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Young AP. Violations of Hyperscaling in Finite-Size Scaling above the Upper Critical Dimension. ENTROPY (BASEL, SWITZERLAND) 2024; 26:509. [PMID: 38920518 PMCID: PMC11202775 DOI: 10.3390/e26060509] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/04/2024] [Revised: 06/03/2024] [Accepted: 06/03/2024] [Indexed: 06/27/2024]
Abstract
We consider how finite-size scaling (FSS) is modified above the upper critical dimension, du=4, due to hyperscaling violations, which in turn arise from a dangerous irrelevant variable. In addition to the commonly studied case of periodic boundary conditions, we also consider new effects that arise with free boundary conditions. Some numerical results are presented in addition to theoretical arguments.
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Affiliation(s)
- A Peter Young
- Physics Department, University of California, Santa Cruz, CA 95064, USA
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Fang S, Zhou Z, Deng Y. Geometric scaling behaviors of the Fortuin-Kasteleyn Ising model in high dimensions. Phys Rev E 2023; 107:044103. [PMID: 37198783 DOI: 10.1103/physreve.107.044103] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/19/2022] [Accepted: 03/07/2023] [Indexed: 05/19/2023]
Abstract
Recently, we argued [Chin. Phys. Lett. 39, 080502 (2022)0256-307X10.1088/0256-307X/39/8/080502] that the Ising model simultaneously exhibits two upper critical dimensions (d_{c}=4,d_{p}=6) in the Fortuin-Kasteleyn (FK) random-cluster representation. In this paper, we perform a systematic study of the FK Ising model on hypercubic lattices with spatial dimensions d from 5 to 7, and on the complete graph. We provide a detailed data analysis of the critical behaviors of a variety of quantities at and near the critical points. Our results clearly show that many quantities exhibit distinct critical phenomena for 4<d<6 and d≥6, and thus strongly support the argument that 6 is also an upper critical dimension. Moreover, for each studied dimension, we observe the existence of two configuration sectors, two lengthscales, as well as two scaling windows, and thus two sets of critical exponents are needed to describe these behaviors. Our finding enriches the understanding of the critical phenomena in the Ising model.
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Affiliation(s)
- Sheng Fang
- MinJiang Collaborative Center for Theoretical Physics, College of Physics and Electronic Information Engineering, Minjiang University, Fuzhou 350108, China
- Hefei National Research Center for Physical Sciences at the Microscales, University of Science and Technology of China, Hefei 230026, China
| | - Zongzheng Zhou
- ARC Centre of Excellence for Mathematical and Statistical Frontiers (ACEMS), School of Mathematics, Monash University, Clayton, Victoria 3800, Australia
| | - Youjin Deng
- MinJiang Collaborative Center for Theoretical Physics, College of Physics and Electronic Information Engineering, Minjiang University, Fuzhou 350108, China
- Hefei National Research Center for Physical Sciences at the Microscales, University of Science and Technology of China, Hefei 230026, China
- Shanghai Research Center for Quantum Sciences, Shanghai 201315, China
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Fang S, Deng Y, Zhou Z. Logarithmic finite-size scaling of the self-avoiding walk at four dimensions. Phys Rev E 2021; 104:064108. [PMID: 35030932 DOI: 10.1103/physreve.104.064108] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/08/2021] [Accepted: 11/19/2021] [Indexed: 06/14/2023]
Abstract
The n-vector spin model, which includes the self-avoiding walk (SAW) as a special case for the n→0 limit, has an upper critical dimensionality at four spatial dimensions (4D). We simulate the SAW on 4D hypercubic lattices with periodic boundary conditions by an irreversible Berretti-Sokal algorithm up to linear size L=768. From an unwrapped end-to-end distance, we obtain the critical fugacity as z_{c}=0.147622380(2), improving over the existing result z_{c}=0.1476223(1) by 50 times. Such a precisely estimated critical point enables us to perform a systematic study of the finite-size scaling of 4D SAW for various quantities. Our data indicate that near z_{c}, the scaling behavior of the free energy simultaneously contains a scaling term from the Gaussian fixed point and the other accounting for multiplicative logarithmic corrections. In particular, it is clearly observed that the critical magnetic susceptibility and the specific heat logarithmically diverge as χ∼L^{2}(lnL)^{2y[over ̂]_{h}} and C∼(lnL)^{2y[over ̂]_{t}}, and the logarithmic exponents are determined as y[over ̂]_{h}=0.251(2) and y[over ̂]_{t}=0.25(3), in excellent agreement with the field theoretical prediction y[over ̂]_{h}=y[over ̂]_{t}=1/4. Our results provide a strong support for the recently conjectured finite-size scaling form for the O(n) universality classes at 4D.
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Affiliation(s)
- Sheng Fang
- MinJiang Collaborative Center for Theoretical Physics, College of Physics and Electronic Information Engineering, Minjiang University, Fuzhou 350108, China
- Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Youjin Deng
- MinJiang Collaborative Center for Theoretical Physics, College of Physics and Electronic Information Engineering, Minjiang University, Fuzhou 350108, China
- Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Zongzheng Zhou
- ARC Centre of Excellence for Mathematical and Statistical Frontiers (ACEMS), School of Mathematics, Monash University, Clayton, Victoria 3800, Australia
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Hu M, Deng Y, Lv JP. Extraordinary-Log Surface Phase Transition in the Three-Dimensional XY Model. PHYSICAL REVIEW LETTERS 2021; 127:120603. [PMID: 34597095 DOI: 10.1103/physrevlett.127.120603] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/11/2021] [Revised: 05/23/2021] [Accepted: 08/11/2021] [Indexed: 06/13/2023]
Abstract
Universality is a pillar of modern critical phenomena. The standard scenario is that the two-point correlation algebraically decreases with the distance r as g(r)∼r^{2-d-η}, with d the spatial dimension and η the anomalous dimension. Very recently, a logarithmic universality was proposed to describe the extraordinary surface transition of the O(N) system. In this logarithmic universality, g(r) decays in a power of logarithmic distance as g(r)∼(lnr)^{-η[over ^]}, dramatically different from the standard scenario. We explore the three-dimensional XY model by Monte Carlo simulations, and provide strong evidence for the emergence of logarithmic universality. Moreover, we propose that the finite-size scaling of g(r,L) has a two-distance behavior: simultaneously containing a large-distance plateau whose height decays logarithmically with L as g(L)∼(lnL)^{-η[over ^]^{'}} as well as the r-dependent term g(r)∼(lnr)^{-η[over ^]}, with η[over ^]^{'}≈η[over ^]-1. The critical exponent η[over ^]^{'}, characterizing the height of the plateau, obeys the scaling relation η[over ^]^{'}=(N-1)/(2πα) with the RG parameter α of helicity modulus. Our picture can also explain the recent numerical results of a Heisenberg system. The advances on logarithmic universality significantly expand our understanding of critical universality.
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Affiliation(s)
- Minghui Hu
- Department of Physics and Anhui Key Laboratory of Optoelectric Materials Science and Technology, Key Laboratory of Functional Molecular Solids, Ministry of Education, Anhui Normal University, Wuhu, Anhui 241000, China
| | - Youjin Deng
- Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
- MinJiang Collaborative Center for Theoretical Physics, College of Physics and Electronic Information Engineering, Minjiang University, Fuzhou 350108, China
| | - Jian-Ping Lv
- Department of Physics and Anhui Key Laboratory of Optoelectric Materials Science and Technology, Key Laboratory of Functional Molecular Solids, Ministry of Education, Anhui Normal University, Wuhu, Anhui 241000, China
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Fang S, Zhou Z, Deng Y. Percolation effects in the Fortuin-Kasteleyn Ising model on the complete graph. Phys Rev E 2021; 103:012102. [PMID: 33601530 DOI: 10.1103/physreve.103.012102] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/17/2020] [Accepted: 11/16/2020] [Indexed: 11/07/2022]
Abstract
The Fortuin-Kasteleyn (FK) random-cluster model, which can be exactly mapped from the q-state Potts spin model, is a correlated bond percolation model. By extensive Monte Carlo simulations, we study the FK bond representation of the critical Ising model (q=2) on a finite complete graph, i.e., the mean-field Ising model. We provide strong numerical evidence that the configuration space for q=2 contains an asymptotically vanishing sector in which quantities exhibit the same finite-size scaling as in the critical uncorrelated bond percolation (q=1) on the complete graph. Moreover, we observe that, in the full configuration space, the power-law behavior of the cluster-size distribution for the FK Ising clusters except the largest one is governed by a Fisher exponent taking the value for q=1 instead of q=2. This demonstrates the percolation effects in the FK Ising model on the complete graph.
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Affiliation(s)
- Sheng Fang
- Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Zongzheng Zhou
- ARC Centre of Excellence for Mathematical and Statistical Frontiers (ACEMS), School of Mathematics, Monash University, Clayton, Victoria 3800, Australia
| | - Youjin Deng
- Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China.,MinJiang Collaborative Center for Theoretical Physics, College of Physics and Electronic Information Engineering, Minjiang University, Fuzhou 350108, China
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Lv JP, Xu W, Sun Y, Chen K, Deng Y. Finite-size scaling of O( n) systems at the upper critical dimensionality. Natl Sci Rev 2020; 8:nwaa212. [PMID: 34691596 PMCID: PMC8288422 DOI: 10.1093/nsr/nwaa212] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/25/2020] [Revised: 06/29/2020] [Accepted: 06/29/2020] [Indexed: 11/14/2022] Open
Abstract
Logarithmic finite-size scaling of the O(n) universality class at the upper critical dimensionality (d c = 4) has a fundamental role in statistical and condensed-matter physics and important applications in various experimental systems. Here, we address this long-standing problem in the context of the n-vector model (n = 1, 2, 3) on periodic four-dimensional hypercubic lattices. We establish an explicit scaling form for the free-energy density, which simultaneously consists of a scaling term for the Gaussian fixed point and another term with multiplicative logarithmic corrections. In particular, we conjecture that the critical two-point correlation g(r, L), with L the linear size, exhibits a two-length behavior: follows [Formula: see text] governed by the Gaussian fixed point at shorter distances and enters a plateau at larger distances whose height decays as [Formula: see text] with [Formula: see text] a logarithmic correction exponent. Using extensive Monte Carlo simulations, we provide complementary evidence for the predictions through the finite-size scaling of observables, including the two-point correlation, the magnetic fluctuations at zero and nonzero Fourier modes and the Binder cumulant. Our work sheds light on the formulation of logarithmic finite-size scaling and has practical applications in experimental systems.
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Affiliation(s)
- Jian-Ping Lv
- Department of Physics, Anhui Key Laboratory of Optoelectric Materials Science and Technology, Key Laboratory of Functional Molecular Solids, Ministry of Education, Anhui Normal University, Wuhu 241000, China
| | - Wanwan Xu
- Department of Physics, Anhui Key Laboratory of Optoelectric Materials Science and Technology, Key Laboratory of Functional Molecular Solids, Ministry of Education, Anhui Normal University, Wuhu 241000, China
| | - Yanan Sun
- Department of Physics, Anhui Key Laboratory of Optoelectric Materials Science and Technology, Key Laboratory of Functional Molecular Solids, Ministry of Education, Anhui Normal University, Wuhu 241000, China
| | - Kun Chen
- Department of Physics and Astronomy, Rutgers, The State University of New Jersey, Piscataway, NJ 08854-8019, USA
| | - Youjin Deng
- National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China
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