Finite-size scaling and corrections in the Ising model with Brascamp-Kunz boundary conditions

Janke, Wolfhard; Kenna, Ralph

doi:10.1103/physrevb.65.064110

Cited by 62 publications

(74 citation statements)

References 26 publications

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“…Higher order terms are straightforward to determine [15]. From the leading term in (8) and from (2), the correlation length critical exponent is indeed ν = 1.…”

Section: Fisher Zeroesmentioning

confidence: 99%

“…These can be performed exactly (we refer the reader to [15] for details) and one finds the following results. Specific Heat at the Critical Point: At the critical temperature the specific heat is, from (5),…”

Section: Specific Heatmentioning

confidence: 99%

“…The coefficients c k can easily be determined exactly and those up to c 3 are explicitly given in [15]. So for the critical specific heat on…”

Section: Specific Heatmentioning

confidence: 99%

“…a Brascamp-Kunz lattice, apart from a trivial ln M/M term (which could be removed by a redefinition of M [15]), the FSS is qualitatively the same as (but quantitatively different to) that of the torus topology in (4). Specific Heat near the Critical Point: The pseudocritical point of the specific heat, z pseudo M,2N , can be determined as the point where the derivative of C M,2N (z) vanishes.…”

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confidence: 99%

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Exact finite-size scaling with corrections in the two-dimensional Ising model with special boundary conditions

Janke

Kenna

2002

Nuclear Physics B - Proceedings Supplements

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The two-dimensional Ising model with Brascamp-Kunz boundary conditions has a partition function more amenable to analysis than its counterpart on a torus. This fact is exploited to exactly determine the full finite-size scaling behaviour of the Fisher zeroes of the model. Moreover, exact results are also determined for the scaling of the specific heat at criticality, for the specific-heat peak and for the pseudocritical points. All corrections to scaling are found to be analytic and the shift exponent λ does not coincide with the inverse of the correlation length exponent 1/ν.

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“…Higher order terms are straightforward to determine [15]. From the leading term in (8) and from (2), the correlation length critical exponent is indeed ν = 1.…”

Section: Fisher Zeroesmentioning

confidence: 99%

Section: Specific Heatmentioning

confidence: 99%

“…The coefficients c k can easily be determined exactly and those up to c 3 are explicitly given in [15]. So for the critical specific heat on…”

Section: Specific Heatmentioning

confidence: 99%

mentioning

confidence: 99%

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Exact finite-size scaling with corrections in the two-dimensional Ising model with special boundary conditions

Janke

Kenna

2002

Nuclear Physics B - Proceedings Supplements

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“…Periodic mixed spin cell, [4]. also known exactly for comparable models such as the 2D Ising model with Brascamp-Kunz boundary value conditions [5]. We compare the FSS of these indicators to evaluate the critical exponents for the VBS transitions, comparing with nonlinear σ-model predictions.…”

Section: Vbs Phase Transitionsmentioning

confidence: 99%

Finite Size Scaling, Fisher Zeroes and Super Yang-Mills

Crompton

Janke

et al. 2005

Nuclear Physics B - Proceedings Supplements

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We investigate critical slowing down in the local updating continuous-time Quantum Monte Carlo method by relating the finite size scaling of Fisher Zeroes to the dynamically generated gap, through the scaling of their respective critical exponents. As we comment, the nonlinear sigma model representation derived through the hamiltonian of our lattice spin model can also be used to give a effective treatment of planar anomalous dimensions in N =4 SYM. We present scaling arguments from our FSS analysis to discuss quantum corrections and recent 2-loop results, and further comment on the prospects of extending this approach for calculating higher twist parton distributions. VBS PHASE TRANSITIONSWe investigate the critical behavior of ValenceBond-Solid (VBS) states in quantum spin chains by means of Quantum Monte Carlo (QMC) simulation using the continuous-time loop cluster algorithm [1]. Following Haldane's conjecture a variety of exotic magnetic phenomena can be attributed to the formation of ground states with an energy-gap in integer spin systems at low temperatures. These gapped states are predicted to have transitions with a massless excitation between phases, and can be effectively expressed in terms of a VBS state (spin-singlet) picture. The Lieb-Schultz-Mattis argument to this conjecture has led to the recent introduction of a string exact order parameter to characterise these VBS transitions. We investigate the Finite Size Scaling (FSS) properties of a generalised twist order parameter for a periodic mixed-spin chain [2] with a unit cell of the form (1, 1, 1 2 , 1 2 ). Also determining the FSS properties of the complex-temperature (Fisher) zeroes of the partition function [3], evaluated in a new scheme through knowledge of the QMC transfer matrix. This enables us to separate pseudocritical and critical point scaling behavior relating to the correlation length exponent and also to locate the VBS state transition points through an independent and expeditious means. Leading corrections to the FSS of the zeroes are sa sb sb sa Jab Jbb Jab Jaa Figure 1. Periodic mixed spin cell, [4]. also known exactly for comparable models such as the 2D Ising model with Brascamp-Kunz boundary value conditions [5]. We compare the FSS of these indicators to evaluate the critical exponents for the VBS transitions, comparing with nonlinear σ-model predictions.The Hamiltonian for the AHFM two-spin chain with spins S a and S b of period-4 is given as,with J aa =J bb =1 and the coupling anisotropy α = J ab /J aa .We anticipate competing low-temperature VBS dimer gap states from nonlinear sigma model treatment [4], separated via a quantum phase 1

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Accurate Expansions of Internal Energy and Specific Heat of Critical Two-Dimensional Ising Model with Free Boundaries

Zheng

Измаилян

et al. 2014

J Stat Phys

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The bond-propagation (BP) algorithm for the specific heat of the two dimensional Ising model is developed and that for the internal energy is completed. Using these algorithms, we study the critical internal energy and specific heat of the model on the square lattice and triangular lattice with free boundaries. Comparing with previous works [X.-T. Wu et al Phys. Rev. E 86, 041149 (2012) and Phys. Rev. E 87, 022124 (2013)], we reach much higher accuracy (10 −26 ) of the internal energy and specific heat,compared to the accuracy 10 −11 of the internal energy and 10 −9 of the specific heat reached in the previous works. This leads to much more accurate estimations of the edge and corner terms. The exact values of some edge and corner terms are therefore conjectured. The accurate forms of finite-size scaling for the internal energy and specific heat are determined for the rectangle-shaped square lattice with various aspect ratios and various shaped triangular lattice. For the rectangle-shaped square and triangular lattices and the triangle-shaped triangular lattice, there is no logarithmic correction terms of order higher than 1/S, with S the area of the system. For the triangular lattice in rhombus, trapezoid and hexagonal shapes, there exist logarithmic correction terms of order higher than 1/S for the internal energy, and logarithmic correction terms of all orders for the specific heat.

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Finite-size scaling and corrections in the Ising model with Brascamp-Kunz boundary conditions

Cited by 62 publications

References 26 publications

Exact finite-size scaling with corrections in the two-dimensional Ising model with special boundary conditions

Exact finite-size scaling with corrections in the two-dimensional Ising model with special boundary conditions

Finite Size Scaling, Fisher Zeroes and Super Yang-Mills

Accurate Expansions of Internal Energy and Specific Heat of Critical Two-Dimensional Ising Model with Free Boundaries

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