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arXiv:0801.3238 · Physical Review B

Rapidly converging methods for the location of quantum critical points from finite-size data

M. Roncaglia, Lorenzo Campos Venuti, C. Degli Esposti Boschi

Published Apr 9, 2008• 11 citations
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Abstract

We analyze in detail, beyond the usual scaling hypothesis, the finite-size convergence of static quantities toward the thermodynamic limit. In this way, we are able to obtain sequences of pseudo-critical points, which display a faster convergence rate as compared to currently used methods. The approaches are valid in any spatial dimension and for any value of the dynamic exponent. We demonstrate the effectiveness of our methods both analytically, on the basis of the one dimensional $XY$ model, and numerically, considering $c=1$ transitions occurring in nonintegrable spin models. In particular, we show that these general methods are able to precisely locate the onset of the Berezinskii--Kosterlitz--Thouless transition making only use of ground-state properties on relatively small systems.

DOI
10.1103/physrevb.77.155413

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We investigate the short-time quantum critical dynamics in the imaginary-time relaxation processes of finite-size systems. Universal scaling behaviors exist in the imaginary-time evolution. In particular, the system undergoes a critical initial slip stage characterized by an exponent $\ensuremath{\theta}$, in which an initial power-law increase emerges in the imaginary-time correlation function when the initial state has a zero order parameter and a vanishing correlation length. Under different initial conditions, the quantum critical point and critical exponents can be determined from the universal scaling behaviors. We apply the method to the one- and two-dimensional transverse field Ising models using quantum Monte Carlo (QMC) simulations. In the one-dimensional case, we locate the quantum critical point at ${(h/J)}_{c}=1.000\phantom{\rule{0.16em}{0ex}}03(8)$ in the thermodynamic limit, and we estimate the critical initial slip exponent $\ensuremath{\theta}=0.3734(2)$ and the static exponent $\ensuremath{\beta}/\ensuremath{\nu}=0.1251(2)$ by analyzing data on chains of length $L=32--256$ and 48--256, respectively. For the two-dimensional square-lattice system, the critical coupling ratio is given by $3.044\phantom{\rule{0.16em}{0ex}}51(7)$ in the thermodynamic limit, while the critical exponents are $\ensuremath{\theta}=0.209(4)$ and $\ensuremath{\beta}/\ensuremath{\nu}=0.518(1)$ estimated by data on systems of size $L=24$--64 and 32--64, respectively. Remarkably, the critical initial slip exponents obtained in both models are notably distinct from their classical counterparts due to the essential differences between classical and quantum dynamics. The short-time critical dynamics and the imaginary-time relaxation QMC approach can be readily adapted to various models.

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We analyze in detail, beyond the usual scaling hypothesis, the finite-size convergence of static quantities toward the thermodynamic limit. In this way, we are able to obtain sequences of pseudo-critical points, which display a faster convergence rate as compared to currently used methods. The approaches are valid in any spatial dimension and for any value of the dynamic exponent. We demonstrate the effectiveness of our methods both analytically, on the basis of the one dimensional XYXYXY model, and numerically, considering c=1c=1c=1 transitions occurring in nonintegrable spin models. In particular, we show that these general methods are able to precisely locate the onset of the Berezinskii--Kosterlitz--Thouless transition making only use of ground-state properties on relatively small systems.
Analyzing in detail the first corrections to the scaling hypothesis, we develop accelerated methods for the determination of critical points from finite size data. The output of these procedures is sequences of pseudo-critical points which rapidly converge towards the true critical points. The convergence is faster than that obtained with the fastest method available to date, which consists of estimating the location of the gap's closure (the so called phenomenological renormalization group). Having fast converging sequences at our disposal allows us to draw conclusions on the basis of shorter system sizes. This can be extremely important in particularly hard cases such as two-dimensional quantum systems with frustrations, or in Monte Carlo simulations when the sign problem occurs. After reviewing the most efficient techniques available to date, we test the effectiveness of the proposed methods both analytically on the basis of the one-dimensional XY model and numerically at phase transitions occurring in non-integrable spin models. In particular, we show how a new Homogeneity Condition Method is able to produce fast converging sequences in correspondence to the Berezinskii–Kosterlitz–Thouless (BKT) transition simply by making use of ground-state quantities on relatively small systems. Remarkably, our method tested on the frustrated spin-1/2 Heisenberg model gives a BKT critical point which is incompatible with the ones present in the past literature based on different methods. This discrepancy raises the fundamental question of determining the correct renormalization group approach and scaling assumptions that yield to the sequences converging to the true critical point. Finally, we formulate a general prescription that allows us to analyze and efficiently locate critical points in a variety of cases, without knowing in advance the universality class of the tested transition. Even if our methods are tested here in one dimension, we expect them to be valid in any spatial dimensionality and both for quantum and classical statistical systems.
We discuss how to locate critical points in the Berezinskii-Kosterlitz-Thouless (BKT) universality class by means of gap-scaling analyses. While accurately determining such points using gap extrapolation procedures is usually challenging and inaccurate due to the exponentially small value of the gap in the vicinity of the critical point, we show that a generic gap-scaling analysis, including the effects of logarithmic corrections, provides very accurate estimates of BKT transition points in a variety of spin and fermionic models. As a first example, we show how the scaling procedure, combined with density-matrix-renormalization-group simulations, performs extremely well in a nonintegrable spin-3/23/23/2 XXZ model, which is known to exhibit strong finite-size effects. We then analyze the extended Hubbard model, whose BKT transition has been debated, finding results that are consistent with previous studies based on the scaling of the Luttinger-liquid parameter. Finally, we investigate an anisotropic extended Hubbard model, for which we present the first estimates of the BKT transition line based on large-scale density-matrix-renormalization-group simulations. Our work demonstrates how gap-scaling analyses can help to locate accurately and efficiently BKT critical points, without relying on model-dependent scaling assumptions.
We test an improved finite-size scaling method for reliably extracting the critical temperature T BKT of a Berezinskii–Kosterlitz–Thouless (BKT) transition. Using known single-parameter logarithmic corrections to the spin stiffness ρ s at T BKT in combination with the Kosterlitz–Nelson relation between the transition temperature and the stiffness, ρ s ( T BKT ) = 2 T BKT /π, we define a size-dependent transition temperature T BKT ( L 1 , L 2 ) based on a pair of system sizes L 1 , L 2 , e.g., L 2 = 2 L 1 . We use Monte Carlo data for the standard two-dimensional classical XY model to demonstrate that this quantity is well behaved and can be reliably extrapolated to the thermodynamic limit using the next expected logarithmic correction beyond the ones included in defining T BKT ( L 1 , L 2 ). For the Monte Carlo calculations we use GPU (graphical processing unit) computing to obtain high-precision data for L up to 512. We find that the sub-leading logarithmic corrections have significant effects on the extrapolation. Our result T BKT = 0.8935(1) is several error bars above the previously best estimates of the transition temperature, T BKT ≈ 0.8929. If only the leading log-correction is used, the result is, however, consistent with the lower value, suggesting that previous works have underestimated T BKT because of the neglect of sub-leading logarithms. Our method is easy to implement in practice and should be applicable to generic BKT transitions.
Finite-size scaling and the renormalization group form the central toolkit for analyzing quantum phase transitions (QPTs). In this Letter, we introduce a novel Hilbert-space tailoring scheme to probe quantum critical phenomena. Applied to the second-order QPT of the one-dimensional (1D) XY model, our method yields precise critical points and exponents on lattices containing merely 50 unit cells. We further establish the universal applicability of this framework via investigations of the Berezinskii-Kosterlitz-Thouless transition in the 1D XXZ chain: critical parameters are recovered with as few as 12 lattice sites. This technique may open an alternative, efficient route to universally characterize QPT across many-body lattice systems.
Finite-size scaling analysis turns out to be a powerful tool to calculate the phase diagram as well as the critical properties of two-dimensional classical statistical mechanics models and quantum Hamiltonians in one dimension. The most used method to locate quantum critical points is the so-called crossing method, where the estimates are obtained by comparing the mass gaps of two distinct lattice sizes. The success of this method is due to its simplicity and the ability to provide accurate results even considering relatively small lattice sizes. In this paper, we introduce an estimator that locates quantum critical points by exploring the known distinct behavior of the entanglement entropy in critical and noncritical systems. As a benchmark test, we use this new estimator to locate the critical point of the quantum Ising chain and the critical line of the spin-1 Blume-Capel quantum chain. The tricritical point of this last model is also obtained. Comparison with the standard crossing method is also presented. The method we propose is simple to implement in practice, particularly in density matrix renormalization group calculations, and provides us, like the crossing method, amazingly accurate results for quite small lattice sizes. Our applications show that the proposed method has several advantages, as compared with the standard crossing method, and we believe it will become popular in future numerical studies.
We investigate the short-time quantum critical dynamics in the imaginary-time relaxation processes of finite-size systems. Universal scaling behaviors exist in the imaginary-time evolution. In particular, the system undergoes a critical initial slip stage characterized by an exponent $\ensuremath{\theta}$, in which an initial power-law increase emerges in the imaginary-time correlation function when the initial state has a zero order parameter and a vanishing correlation length. Under different initial conditions, the quantum critical point and critical exponents can be determined from the universal scaling behaviors. We apply the method to the one- and two-dimensional transverse field Ising models using quantum Monte Carlo (QMC) simulations. In the one-dimensional case, we locate the quantum critical point at (h/J)c=1.00003(8){(h/J)}_{c}=1.000\phantom{\rule{0.16em}{0ex}}03(8)(h/J)c​=1.00003(8) in the thermodynamic limit, and we estimate the critical initial slip exponent $\ensuremath{\theta}=0.3734(2)$ and the static exponent $\ensuremath{\beta}/\ensuremath{\nu}=0.1251(2)$ by analyzing data on chains of length L=32−−256L=32--256L=32−−256 and 48--256, respectively. For the two-dimensional square-lattice system, the critical coupling ratio is given by 3.04451(7)3.044\phantom{\rule{0.16em}{0ex}}51(7)3.04451(7) in the thermodynamic limit, while the critical exponents are $\ensuremath{\theta}=0.209(4)$ and $\ensuremath{\beta}/\ensuremath{\nu}=0.518(1)$ estimated by data on systems of size L=24L=24L=24--64 and 32--64, respectively. Remarkably, the critical initial slip exponents obtained in both models are notably distinct from their classical counterparts due to the essential differences between classical and quantum dynamics. The short-time critical dynamics and the imaginary-time relaxation QMC approach can be readily adapted to various models.