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ABSTRACT: The transverse-field Ising model with random exchange interactions in finite
dimensions are investigated by means of a real-space renormalization group
method. The scheme yields the exact values of the critical point and critical
exponent nu in one dimension, and some previous results in the case of random
ferromagnetic interactions are reproduced in two and three dimensions. We apply
the scheme to spin glasses in transverse fields in two and three dimensions,
which have not been analyzed very extensively. The phase diagrams and the
critical exponent nu are obtained, and evidence for the existence of an
infinite-randomness fixed point in these models is found.
10/2012;
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ABSTRACT: The two- and three-dimensional transverse-field Ising models with ferromagnetic exchange interactions are analyzed by means of the real-space renormalization-group method. The basic strategy is a generalization of a method developed for the one-dimensional case, which exploits the exact invariance of the model under renormalization and is known to give the exact values of the critical point and critical exponent ν. The resulting values of the critical exponent ν in two and three dimensions are in good agreement with those for the classical Ising model in three and four dimensions. To the best of our knowledge, this is the first example in which a real-space renormalization group on (2+1)- and (3+1)-dimensional Bravais lattices yields accurate estimates of the critical exponents.
Physical Review E 05/2011; 83(5 Pt 1):051103. · 2.26 Impact Factor
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ABSTRACT: We derive a number of exact relations between equilibrium and nonequilibrium
quantities for spin glasses in external fields using the Jarzynski equality and
gauge symmetry. For randomly-distributed longitudinal fields, a lower bound is
established for the work done on the system in nonequilibrium processes, and
identities are proven to relate equilibrium and nonequilibrium quantities. In
the case of uniform transverse fields, identities are proven between physical
quantities and exponentiated work done to the system at different parts of the
phase diagram with the context of quantum annealing in mind. Additional
relations are given, which relate the exponentiated work in quantum and
simulated (classical) annealing. It is also suggested that the Jarzynski
equality may serve as a guide to develop a method to perform quantum annealing
under non-adiabatic conditions.
04/2011;
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ABSTRACT: We investigate in detail the phase diagrams of the p-body +/-J Ising model
with and without random fields on random graphs with fixed connectivity. One of
our most interesting findings is that a thermodynamic spin glass phase is
present in the three-body purely ferromagnetic model in random fields, unlike
for the canonical two-body interaction random-field Ising model. We also
discuss the location of the phase boundary between the paramagnetic and spin
glass phases that does not depend on the change of the ferromagnetic bias. This
behavior is explained by a gauge transformation, which shows that
gauge-invariant properties generically do not depend on the strength of the
ferromagnetic bias for the +/-J Ising model on regular random graphs.
01/2011;
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ABSTRACT: We study the effect that randomness has on long-range interacting systems by
using the ferromagnetic Ising model with $p$-body interactions in random
fields. The case with p=2 yields a phase diagram similar to that of previously
studied models and shows known features that inequivalence of the canonical and
microcanonical ensembles brings with it, for example negative specific heat in
a narrow region of the phase diagram. When p>2, however, the canonical phase
diagram is completely different from the microcanonical one. The temperature
does not necessarily determine the microcanonical phases uniquely, and thus the
ferromagnetic and paramagnetic phases are not separated in such a region of a
conventional phase diagram drawn with the temperature and field strength as the
axes. Below a certain value of the external field strength, part of the
ferromagnetic phase has negative specific heat. For large values of the
external field strength the ergodicity is broken before the phase transition
occurs for p>2. Moreover, for p>2, the Maxwell construction cannot be derived
in a consistent manner and therefore, in contrast to previous cases with
negative specific heat, the Maxwell construction does not bridge the gap
between the ensembles.
01/2011;
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ABSTRACT: We propose a method to reduce the relaxation time toward equilibrium in stochastic sampling of complex energy landscapes in statistical systems with discrete degrees of freedom by generalizing the platform previously developed for continuous systems. The method starts from a master equation, in contrast to the Fokker-Planck equation for the continuous case. The master equation is transformed into an imaginary-time Schrödinger equation. The Hamiltonian of the Schrödinger equation is modified by adding a projector to its known ground state. We show how this transformation decreases the relaxation time and propose a way to use it to accelerate simulated annealing for optimization problems. We implement our method in a simplified kinetic Monte Carlo scheme and show acceleration by one order of magnitude in simulated annealing of the symmetric traveling salesman problem. Comparisons of simulated annealing are made with the exchange Monte Carlo algorithm for the three-dimensional Ising spin glass. Our implementation can be seen as a step toward accelerating the stochastic sampling of generic systems with complex landscapes and long equilibration times.
Physical Review E 11/2010; 82(5 Pt 2):056704. · 2.26 Impact Factor
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ABSTRACT: Quantum annealing is a generic algorithm using quantum-mechanical fluctuations to search for the solution of an optimization problem. The present paper first reviews the fundamentals of quantum annealing and then reports on preliminary results for an alternative method. The review part includes the relationship of quantum annealing with classical simulated annealing. We next propose a novel quantum algorithm which might be available for hard optimization problems by using a classical-quantum mapping as well as the Jarzynski equality introduced in nonequilibrium statistical physics. Comment: 9 pages, 6 figures, to appear in a Special Issue on Foundations of Computational and Theoretical Nanoscience on Journal of Computational and Theoretical Nanoscience
06/2010;
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ABSTRACT: We study the applications of non-equilibrium relations such as the Jarzynski equality and fluctuation theorem to spin glasses with gauge symmetry. It is shown that the exponentiated free-energy difference appearing in the Jarzynski equality reduces to a simple analytic function written explicitly in terms of the initial and final temperatures if the temperature satisfies a certain condition related to gauge symmetry. This result is used to derive a lower bound on the work done during the non-equilibrium process of temperature change. We also prove identities relating equilibrium and non-equilibrium quantities. These identities suggest a method to evaluate equilibrium quantities from non-equilibrium computations, which may be useful to avoid the problem of slow relaxation in spin glasses. Comment: 8 pages, 2 figures, submitted to JPSJ
04/2010;
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ABSTRACT: The distribution of partition function zeros is studied for the $\pm J$ model of spin glasses on the Bethe lattice. We find a relation between the distribution of complex cavity fields and the density of zeros, which enables us to obtain the density of zeros for the infinite system size by using the cavity method. The phase boundaries thus derived from the location of the zeros are consistent with the results of direct analytical calculations. This is the first example in which the spin glass transition is related to the distribution of zeros directly in the thermodynamical limit. We clarify how the spin glass transition is characterized by the zeros of the partition function. It is also shown that in the spin glass phase a continuous distribution of singularities touches the axes of real field and temperature. Comment: 23 pages, 12 figures
01/2010;
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ABSTRACT: We show a practical application of an well-known nonequilibrium relation, the Jarzynski equality, in quantum computation. Its implementation may open a way to solve combinatorial optimization problems, minimization of a real single-valued function, cost function, with many arguments. It has been disclosed that the ordinary quantum computational algorithm to solve a kind of hard optimization problems, has a bottleneck that its computational time is restricted to be extremely slow without relevant errors. However, by our novel strategy shown in the present study, we might overcome such a difficulty. Comment: 3 pages, 1 figure, Proceeding of Conference on Computational Physics 2009, Taiwan
01/2010;
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ABSTRACT: We present a theoretical framework to accurately calculate the location of the multicritical point in the phase diagram of spin glasses. The result shows excellent agreement with numerical estimates. The basic idea is a combination of the duality relation, the replica method, and the gauge symmetry. An additional element of the renormalization group, in particular in the context of hierarchical lattices, leads to impressive improvements of the predictions. Comment: 6 pages, 3 figures, Dedicated to Prof. A. Nihat Berker on the occasion of his sixtieth birthday. to appear in Physica A
11/2009;
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ABSTRACT: We show strong evidence for the absence of a finite-temperature spin glass
transition for the random-bond Ising model on self-dual lattices. The analysis
is performed by an application of duality relations, which enables us to derive
a precise but approximate location of the multicritical point on the Nishimori
line. This method can be systematically improved to presumably give the exact
result asymptotically. The duality analysis, in conjunction with the
relationship between the multicritical point and the spin glass transition
point for the symmetric distribution function of randomness, leads to the
conclusion of the absence of a finite-temperature spin glass transition for the
case of symmetric distribution. The result is applicable to the random bond
Ising model with $\pm J$ or Gaussian distribution and the Potts gauge glass on
the square, triangular and hexagonal lattices as well as the random three-body
Ising model on the triangular and the Union-Jack lattices and the four
dimensional random plaquette gauge model. This conclusion is exact provided
that the replica method is valid and the asymptotic limit of the duality
analysis yields the exact location of the multicritical point
05/2009;
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ABSTRACT: We study the performance of quantum annealing for systems with ground-state degeneracy by directly solving the Schrödinger equation for small systems and quantum Monte Carlo simulations for larger systems. The results indicate that quantum annealing may not be well suited to identify all degenerate ground-state configurations, although the value of the ground-state energy is often effciently estimated. The strengths and weaknesses of quantum annealing for problems with degenerate ground states are discussed in comparison with classical simulated annealing.
Journal of Physics Conference Series 01/2009; 143(1):012003.
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ABSTRACT: Zeros of the $n$th moment of the partition function $[Z^n]$ are investigated in a vanishing temperature limit $\beta \to \infty$, $n \to 0$ keeping $y=\beta n \sim O(1)$. In this limit, the moment parameterized by $y$ characterizes the distribution of the ground-state energy. We numerically investigate the zeros for $\pm J$ Ising spin glass models with several ladder and tree systems, which can be carried out with a feasible computational cost by a symbolic operation based on the Bethe--Peierls method. For several tree systems we find that the zeros tend to approach the real axis of $y$ in the thermodynamic limit implying that the moment cannot be described by a single analytic function of $y$ as the system size tends to infinity, which may be associated with breaking of the replica symmetry. However, examination of the analytical properties of the moment function and assessment of the spin-glass susceptibility indicate that the breaking of analyticity is relevant to neither one-step or full replica symmetry breaking. Comment: 27 pages, 13 figures. Added references, some comments, and corrections to minor errors
09/2008;
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ABSTRACT: We study the performance of quantum annealing for systems with ground-state degeneracy by directly solving the Schr\"odinger equation for small systems and quantum Monte Carlo simulations for larger systems. The results indicate that naive quantum annealing using a transverse field may not be well suited to identify all degenerate ground-state configurations, although the value of the ground-state energy is often efficiently estimated. An introduction of quantum transitions to all states with equal weights is shown to greatly improve the situation but with a sacrifice in the annealing time. We also clarify the relation between the spin configurations in the degenerate ground states and the probabilities that those states are obtained by quantum annealing. The strengths and weaknesses of quantum annealing for problems with degenerate ground states are discussed in comparison with classical simulated annealing. Comment: 12 pages, 16 epsfiles
08/2008;
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ABSTRACT: Quantum annealing is a generic name of quantum algorithms to use quantum-mechanical fluctuations to search for the solution of optimization problem. It shares the basic idea with quantum adiabatic evolution studied actively in quantum computation. The present paper reviews the mathematical and theoretical foundation of quantum annealing. In particular, theorems are presented for convergence conditions of quantum annealing to the target optimal state after an infinite-time evolution following the Schroedinger or stochastic (Monte Carlo) dynamics. It is proved that the same asymptotic behavior of the control parameter guarantees convergence both for the Schroedinger dynamics and the stochastic dynamics in spite of the essential difference of these two types of dynamics. Also described are the prescriptions to reduce errors in the final approximate solution obtained after a long but finite dynamical evolution of quantum annealing. It is shown there that we can reduce errors significantly by an ingenious choice of annealing schedule (time dependence of the control parameter) without compromising computational complexity qualitatively. A review is given on the derivation of the convergence condition for classical simulated annealing from the view point of quantum adiabaticity using a classical-quantum mapping. Comment: 51pages, 8 figures
06/2008;
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ABSTRACT: The locations of multicritical points on many hierarchical lattices are numerically investigated by the renormalization group analysis. The results are compared with an analytical conjecture derived by using the duality, the gauge symmetry, and the replica method. We find that the conjecture does not give the exact answer but leads to locations slightly away from the numerically reliable data. We propose an improved conjecture to give more precise predictions of the multicritical points than the conventional one. This improvement is inspired by a different point of view coming from the renormalization group and succeeds in deriving very consistent answers with many numerical data.
Physical Review E 06/2008; 77(6 Pt 1):061116. · 2.26 Impact Factor
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ABSTRACT: The spatially uniaxially anisotropic d=3 Ising spin glass is solved exactly on a hierarchical lattice. Five different ordered phases, namely, ferromagnetic, columnar, layered, antiferromagnetic, and spin-glass phases, are found in the global phase diagram. The spin-glass phase is more extensive when randomness is introduced within the planes than when it is introduced in lines along one direction. Phase diagram cross sections, with no Nishimori symmetry, with Nishimori symmetry lines, or entirely imbedded into Nishimori symmetry, are studied. The boundary between the ferromagnetic and spin-glass phases can be either reentrant or forward, that is either receding from or penetrating into the spin-glass phase, as temperature is lowered. However, this boundary is always reentrant when the multicritical point terminating it is on the Nishimori symmetry line.
Physical Review E 06/2008; 77(6 Pt 1):061110. · 2.26 Impact Factor
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ABSTRACT: We investigate the distribution of zeros of the partition function of the two- and three-dimensional symmetric $\pm J$ Ising spin glasses on the complex field plane. We use the method to analytically implement the idea of numerical transfer matrix which provides us with the exact expression of the partition function as a polynomial of fugacity. The results show that zeros are distributed in a wide region in the complex field plane. Nevertheless we observe that zeros on the imaginary axis play dominant roles in the critical behaviour since zeros on the imaginary axis are in closer proximity to the real axis. We estimate the density of zeros on the imaginary axis by an importance-sampling Monte Carlo algorithm, which enables us to sample very rare events. Our result suggests that the density has an essential singularity at the origin. This observation is consistent with the existence of Griffiths singularities in the present systems. This is the first evidence for Griffiths singularities in spin glass systems in equilibrium.
01/2008;
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ABSTRACT: Convergence conditions for quantum annealing are derived for optimization problems represented by the Ising model of a general form. Quantum fluctuations are introduced as a transverse field and/or transverse ferromagnetic interactions, and the time evolution follows the real-time Schrodinger equation. It is shown that the system stays arbitrarily close to the instantaneous ground state, finally reaching the target optimal state, if the strength of quantum fluctuations decreases sufficiently slowly, in particular inversely proportionally to the power of time in the asymptotic region. This is the same condition as the other implementations of quantum annealing, quantum Monte Carlo and Green's function Monte Carlo simulations, in spite of the essential difference in the type of dynamics. The method of analysis is an application of the adiabatic theorem in conjunction with an estimate of a lower bound of the energy gap based on the recently proposed idea of Somma et. al. for the analysis of classical simulated annealing using a classical-quantum correspondence.
03/2007;
