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arXiv:1605.09496 · Physical review. E

Upper and lower critical decay exponents of Ising ferromagnets with long-range interaction

俊樹 堀田, Hidemaro Suwa, Synge Todo

Published Jan 23, 2017• 52 citations
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Abstract

We investigate the universality class of the finite-temperature phase transition of the two-dimensional Ising model with the algebraically decaying ferromagnetic long-range interaction, J_{ij}=|r[over ⃗]_{i}-r[over ⃗]_{j}|^{-(d+σ)}, where d (=2) is the dimension of the system and σ is the decay exponent, by means of the order-N cluster-algorithm Monte Carlo method. In particular, we focus on the upper and lower critical decay exponents, the boundaries between the mean-field-universality, intermediate, and short-range-universality regimes. At the critical decay exponents, it is found that the standard Binder ratio of magnetization at the critical temperature exhibits extremely slow convergence as a function of the system size. We propose more effective physical quantities, namely the combined Binder ratio and the self-combined Binder ratio, both of which cancel the leading finite-size corrections of the conventional Binder ratio. Utilizing these techniques, we clearly demonstrate that in two dimensions, the lower and upper critical decay exponents are σ=1 and 7/4, respectively, contrary to the recent Monte Carlo and renormalization-group studies [M. Picco, arXiv:1207.1018; T. Blanchard et al., Europhys. Lett. 101, 56003 (2013)EULEEJ0295-507510.1209/0295-5075/101/56003].

DOI
10.1103/physreve.95.012143

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We investigate the universality class of the finite-temperature phase transition of the two-dimensional Ising model with the algebraically decaying ferromagnetic long-range interaction, J_{ij}=|r[over ⃗]{i}-r[over ⃗]{j}|^{-(d+σ)}, where d (=2) is the dimension of the system and σ is the decay exponent, by means of the order-N cluster-algorithm Monte Carlo method. In particular, we focus on the upper and lower critical decay exponents, the boundaries between the mean-field-universality, intermediate, and short-range-universality regimes. At the critical decay exponents, it is found that the standard Binder ratio of magnetization at the critical temperature exhibits extremely slow convergence as a function of the system size. We propose more effective physical quantities, namely the combined Binder ratio and the self-combined Binder ratio, both of which cancel the leading finite-size corrections of the conventional Binder ratio. Utilizing these techniques, we clearly demonstrate that in two dimensions, the lower and upper critical decay exponents are σ=1 and 7/4, respectively, contrary to the recent Monte Carlo and renormalization-group studies [M. Picco, arXiv:1207.1018; T. Blanchard et al., Europhys. Lett. 101, 56003 (2013)EULEEJ0295-507510.1209/0295-5075/101/56003].
We perform a numerical study of the long-range (LR) ferromagnetic Ising model with power law decaying interactions (J∝r{-d-σ}) on both a one-dimensional chain (d=1) and a square lattice (d=2). We use advanced cluster algorithms to avoid the critical slowing down. We first check the validity of the relation connecting the critical behavior of the LR model with parameters (d,σ) to that of a short-range (SR) model in an equivalent dimension D. We then study the critical behavior of the d=2 LR model close to the lower critical σ, uncovering that the spatial correlation function decays with two different power laws: The effect of the subdominant power law is much stronger than finite-size effects and actually makes the estimate of critical exponents very subtle. By including this subdominant power law, the numerical data are consistent with the standard renormalization group (RG) prediction by Sak [Phys. Rev. B 8, 281 (1973)], thus making not necessary (and unlikely, according to Occam's razor) the recent proposal by Picco [arXiv:1207.1018] of having a new set of RG fixed points in addition to the mean-field one and the SR one.
Determining the threshold value σ∗σ_*σ∗​ that separates the short-range (SR) and long-range (LR) universality classes in phase transitions remains a controversial issue. While Sak's criterion, σ∗=2−ηSRσ_* = 2 - η_{\mathrm{SR}}σ∗​=2−ηSR​, has been widely accepted, recent studies of two-dimensional (2D) models with long-range interactions have challenged it. In this work, we focus on the crossover between LR and SR criticality in several classical 2D statistical models, including the XY, Heisenberg, percolation, and Ising models, whose interactions decay as 1/r2+σ1/r^{2+σ}1/r2+σ. Our previous simulations for the XY, Heisenberg, and percolation models consistently indicate a universal boundary at σ∗=2σ_* = 2σ∗​=2. Here, we complete the picture by performing large-scale Monte Carlo simulations of the 2D LR-Ising model, reaching lattice sizes up to L=8192L = 8192L=8192. By analyzing the Fortuin-Kasteleyn critical polynomial RpR_pRp​, the Binder ratio QmQ_mQm​, and the anomalous dimension ηηη, we obtain convergent and self-consistent evidence that the universality class already changes sharply at σ=2σ= 2σ=2. Taken together, these results establish a unified scenario for LR interacting systems: across all studied models, the crossover from LR to SR universality occurs at σ∗=2σ_* = 2σ∗​=2.
We study the criticality of a long-range quantum ferromagnetic Ising chain with algebraically decaying interactions $1/{r}^{\ensuremath{\alpha}}$ via the fidelity susceptibility based on the exact diagonalization and the density-matrix renormalization-group techniques. We find that critical exponents change monotonically from the mean-field universality class to the short-range Ising universality class for intermediate $\ensuremath{\alpha}$, which are consistent with recent results obtained from renormalization-group techniques. In addition, we determine the critical values for $1.8\ensuremath{\le}\ensuremath{\alpha}\ensuremath{\le}3$ from the finite-size scaling of the fidelity susceptibility. Our work provides very nice numerical data from the fidelity susceptibility for the quantum long-range ferromagnetic Ising chain.
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 phase diagram and critical properties of quantum Ising chains with long-range ferromagnetic interactions decaying in a power-law fashion with exponent α \\alpha , in regimes of direct interest for current trapped ion experiments. Using large-scale path integral Monte Carlo simulations, we investigate both the ground-state and the nonzero-temperature regimes. We identify the phase boundary of the ferromagnetic phase and obtain accurate estimates for the ferromagnetic-paramagnetic transition temperatures. We further determine the critical exponents of the respective transitions. Our results are in agreement with existing predictions for interaction exponents α > 1 \\alpha > 1 up to small deviations in some critical exponents. We also address the elusive regime α < 1 \\alpha < 1 , where we find that the universality class of both the ground-state and nonzero-temperature transition is consistent with the mean-field limit at α = 0 \\alpha = 0 . Our work not only contributes to the understanding of the equilibrium properties of long-range interacting quantum Ising models, but can also be important for addressing fundamental dynamical aspects, such as issues concerning the open question of thermalization in such models.
We have studied the critical phenomena of the Ising model with long-range interaction and the Ising model with spatially correlated random field, both of which are expected to change the effective dimension of the system continuously.The investigation was conducted from the numerical viewpoint by using the Markov chain Monte Carlo method.Second order phase transitions can be classified into the universality classes and the dimensionality of the Ising model changes the universality class: the one-dimensional Ising model has no long-range order at finite temperatures (lower critical dimension), four or higher dimensional model has the mean-field universality (upper critical dimension), two-dimensional Ising model belongs to different universality from either of them, and so on.Although both of the Ising models with long-range interaction and the Ising model with spatially correlated random field are believed to change their critical behavior from its real spatial dimension into different effective dimensions, their detailed behavior has not been known well because of the notorious difficulties in analysis.In this work, we have obtained the detailed behavior of these models by introducing and developing the various methods.To investigate the Ising model with long-range interaction, we introduced the Fukui-Todo method, which drastically reduces the computational cost from O(N 2 ) to O(N ), where N is the number of spins, and developed the generalized Ewald summation, the improved estimator, and the combined Binder ratio technique, which improves the convergence of the estimation.These techniques lead to the successful estimation of the boundary of the universality classes.We also investigated the Ising model with correlated random field by introducing the method for correlated random number generation and the various techniques for the precise analysis.Although the present numerical investigations suggested that the correlated random field causes a shift of effective dimensionality, they also revealed the difficulty for stronger random-field correlations because of the huge finite-size effects.i Prof. Synge