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arXiv:2310.14756 · Physical review. B./Physical review. B

Finite-size scaling analysis of the two-dimensional random transverse-field Ising ferromagnet

Jiwon Choi, Seung Ki Baek

Published Oct 20, 2023• 3 citations
arXivHTMLPDF

Abstract

The random transverse-field Ising ferromagnet (RTFIF) is a highly disordered quantum system which contains randomness in the coupling strengths as well as in the transverse-field strengths. In one dimension, the critical properties are governed by an infinite-randomness fixed point (IRFP), and renormalization-group studies argue that the two-dimensional (2D) model is also governed by an IRFP. However, even the location of the critical point remains unsettled among quantum Monte Carlo (QMC) studies. In this work, we perform extensive QMC simulations to locate the quantum critical point and attempt a finite-size scaling analysis to observe the critical behavior. We estimate the critical field strength of the 2D RTFIF as ${\mathrm{\ensuremath{\Gamma}}}_{c}=7.52(2)$, together with critical exponents such as $\ensuremath{\beta}=1.5(3), \ensuremath{\nu}=1.6(3)$, and $z=3.3(3)$ or $\ensuremath{\psi}=0.50(3)$. We have also considered the McCoy-Wu model, which has randomness in the ferromagnetic coupling strengths but not in the transverse-field strength. Our QMC calculation shows that the critical behavior of the 2D McCoy-Wu model is closer to that of the 2D transverse-field Ising spin glass than to that of the 2D RTFIF. These numerical findings enhance our understanding of disordered 2D quantum systems.

DOI
10.1103/physrevb.108.144204

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The random transverse-field Ising ferromagnet (RTFIF) is a highly disordered quantum system which contains randomness in the coupling strengths as well as in the transverse-field strengths. In one dimension, the critical properties are governed by an infinite-randomness fixed point (IRFP), and renormalization-group studies argue that the two-dimensional (2D) model is also governed by an IRFP. However, even the location of the critical point remains unsettled among quantum Monte Carlo (QMC) studies. In this work, we perform extensive QMC simulations to locate the quantum critical point and attempt a finite-size scaling analysis to observe the critical behavior. We estimate the critical field strength of the 2D RTFIF as \ensuremathΓc=7.52(2){\mathrm{\ensuremath{\Gamma}}}_{c}=7.52(2)\ensuremathΓc​=7.52(2), together with critical exponents such as \ensuremathβ=1.5(3),\ensuremathν=1.6(3)\ensuremath{\beta}=1.5(3), \ensuremath{\nu}=1.6(3)\ensuremathβ=1.5(3),\ensuremathν=1.6(3), and z=3.3(3)z=3.3(3)z=3.3(3) or \ensuremathψ=0.50(3)\ensuremath{\psi}=0.50(3)\ensuremathψ=0.50(3). We have also considered the McCoy-Wu model, which has randomness in the ferromagnetic coupling strengths but not in the transverse-field strength. Our QMC calculation shows that the critical behavior of the 2D McCoy-Wu model is closer to that of the 2D transverse-field Ising spin glass than to that of the 2D RTFIF. These numerical findings enhance our understanding of disordered 2D quantum systems.
The critical behavior of the random transverse-field Ising model in finite-dimensional lattices is governed by infinite disorder fixed points, several properties of which have already been calculated by the use of the strong disorder renormalization-group (SDRG) method. Here we extend these studies and calculate the connected transverse-spin correlation function by a numerical implementation of the SDRG method in d=1,2d=1,2d=1,2, and 3 dimensions. At the critical point an algebraic decay of the form $\ensuremath{\sim}{r}^{\ensuremath{-}{\ensuremath{\eta}}_{t}}$ is found, with a decay exponent being approximately ${\ensuremath{\eta}}_{t}\ensuremath{\approx}2+2d$. In d=1d=1d=1 the results are related to dimer-dimer correlations in the random antiferromagnetic XXXXXX chain and have been tested by numerical calculations using free-fermionic techniques.
We study the ferromagnetic transverse-field Ising model with quenched disorder at T = 0 T = 0 in one and two dimensions by means of stochastic series expansion quantum Monte Carlo simulations using a rigorous zero-temperature scheme. Using a sample-replication method and averaged Binder ratios, we determine the critical shift and width exponents \nu_\mathrm{s} ν s and \nu_\mathrm{w} ν w as well as unbiased critical points by finite-size scaling. Further, scaling of the disorder-averaged magnetisation at the critical point is used to determine the order-parameter critical exponent \beta β and the critical exponent \nu_{\mathrm{av}} ν a v of the average correlation length. The dynamic scaling in the Griffiths phase is investigated by measuring the local susceptibility in the disordered phase and the dynamic exponent z' z ′ is extracted. By applying various finite-size scaling protocols, we provide an extensive and comprehensive comparison between the different approaches on equal footing. The emphasis on effective zero-temperature simulations resolves several inconsistencies in existing literature.
Quantum multicritical points (QMCPs) emerge at the junction of two or more quantum phase transitions due to the interplay of disparate fluctuations, leading to novel universality classes. While quantum critical points have been well characterized, our understanding of QMCPs is much more limited, even though they might be less elusive to study experimentally than quantum critical points. Here, we characterize the QMCP of an interacting heterogeneous quantum system in two and three dimensions, the ferromagnetic random transverse-field Ising model (RTIM). The QMCP of the RTIM emerges due to both geometric and quantum fluctuations, studied here numerically by the strong disorder renormalization group method. The QMCP of the RTIM is found to exhibit ultraslow, activated dynamic scaling, governed by an infinite disorder fixed point. This ensures that the obtained multicritical exponents tend to the exact values at large scales, while also being universal---i.e., independent of the form of disorder---providing a solid theoretical basis for future experiments.
Quantum tricriticality of a ${J}_{1}\ensuremath{-}{J}_{2}$ antiferromagnetic Ising model on a square lattice is studied using the mean-field (MF) theory, scaling theory, and the unbiased worldline quantum Monte-Carlo (QMC) method based on the Feynman path integral formula. The critical exponents of the quantum tricritical point (QTCP) and the qualitative phase diagram are obtained from the MF analysis. By performing the unbiased QMC calculations, we provide the numerical evidence for the existence of the QTCP and numerically determine the location of the QTCP in the case of J1=J2{J}_{1}={J}_{2}J1​=J2​. From the systematic finite-size scaling analysis, we conclude that the QTCP is located at HQTCP/J1=3.260(2){H}_{\mathrm{QTCP}}/{J}_{1}=3.260(2)HQTCP​/J1​=3.260(2) and ${\mathrm{\ensuremath{\Gamma}}}_{\mathrm{QTCP}}/{J}_{1}=4.10(5)$. We also show that the critical exponents of the QTCP are identical to those of the MF theory, because the QTCP in this model is in the upper critical dimension. The QMC simulations reveal that unconventional proximity effects of the ferromagnetic susceptibility appear close to the antiferromagnetic QTCP, and the proximity effects survive for the conventional quantum critical point. We suggest that the momentum dependence of the dynamical and static spin structure factors is useful for identifying the QTCP in experiments.
At low temperatures, the classical two-dimensional random bond Ising model undergoes a frustration-driven ferromagnet-to-paramagnet transition controlled by a zero-temperature fixed point separating ferromagnet and spin glass phases. We show that this critical point can be understood through a renormalization group transformation that constructs the ground state of the Ising model through a sequence of Hamiltonians that, starting with an unfrustrated model, iteratively adds in frustration until the target Hamiltonian is reached. Via a mapping of the thermodynamics of the 2d Ising model to the spectral properties of a related Hermitian matrix -- the Hamiltonian of a noninteracting quantum problem -- this RG procedure corresponds to an iterative diagonalization of the quantum Hamiltonian. The flow toward zero temperature in the Ising picture manifests as a flow toward infinite randomness in the spectrum of the quantum Hamiltonian, with the log gap of the Hamiltonian scaling as a power of the system size: log⁡εmin−1∼Lψ\log \varepsilon_{\it min}^{-1} \sim L^ψlogεmin−1​∼Lψ. The tunneling exponent ψψψ is equal to the spin stiffness exponent θcθ_cθc​ characterizing the zero-temperature fixed point.
The edge of a quantum critical system can exhibit multiple distinct types of boundary criticality. We use a numerical real-space renormalization group (RSRG) to study the boundary criticality of a 2d quantum Ising model with random exchange couplings and transverse fields, whose bulk exhibits an infinite randomness critical point. This approach enables an asymptotically numerically exact extraction of universal scaling data from very large systems with many thousands of spins that cannot be efficiently simulated directly. We identify three distinct classes of boundary criticality, and extract key scaling exponents governing boundary-boundary and boundary-bulk correlations and dynamics. We anticipate that this approach can be generalized to studying a broad class of (disordered) boundary criticality, including symmetry-enriched criticality and edge modes of gapless symmetry-protected topological states, in contexts were other numerical methods are restricted to one-dimensional chains.