Universal critical wrapping probabilities in the canonical ensemble

Hu, Hao; Deng, Youjin

doi:10.1016/j.nuclphysb.2015.06.025

Cited by 9 publications

(4 citation statements)

References 42 publications

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“…The equilibrium SSB transition of the Potts model in the canonical ensemble, where inverse temperature β is the control parameter, has been extensively investigated (see Refs. [5][6][7][8][9][10][11][12] for some of the recent results). Driven by energy-entropy competitions, this transition is a discontinuous phenomenon when Q is sufficiently large, with the density ρ 1 of the dominant color jumps from 1/Q to a much larger value at the critical inverse temperature β c .…”

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confidence: 89%

Kinked Entropy and Discontinuous Microcanonical Spontaneous Symmetry Breaking

Zhou

2019

Phys. Rev. Lett.

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Spontaneous symmetry breaking (SSB) in statistical physics is a macroscopic collective phenomenon. For the paradigmatic Q-state Potts model it means a transition from the disordered color-symmetric phase to an ordered phase in which one color dominates. Existing mean field theories imply that SSB in the microcanonical statistical ensemble (with energy being the control parameter) should be a continuous process. Here we study microcanonical SSB on the random-graph Potts model, and discover that the entropy is a kinked function of energy. This kink leads to a discontinuous phase transition at certain energy density value, characterized by a jump in the density of the dominant color and a jump in the microcanonical temperature. This discontinuous SSB in random graphs is confirmed by microcanonical Monte Carlo simulations, and it is also observed in bond-diluted finite-size lattice systems.Spontaneous symmetry breaking (SSB) is a fundamental concept of physics and is tightly linked to the origin of mass in particle physics, the emergence of superconductivity in condensed-matter system, and the ferromagnetic phase transition in statistical mechanics, to name just a few eminent examples [1]. In statistical physics a theoretical paradigm for SSB is the Potts model, a simple twobody interaction graphical system in which each vertex has Q discrete color states [2][3][4]. The equilibrium SSB transition of the Potts model in the canonical ensemble, where inverse temperature β is the control parameter, has been extensively investigated (see for some of the recent results). Driven by energy-entropy competitions, this transition is a discontinuous phenomenon when Q is sufficiently large, with the density ρ 1 of the dominant color jumps from 1/Q to a much larger value at the critical inverse temperature β c . To compensate for the extensive loss of entropy, such a discontinuous transition is always accompanied by a discontinuous decrease of the system's energy density u [3, 4].When the system is isolated and cannot exchange energy with the environment (the microcanonincal ensemble [13-16]), it is generally believed that the SSB transition will occur gradually, with the dominant color density ρ 1 deviating from 1/Q continuously at certain critical energy density u mic . Indeed if the microscopic entropy density s(u) is a C 1 -continuous function of energy density u (i.e., both s(u) and its first derivative are continuous), there is no reason to expect a discontinuity of the order parameter ρ 1 . The C 1 -continuity of s(u) can be easily verified for the mean field Potts model on a complete graph [17]. For finite-dimensional lattices the phase separation mechanism (the nucleation and expansion of droplets [18-21]) will guarantee a C 1 -continuous entropy profile in the thermodynamic limit. For random graph systems one would naïvely expect u mic to be an inflection point of s(u) [22], which ensures C 1 -continuity.In this Letter we investigate the microcanonical Potts model on random graphs using the Bethe-Peierls mean field theo...

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confidence: 89%

Kinked Entropy and Discontinuous Microcanonical Spontaneous Symmetry Breaking

Zhou

2019

Phys. Rev. Lett.

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“…Arguin extended Pinson's work to the case of 2D RC models with 1 ≤ q ≤ 4 and derived the closed forms of wrapping probabilities in terms of Jacobi θ functions [10]. Finite-size corrections of wrapping probabilities for the RC model in the canonical ensemble (where the total number of the occupied bonds is fixed) were also studied recently [21]. In 3D, there also exist a few studies on the wrapping probabilities [13,22].…”

Section: Wrapping Probabilities and Critical Point In 3dmentioning

confidence: 99%

Geometric properties of the Fortuin-Kasteleyn representation of the Ising model

et al. 2019

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We present a Monte Carlo study of the Fortuin-Kasteleyn (FK) clusters of the Ising model on the square (2D) and simple-cubic (3D) lattices. The wrapping probability, a dimensionless quantity characterizing the topology of the FK clusters on a torus, is found to suffer from smaller finitesize corrections than the well-known Binder ratio, and yields a high-precision critical coupling as Kc(3D) = 0.221 654 631(8). We then study geometric properties of the FK clusters at criticality. It is demonstrated that the distribution of the critical largest-cluster size C1 follows a single-variable function as P (C1, L)dC1 =P (x)dx with x ≡ C1/L d F (L is the linear size), and that the fractal dimension dF is identical to the magnetic exponent. An interesting bimodal feature is observed in distributionP (x) in 3D, and attributed to the different approaching behaviors for K → Kc + 0 ± . For a critical FK configuration, the cluster number per site n(s, L) of size s is confirmed to obey the standard scaling form n(s, L) ∼ s −τñ (s/L d F ), with hyper-scaling relation τ = 1 + d/dF and the spatial dimension d. To further characterize the compactness of the FK clusters, we measure their graph distances and determine the shortest-path exponents as dmin(3D) = 1.259 4(2) and dmin(2D) = 1.094 0(3). Further, by excluding all the bridges from the occupied bonds, we obtain bridge-free configurations and determine the backbone exponents as dB(3D) = 2.167 3(15) and dB(2D) = 1.732 1(4). The estimates of the universal wrapping probabilities for the 3D Ising model and of the geometric critical exponents dmin and dB either improve over the existing results or have not been reported yet.

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“…For many statistical-mechanical systems, much insight can be gained by exploring geometric properties of the systems [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32]. For the Ising and Potts model, geometric clusters in the Fortuin-Kasteleyn bond representation have a percolation threshold coinciding with the thermodynamic phase transition, and exhibit rich fractal properties, some of which have no thermodynamic correspondence.…”

Section: Introductionmentioning

confidence: 99%

Percolation of the two-dimensional XY model in the flow representation

Wang,

Hou,

Huang

et al. 2020

Preprint

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We simulate the two-dimensional XY model in the flow representation by a worm-type algorithm, up to linear system size L = 4096, and study the geometric properties of the flow configurations. As the coupling strength K increases, we observe that the system undergoes a percolation transition Kperc from a disordered phase consisting of small clusters into an ordered phase containing a giant percolating cluster. Namely, in the low-temperature phase, there exhibits a long-ranged order regarding the flow connectivity, in contrast to the qusi-long-range order associated with spin properties. Near Kperc, the scaling behavior of geometric observables is well described by the standard finite-size scaling ansatz for a second-order phase transition. The estimated percolation threshold Kperc = 1.105 3(4) is close to but obviously smaller than the Berezinskii-Kosterlitz-Thouless (BKT) transition point KBKT = 1.119 3(10), which is determined from the magnetic susceptibility and the superfluid density. Various interesting questions arise from these unconventional observations, and their solutions would shed lights on a variety of classical and quantum systems of BKT phase transitions.

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Universal critical wrapping probabilities in the canonical ensemble

Cited by 9 publications

References 42 publications

Kinked Entropy and Discontinuous Microcanonical Spontaneous Symmetry Breaking

Kinked Entropy and Discontinuous Microcanonical Spontaneous Symmetry Breaking

Geometric properties of the Fortuin-Kasteleyn representation of the Ising model

Percolation of the two-dimensional XY model in the flow representation

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