2Q=2 and 333. Similarly, the bulk and surface critical exponents of the models are in very good agreement with the conformal and exact values: their accuracy has reached or even exceeded the accuracy of other available numerical methods. For the Q=4Q=4Q=4 model both the profiles and the critical exponents show strong logarithmic corrections, which are also studied.
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The density-matrix renormalization-group (DMRG) method with periodic boundary conditions is introduced for two-dimensional (2D) classical spin models. It is shown that this method is more suitable for derivation of the properties of infinite 2D systems than the DMRG with open boundary conditions, despite the fact that the latter describes much better strips of finite width. For calculation at criticality, phenomenological renormalization at finite strips is used together with a criterion for optimum strip width for a given order of approximation. For this width the critical temperature of the 2D Ising model is estimated with seven-digit accuracy for a not too large order of approximation. Similar precision is reached for critical exponents. These results exceed the accuracy of similar calculations for the DMRG with open boundary conditions by several orders of magnitude. The method is applied to the calculation of critical exponents of the q=3,q=3,q=3, 4 Potts model, as well.
The analysis of renormalisation group equations for the q-state Potts model in two dimensions is generalised to include the presence of a surface field H s besides the bulk one, H B . Combined with conformal invariance results, this allows prediction of a surface susceptibility behaviour delta 2/f delta H s delta H B approximately L -1/8 (1-C'lnL) 15/16 for a q=4 system of size L at the critical temperature. This prediction is checked by a Monte Carlo based finite-size scaling analysis, which also nicely reproduces the exact and conjectured magnetic surface exponents for q=2 and 3, respectively. For the Baxter-Wu model similar methods give results consistent with the q=4 behaviour without logarithmic corrections (c'=0).
The conformal-invariance approach is extended to two-dimensional critical systems with mixed ab boundary conditions. The profiles 〈\ensuremath{\varphi}(r)〉ab{\mathrm{〉}}_{\mathit{a}\mathit{b}}〉ab​ of the magnetization and the energy density in Ising and Q-state Potts strips with +- and fixed-free boundary conditions are derived. The profile of an arbritrary operator degenerate at level 2 is expressed in terms of hypergeometric functions. The universal amplitudes Aab{\mathit{A}}_{\mathit{a}\mathit{b}}Aab​Bab{\mathit{B}}_{\mathit{a}\mathit{b}}Bab​ of the Casimir force and the Fisher--de Gennes distant-wall correction to the profile satisfy Bab{\mathit{B}}_{\mathit{a}\mathit{b}}Bab​=-4\ensuremath{\pi}${\mathit{x}}_{\mathrm{\ensuremath{\varphi}}}$${\mathit{c}}^{\mathrm{\ensuremath{-}}1}$Aab{\mathit{A}}_{\mathit{a}\mathit{b}}Aab​. Here ${\mathit{x}}_{\mathrm{\ensuremath{\varphi}}}$ is the scaling dimension of \ensuremath{\varphi}(r) and c is the central charge.
We study the critical behavior of the random q-state Potts quantum chain by density-matrix renormalization techniques. Critical exponents are calculated by scaling analysis of finite lattice data of short chains (L< 16)(L<~16)(L< 16) averaging over all possible realizations of disorder configurations chosen according to a binary distribution. Our numerical results show that the critical properties of the model are independent of q in agreement with a renormalization group analysis of Senthil and Majumdar [Phys. Rev. Lett. 76, 3001 (1996)]. We show how an accurate analysis of moments of the distribution of magnetizations allows a precise determination of critical exponents, circumventing some problems related to binary disorder. Multiscaling properties of the model and dynamical correlation functions are also investigated.