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arXiv:1401.0788 · Physical Review B

Finite-size scaling at quantum transitions

Massimo Campostrini B, Andrea Pelissetto C, Ettore Vicari

Published Mar 21, 2014• 107 citations
arXivHTMLPDF

Abstract

We develop the finite-size scaling (FSS) theory at quantum transitions. We consider various boundary conditions, such as open and periodic boundary conditions, and characterize the corrections to the leading FSS behavior. Using renormalization-group (RG) theory, we generalize the classical scaling ansatz to describe FSS in the quantum case, classifying the different sources of scaling corrections. We identify nonanalytic corrections due to irrelevant (bulk and boundary) RG perturbations and analytic contributions due to regular backgrounds and analytic expansions of the nonlinear scaling fields. To check the general predictions, we consider the quantum $XY$ chain in a transverse field. For this model exact or numerically accurate results can be obtained by exploiting its fermionic quadratic representation. We study the FSS of several observables, such as the free energy, the energy differences between low-energy levels, correlation functions of the order parameter, etc., confirming the general predictions in all cases. Moreover, we consider bipartite entanglement entropies, which are characterized by the presence of additional scaling corrections, as predicted by conformal field theory.

DOI
10.1103/physrevb.89.094516

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We develop the finite-size scaling (FSS) theory at quantum transitions. We consider various boundary conditions, such as open and periodic boundary conditions, and characterize the corrections to the leading FSS behavior. Using renormalization-group (RG) theory, we generalize the classical scaling ansatz to describe FSS in the quantum case, classifying the different sources of scaling corrections. We identify nonanalytic corrections due to irrelevant (bulk and boundary) RG perturbations and analytic contributions due to regular backgrounds and analytic expansions of the nonlinear scaling fields. To check the general predictions, we consider the quantum XYXYXY chain in a transverse field. For this model exact or numerically accurate results can be obtained by exploiting its fermionic quadratic representation. We study the FSS of several observables, such as the free energy, the energy differences between low-energy levels, correlation functions of the order parameter, etc., confirming the general predictions in all cases. Moreover, we consider bipartite entanglement entropies, which are characterized by the presence of additional scaling corrections, as predicted by conformal field theory.
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Using free-fermionic techniques we study the entanglement entropy of a block of contiguous spins in a large finite quantum Ising chain in a transverse field, with couplings of different types: homogeneous, periodically modulated and random. We carry out a systematic study of finite-size effects at the quantum critical point, and evaluate subleading corrections both for open and for periodic boundary conditions. For a block corresponding to a half of a finite chain, the position of the maximum of the entropy as a function of the control parameter (e.g. the transverse field) can define the effective critical point in the finite sample. On the basis of homogeneous chains, we demonstrate that the scaling behavior of the entropy near the quantum phase transition is in agreement with the universality hypothesis, and calculate the shift of the effective critical point, which has different scaling behaviors for open and for periodic boundary conditions.
The half-chain entanglement entropy (EE) of the one-dimensional XXXXXX (isotropic XYXYXY) chain and quantum Ising chain (QIC) at the critical point are calculated numerically for large lattices. The periodic, free-free, fixed-fixed (++++++) and ($+\ensuremath{-}$), and fixed-free boundary conditions (BCs) are studied. The models are transformed into a fermion system with a Jordan-Wigner transformation. The correlation functions of these fermions are obtained numerically. With the correlation functions, the entanglement entropy is obtained. The numerical calculation is performed with quadruple precision. The numerical results agree with the exact relationship with an accuracy of ${10}^{\ensuremath{-}28}$ between the EEs of the XXXXXX chain and QICs: (1) the EE of the XXXXXX chain with periodic BCs is the sum of two QICs with periodic BCs; (2) the EE of the XXXXXX chain with free-free BCs is the sum of two QICs with free-free BCs; (3) the EE of the XXXXXX chain with fixed-free BCs is the sum of the EE of a QIC with fixed-free BCs and that with free-free BCs; (4) the EE of the XXXXXX chain with fixed-fixed (++++++) BCs is the sum of the EE of a QIC with fixed-fixed (++++++) BCs and that with free-free BCs; and (5) the EE of the XXXXXX chain with fixed-fixed ($+\ensuremath{-}$) BCs is the sum of the EE of a QIC with fixed-fixed (++++++) BCs and that of the first excited state of a QIC with free-free BCs. We fit the half-chain EE with finite-size scaling expansion of terms ${L}^{\ensuremath{-}n}$ and ${L}^{\ensuremath{-}n}lnL$, where LLL is the length of the spin chain. We find that the fitted parameters for the leading term lnLlnLlnL and the constant term are consistent with conformal field theory and that the exact solutions are highly accurate. The fitting accuracy is so high that the fitted central charge of the Ising model is $0.5\ifmmode\pm\else\textpm\fi{}1\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}26}$ for the periodic boundary condition. From the higher-order corrections in the fitted expansion, we find that there exist ${L}^{\ensuremath{-}4}lnL$ and higher-order corrections for the fixed-fixed ($+\ensuremath{-}$) boundary conditions, whereas there exist ${L}^{\ensuremath{-}3}lnL$ and higher-order corrections for the fixed-free boundary conditions. These types of corrections mean corrections of the central charge.
We analyze the scaling behavior of the fidelity, and the corresponding susceptibility, emerging in finite-size many-body systems whenever a given control parameter $\ensuremath{\lambda}$ is varied across a quantum phase transition. For this purpose we consider a finite-size scaling (FSS) framework. Our working hypothesis is based on a scaling assumption of the fidelity in terms of the FSS variables associated with $\ensuremath{\lambda}$ and its variation $\ensuremath{\delta}\ensuremath{\lambda}$. This framework entails the FSS predictions for continuous transitions and enables one to extend them to first-order transitions, where the FSS becomes qualitatively different. The latter is supported by analytical and numerical analyses of the quantum Ising chain along its first-order quantum transition line, driven by an external longitudinal field.
Finite-size scaling (FSS) for a critical phase transition (t=0t=0t=0) states that within a window of size ∣t∣∼L−1/ν|t|\sim L^{-1/ν}∣t∣∼L−1/ν, the scaling behavior of any observable QQQ in a system of linear size LLL asymptotically follows a scaling form as Q(t,L)=LYQQ~(tL1/ν)Q(t,L)=L^{Y_Q}\tilde{Q}(tL^{1/ν})Q(t,L)=LYQ​Q~​(tL1/ν), where ννν is the correlation-length exponent, YQY_QYQ​ is an FSS exponent and Q~(x){\tilde Q}(x)Q~​(x) is a function of the scaled distance-to-criticality x≡tL1/νx \equiv tL^{1/ν}x≡tL1/ν. We systematically study the asymptotic scaling behavior of Q~(∣x∣→∞){\tilde Q}(|x|\to\infty)Q~​(∣x∣→∞) for a broad variety of observables by requiring that the FSS and infinite-system critical behaviors match with each other in the crossover critical regime with t→0t \to 0t→0 and ∣x∣→∞|x|\to\infty∣x∣→∞. This crossover FSS theory predicts that when the criticality is approached at a slower speed as ∣t∣∼L−λ|t|\sim L^{-λ}∣t∣∼L−λ with λ<1/νλ<1/νλ<1/ν, the FSS becomes λλλ-dependent and the exponent can be derived. As applications, explosive percolation and high-dimensional percolation are considered. For the former, it is shown that the widely observed anomalous phenomena at the infinite-system criticality t=0t=0t=0 can be attributed to the mixing effects of the standard FSS behaviors around the pseudocritical point in an event-based ensemble. For the latter, FSS exponents are found to be different at the infinite-system critical and the pseudocritical point if free boundary conditions are used, and they are related to each other by using the crossover FSS theory. From these observations, the FSS of percolation systems falls into three classifications. Extensive simulations are carried out to affirm these predictions.