Quasi-Long Range Order in Glass States of Impure Liquid Crystals, Magnets, and Superconductors

Feldman, D. E.

doi:10.1142/s0217979201006641

Cited by 63 publications

(78 citation statements)

References 77 publications

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“…Similar to Ref. [9] we do not expect that the defects are relevant at weak disorder. At strong disorder, dislocations also do not proliferate deep in the bulk.…”

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confidence: 85%

Destruction of Bulk Ordering by Surface Randomness

Vinokur²

2002

Phys. Rev. Lett.

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We demonstrate that the arbitrarily weak quenched disorder on the surface of a system of continuous symmetry destroys long range order in the bulk, and, instead, quasi-long range order emerges. Correlation functions are calculated exactly for the two-and three-dimensional XY model with surface randomness via the functional renormalization group. Even at strong quenched disorder the three-dimensional XY model possesses topological order. We also determine roughness of a domain wall in the presence of surface disorder. 75.10.Nr, 74.60.Ge, 05.50.+q, 64.60.Ak The arbitrarily weak quenched disorder in the bulk of a system of continuous symmetry destroys long range order inherent to the pure system [1] provided that disorder breaks not only the translational symmetry but also the symmetry with respect to transformations of the order parameter, as e.g. random anisotropy in amorphous magnets does. This fundamental fact governs all the physics of condensed matter and results in a wealth of observed static and dynamics behaviors of real solids.In many cases noticeable disorder presents only at the surface. Not surprisingly, surface randomness modifies the critical behavior near the surface [2], yet the common expectation is for the bulk properties to remain intact. In this Letter we show that arbitrarily weak surface disorder destroys long range order in the bulk of a system of continuous symmetry at the arbitrarily low temperature.The predicted effect occurs in a rich variety of systems. Examples include crystal ordering in solids grown on a disordered substrate, liquid crystals interacting with an inhomogeneous surface, superconducting vortices pinned by surface impurities, etc. There are also many two-dimensional systems with edge randomness, e.g. superconducting films with columnar defects in a part of the film or films with a rough edge [3].The reason as to why surface impurities, however weak, break long range bulk order is that the bulk contributes little to the energy of long-wave Goldstone modes: the surface random energy of long-wave excitations turns out to be greater than the corresponding bulk energy. As a result, the inhomogeneous state becomes favorable energetically. Note that ordering survives in the regions of the size less than the distance of these regions from the surface. In other words, if the distance between the two points is greater than their separation from the surface, the order parameter is different in those points. While long range order breaks down, topological order survives and quasi-long range order emerges. This means that the correlation length is infinite and that the correlation functions obey a slow logarithmic dependence of the distance.An easy way to understand the main result of the Letter is based on Imry-Ma arguments [1,4]. Let us consider a region of size L near the surface and compare the energies of ordered and disordered states of the region. If long range order is broken on the scales of the order of L the loss in the bulk (elastic) energy is E bulk ∼ L D /L 2 , where ...

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“…Similar to Ref. [9] we do not expect that the defects are relevant at weak disorder. At strong disorder, dislocations also do not proliferate deep in the bulk.…”

supporting

confidence: 85%

Destruction of Bulk Ordering by Surface Randomness

Vinokur²

2002

Phys. Rev. Lett.

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“…Such power laws are typical for weakly disordered systems [23] exhibiting mid-range order [24]. In that case, the local stability index (5) behaves as σ(t) ∼ β ln t, which can be either positive or negative depending on the sign of β.…”

Section: Stability Of Stochastic Systemsmentioning

confidence: 95%

Stochastic instability of quasi-isolated systems

2002

Phys. Rev. E

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The stability of solutions to evolution equations with respect to small stochastic perturbations is considered. The stability of a stochastic dynamical system is characterized by the local stability index. The limit of this index with respect to infinite time describes the asymptotic stability of a stochastic dynamical system. Another limit of the stability index is given by the vanishing intensity of stochastic perturbations. A dynamical system is stochastically unstable when these two limits do not commute with each other. Several examples illustrate the thesis that there always exist such stochastic perturbations which render a given dynamical system stochastically unstable. The stochastic instability of quasi-isolated systems is responsible for the irreversibility of time arrow.02.50.Ey, 02.30.Jr, 05.70.Ln

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“…However, this leaves aside two questions: first, the nature of the critical behavior in random field models, a question connected to the breakdown of the so-called "dimensional reduction" (DR) property that relates the critical exponents of the RFO(N )M to those of the pure O(N ) model in two dimensions less [5]; and second, the possible occurence of a low-temperature phase with quasi-long range order (QLRO), i.e. , a phase characterized by no magnetization and a power-law decrease of the correlation fuctions, in models with a continuous symmetry [6,7]. Progress has been made to better circumscribe this latter point.…”

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Unified Picture of Ferromagnetism, Quasi-Long-Range Order, and Criticality in Random-Field Models

2006

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By applying the recently developed nonperturbative functional renormalization group (FRG) approach, we study the interplay between ferromagnetism, quasi-long range order (QLRO) and criticality in the d-dimensional random field O(N ) model in the whole (N , d) diagram. Even though the "dimensional reduction" property breaks down below some critical line, the topology of the phase diagram is found similar to that of the pure O(N ) model, with however no equivalent of the Kosterlitz-Thouless transition. In addition, we obtain that QLRO, namely a topologically ordered "Bragg glass" phase, is absent in the 3-dimensional random field XY model. The nonperturbative results are supplemented by a perturbative FRG analysis to two loops around d = 4.How the phase behavior and ordering transitions of a system are affected by the presence of a weak random field remains in part an unsettled problem. Heuristic and rigorous arguments show that the lower critical dimension below which no long-range order is possible is 2 for the random field Ising model (RFIM) [1,2] and 4 for models with a continuous symmetry (RFO(N )M with N > 1) [1,3,4]. However, this leaves aside two questions: first, the nature of the critical behavior in random field models, a question connected to the breakdown of the so-called "dimensional reduction" (DR) property that relates the critical exponents of the RFO(N )M to those of the pure O(N ) model in two dimensions less [5]; and second, the possible occurence of a low-temperature phase with quasi-long range order (QLRO), i.e. , a phase characterized by no magnetization and a power-law decrease of the correlation fuctions, in models with a continuous symmetry [6,7]. Progress has been made to better circumscribe this latter point. It has indeed been shown that QLRO is absent for N = 2 when disorder is strong and for N > 3 for arbitrarily weak random field [8]; but this still keeps open the cases N = 2, 3 in the physical dimensions d = 2, 3.Those questions are important because, on top of purely theoretical motivations, they concern the behavior of the known experimental realizations of random field models. This is the case for instance of vortex lattices in disordered type-II superconductors [7,9]. In such systems, the randomly pinned lattice of vortices can be mapped onto an "elastic glass model" [7,9], whose simplest realization is the N = 2 RFXY M. The occurence of a phase with QLRO, termed "Bragg glass", has been predicted for the 3 − d version of the model [7]. Further theoretical support for this prediction has been given by a Monte Carlo simulation of the RFXY M [10] and by analyses of the energetics of dislocation loops [7,11].In this letter, we apply our recently developed nonperturbative FRG approach of the RFO(N )M [12] to provide a unified picture of ferromagnetism, QLRO and criticality in the whole (N , d) diagram. We find that below a critical value N c = 2.83.. and for d < 4 the model has a transition to a QLRO phase, both this phase and the transition being governed by zero-temperature nonana...

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Quasi-Long Range Order in Glass States of Impure Liquid Crystals, Magnets, and Superconductors

Cited by 63 publications

References 77 publications

Destruction of Bulk Ordering by Surface Randomness

Destruction of Bulk Ordering by Surface Randomness

Stochastic instability of quasi-isolated systems

Unified Picture of Ferromagnetism, Quasi-Long-Range Order, and Criticality in Random-Field Models

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