Nishimori Point in Random-Bond Ising and Potts Models in 2D

Honecker, A.; Jacobsen, Jesper Lykke; Picco, Marco; Pujol, Pierre

doi:10.1007/978-94-010-0514-2_23

Cited by 8 publications

(16 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…This m eans that even when 15.8% o f the qutrits in a topological m em ory are faulty, error correction is still possible and the encoded inform ation is retained. Notably, this num erical estim ate agrees w ith the result previously found by H onecker et al [38], w ith the disorder defined as p ' = p/(d -1). Since the m ethod based on the dom ain-w all free energy they em ploy detects the transition at T = 0, this also indicates that a possible reentrance effect is small.…”

Section: Resultssupporting

confidence: 91%

Error thresholds for Abelian quantum double models: Increasing the bit-flip stability of topological quantum memory

2015

View full text Add to dashboard Cite

Current approaches for building quantum computing devices focus on two-level quantum systems which nicely mimic the concept of a classical bit, albeit enhanced with additional quantum properties. However, rather than artificially limiting the number of states to two, the use of r/-level quantum systems (qudits) could provide advantages for quantum information processing. Among other merits, it has recently been shown that multilevel quantum systems can offer increased stability to external disturbances. In this study we demonstrate that topological quantum memories built from qudits, also known as Abelian quantum double models, exhibit a substantially increased resilience to noise. That is, even when taking into account the multitude of errors possible for multilevel quantum systems, topological quantum error-correction codes employing qudits can sustain a larger error rate than their two-level counterparts. In particular, we find strong numerical evidence that the thresholds of these error-correction codes are given by the hashing bound. Considering the significantly increased error thresholds attained, this might well outweigh the added complexity of engineering and controlling higher-dimensional quantum systems.

show abstract

Section: Resultssupporting

confidence: 91%

Error thresholds for Abelian quantum double models: Increasing the bit-flip stability of topological quantum memory

2015

View full text Add to dashboard Cite

show abstract

“…In this case, the exponents depend less on the precise value of p N , but the finite-size corrections are larger. Extrapolating, we find a value of roughly η e 1 = 2.75 − 2.85, rather close to the one obtained for the RBIM Nishimori point η e 1 = 2.83(2) using a similar fit [19]. Discarding data with small ∆x leads to larger error bars, but is still consistent with η e 1 ∼ 2.85.…”

supporting

confidence: 84%

Phase diagram and critical exponents of a Potts gauge glass

Jacobsen

Picco

2002

Phys. Rev. E

Self Cite

View full text Add to dashboard Cite

The two-dimensional q-state Potts model is subjected to a Zq symmetric disorder that allows for the existence of a Nishimori line. At q = 2, this model coincides with the ±J randombond Ising model. For q > 2, apart from the usual pure and zero-temperature fixed points, the ferro/paramagnetic phase boundary is controlled by two critical fixed points: a weak disorder point, whose universality class is that of the ferromagnetic bond-disordered Potts model, and a strong disorder point which generalizes the usual Nishimori point. We numerically study the case q = 3, tracing out the phase diagram and precisely determining the critical exponents. The universality class of the Nishimori point is inconsistent with percolation on Potts clusters.During the last decade, the study of disordered systems has attracted much interest. This is true in particular in two dimensions, where the possible types of critical behavior for the corresponding pure models can be classified using conformal field theory [1]. Recently, similar classification issues for disordered models have been addressed through the study of various random matrix ensembles [2], but many fundamental questions remain open.An important category of 2D disordered systems is given by models where the disorder couples to the local energy density. Two paradigmatic members of this class are the ±J random-bond Ising model, and the q-state ferromagnetic random-bond Potts model. The model to be studied in the present Letter can be thought of as an interpolation between these two members; we shall therefore begin by recalling some of their basic properties. The random-bond Ising model (RBIM) is defined by the energy functionalwhere the sum is over the edges of the square lattice, S i = ±1 are Ising spins, and δ(., .) is the Kronecker delta function. The random bonds take the values J ij = ±1 according to the probability distributionThe salient feature of this model is that it marries disorder with frustration, leading to the possibility of spin glass order.Its phase diagram is generally believed to be as in Fig. 1.a [3]. The boundary FP between the ferromagnetic and the paramagnetic phases is controlled by three fixed points. The attractive fixed points at either end of the phase boundary are respectively the critical point of the pure Ising model and a zero-temperature fixed point. Between these two we find the multicritical point N, intersecting the so-called Nishimori line [4] e β = (1 − p)/p .On this line, the replicated version of the model possesses a local Z 2 gauge symmetry that, among other things, allows for exactly computing the internal energy and for establishing the pairwise equality of correlation functions

show abstract

“…A convenient choice is to consider exchange interactions with fixed magnitude which are independently ferromagnetic or antiferromagnetic, with probabilities 1 − p and p respectively. In this case, it is known from a variety of approaches [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29] that the Curie temperature is depressed with increasing p, reaching zero at a critical disorder strength, p c . Moreover, while the scaling flow at the transition is controlled for small p by the critical fixed point of the pure system, at larger p it is determined by a disorder-dominated multicritical point, known as the Nishimori point.…”

Section: Introductionmentioning

confidence: 99%

Two-dimensional random-bond Ising model, free fermions, and the network model

2002

View full text Add to dashboard Cite

We develop a recently-proposed mapping of the two-dimensional Ising model with random exchange (RBIM), via the transfer matrix, to a network model for a disordered system of non-interacting fermions. The RBIM transforms in this way to a localisation problem belonging to one of a set of non-standard symmetry classes, known as class D; the transition between paramagnet and ferromagnet is equivalent to a delocalisation transition between an insulator and a quantum Hall conductor. We establish the mapping as an exact and efficient tool for numerical analysis: using it, the computational effort required to study a system of width M is proportional to M 3 , and not exponential in M as with conventional algorithms. We show how the approach may be used to calculate for the RBIM: the free energy; typical correlation lengths in quasi-one dimension for both the spin and the disorder operators; even powers of spin-spin correlation functions and their disorder-averages. We examine in detail the square-lattice, nearest-neighbour ±J RBIM, in which bonds are independently antiferromagnetic with probability p, and ferromagnetic with probability 1 − p. Studying temperatures T ≥ 0.4J, we obtain precise coordinates in the p − T plane for points on the phase boundary between ferromagnet and paramagnet, and for the multicritical (Nishimori) point. We demonstrate scaling flow towards the pure Ising fixed point at small p, and determine critical exponents at the multicritical point. i i+1 i+2 J (n,i) J (n+1,i) J (n,i+1) v J (n,i-1) J (n,i)

show abstract

Nishimori Point in Random-Bond Ising and Potts Models in 2D

Cited by 8 publications

References 24 publications

Error thresholds for Abelian quantum double models: Increasing the bit-flip stability of topological quantum memory

Error thresholds for Abelian quantum double models: Increasing the bit-flip stability of topological quantum memory

Phase diagram and critical exponents of a Potts gauge glass

Two-dimensional random-bond Ising model, free fermions, and the network model

Contact Info

Product

Resources

About