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arXiv:2606.24312

Universal Extraction of Quantum Critical Exponents and Phase Transitions via Tailored Hilbert Space

Ye Xiong

cond-mat.stat-mechcond-mat.str-elphysics.comp-phquant-phSubmitted Jun 23, 2026• Announced Jun 24, 2026• 0 citations
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Abstract

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.

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Frontiers in Physics•Feb 10, 2022•4 citations
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.
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.
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 apply the generalized Wigner function formalism to detect and characterize a range of quantum phase transitions in several cyclic, finite-length, spin-12 one-dimensional spin-chain models, viz., the Ising and anisotropic XY models in a transverse field, and the XXZ anisotropic Heisenberg model. We make use of the finite system size to provide an exhaustive exploration of each system’s single-site, bipartite and multi-partite correlation functions. In turn, we are able to demonstrate the utility of phase-space techniques in detecting and characterizing first-, second- and infinite-order quantum phase transitions, while also enabling an in-depth analysis of the correlations present within critical systems. We also highlight the method’s ability to capture other features of spin systems such as ground-state factorization and critical system scaling. Finally, we demonstrate the generalized Wigner function’s utility for state verification by determining the state of each system and their constituent sub-systems at points of interest across the quantum phase transitions, enabling interesting features of critical systems to be readily analyzed.
We derive a renormalization group to calculate the nontrivial critical exponent of the divergent correlation length, thereby providing a universality classification of essential singularities in infinite-order phase transitions. This method thus resolves the vanishing scaling matrix problem. The exponent is obtained from the maximal eigenvalue of a scaling matrix in this renormalization group, as in the case of ordinary second-order phase transitions. We exhibit several nontrivial universality classes in infinite-order transitions different from the well known Berezinskiĭ-Kosterlitz-Thouless transition.
Up to now, probing the quantum phase transition (QPT) and quantum critical (QC) phenomena at finite temperatures in one-dimensional (1D) spin systems still lacks an in-depth understanding. Herein, we study the QPT and thermodynamics of 1D spin-1/2 anisotropic Heisenberg antiferromagnetic chains by Green’s function theory. The quantum phase diagram is renormalized by the anisotropy (∆), which manifests a quantum critical point (QCP) hc = 1 + ∆ signaling the transition from gapless Tomonaga–Luttinger liquid (TLL) to gapped ferromagnetic (FM) state, demonstrated by the magnetic entropy and thermal Drude weight. At low temperatures, it is shown that two crossover temperatures fan out a QC regime and capture the QCP via the linear extrapolation to zero temperature. In addition, around QCP, the QC scaling is performed by analyzing the entropy and thermal Drude weight to extract the critical exponents (α, δ, and β) that fulfill the Essamm–Fisher scaling law, which provides a novel thermodynamic means to detect QPT for experiment. Furthermore, scaling hypothesis equations with two rescaled manners are proposed to testify the scaling analysis, for which all the data points fall on a universal curve or two independent branches for the plot against rescaled field or temperature, implying the self-consistency and reliability of the obtained critical exponents.