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:1507.08741 · Physical Review B

Quantum tricriticality in antiferromagnetic Ising model with transverse field: A quantum Monte Carlo study

Yasuyuki Kato, Takahiro Misawa

Published Nov 23, 2015• 25 citations
arXivHTMLPDF

Abstract

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 ${J}_{1}={J}_{2}$. From the systematic finite-size scaling analysis, we conclude that the QTCP is located at ${H}_{\mathrm{QTCP}}/{J}_{1}=3.260(2)$ 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.

DOI
10.1103/physrevb.92.174419

Related papers

Quantum critical behavior of the quantum Ising model on fractal lattices

Hangmo Yi

I study the properties of the quantum critical point of the transverse-field quantum Ising model on various fractal lattices such as the Sierpiński carpet, Sierpiński gasket, and Sierpiński tetrahedron. Using a continuous-time quantum Monte Carlo simulation method and finite-size scaling analysis, I identify the quantum critical point and investigate its scaling properties. Among others, I calculate the dynamic critical exponent and find that it is greater than one for all three structures. The fact that it deviates from one is a direct consequence of the fractal structures not being integer-dimensional regular lattices. Other critical exponents are also calculated. The exponents are different from those of the classical critical point and satisfy the quantum scaling relation, thus confirming that I have indeed found the quantum critical point. I find that the Sierpiński tetrahedron, of which the dimension is exactly 2, belongs to a different universality class than that of the two-dimensional square lattice. I conclude that the critical exponents depend on more details of the structure than just the dimension and the symmetry.

arXiv:1408.5595•Physical Review E•Jan 9, 2015•9 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

Effects of third-neighbour interactions on the criticality of the quantum Ising model on the frustrated square lattice

Matheus José Passos Bento Vianna de [UNIFESP] Oliveira

In the present work, we investigate the phase transitions of the Ising model with interactions between first (J1), second (J2) and third (J3) neighbours in the square lattice with transverse magnetic field. In this study, we adopt antiferromagnetic interactions between first and second neighbours and consider third-neighbours interactions to be both ferromagnetic and antiferromagnetic. The description of the classical and quantum phase transitions of the model is carried out by adopting the cluster mean-field approximation with four sites. As a result, we identified that strong enough third-neighbor interactions lead to the disappearance of tricriticality at the boundary between the superantiferromagnetic (SAF) and paramagnetic (PM) phases. In particular, tricriticality is more sensitive to the presence of ferromagnetic interactions, disappearing for |J3| ă 0.3|J1| in the absence of transverse field. In the presence of antiferromagnetic J3, we find first-order transitions between the degenerate staggered dimer phase and the PM phase. Furthermore, we find a change in the nature of the SAF-PM phase transitions introduced by quantum fluctuations. This phenomenon, called quantum annealed criticality (QAC), consists of classical first-order phase transitions that become second-order phase transitions in the presence of a strong enough transverse field. Our results allow establishing a range of parameters J2 and J3 for which it is possible to find QAC. The analysis of the model criticality allows concluding that a strong enough J3 interaction eliminates the QAC phenomenon. Therefore, our results suggest that attempts to realize QAC in a square lattice system should avoid strong interactions between third neighbors, mainly if these interactions are ferromagnetic.

•Feb 14, 2023•0 citations

Quantum Monte Carlo study of the bond- and site-diluted transverse-field Ising model

Calvin Krämer, Max Hörmann, Kai Phillip Schmidt

We study the transverse-field Ising model on a square lattice with bond- and site-dilution at zero temperature by stochastic series expansion quantum Monte Carlo simulations. Tuning the transverse field $h$ and the dilution $p$, the quantum phase diagram of both models is explored. Both quantum phase diagrams show long-range order for small $h$ and small $p$. The ordered phase of each is separated from the disordered (quantum) Griffiths phase by second-order phase transitions on two critical lines touching at a multi-critical point. Using Binder ratios we locate quantum critical points with high accuracy. The order-parameter critical exponent $β$ and the average correlation-length exponent $ν_{\mathrm{av}}$ are determined along the critical lines and at the multi-critical points via finite-size scaling. We find three internally consistent sets of critical exponents and compare them with potentially connected universality classes. The quantum Griffiths phase in the vicinity of the phase transition lines is analyzed through the local susceptibility. Our results indicate that activated scaling occurs not only at the percolation transition, but also at the phase transition line for $p$ smaller than the percolation threshold $p_c$.

arXiv:2505.07627•Physical review. B./Physical review. B•Jul 8, 2025•1 citation

Quantum-critical properties of the long-range transverse-field Ising model from quantum Monte Carlo simulations

Jan Alexander Koziol, Anja Langheld, Sebastian C. Kapfer +1 more

The quantum-critical properties of the transverse-field Ising model with algebraically decaying interactions are investigated by means of stochastic series expansion quantum Monte Carlo, on both the one-dimensional linear chain and the two-dimensional square lattice. We extract the critical exponents $\ensuremath{\nu}$ and $\ensuremath{\beta}$ as a function of the decay exponent of the long-range interactions. For ferromagnetic Ising interactions, we resolve the limiting regimes known from field theory, ranging from the nearest-neighbor Ising to the long-range Gaussian universality classes, as well as the intermediate regime with continuously varying critical exponents. In the long-range Gaussian regime, we treat the effect of dangerous irrelevant variables on finite-size scaling forms. For antiferromagnetic and therefore competing Ising interactions, the stochastic series expansion algorithm displays growing autocorrelation times leading to a reduced performance. Nevertheless, our results are consistent with the nearest-neighbor Ising universality for all investigated interaction ranges both on the linear chain and the square lattice.

arXiv:2103.09469•Physical review. B./Physical review. B•Jun 23, 2021•47 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
Quantum tricriticality of a J1\ensuremath−J2{J}_{1}\ensuremath{-}{J}_{2}J1​\ensuremath−J2​ 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 \ensuremathΓQTCP/J1=4.10(5){\mathrm{\ensuremath{\Gamma}}}_{\mathrm{QTCP}}/{J}_{1}=4.10(5)\ensuremathΓQTCP​/J1​=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.
I study the properties of the quantum critical point of the transverse-field quantum Ising model on various fractal lattices such as the Sierpiński carpet, Sierpiński gasket, and Sierpiński tetrahedron. Using a continuous-time quantum Monte Carlo simulation method and finite-size scaling analysis, I identify the quantum critical point and investigate its scaling properties. Among others, I calculate the dynamic critical exponent and find that it is greater than one for all three structures. The fact that it deviates from one is a direct consequence of the fractal structures not being integer-dimensional regular lattices. Other critical exponents are also calculated. The exponents are different from those of the classical critical point and satisfy the quantum scaling relation, thus confirming that I have indeed found the quantum critical point. I find that the Sierpiński tetrahedron, of which the dimension is exactly 2, belongs to a different universality class than that of the two-dimensional square lattice. I conclude that the critical exponents depend on more details of the structure than just the dimension and the symmetry.
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.
In the present work, we investigate the phase transitions of the Ising model with interactions between first (J1), second (J2) and third (J3) neighbours in the square lattice with transverse magnetic field. In this study, we adopt antiferromagnetic interactions between first and second neighbours and consider third-neighbours interactions to be both ferromagnetic and antiferromagnetic. The description of the classical and quantum phase transitions of the model is carried out by adopting the cluster mean-field approximation with four sites. As a result, we identified that strong enough third-neighbor interactions lead to the disappearance of tricriticality at the boundary between the superantiferromagnetic (SAF) and paramagnetic (PM) phases. In particular, tricriticality is more sensitive to the presence of ferromagnetic interactions, disappearing for |J3| ă 0.3|J1| in the absence of transverse field. In the presence of antiferromagnetic J3, we find first-order transitions between the degenerate staggered dimer phase and the PM phase. Furthermore, we find a change in the nature of the SAF-PM phase transitions introduced by quantum fluctuations. This phenomenon, called quantum annealed criticality (QAC), consists of classical first-order phase transitions that become second-order phase transitions in the presence of a strong enough transverse field. Our results allow establishing a range of parameters J2 and J3 for which it is possible to find QAC. The analysis of the model criticality allows concluding that a strong enough J3 interaction eliminates the QAC phenomenon. Therefore, our results suggest that attempts to realize QAC in a square lattice system should avoid strong interactions between third neighbors, mainly if these interactions are ferromagnetic.
We study the transverse-field Ising model on a square lattice with bond- and site-dilution at zero temperature by stochastic series expansion quantum Monte Carlo simulations. Tuning the transverse field hhh and the dilution ppp, the quantum phase diagram of both models is explored. Both quantum phase diagrams show long-range order for small hhh and small ppp. The ordered phase of each is separated from the disordered (quantum) Griffiths phase by second-order phase transitions on two critical lines touching at a multi-critical point. Using Binder ratios we locate quantum critical points with high accuracy. The order-parameter critical exponent βββ and the average correlation-length exponent νavν_{\mathrm{av}}νav​ are determined along the critical lines and at the multi-critical points via finite-size scaling. We find three internally consistent sets of critical exponents and compare them with potentially connected universality classes. The quantum Griffiths phase in the vicinity of the phase transition lines is analyzed through the local susceptibility. Our results indicate that activated scaling occurs not only at the percolation transition, but also at the phase transition line for ppp smaller than the percolation threshold pcp_cpc​.
The quantum-critical properties of the transverse-field Ising model with algebraically decaying interactions are investigated by means of stochastic series expansion quantum Monte Carlo, on both the one-dimensional linear chain and the two-dimensional square lattice. We extract the critical exponents $\ensuremath{\nu}$ and $\ensuremath{\beta}$ as a function of the decay exponent of the long-range interactions. For ferromagnetic Ising interactions, we resolve the limiting regimes known from field theory, ranging from the nearest-neighbor Ising to the long-range Gaussian universality classes, as well as the intermediate regime with continuously varying critical exponents. In the long-range Gaussian regime, we treat the effect of dangerous irrelevant variables on finite-size scaling forms. For antiferromagnetic and therefore competing Ising interactions, the stochastic series expansion algorithm displays growing autocorrelation times leading to a reduced performance. Nevertheless, our results are consistent with the nearest-neighbor Ising universality for all investigated interaction ranges both on the linear chain and the square lattice.
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.