The partition function of Z(2) and Z(8) lattice gauge theory in four dimensions, a novel approach to simulations of lattice systems

Bhanot, Gyan; Bitar, Khalil M.; Black, Steve; Carter, Paul; Salvador, Román

doi:10.1016/0370-2693(87)91114-2

Cited by 49 publications

(11 citation statements)

References 10 publications

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“…Moreover, the final algorithm must be ergodic: any energy value in a window should be obtained from any other value in the same window after a sufficient number of spin flips. The same basic idea has been used in other short-range Hamiltonians [14][15][16][17]22]. We are concerned now with the extension of this method to consider long-range interactions, in particular the LRIM introduced before.…”

Section: The Energy Histogram Methods Using Overlapping Windowsmentioning

confidence: 99%

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Thermostatistics of extensive and non-extensive systems using generalized entropies

Salazar¹,

Toral²

2001

Physica A: Statistical Mechanics and its Applications

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We describe in detail two numerical simulation methods valid to study systems whose thermostatistics is described by generalized entropies, such as Tsallis. The methods are useful for applications to non-trivial interacting systems with a large number of degrees of freedom, and both short-range and long-range interactions. The first method is quite general and it is based on the numerical evaluation of the density of states with a given energy. The second method is more specific for Tsallis thermostatistics and it is based on a standard Monte Carlo Metropolis algorithm along with a numerical integration procedure. We show here that both methods are robust and efficient. We present results of the application of the methods to the one-dimensional Ising model both in a short-range case and in a long-range (non-extensive) case. We show that the thermodynamic potentials for different values of the system size N and different values of the non-extensivity parameter q can be described by scaling relations which are an extension of the ones holding for the Boltzmann-Gibbs statistics (q = 1). Finally, we discuss the differences in using standard or non-standard mean value definitions in the Tsallis thermostatistics formalism and present a microcanonical ensemble calculation approach of the averages.

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Section: The Energy Histogram Methods Using Overlapping Windowsmentioning

confidence: 99%

“…(16). In this equation, we can use the Monte Carlo averages for the internal energy U q = H , but the entropy is yet unknown.…”

Section: The Monte Carlo Methodsmentioning

confidence: 99%

Thermostatistics of extensive and non-extensive systems using generalized entropies

Salazar¹,

Toral²

2001

Physica A: Statistical Mechanics and its Applications

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“…E is integer valued, ranging from zero for the ordered state to a number of the order of the volume for the maximally frustrated state. The partition function is Em Z(u)= J^P(E)u E 9 (2) where P(E) is the number of states of the system at energy £", u =e ~4 fi , and E m =dL d /2. To make our simulation fast, we found it useful to update several independent lattices simultaneously.…”

Section: Uj)mentioning

confidence: 99%

Accurate estimate ofνfor the three-dimensional Ising model from a numerical measurement of its partition function

et al. 1987

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We give results from an accurate determination of the partition function of the three-dimensional Ising model on lattices of size up to 10 3 . We compute the two complex zeros of Z closest to the real temperature axis. From a finite-size-scaling analysis of these, we get the estimates v =0.6295 (10) for the correlation-length exponent and 0 = 52.2(7) for the angle between the first two zeros and the real u =e ~4 /3 axis.PACS numbers: 75.10.Hk, 75.40.CxWe have recently developed 1 " 3 a new technique that directly measures the partition function of a statisticalmechanical system by numerical methods. In the present paper, we give results for some critical properties of the three-dimensional Ising model by accurately measuring its partition function on cubic lattices (with periodic boundary conditions) of size up to 10 3 using our method. The details of our numerical simulation and analysis will be published elsewhere. 4 Here we will describe our method, give results for the two zeros closest to the critical point in the u =e ~4& plane and, from these, determine the correlation-length exponent v and the angle 0 between the first two zeros and the real u =e ~4P axis.The energy of a lattice configuration is Uj)Here the sum is over all nearest-neighbor pairs of lattice sites. E is integer valued, ranging from zero for the ordered state to a number of the order of the volume for the maximally frustrated state. The partition function iswhere P(E) is the number of states of the system at energy £", u =e ~4 fi , and E m =dL d /2. To make our simulation fast, we found it useful to update several independent lattices simultaneously. Our method to compute Z went as follows: We divided up the range of E values into sets containing four consecutive energies each. The last E value of one set was the first of the next set. Consider one of the sets. To initialize a lattice into that energy set, it was started in a disordered or ordered configuration and randomly chosen spins were flipped, the flip being accepted if the energy went in the desired direction. The independent lattices to be processed together were initialized independently.Once all the lattices were initialized, they were updated by flipping spins at sites chosen randomly (but the same site on all the lattices in a single trial flip). If the spin flip kept a lattice within the range of allowed energy values, it was accepted. These spin-flip attempts were repeated a large number of times, and the number of times the lattice energy had a given E value was recorded. This experiment was repeated over all sets. The relative probability for the system to be in one (E) or the other (£") energy state in the set is an unbiased estimator of the relative number of states P(E)/P(E') at these energy values. From the overlap in E between sets, one gets the complete partition function, apart from an irrelevant normalization factor.The choice of four energy values in a set is to allow the system to reach any local spin configuration from any other. One could have picked more than fo...

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“…Several methods have been invented to circumvent this problem (for a review, see [1] and [2]): the reweighting from the ensemble at µ = 0 [3][4][5][6], the Taylor expansion method [7][8][9][10][11][12][13][14][15], the canonical approach [16][17][18][19], the density of states method [20][21][22][23][24][25][26] and the method of analytic continuation from an imaginary chemical potential . Their application has allowed to get relevant information on the critical line separating the hadronic phase from the quark-gluon plasma phase in the region µ/T 1.…”

Section: Introductionmentioning

confidence: 99%

Analytic continuation from imaginary to real chemical potential in two-color QCD

Cea¹,

Cosmai²,

D’Elia³

et al. 2007

J. High Energy Phys.

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The method of analytic continuation from imaginary to real chemical potential is one of the most powerful tools to circumvent the sign problem in lattice QCD. Here we test this method in a theory, two-color QCD, which is free from the sign problem. We find that the method gives reliable results, within appropriate ranges of the chemical potential, and that a considerable improvement can be achieved if suitable functions are used to interpolate data with imaginary chemical potential.

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The partition function of Z(2) and Z(8) lattice gauge theory in four dimensions, a novel approach to simulations of lattice systems

Cited by 49 publications

References 10 publications

Thermostatistics of extensive and non-extensive systems using generalized entropies

Thermostatistics of extensive and non-extensive systems using generalized entropies

Accurate estimate ofνfor the three-dimensional Ising model from a numerical measurement of its partition function

Analytic continuation from imaginary to real chemical potential in two-color QCD

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