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arXiv:1503.03727 · Journal of Statistical Mechanics Theory and Experiment

Dyson hierarchical quantum ferromagnetic Ising chain with pure or random transverse fields

Cécile Monthus

Published May 20, 2015• 11 citations
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Abstract

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.

DOI
10.1088/1742-5468/2015/05/p05026

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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.
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 .
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.
We study the one-dimensional Dyson hierarchical model in the presence of a random field. This is a long range model where the interaction scales with the distance in a power-law-like form, $J(r)\ensuremath{\sim}{r}^{\ensuremath{-}\ensuremath{\rho}}$, and we can explore mean-field and non-mean-field behavior by changing $\ensuremath{\rho}$. We analyze the model at T=0T=0T=0 and we numerically compute the non-mean-field critical exponents for Gaussian disorder. We also compute an analytic expression for the critical exponent $\ensuremath{\delta}$, and give an interesting relation between the critical exponents of the disordered model and the ones of the pure model, which seems to break down in the non-mean-field region. We finally compare our results for the critical exponents with the expected ones in DDD-dimensional short range models and with the ones of the straightforward one-dimensional long range model.
A real-space renormalization-group treatment of random transverse-field Ising spin chains that was introduced previously is developed and extensively analyzed. It yields results that are asymptotically exact in the critical region near the zero-temperature para-to-ferromagnetic quantum phase transition. In particular, the exact scaling function is obtained for the magnetization as a function of a uniform applied magnetic field and the distance to the critical point, and up to the solution of a linear ordinary differential equation whose solution can be exhaustively analyzed, the scaling function of the average spin-spin correlation function is also obtained. Thus more exact information is obtainable about the critical behavior for this random model than is known for the pure version which is equivalent to the two-dimensional Ising model. The basic reason for this is the extremely broad distribution of energy scales that occurs at low energies near the critical point of the random system. For the random chain the distribution of the magnetization of the first spin in a semi-infinite system is also studied and the results found to agree in the scaling limit with results of McCoy obtained from the exact solution of the closely related McCoy-Wu Ising model; this provides strong justification for the validity of the present approach. The singular properties of the weakly ordered and weakly disordered ``Griffiths' phases'' that occur at zero temperature near the critical point are also studied, as well as the behavior at low but nonzero temperature. Possible extensions of the results and general lessons drawn from them for other random systems are briefly discussed.
Fisher's [Phys. Rev. B 51,51,51, 6411 (1995)] real-space renormalization-group (RG) treatment of random transverse-field Ising spin chains is extended into the strongly ordered and strongly disordered Griffiths phases, and asymptotically exact results are obtained. In the noncritical region the asymmetry of the renormalization of the couplings and the transverse fields is related to a nonlinear quantum control parameter \ensuremath{\Delta}, which is a natural measure of the distance from the quantum critical point. \ensuremath{\Delta}, which is found to stay invariant along the RG trajectories, and has been expressed by the initial disorder distributions, stands in the singularity exponents of different physical quantities (magnetization, susceptibility, specific heat, etc.), which are exactly calculated. In this way we have observed a weak-universality scenario: the Griffiths-McCoy singularities do not depend on the form of the disorder, provided the nonlinear quantum control parameter has the same value. The exact scaling function of the magnetization with a small applied magnetic field is calculated, and the critical point magnetization singularity is determined in a simple, direct way.
We study the low-energy properties of the long-range random transverse-field Ising chain with ferromagnetic interactions decaying as a power of the distance. Using variants of the strong-disorder renormalization group method, the critical behavior is found to be controlled by a strong-disorder fixed point with a finite dynamical exponent . Approaching the critical point, the correlation length diverges exponentially. In the critical point, the magnetization shows an -independent logarithmic finite-size scaling and the entanglement entropy satisfies the area law. These observations are argued to hold for other systems with long-range interactions, even in higher dimensions.