Infinite-randomness quantum Ising critical fixed points

Motrunich, Olexei I.; Mau, Siun-Chuon; Huse, David A.; Fisher, Daniel S.

doi:10.1103/physrevb.61.1160

Cited by 275 publications

(474 citation statements)

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“…The critical behavior of this system is governed by an IRFP [10,11] implying that the disorder strength grows without limit as the system is coarse grained in the renormalization group (RG) sense. In our study, the ground state of the system and the entanglement entropy are numerically calculated using a strong-disorder RG method [18,19], which yields asymptotically exact results at an IRFP.…”

Section: Pacs Numbers: Valid Pacs Appear Herementioning

confidence: 99%

Entanglement Entropy at Infinite-Randomness Fixed Points in Higher Dimensions

2007

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The entanglement entropy of the two-dimensional random transverse Ising model is studied with a numerical implementation of the strong disorder renormalization group. The asymptotic behavior of the entropy per surface area diverges at, and only at, the quantum phase transition that is governed by an infinite randomness fixed point. Here we identify a double-logarithmic multiplicative correction to the area law for the entanglement entropy. This contrasts with the pure area law valid at the infinite randomness fixed point in the diluted transverse Ising model in higher dimensions. PACS numbers: Valid PACS appear hereExtensive studies have been devoted recently to understand ground state entanglement in quantum many-body systems [1]. In particular, the behavior of various entanglement measures at/near quantum phase transitions has been of special interest. One of the widely used entanglement measures is the von Neumann entropy, which quantifies entanglement of a pure quantum state in a bipartite system. Critical ground states in one dimension (1D) are known to have entanglement entropy that diverges logarithmically in the subsystem size with a universal coefficient determined by the central charge of the associated conformal field theory [2]. Away from the critical point, the entanglement entropy saturates to a finite value, which is related to the finite correlation length.In higher dimensions, the scaling behavior of the entanglement entropy is far less clear. A standard expectation is that non-critical entanglement entropy scales as the area of the boundary between the subsystems, known as the "area law" [3,4]. This area-relationship is known to be violated for gapless fermionic systems [5] in which a logarithmic multiplicative correction is found. One might suspect that whether the area law holds or not depends on whether the correlation length is finite or diverges. However, it has turned out that the situation is more complex: numerical findings [7] and a recent analytical study [8] have shown that the area law holds even for critical bosonic systems, despite a divergent correlation length. This indicates that the length scale associated with entanglement may differ from the correlation length. Another ongoing research activity for entanglement in higher spatial dimensions is to understand topological contributions to the entanglement entropy [9].The nature of quantum phase transitions with quenched randomness is in many systems quite different from the pure case. For instance, in a class of systems the critical behavior is governed by a so-called infiniterandomness fixed point (IRFP), at which the energy scale ǫ and the length scale L are related as: ln ǫ ∼ L ψ with 0 < ψ < 1. In these systems the off-critical regions are also gapless and the excitation energies in these socalled Griffiths phases scale as ǫ ∼ L −z with a nonuniversal dynamical exponent z < ∞. Even so, certain random critical points in 1D are shown to have logarithmic divergences of entanglement entropy with universal coefficients, as in the...

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Section: Pacs Numbers: Valid Pacs Appear Herementioning

confidence: 99%

Entanglement Entropy at Infinite-Randomness Fixed Points in Higher Dimensions

2007

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“…They probably control phase discretesymmetry-breaking transitions in all random quantum systems -in any dimension 12 -and, in addition to the spin-1 2 AFM chain, also control the low-temperature properties of a range of random quantum phases. 15 Because of their ubiquity, further investigation of what types of random quantum phases and transitions can occur should shed light more generally on the combined roles of randomness and quantum fluctuations.…”

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“…At infinite randomness fixed points, also controls the decay of typical correlations: 11,12 ln͉͑͗S i •S j ͉͒͘ϷϪC i j ͉iϪ j͉ ͑7͒ with the random coefficient C i j having a universal distribution. The average correlations will, however, decay as 1/͉i Ϫ j͉ 2 at the critical point as in both phases.…”

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Spin reduction transition in spin-32random Heisenberg chains

2002

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Random spin-3 2 antiferromagnetic Heisenberg chains are investigated using an asymptotically exact renormalization group. Randomness is found to induce a quantum phase transition between two random-singlet phases. In the strong randomness phase the effective spins at low energies are S e f f ϭ 3 2 , while in the weak randomness phase the effective spins are S e f f ϭ 1 2 . Separating them is a quantum critical point near which there is a nontrivial mixture of spin-1 2 , spin-1, and spin-3 2 effective spins at low temperatures. Some of the most dramatic effects of randomness in solids appear in the low-temperature behavior of quantum systems. A ͑deceptively͒ simple class of such systems are random quantum spin chains, in particular, Heisenberg antiferromagnetic chains with the HamiltonianFrom a real-space renormalization-group ͑RG͒ analysis, 1 it has been shown that the spin-1 2 random antiferromagnetic ͑AFM͒ chain is strongly dominated by randomness at low temperatures even when the disorder is weak. 2 Its ground state is a random-singlet ͑RS͒ phase in which pairs of spins-mostly close together but occasionally arbitrarily far apart-form singlets. As the temperature is lowered, some of these singlets form at temperatures of order of the typical exchange and become inactive. But their neighboring spins will interact weakly across them via virtual triplet excitations. At lower temperatures, such further neighbors can form singlets and the process repeats. Concomitantly, the distribution of effective coupling strengths broadens rapidly. Eventually, singlets form on all length scales and the ground state is controlled by an RG fixed point with extremely strong disorder: an infinite randomness fixed point.This low-temperature behavior is in striking contrast to that of the pure spin-1 2 AFM chain, in which spin-spin correlations decay as x Ϫ1 because of long-wavelength lowenergy spin-wave ͑or spinon͒ modes. In the random-singlet phase, the average correlations decay as a power of distance -as x Ϫ2 -but for a very different reason: A typical pair of widely spaced spins will have only exponentially ͑in the square root of their separation͒ small correlations. But a small fraction, those that form a singlet pair, will have correlations of order unity independent of their separation; these rare pairs completely dominate the average correlations as well as the other low-temperature properties of the random system.Infinite randomness fixed points are ubiquitous in random quantum systems. They probably control phase discretesymmetry-breaking transitions in all random quantum systems -in any dimension 12 -and, in addition to the spin-1 2 AFM chain, also control the low-temperature properties of a range of random quantum phases. 15 Because of their ubiquity, further investigation of what types of random quantum phases and transitions can occur should shed light more generally on the combined roles of randomness and quantum fluctuations. The simplest cases to analyze are one dimensional because asymptotically exact RG's can be ...

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“…Using a real-space renormalization group, some examples have been found of infinite disorder quantum critical points two or more dimensions [16]. However, these are interacting quantum systems and can only be described by 2+1 or 3+1 dimensional theories with a privileged time coordinate, so they would not be expected to have conformal invariance, although the question of scale invariance in these systems remains an interesting problem for future work.…”

Section: A Model System In Three Dimensionsmentioning

confidence: 99%

Breakdown of conformal invariance at strongly random critical points

Hastings

Sondhi

2001

Phys. Rev. B

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We consider the breakdown of conformal and scale invariance in random systems with strongly random critical points. Extending previous results on one-dimensional systems, we provide an example of a three-dimensional system which has a strongly random critical point. The average correlation functions of this system demonstrate a breakdown of conformal invariance, while the typical correlation functions demonstrate a breakdown of scale invariance. The breakdown of conformal invariance is due to the vanishing of the correlation functions at the infinite disorder fixed point, causing the critical correlation functions to be controlled by a dangerously irrelevant operator describing the approach to the fixed point. We relate the computation of average correlation functions to a problem of persistence in the RG flow. I. INTRODUCTIONThe principle of conformal invariance [1] has proven immensely powerful in the study of continuous phase transitions. While it is believed to hold at critical points of systems with short ranged interactions in all dimensions [2], it is especially powerful in two dimensions, where it has enabled a substantial classification of critical theories based on the representation theory of the Virasoro algebra and its extensions [3].Recent interest has focussed on applying this analysis to systems with quenched randomness in two dimensions; see for examples Refs. [4][5][6]. The formal device that makes this possible is the construction of translationally invariant field theories whose correlation functions are the disorder averaged correlators of the random problem, quite generally via the replica trick and in Gaussian problems, via the supersymmetry method. (There is at least one notable example where neither is needed [7], and conformal invariance follows straightforwardly.)Our interest in this paper, is in asking whether the averaged field theories are necessarily conformally invariant at their critical points. We will find that this is not the case in two instances of what we term strongly random critical points, a category that may well have considerable overlap with the "infinite disorder" critical points studied by means of real space renormalization group techniques [8]. Both are localization problems, the first a well studied problem in d = 1 where conformal invariance is even more powerful than in d = 2, and the second is an analogous construction on our part in d = 3. We also comment on another well studied member of this family of problems in d = 2 which does appear to have a region where conformal invariance holds. We should note before proceeding further, that while we were motivated by the absence of conformal invariance in the one dimensional example, what we find is a breakdown of scale invariance itself in that the operator product expansions is anomalous even though two point correlators are algebraic.Another question appears to be closely connected to the above issues: namely, how one might recover the distributions for correlation functions that should characterize the universa...

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Infinite-randomness quantum Ising critical fixed points

Cited by 275 publications

References 26 publications

Entanglement Entropy at Infinite-Randomness Fixed Points in Higher Dimensions

Entanglement Entropy at Infinite-Randomness Fixed Points in Higher Dimensions

Spin reduction transition in spin-32random Heisenberg chains

Breakdown of conformal invariance at strongly random critical points

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