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arXiv:1412.5624 · Physical Review B

Gap scaling at Berezinskii-Kosterlitz-Thouless quantum critical points in one-dimensional Hubbard and Heisenberg models

Marcello Dalmonte, Juan Carrasquilla, Luca Taddia, Elisa Ercolessi, Marcos Rigol

Published Apr 29, 2015• 35 citations
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Abstract

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-$3/2$ 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.

DOI
10.1103/physrevb.91.165136

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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.
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 present a controlled numerical study of the Berezinskii-Kosterlitz-Thouless (BKT) transition in the one-dimensional Bose-Hubbard model at unit filling, providing evidence of the characteristic logarithmic finite-size scaling of the BKT transition. Employing density matrix renormalization group and quantum Monte Carlo simulations under periodic boundary conditions, together with a systematic finite-size scaling analysis of bipartite particle number fluctuations, we resolve boundary-induced complications that previously obscured critical scaling. We demonstrate that a suitably chosen central region under open boundaries reproduces universal renormalization group signatures, reconciling earlier discrepancies. Finally, leveraging a nonparametric Bayesian analysis, we determine the critical interaction strength with high precision to be U_{c}/J=3.275(2), establishing a benchmark for BKT physics in one-dimensional quantum models.
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.
Abstract Characterizing the superconducting and superfluid transitions in two-dimensional (2D) many-body systems is of broad interest and remains a fundamental issue. In this study, we establish the condensate fraction scaling as a highly efficient tool to achieve that and accordingly propose efficient schemes to accurately determine the associated Berezinskii−Kosterlitz−Thouless (BKT) transitions. Using the 2D attractive Fermi-Hubbard model as a testbed and applying numerically exact auxiliary-field quantum Monte Carlo simulations, we access unprecedented system sizes (up to 64 × 64 = 4096 lattice sites) and perform a comprehensive analysis for the temperature dependence and finite-size scaling of condensate fraction across the BKT transition. We demonstrate that this quantity exhibits algebraic scaling below the transition and exponential scaling above it, with significantly reduced finite-size effects comparing to the extensively studied on-site pairing correlator. This greatly improves the determination of BKT transition using moderate system sizes. We also extract finite-size BKT transition temperature from condensate fraction, and confirm its logarithmic correction on system size. Based on the accurately determined transition, we reveal that the specific heat displays an anomaly, showing a peak at a temperature slightly above BKT transition. Our findings should be generally applicable to 2D fermionic and bosonic systems hosting superconductivity or superfluidity.
We study the ground state of the one-dimensional extended Hubbard model at half filling using the entanglement entropy calculated by density matrix renormalization-group techniques. We apply curve fitting and scaling methods to accurately identify a second-order critical point as well as a Berezinskii-Kosterlitz-Thouless critical point. Using open boundary conditions and medium-sized lattices with very small truncation errors, we are able to achieve similar accuracy to that of previous authors. We also report observations of finite-size and boundary effects that can be remedied with careful pinning.
We investigate Berezinskii-Kosterlitz-Thouless (BKT) transitions in one-dimensional Rydberg chains, where commensurate critical regimes associated with the melting of crystalline orders with period larger than five and incommensurate floating phases are both described by Luttinger liquid theory. The central quantity is the Luttinger liquid parameter KKK, which characterizes the universal low-energy theory and controls the relevance of perturbations driving BKT transitions. We extract KKK using Friedel oscillations and the recently developed crosscap method introduced in Phys. Rev. Lett. 134, 076501 (2025). As benchmarks, we first apply the crosscap method to a Z3\mathbb{Z}_3Z3​ dual hard-core boson chain and a spin-1 XY chain with single-ion anisotropy, obtaining BKT transition points consistent with previous results after finite-size extrapolation. We then compute KKK in the Rydberg chain along lines with fixed correlation-oscillation period near BKT transitions. Along the commensurate period-five line, the critical values of KKK predicted by sine-Gordon theory reveal two BKT transitions separating the disordered phase, the critical phase, and the Z5\mathbb{Z}_5Z5​ crystalline phase. The results from Friedel oscillations and the crosscap method agree with each other and are further supported by energy-gap scaling and Binder-cumulant analysis. To obtain reliable values of KKK from Friedel oscillations, we use a multi-harmonic fitting scheme throughout the analysis. Along incommensurate lines, the BKT points obtained from Friedel oscillations agree with those extracted from entanglement entropy. Finally, we show that the values of KKK obtained from the two methods are mutually consistent inside the incommensurate floating phase.