Wolff-type embedding algorithms for general nonlinear σ-models

Caracciolo, Sergio; Edwards, Robert G.; Pelissetto, Andrea; Sokal, Alan D.

doi:10.1016/0550-3213(93)90044-p

Cited by 78 publications

(79 citation statements)

References 81 publications

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“…ξ 2 does not equal ξ = 1/ma but scales as ξ in the β → ∞ limit [44,46,47], and in infinite volume, the ratio ξ/ξ 2 is very close to 1; it is 1.000826(1) [48]. The advantage of using ξ 2 is that one does not need to fit any correlators this way.…”

Section: Scale Settingmentioning

confidence: 99%

Comparison of algorithms for solving the sign problem in the O(3) model in 1+1 dimensions at finite chemical potential

et al. 2017

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We study three possible ways to circumvent the sign problem in the O(3) nonlinear sigma model in 1+1 dimensions. We compare the results of the worm algorithm to complex Langevin and multiparameter reweighting. Using the worm algorithm, the thermodynamics of the model is investigated, and continuum results are shown for the pressure at different µ/T values in the range 0 − 4. By performing T = 0 simulations using the worm algorithm, the Silver Blaze phenomenon is reproduced. Regarding the complex Langevin, we test various implementations of discretizing the complex Langevin equation. We found that the exponentialized Euler discretization of the Langevin equation gives wrong results for the action and the density at low T /m. By performing a continuum extrapolation, we found that this discrepancy does not disappear and depends slightly on temperature. The discretization with spherical coordinates performs similarly at low µ/T but breaks down also at some higher temperatures at high µ/T . However, a third discretization that uses a constraining force to achieve the φ 2 = 1 condition gives correct results for the action but wrong results for the density at low µ/T .

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Section: Scale Settingmentioning

confidence: 99%

Comparison of algorithms for solving the sign problem in the O(3) model in 1+1 dimensions at finite chemical potential

et al. 2017

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“…We have also measured the second momentum correlation length, which is expected to have the same scaling behavior at the critical point as the exponential (physical) one, but it is much easier to measure [10] …”

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confidence: 99%

New universality class in three dimensions?: the antiferromagnetic RP2 model

et al. 1996

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We present the results of a Monte Carlo simulation of the RP 2 model in three dimensions with negative coupling. We observe a second order phase transition between the disordered phase and an antiferromagnetic, unfrustrated, ordered one. We measure, with a Finite Size Scaling analysis, the thermal exponent, obtaining ν = 0.784(8). We have found two magnetic-type relevant operators whose related η exponents are 0.038(2) and 1.338(8) respectively.

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“…Again, in contrast to O(N ) models which can be studied with the efficient Wolff cluster algorithm [8], no efficient cluster algorithm is available for CP (N − 1) models [9]. There is even a no-go theorem that forbids the construction of an efficient Wolff-type embedding algorithm for these models [10]. Still, at θ = 0 a rather efficient multigrid algorithm was developed in [11].…”

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confidence: 99%

“…In order to demonstrate that we approach the continuum limit of an asymptotically free field theory we have determined the correlation length ξ(n) (defined using the second moment method [10]) in the CP (2) case as a function of n = L ′ /a which controls the coupling 1/g 2 = L ′ ρ s /c. One obtains ξ(2) = 4.82(4)a, ξ(4) = 17.6(2)a, and ξ(6) = 61(2)a, which indeed shows the exponential increase of the correlation length characteristic for an asymptotically free theory.…”

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Study ofCP(N−1)θ-Vacua by Cluster Simulation of
Beard
¹
,
Riederer
²
,
Wiese
³

2005
Phys. Rev. Lett.
50
3
50
0
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D-theory provides an alternative lattice regularization of the (1 + 1)-d CP (N − 1) quantum field theory. In this formulation the continuous classical CP (N − 1) fields emerge from the dimensional reduction of discrete SU (N ) quantum spins. In analogy to Haldane's conjecture, ladders consisting of an even number of transversely coupled spin chains lead to a CP (N − 1) model with vacuum angle θ = 0, while an odd number of chains yields θ = π. In contrast to Wilson's formulation of lattice field theory, in D-theory no sign problem arises at θ = π, and an efficient cluster algorithm is used to investigate the θ-vacuum effects. At θ = π there is a first order phase transition with spontaneous breaking of charge conjugation symmetry for CP (N − 1) models with N > 2.PACS numbers: 05.50.+q,75.10.Jm CP (N − 1) models are interesting (1 + 1)-d quantum field theories [1] which share a number of important features with (3 + 1)-d QCD. These include asymptotic freedom, the dynamical generation of a mass-gap, and an instanton topological charge leading to non-trivial θ-vacuum effects. Despite the fact that Nature has chosen θ = 0, it is an interesting challenge to understand the structure of θ-vacua, which in QCD may, for example, lead to 't Hooft's oblique confinement phases [2]. It has been conjectured that CP (N − 1) models have a phase transition at θ = π. In the CP (1) = O(3) case this phase transition is known to be second order with a vanishing mass-gap [3][4][5]. For N > 2, on the other hand, the transition is conjectured to be first order [6], which is consistent with large N analytic results [7]. For finite N , however, the investigation of these nonperturbative problems is highly nontrivial. In particular, in contrast to O(N ) models and other (1 + 1)-d quantum field theories, CP (N − 1) models with N > 2 cannot be solved with the Bethe ansatz. This is because in the CP (N − 1) case an infinite set of classical symmetries is anomalous and can thus not be used to solve the quantum theory analytically.The numerical investigation of lattice CP (N −1) models is also far from being straightforward, even at θ = 0. Again, in contrast to O(N ) models which can be studied with the efficient Wolff cluster algorithm [8], no efficient cluster algorithm is available for CP (N − 1) models [9]. There is even a no-go theorem that forbids the construction of an efficient Wolff-type embedding algorithm for these models [10]. Still, at θ = 0 a rather efficient multigrid algorithm was developed in [11]. However, at θ = π the situation is much worse due to a very severe sign problem: the contributions from odd topological charge sectors almost completely cancel those from even charge sectors. This makes it exponentially hard to access large lattices which is necessary for reaching reliable conclusions about the phase structure. For this reason, previous numerical studies of θ-vacua were limited to mod- erate volumes [12,13] or rely on additional assumptions [14]. In the CP (1) = O(3) case, a Wolff-type meroncluster algorithm allows...

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Wolff-type embedding algorithms for general nonlinear σ-models

Cited by 78 publications

References 81 publications

Comparison of algorithms for solving the sign problem in the O(3) model in 1+1 dimensions at finite chemical potential

Comparison of algorithms for solving the sign problem in the O(3) model in 1+1 dimensions at finite chemical potential

New universality class in three dimensions?: the antiferromagnetic RP2 model

Study ofCP(N−1)θ-Vacua by Cluster Simulation of
Beard
¹
,
Riederer
²
,
Wiese
³

2005
Phys. Rev. Lett.
50
3
50
0
View full text Add to dashboard Cite

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Wolff-type embedding algorithms for general nonlinear σ-models

Cited by 78 publications

References 81 publications

Comparison of algorithms for solving the sign problem in the O(3) model in 1+1 dimensions at finite chemical potential

Comparison of algorithms for solving the sign problem in the O(3) model in 1+1 dimensions at finite chemical potential

New universality class in three dimensions?: the antiferromagnetic RP2 model

Study ofCP(N−1)θ-Vacua by Cluster Simulation ofBeard1, Riederer2, Wiese3 2005Phys. Rev. Lett.503500View full textAdd to dashboardCite

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Study ofCP(N−1)θ-Vacua by Cluster Simulation of
Beard
¹
,
Riederer
²
,
Wiese
³

2005
Phys. Rev. Lett.
50
3
50
0
View full text Add to dashboard Cite