Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/9802104

Application of a continous time cluster algorithm to the Two-dimensional Random Quantum Ising Ferromagnet

Heiko Rieger, Naoki Kawashima

Published Feb 10, 1998• 1 citation
arXivHTMLPDF

Abstract

A cluster algorithm formulated in continuous (imaginary) time is presented for Ising models in a transverse field. It works directly with an infinite number of time-slices in the imaginary time direction, avoiding the necessity to take this limit explicitly. The algorithm is tested at the zero-temperature critical point of the pure two-dimensional (2d) transverse Ising model. Then it is applied to the 2d Ising ferromagnet with random bonds and transverse fields, for which the phase diagram is determined. Finite size scaling at the quantum critical point as well as the study of the quantum Griffiths-McCoy phase indicate that the dynamical critical exponent is infinite as in 1d.

DOI
10.48550/arxiv.cond-mat/9802104

Related papers

Critical Behavior and Griffiths-McCoy Singularities in the Two-Dimensional Random Quantum Ising Ferromagnet

Christian Pich, A. P. Young, Heiko Rieger +1 more

We study the quantum phase transition in the two-dimensional random Ising model in a transverse field by Monte Carlo simulations. We find results similar to those known analytically in one dimension. At the critical point the dynamical exponent is infinite and the typical correlation function decays with a stretched exponential dependence on distance. Away from the critical point there are Griffiths-McCoy singularities, characterized by a single continuously varying exponent, ${z}^{\ensuremath{'}}$, which diverges at the critical point, as in one dimension. Consequently, the zero temperature susceptibility diverges for a range of parameters about the transition.

arXiv:cond-mat/9812414•Physical Review Letters•Dec 28, 1998•141 citations

Quantum-critical properties of the one- and two-dimensional random transverse-field Ising model from large-scale quantum Monte Carlo simulations

Calvin Krämer, Jan Alexander Koziol, Anja Langheld +2 more

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.

arXiv:2403.05223•SciPost Physics•Aug 23, 2024•5 citations

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

Jiwon Choi, Seung Ki Baek

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.

arXiv:2310.14756•Physical review. B./Physical review. B•Oct 20, 2023•3 citations

Low-temperature transition of 2d random-bond Ising model and quantum infinite randomness

Akshat Pandey, Aditya Mahadevan, A. Alan Middleton +1 more

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: $\log \varepsilon_{\it min}^{-1} \sim L^ψ$. The tunneling exponent $ψ$ is equal to the spin stiffness exponent $θ_c$ characterizing the zero-temperature fixed point.

cond-mat.stat-mechCross-listedarXiv:2603.02308•Mar 4, 2026•0 citations

Critical behavior of random transverse-field Ising spin chains

Daniel S. Fisher

A real-space renormalization-group treatment of random transverse-field Ising spin chains that was introduced previously is developed and extensively analyzed. It yields results that are asymptotically exact in the critical region near the zero-temperature para-to-ferromagnetic quantum phase transition. In particular, the exact scaling function is obtained for the magnetization as a function of a uniform applied magnetic field and the distance to the critical point, and up to the solution of a linear ordinary differential equation whose solution can be exhaustively analyzed, the scaling function of the average spin-spin correlation function is also obtained. Thus more exact information is obtainable about the critical behavior for this random model than is known for the pure version which is equivalent to the two-dimensional Ising model. The basic reason for this is the extremely broad distribution of energy scales that occurs at low energies near the critical point of the random system. For the random chain the distribution of the magnetization of the first spin in a semi-infinite system is also studied and the results found to agree in the scaling limit with results of McCoy obtained from the exact solution of the closely related McCoy-Wu Ising model; this provides strong justification for the validity of the present approach. The singular properties of the weakly ordered and weakly disordered ``Griffiths' phases'' that occur at zero temperature near the critical point are also studied, as well as the behavior at low but nonzero temperature. Possible extensions of the results and general lessons drawn from them for other random systems are briefly discussed.

Physical review. B, Condensed matter•Mar 1, 1995•828 citations

Finite-temperature and dynamical properties of the random transverse-field Ising spin chain

A. P. Young

We study numerically the paramagnetic phase of the spin-$1/2$ random transverse-field Ising chain, using a mapping to noninteracting fermions. We extend our earlier work, Phys. Rev. B 53, 8486 (1996), to finite temperatures and to dynamical properties. Our results are consistent with the idea that there are Griffiths-McCoy singularities in the paramagnetic phase described by a continuously varying exponent $z(\ensuremath{\delta}),$ where \ensuremath{\delta} measures the deviation from criticality. There are some discrepancies between the values of $z(\ensuremath{\delta})$ obtained from different quantities, but this may be due to corrections to scaling. The average on-site time dependent correlation function decays with a power law in the paramagnetic phase, namely, ${\ensuremath{\tau}}^{\ensuremath{-}1/z(\ensuremath{\delta})},$ where \ensuremath{\tau} is imaginary time. However, the typical value decays with a stretched exponential behavior, $\mathrm{exp}(\ensuremath{-}c{\ensuremath{\tau}}^{1/\ensuremath{\mu}}),$ where \ensuremath{\mu} may be related to $z(\ensuremath{\delta}).$ We also obtain results for the full probability distribution of time dependent correlation functions at different points in the paramagnetic phase.

arXiv:cond-mat/9707060•Physical review. B, Condensed matter•Nov 1, 1997•58 citations
A cluster algorithm formulated in continuous (imaginary) time is presented for Ising models in a transverse field. It works directly with an infinite number of time-slices in the imaginary time direction, avoiding the necessity to take this limit explicitly. The algorithm is tested at the zero-temperature critical point of the pure two-dimensional (2d) transverse Ising model. Then it is applied to the 2d Ising ferromagnet with random bonds and transverse fields, for which the phase diagram is determined. Finite size scaling at the quantum critical point as well as the study of the quantum Griffiths-McCoy phase indicate that the dynamical critical exponent is infinite as in 1d.
We study the quantum phase transition in the two-dimensional random Ising model in a transverse field by Monte Carlo simulations. We find results similar to those known analytically in one dimension. At the critical point the dynamical exponent is infinite and the typical correlation function decays with a stretched exponential dependence on distance. Away from the critical point there are Griffiths-McCoy singularities, characterized by a single continuously varying exponent, ${z}^{\ensuremath{'}}$, which diverges at the critical point, as in one dimension. Consequently, the zero temperature susceptibility diverges for a range of parameters about the transition.
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.
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)z=3.3(3)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.
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.
A real-space renormalization-group treatment of random transverse-field Ising spin chains that was introduced previously is developed and extensively analyzed. It yields results that are asymptotically exact in the critical region near the zero-temperature para-to-ferromagnetic quantum phase transition. In particular, the exact scaling function is obtained for the magnetization as a function of a uniform applied magnetic field and the distance to the critical point, and up to the solution of a linear ordinary differential equation whose solution can be exhaustively analyzed, the scaling function of the average spin-spin correlation function is also obtained. Thus more exact information is obtainable about the critical behavior for this random model than is known for the pure version which is equivalent to the two-dimensional Ising model. The basic reason for this is the extremely broad distribution of energy scales that occurs at low energies near the critical point of the random system. For the random chain the distribution of the magnetization of the first spin in a semi-infinite system is also studied and the results found to agree in the scaling limit with results of McCoy obtained from the exact solution of the closely related McCoy-Wu Ising model; this provides strong justification for the validity of the present approach. The singular properties of the weakly ordered and weakly disordered ``Griffiths' phases'' that occur at zero temperature near the critical point are also studied, as well as the behavior at low but nonzero temperature. Possible extensions of the results and general lessons drawn from them for other random systems are briefly discussed.
We study numerically the paramagnetic phase of the spin-1/21/21/2 random transverse-field Ising chain, using a mapping to noninteracting fermions. We extend our earlier work, Phys. Rev. B 53, 8486 (1996), to finite temperatures and to dynamical properties. Our results are consistent with the idea that there are Griffiths-McCoy singularities in the paramagnetic phase described by a continuously varying exponent $z(\ensuremath{\delta}),$ where \ensuremath{\delta} measures the deviation from criticality. There are some discrepancies between the values of $z(\ensuremath{\delta})$ obtained from different quantities, but this may be due to corrections to scaling. The average on-site time dependent correlation function decays with a power law in the paramagnetic phase, namely, ${\ensuremath{\tau}}^{\ensuremath{-}1/z(\ensuremath{\delta})},$ where \ensuremath{\tau} is imaginary time. However, the typical value decays with a stretched exponential behavior, $\mathrm{exp}(\ensuremath{-}c{\ensuremath{\tau}}^{1/\ensuremath{\mu}}),$ where \ensuremath{\mu} may be related to $z(\ensuremath{\delta}).$ We also obtain results for the full probability distribution of time dependent correlation functions at different points in the paramagnetic phase.