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arXiv:1506.06012 · Journal of Statistical Mechanics Theory and Experiment

Dyson hierarchical long-ranged quantum spin-glass via real-space renormalization

Cécile Monthus

Published Oct 21, 2015• 11 citations
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Abstract

We consider the Dyson hierarchical version of the quantum spin-glass with random Gaussian couplings characterized by the power-law decaying variance and a uniform transverse field h . The ground state is studied via real-space renormalization to characterize the spin-glass-paramagnetic zero temperature quantum phase transition as a function of the control parameter h . In the spin-glass phase , the typical renormalized coupling grows with the length scale L as the power-law with the classical droplet exponent , where the stiffness modulus vanishes at criticality , whereas the typical renormalized transverse field decays exponentially in terms of the diverging correlation length . At the critical point , the typical renormalized coupling and the typical renormalized transverse field display the same power-law behavior L − z with a finite dynamical exponent z . The RG rules are applied numerically to chains containing L = 2 12 = 4096 spins in order to measure these critical exponents for various values of σ in the region .

DOI
10.1088/1742-5468/2015/10/p10024

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We consider the Dyson hierarchical version of the quantum spin-glass with random Gaussian couplings characterized by the power-law decaying variance and a uniform transverse field h . The ground state is studied via real-space renormalization to characterize the spin-glass-paramagnetic zero temperature quantum phase transition as a function of the control parameter h . In the spin-glass phase , the typical renormalized coupling grows with the length scale L as the power-law with the classical droplet exponent , where the stiffness modulus vanishes at criticality , whereas the typical renormalized transverse field decays exponentially in terms of the diverging correlation length . At the critical point , the typical renormalized coupling and the typical renormalized transverse field display the same power-law behavior L − z with a finite dynamical exponent z . The RG rules are applied numerically to chains containing L = 2 12 = 4096 spins in order to measure these critical exponents for various values of σ in the region .
For the one-dimensional long-ranged Ising spin-glass with random couplings decaying with the distance r as J ( r ) ∼ r − σ and distributed with the Lévy symmetric stable distribution of index 1 < μ ≤ 2 (including the usual Gaussian case μ = 2), we consider the region σ > 1/ μ where the energy is extensive. We study two real space renormalization procedures at zero temperature, namely a simple box decimation that leads to explicit calculations, and a strong disorder decimation that can be studied numerically on large sizes. The droplet exponent governing the scaling of the renormalized couplings J L ∝ L θ μ ( σ ) is found to be whenever the long-ranged couplings are relevant θ μ ( σ ) ≥ −1. For the statistics of the ground state energy over disordered samples, we obtain that the droplet exponent θ μ ( σ ) governs the leading correction to extensivity of the averaged value . The characteristic scale of the fluctuations around this average is of order , and the rescaled variable is Gaussian distributed for μ = 2, or displays the negative power-law tail in 1/(− u ) 1+ μ for u → − ∞ in the Lévy case 1 < μ < 2. Finally we apply the zero-temperature renormalization procedure to the related Dyson hierarchical spin-glass model where the same droplet exponent appears.
The Dyson hierarchical version of the quantum Ising chain with Long-Ranged power-law ferromagnetic couplings J ( r ) ∝ r −1− σ and pure or random transverse fields is studied via real-space renormalization. For the pure case, the critical exponents are explicitly obtained as a function of the parameter σ , and are compared with previous results of other approaches. For the random case, the RG rules are numerically applied and the critical behaviors are compared with previous strong disorder renormalization results.
The Dyson hierarchical one-dimensional Ising model of parameter σ > 0 contains long-ranged ferromagnetic couplings decaying as 1/ r 1+σ in terms of the distance r . We study the stochastic dynamics near zero temperature via the real space renormalization introduced in our previous work (Monthus and Garel, 2012 arXiv: 1212.0643 ) in order to compute explicitly the equilibrium time t eq ( L ) as a function of the system size L . For σ < 1 where the static critical temperature for the ferromagnetic transition is finite, T c > 0, we obtain that dynamical barriers grow as the power law ln t eq ( L ) ≃ β(4 J 0 /3(2 1−σ − 1)) L 1−σ . For σ = 1 where the static critical temperature vanishes, T c = 0, we obtain that dynamical barriers grow logarithmically as ln t eq ( L ) ≃ [β(4 J 0 /3ln2) − 1]ln L . We also compute finite contributions to the dynamical barriers that can depend on the choice of transition rates satisfying detailed balance.
Abstract The quantum Ising model with random couplings and random transverse fields on the Cayley tree is studied by real-space-renormalization in order to construct the whole set of eigenstates. The renormalization rules are analyzed via large deviations. The phase transition between the paramagnetic and the spin-glass many-body-localized phases involves the activated exponent ψ = 1 and the correlation length exponent ν = 1 . The spin-glass-ordered cluster containing N SG spins is found to be extremely sparse with respect to the total number N of spins: its size grows only logarithmically at the critical point N S G c r i t i ∝ ln ⁡ N , and it is sub-extensive N S G ∝ N θ in the finite region of the spin-glass phase where the continuously varying exponent θ remains in the interval 0 < θ < 1 .
We consider quantum rotors or Ising spins in a transverse field on a d-dimensional lattice, with random, frustrating, short-range, exchange interactions. The quantum dynamics are associated with a finite moment of inertia for the rotors, and with the transverse field for the Ising spins. For a suitable distribution of exchange constants, these models display spin-glass and quantum paramagnet phases and a zero-temperature (T) quantum transition between them. An earlier exact solution for the critical properties of a model with infinite-range interactions cna be reproduced by minimization of a Landau effective-action functional for the model in finite d with short-range interactions. The functional is expressed in terms of a composite spin field which is bilocal in time. The mean-field phase diagram near the T=0 critical point is mapped out as a function of T, strength of the quantum coupling, and applied fields. The spin-glass phase has replica symmetry breaking; but, as in the classical Ising spin glass, the order parameter becomes replica symmetric as T\ensuremath{\rightarrow}0. Next we examine the consequences of fluctuations about the mean field for the critical properties. Above d=8, and with certain restrictions on the values of the Landau couplings, we find that the transition is controlled by a Gaussian fixed point with mean-field critical exponents. For couplings not attracted by the Gaussian fixed point above d=8, and for all physical couplings below d=8, we find runaway renormalization-group flows to strong coupling. General scaling relations that should be valid even at the strong-coupling fixed point are proposed and compared with Monte Carlo simulations.
For the Ising model with Gaussian random coupling of average J0{J}_{0}J0​ and unit variance, the zero-temperature spin-glass--ferromagnetic transition as a function of the control parameter J0{J}_{0}J0​ can be studied via the size-LLL-dependent renormalized coupling defined as the domain-wall energy ${J}^{R}(L)\ensuremath{\equiv}{E}_{\text{GS}}^{(\text{AF})}(L)\ensuremath{-}{E}_{\text{GS}}^{(\text{F})}(L)$ (i.e., the difference between the ground-state energies corresponding to antiferromagnetic and ferromagnetic boundary conditions in one direction). We study numerically the critical exponents of this zero-temperature transition within the Migdal-Kadanoff approximation as a function of the dimension d=2,3,4,5,6d=2,3,4,5,6d=2,3,4,5,6. We then compare with the mean-field spherical model. Our main conclusion is that in low dimensions, the critical stiffness exponent ${\ensuremath{\theta}}^{c}$ is clearly bigger than the spin-glass stiffness exponent ${\ensuremath{\theta}}^{\text{SG}}$, but that they turn out to coincide in high enough dimension and in the mean-field spherical model. We also discuss the finite-size scaling properties of the averaged value and of the width of the distribution of the renormalized couplings.