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arXiv:cond-mat/0505002 · Physical Review B

Absence of domain wall roughening in a transverse-field Ising model with long-range interactions

George I. Mias, S. M. Girvin

Published Aug 5, 2005• 6 citations
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Abstract

We investigate roughening transitions in the context of transverse-field Ising models. As a modification of the transverse Ising model with short range interactions, which has been shown to exhibit domain wall roughening, we have looked into the possibility of a roughening transition for the case of long-range interactions, since such a system is physically realized in the insulator $\mathrm{Li}\mathrm{Ho}{\mathrm{F}}_{4}$. The combination of strong Ising anisotropy and long-range forces lead naturally to the formation of domain walls but we find that the long-range forces destroy the roughening transition.

DOI
10.1103/physrevb.72.064411

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We investigate roughening transitions in the context of transverse-field Ising models. As a modification of the transverse Ising model with short range interactions, which has been shown to exhibit domain wall roughening, we have looked into the possibility of a roughening transition for the case of long-range interactions, since such a system is physically realized in the insulator LiHoF4\mathrm{Li}\mathrm{Ho}{\mathrm{F}}_{4}LiHoF4​. The combination of strong Ising anisotropy and long-range forces lead naturally to the formation of domain walls but we find that the long-range forces destroy the roughening transition.
The one-dimensional Ising model in a random field and with power-law decaying ferromagnetic bonds is studied at zero temperature. Comparing the scaling of the energy contributions of the ferromagnetic domain wall flip and of the random field \`a la Imry-Ma, a threshold value for the power $\ensuremath{\rho}$ of the long-range interaction can be determined, beyond which no critical behavior occurs. The critical threshold value is ${\ensuremath{\rho}}_{c}=3/2$, at a difference with the zero field model in which ${\ensuremath{\rho}}_{c}=2$. This prediction is analyzed by numerical computation of the ground states below, at, and above this threshold value. The analogy between the critical behavior of long-range one-dimensional systems and in short-range DDD-dimensional systems is investigated. At the critical threshold value of $\ensuremath{\rho}$, corresponding to the lower critical dimension, numerical evidence is found for a zero temperature transition at a finite critical field. Possible finite size crossover effects are discussed in this case. Some implications for the critical behavior of spin glasses in a field are conjectured.
Randomly placed impurities that alter the local exchange couplings, but do not generate random fields or destroy the long-range order, roughen domain walls in Ising systems for dimensionality 53<d<5\frac{5}{3}<d<535​<d<5. They also pin (localize) the walls in energetically favorable positions. This drastically slows down the kinetics of ordering. The pinned domain wall is a new critical phenomenon governed by a zero-temperature fixed point. For d=2d=2d=2, the critical exponents for domain-wall pinning energies and roughness as a function of length scale are estimated from numerically generated ground states.
We have performed quantum Monte Carlo simulations to investigate the superfluid behavior of one- and two-dimensional interfaces separating checkerboard solid domains. The system is described by the hard-core Bose-Hubbard Hamiltonian with nearest-neighbor interaction. In accordance with Burovski et al. [Phys. Rev. Lett. 94, 165301 (2005)] we find that (i) the interface remains superfluid in a wide range of interaction strength before it undergoes a superfluid-insulator transition; (ii) in one dimension, the transition is of the Kosterlitz-Thouless type and is accompanied by the roughening transition, driven by proliferation of charge-1∕21∕21∕2 quasiparticles; (iii) in two dimensions, the transition belongs to the three-dimensional U(1)\mathrm{U}(1)U(1) universality class and the interface remains smooth. Similar phenomena are expected for domain walls in quantum antiferromagnets.
The properties of interfaces are key to understanding the physics of matter. However, the study of quantum interface dynamics has remained an outstanding challenge. Here, we use large-scale tree tensor network simulations to identify the dynamical signature of an interface roughening transition within the ferromagnetic phase of the 2D quantum Ising model. For initial domain wall profiles we find extended prethermal plateaus for smooth interfaces, whereas above the roughening transition the domain wall decays quickly. Our results can be readily explored experimentally in Rydberg atomic systems.
The distribution of energy and free energy of domain walls near the spin-glass transition is studied numerically in the three-dimensional Ising spin-glass model with discrete $\ifmmode\pm\else\textpm\fi{}J$ interactions. Temperature dependence of the width of the free-energy distribution clearly shows a phase transition to a low-temperature phase with long-range spin correlations at $\frac{k{T}_{g}}{J}\ensuremath{\simeq}1.2$. Simple scaling analysis is consistent with previous studies of this system.