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arXiv:1606.07756 · Physical review. B./Physical review. B

Anisotropic long-range spin systems

Nicolò Defenu, Andrea Trombettoni, Stefano Ruffo

Published Dec 9, 2016• 38 citations
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Abstract

We consider anisotropic long-range interacting spin systems in $d$ dimensions. The interaction between the spins decays with the distance as a power law with different exponents in different directions: We consider an exponent ${d}_{1}+{\ensuremath{\sigma}}_{1}$ in ${d}_{1}$ directions and another exponent ${d}_{2}+{\ensuremath{\sigma}}_{2}$ in the remaining ${d}_{2}\ensuremath{\equiv}d\ensuremath{-}{d}_{1}$ ones. We introduce a low energy effective action with nonanalytic power of the momenta. As a function of the two exponents ${\ensuremath{\sigma}}_{1}$ and ${\ensuremath{\sigma}}_{2}$ we show the system to have three different regimes at criticality, two where it is actually anisotropic and one where the isotropy is finally restored. We determine the phase diagram and provide estimates of the critical exponents as a function of the parameters of the system, in particular considering the case where one of the two $\ensuremath{\sigma}$'s is fixed and the other varying. A discussion of the physical relevance of our results is also presented.

DOI
10.1103/physrevb.94.224411

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We consider anisotropic long-range interacting spin systems in ddd dimensions. The interaction between the spins decays with the distance as a power law with different exponents in different directions: We consider an exponent d1+\ensuremathσ1{d}_{1}+{\ensuremath{\sigma}}_{1}d1​+\ensuremathσ1​ in d1{d}_{1}d1​ directions and another exponent d2+\ensuremathσ2{d}_{2}+{\ensuremath{\sigma}}_{2}d2​+\ensuremathσ2​ in the remaining d2\ensuremath≡d\ensuremath−d1{d}_{2}\ensuremath{\equiv}d\ensuremath{-}{d}_{1}d2​\ensuremath≡d\ensuremath−d1​ ones. We introduce a low energy effective action with nonanalytic power of the momenta. As a function of the two exponents \ensuremathσ1{\ensuremath{\sigma}}_{1}\ensuremathσ1​ and \ensuremathσ2{\ensuremath{\sigma}}_{2}\ensuremathσ2​ we show the system to have three different regimes at criticality, two where it is actually anisotropic and one where the isotropy is finally restored. We determine the phase diagram and provide estimates of the critical exponents as a function of the parameters of the system, in particular considering the case where one of the two \ensuremathσ\ensuremath{\sigma}\ensuremathσ's is fixed and the other varying. A discussion of the physical relevance of our results is also presented.
Several recent experiments in atomic, molecular, and optical systems motivated a huge interest in the study of quantum long-range systems. Our goal in this paper is to present a general description of their critical behavior and phases, devising a treatment valid in ddd dimensions, with an exponent $d+\ensuremath{\sigma}$ for the power-law decay of the couplings in the presence of an O(N)\text{O}(N)O(N) symmetry. By introducing a convenient ansatz for the effective action, we determine the phase diagram for the NNN-component quantum rotor model with long-range interactions, with N=1N=1N=1 corresponding to the Ising model. The phase diagram in the $\ensuremath{\sigma}\text{\ensuremath{-}}d$ plane shows a nontrivial dependence on $\ensuremath{\sigma}$. As a consequence of the fact that the model is quantum, the correlation functions are anisotropic in the spatial and time coordinates for $\ensuremath{\sigma}$ smaller than a critical value, and in this region the isotropy is not restored even at criticality. Results for the correlation length exponent $\ensuremath{\nu}$, the dynamical critical exponent zzz, and a comparison with numerical findings for them are presented.
We present analytical results for the finite-size scaling in ddd-dimensional O(N)O(N)O(N) systems with strong anisotropy where the critical exponents (e.g., ${\ensuremath{\nu}}_{\ensuremath{\Vert}}$ and ${\ensuremath{\nu}}_{\ensuremath{\perp}}$) depend on the direction. Prominent examples are systems with long-range interactions, decaying with the interparticle distance rrr as ${r}^{\ensuremath{-}d\ensuremath{-}\ensuremath{\sigma}}$ with different exponents $\ensuremath{\sigma}$ in corresponding spatial directions, systems with space-``time'' anisotropy near a quantum critical point, and systems with Lifshitz points. The anisotropic properties involve also the geometry of the systems. We consider O(N)O(N)O(N) systems in the $N\ensuremath{\rightarrow}\ensuremath{\infty}$ limit, confined to a ddd-dimensional layer with geometry ${L}^{m}\ifmmode\times\else\texttimes\fi{}{\ensuremath{\infty}}^{n};m+n=d$ and periodic boundary conditions across the finite mmm dimensions. The arising difficulties are avoided using a technique of calculations based on the analytical properties of the generalized Mittag-Leffler functions.
Dimensional correspondences have a long history in critical phenomena. Here, we review the effective dimension approach, which relates the scaling exponents of a critical system in d spatial dimensions with power-law decaying interactions r^{d+σ} to a local system, i.e., with finite-range interactions, in an effective fractal dimension d_{eff}. This method simplifies the study of long-range models by leveraging known results from their local counterparts. While the validity of this approximation beyond the mean-field level has been long debated, we demonstrate that the effective dimension approach, while approximate for non-Gaussian fixed points, accurately estimates the critical exponents of long-range models with an accuracy typically larger than 97%. To do so, we review perturbative renormalization-group (RG) results, extend the approximation's validity using functional RG techniques, and compare our findings with precise numerical data from conformal bootstrap for the two-dimensional Ising model with long-range interactions.
Ferromagnets with the number of spin components n>2n>2n>2 and with power-law interactions ${r}^{\ensuremath{-}d\ensuremath{-}\ensuremath{\sigma}}$ are studied in dimensions $d=2+\ensuremath{\epsilon}$ for small $\ensuremath{\epsilon}$. The following theorem is proved: Let ${\ensuremath{\eta}}_{\mathrm{SR}}$ be the value of the critical exponent $\ensuremath{\eta}$ for short-range force (e.g., nearest-neighbor exchange). Then in the presence of the long-range power-law interaction the critical behavior is the same as for short-range interactions provided that the inequality $2\ensuremath{-}\ensuremath{\sigma}<{\ensuremath{\eta}}_{\mathrm{SR}}$ is satisfied. In this case the critical exponents depend on ddd and nnn but not on $\ensuremath{\sigma}$. In the opposite case, the value of $\ensuremath{\eta}$ is ${\ensuremath{\eta}}_{\mathrm{LR}}=2\ensuremath{-}\ensuremath{\sigma}$ and the critical exponent $\ensuremath{\nu}$ depends on $\ensuremath{\sigma}$ in addition to ddd and nnn.
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.
We present the results of extensive Monte Carlo simulations of Ising models with algebraically decaying ferromagnetic interactions in the regime where classical critical behavior is expected for these systems. We corroborate the values for the exponents predicted by renormalization theory for systems in one, two, and three dimensions and accurately observe the predicted logarithmic corrections at the upper critical dimension. We give both theoretical and numerical evidence that above the upper critical dimension the decay of the critical spin-spin correlation function in finite systems consists of two different regimes. For one-dimensional systems our estimates for the critical couplings are more than two orders of magnitude more accurate than existing estimates. In two and three dimensions we are unaware of any other results for the critical couplings.