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arXiv:1704.00528 · Physical review. B./Physical review. B

Criticality and phase diagram of quantum long-range O( N ) models

Nicolò Defenu, Andrea Trombettoni, Stefano Ruffo

Published Sep 22, 2017• 117 citations
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Abstract

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 $d$ dimensions, with an exponent $d+\ensuremath{\sigma}$ for the power-law decay of the couplings in the presence of an $\text{O}(N)$ symmetry. By introducing a convenient ansatz for the effective action, we determine the phase diagram for the $N$-component quantum rotor model with long-range interactions, with $N=1$ 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 $z$, and a comparison with numerical findings for them are presented.

DOI
10.1103/physrevb.96.104432

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We investigate the zero-temperature and finite-temperature phase transitions of quantum Ising and quantum rotor models. We here assume a long-range (falling off as ${1/r}^{d+\ensuremath{\sigma}},$ where r is the distance between two spins/rotors in units of lattice spacing) ferromagnetic interaction among the spins or rotors. We find that the long-range behavior of the interaction drastically modifies the universal critical behavior of the system. The corresponding upper critical dimension and the hyperscaling relation and exponents associated with the quantum transition are modified and, as expected, they attain values of short-range system when $\ensuremath{\sigma}=2.$ The dynamical exponent varies continuously as the parameter \ensuremath{\sigma} and is unity for $\ensuremath{\sigma}=2.$ The one-dimensional long-range quantum Ising system shows a phase transition at T=0T=0T=0 for all values of \ensuremath{\sigma}. The most interesting observation is that the phase diagram for $\ensuremath{\sigma}=d=1$ shows a line of Kosterlitz-Thouless transition at finite temperature even though the T=0T=0T=0 transition is a simple order-disorder transition. These finite temperature transitions are studied near the phase boundary using renormalisation group equations and a region with diverging susceptibility is located. We have also studied one-dimensional quantum rotor model which exhibits a rich and interesting transition behavior depending upon the parameter \ensuremath{\sigma}. We explore the phase diagram extending the short-range quantum nonlinear \ensuremath{\sigma} model renormalisation group equations to the present case.
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 ${d}_{1}+{\ensuremath{\sigma}}_{1}$ in d1{d}_{1}d1​ 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.
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Quantum quenches to or near criticality give rise to the phenomenon of aging, manifested by glassylike dynamics at short times and far from equilibrium. The recent surge of interest in the dynamics of quantum many-body systems has rejuvenated interest in this phenomenon. Motivated by the ubiquitous long-range interactions in emerging experimental platforms, it is vital to study quantum aging in such settings. In this paper, we investigate the dynamical universality and aging in the d-dimensional O(N) model with the long-range coupling 1/x^{d+σ} and in the mean-field limit N→∞ that allows an exact treatment. An immediate consequence of long-range coupling is the emergence of nonlinear light cones. We focus on the correlation and response functions, and identify a rich scaling behavior depending on how the corresponding space-time positions are located relative to each other, via a local light cone, and to the time of the quench via a global quench light cone. We determine the initial-slip exponent that governs the short-time dependence of two-point functions. We highlight the qualitative features of aging due to the long-range coupling, in particular in the region outside the light cones. As an important consequence of long-range coupling, the correlation function decays as 1/x^{d+σ} outside the quench light cone while increasing polynomially with the total time after quench. This is while, for short-time differences, the two-time response function "equilibrates" at all distances even outside this light cone. Our analytic findings are in excellent agreement with exact numerics, and provide a useful benchmark for modern experimental platforms with long-range interactions.
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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.