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Renormalization Group

Giuseppe Mussardo

Published Aug 27, 2009• 0 citations
Publisher

Abstract

Abstract At a critical point, the correlation length diverges: the statistical influctuations extend on all scales of the system and any attempt to solve the dynamics by taking into account only a finite number of degrees of freedom fails. In the absence of an exact solution of the model under consideration, the computation of the critical exponents is often obtained only by numerical methods and Monte Carlo simulations.

DOI
10.1093/oso/9780199547586.003.0008

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We analyze the critical properties of the three-dimensional Ising model with linear parallel extended defects. Such a form of disorder produces two distinct correlation lengths, a parallel correlation length ${\ensuremath{\xi}}_{\ensuremath{\parallel}}$ in the direction along defects and a perpendicular correlation length ${\ensuremath{\xi}}_{\ensuremath{\perp}}$ in the direction perpendicular to the lines. Both ${\ensuremath{\xi}}_{\ensuremath{\parallel}}$ and ${\ensuremath{\xi}}_{\ensuremath{\perp}}$ diverge algebraically in the vicinity of the critical point, but the corresponding critical exponents ${\ensuremath{\nu}}_{\ensuremath{\parallel}}$ and ${\ensuremath{\nu}}_{\ensuremath{\perp}}$ take different values. This property is specific for anisotropic scaling and the ratio ${\ensuremath{\nu}}_{\ensuremath{\parallel}}/{\ensuremath{\nu}}_{\ensuremath{\perp}}$ defines the anisotropy exponent $\ensuremath{\theta}$. Until now, estimates of quantitative characteristics of the critical behavior for such systems have been obtained only within the renormalization group approach. We report a study of the anisotropic scaling in this system via Monte Carlo simulation of the three-dimensional system with Ising spins and nonmagnetic impurities arranged into randomly distributed parallel lines. Several independent estimates for the anisotropy exponent $\ensuremath{\theta}$ of the system are obtained, as well as an estimate of the susceptibility exponent $\ensuremath{\gamma}$. Our results corroborate the renormalization group predictions obtained earlier.
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Ising model has been successful in describing ferromagnetism and its phase transition to paramagnet. At the critical point, the free energy density function and its derivatives diverge. Their behaviour near the critical point are described by power-laws with associated critical exponents. In many cases, the critical exponents can be determined from analytic solutions via conventional renormalization groups methods or from Monte Carlo simulations. However, for quantum many-body systems, very few are tractable to analytical solutions. The quantum many-body wavefunction belongs to large dimensional Hilbert space that increases exponentially with system size. If the Hamiltonian is gapped and only local interaction is considered, then the wavefunction can be efficiently truncated. Tensor Network formalism provides a scheme to truncate the less important degrees of freedom via Singular Value Decomposition (SVD) of the density matrix. In this study, we investigated the thermodynamic properties and phase transition of one-dimensional transverse-field quantum Ising model (1D-tQIM) under the finite-size effect and random coupling strength. Starting with Matrix Product States (MPS) as a wavefunction ansatz, the Density Matrix Renormalization Group algorithm is applied to the MPS. The variational algorithm, which iteratively performs SVDs and truncation at each bond, approximates the ground state MPS wavefunction. All quantum observables are calculated from the contraction of the resultant ground state MPS. Although theoretically, divergence at critical points only happen in an infinite system, one can obtain the critical exponents through simulation of finite-size 1D-tQIM. Using the analytic solution as a benchmark, we compared the finite-size effects of the system using finite-size scaling analysis and MPS methods. The critical exponents of 1D-tQIM are independently calculated and compared with the analytical results. Thermodynamic quantities such as magnetization, susceptibility and correlation function are calculated for system sizes of 20, 40, 60, 80, 100 and 120 spins. We determined the respective critical exponents: