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Table Of Integrals Series And Products

H E Stanley Oxford: University Press 1971 pp xx + 308 price ?5 In the past fifteen years or so there has been a lot of experimental and theoretical work on the nature of critical phenomena in the neighbourhood of second order phase transitions. It has not been easy to get a good overall view of this work without digging through the rather complex original literature, although there are some good review articles covering particular aspects of the work.

https://iopscience.iop.org/article/10.1088/0031-9112/23/4/018/pdf

Introduction to Phase Transitions and Critical Phenomena

(2 + 1) [XTpXpH12, CTH11]. + [Zuc11b]. 0 [Fed17]. 1 [BELP15, CAS11, Cor16, Fed17, GDL10, GBL16, Hau16, JV19, KT12, KM19c, Li19, MN14b, Nak17, Pal11, Pan14, RT14, RBS16b, RY12, SS18c, Sug10, dOP18]. 1 + 1 [Sak18, CP15b]. 1/2 [MD10]. 1/f [FDR12]. 1/n [Per17]. 1/|x− y| [MSV10, MSV13]. 13 [DFL17]. 1 ≤ p ≤ ∞ [Dud13]. 1/f [HPF15]. 2 [AB19, ADS19, BF12, BNT13, DSS15, EKD12, Her13, Ily12, Lan10, Li12, Li19, LZ11, Ny13, Ost16, PSS16, ST14, Sch13b, TJ15, WPB15, dWL10]. 2 + 1 [dWL14]. 2.5 [BC15a]. 2R [WLEC17]. 2× 2 [CLTT13]. 3 [BCF19, BLS17, ESPP14, Kar18, SH16, SWKS14, dCCS19]. 3/2 [DK10]. 38 [Cam13]. 4 [BBS14, Zha14]. 4× 4 [LN19a]. 5/2 [DK10, EKD12]. 6 [EC11]. 8 [Zha14]. 90◦ [YM11]. 3 [Afz12]. 1−x [EFO11]. 13 [CDCL18]. 2 [ML15, QR13, ST11c]. 4 [HBB10]. 6 [BCL10a, BCL10b, EFO11]. x [EFO11].

/pdf/a-complete-bibliography-of-the-journal-of-statistical-2bw4esmzh4.pdf

A Complete Bibliography of the Journal of Statistical Physics: 2000{2009

By using conformal-field theory, we classify the possible irrelevant operators for the Ising model with nearest-neighbour interactions on the square and triangular lattices. We analyse the existing results for the free energy and its derivatives and for the correlation length, showing that they are in agreement with the conformal-field theory predictions. Moreover, these results imply that the nonlinear scaling field of the T operator, where T is the energy–momentum tensor, vanishes at the critical point. Several other peculiar cancellations are explained in terms of a number of general conjectures. We show that all existing results on the square and triangular lattices are consistent with the assumption that only nonzero-spin operators are present.

Irrelevant operators in the two-dimensional Ising model

At the critical point in two dimensions, the number of percolation clusters of enclosed area greater than A is proportional to A−1, with a proportionality constant C that is universal. We show theoretically (based upon Coulomb gas methods), and verify numerically to high precision, that $C = 1/(8\sqrt 3 \pi = 0.022972037 \ldots )$. We also derive, and verify to varying precision, the corresponding constant for Ising spin clusters, and for Fortuin–Kasteleyn clusters of the Q = 2, 3 and 4-state Potts models.

/pdf/exact-results-for-the-universal-area-distribution-of-7ct7jkf44m.pdf

Exact Results for the Universal Area Distribution of Clusters in Percolation, Ising, and Potts Models

Using an iteration technique, we obtain exact expressions for the free energy and the magnetization of an Ising model on a two-layer Bethe lattice with intralayer coupling constants J1 and J2 for the first and the second layer, respectively, and interlayer coupling constant J3 between the two layers; the Ising spins also couple with external magnetic fields, which are different in the two layers. We obtain exact phase diagrams for the system and find that when /J3/-->0, DeltaT(c) identical with[T(c)(J3)-T(c)(0)]/T(c)(0) approximately [J3]/J(1)/(1/psi), where T(c)(J3) is the phase-transition temperature for the system with interlayer coupling constant J3 and the shift exponent psi is 1 for J(1)=J(2) and is 0.5 for J1 not equal to J2. Such results are consistent with predictions of a scaling theory. We also derive equations for DeltaT(c) when /J3/ approaches infinity.

Exact phase diagrams for an Ising model on a two-layer Bethe lattice

Let f(N) and xi(-1)(N) represent, respectively, the free energy per spin and the inverse spin-spin correlation length of the critical Ising model on a N x infinity lattice, with f(N)-->f(infinity) as N-->infinity. We obtain analytic expressions for a(k) and b(k) in the expansions N( f(N)-f(infinity)) = SUM (k = 1)(infinity)a(k)/N(2k-1) and xi(-1)(N) = SUM (k = 1)(infinity)b(k)/N(2k-1) for square, honeycomb, and plane-triangular lattices, and find that b(k)/a(k) = (2(2k)-1)/(2(2k-1)-1) for all of these lattices, i.e., the amplitude ratio b(k)/a(k) is universal. We also obtain similar results for a critical quantum spin chain and find that such results could be understood from a perturbated conformal field theory.

Exact universal amplitude ratios for two-dimensional Ising models and a quantum spin chain.

Using the bond-propagation algorithm, we study the Ising model on a rectangle of size $M\ifmmode\times\else\texttimes\fi{}N$ with free boundaries. For five aspect ratios, $\ensuremath{\rho}=M/N=1$, 2, 4, 8, and 16, the critical free energy, internal energy and specific heat are calculated. The largest size reached is $M\ifmmode\times\else\texttimes\fi{}N=64\ifmmode\times\else\texttimes\fi{}{10}^{6}$. The accuracy of the free energy reaches ${10}^{\ensuremath{-}26}$. Based on these accurate data, we determine exact expansions of the critical free energy, internal energy, and specific heat. With these expansions, we extract the bulk, surface, and corner parts of the free energy, internal energy, and specific heat. The fitted bulk free energy density is given by ${f}_{\ensuremath{\infty}}=0.92969539834161021499(1)$, compared with Onsager's exact result ${f}_{\ensuremath{\infty}}=0.929695398341610214985\ensuremath{\cdots}.$ We confirm the conformal field theory (CFT) prediction of the corner free energy, in which the central charge of the Ising model is found to be $c=0.5\ifmmode\pm\else\textpm\fi{}1\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}10}$, compared with the CFT result $c=0.5$. We find that not only the corner free energy but also the corner internal energy and specific heat are geometry independent, i.e., independent of aspect ratio. The implication of this finding for finite-size scaling is discussed. In the second-order correction of the free energy, we confirm the geometry dependence predicted by CFT and determine a geometry-independent constant beyond CFT. High-order corrections are also obtained.

Finite-size behavior of the critical Ising model on a rectangle with free boundaries.

We establish an intriguing connection between geometry and thermodynamics in the critical q-state Potts model on two-dimensional lattices, using the q-state bond-correlated percolation model ~QBCPM! representation. We find that the number of clusters^Nc& of the QBCPM has an energylike singularity for qfi1, which is reached and supported by exact results, numerical simulation, and scaling arguments. We also establish that the finite-size correction to the number of bonds,^Nb&, has no constant term and explains the divergence of related quantities as q!4, the multicritical point. Similar analyses are applicable to a variety of other systems. @S1063-651X~99!09612-9#

Geometry, thermodynamics, and finite-size corrections in the critical Potts model.

Using a bond-propagation algorithm, we study the finite-size behavior of the critical two-dimensional Ising model on a finite triangular lattice with free boundaries in five shapes: triangular, rhomboid, trapezoid, hexagonal, and rectangular. The critical free energy, internal energy, and specific heat are calculated. The accuracy of the free energy reaches ${10}^{\ensuremath{-}26}$. Based on accurate data on several finite systems with linear size up to $N=2000$, we extract the bulk, surface, and corner parts of the free energy, internal energy, and specific heat accurately. We confirm the conformal field theory prediction that the corner free energy is universal and find logarithmic corrections in higher-order terms in the critical free energy for the rhomboid, trapezoid, and hexagonal systems, which are absent for the triangular and rectangular systems. The logarithmic edge corrections due to edges parallel or perpendicular to the bond directions in the internal energy are found to be identical, while the logarithmic edge corrections due to corresponding edges in the specific heat are different. The corner internal energy and corner specific heat for angles $\ensuremath{\pi}/3$, $\ensuremath{\pi}/2$, and $2\ensuremath{\pi}/3$ are obtained, as well as higher-order corrections. Comparing with the corner internal energy and corner specific heat we previously found on a rectangle of the square lattice [Phys. Rev. E 86, 041149 (2012)], we conclude that the corner internal energy and corner specific heat for the rectangular shape are not universal.

Nickolay Izmailian

Papers

Exact phase diagrams for an Ising model on a two-layer Bethe lattice

Exact universal amplitude ratios for two-dimensional Ising models and a quantum spin chain.

Finite-size behavior of the critical Ising model on a rectangle with free boundaries.

Geometry, thermodynamics, and finite-size corrections in the critical Potts model.

Shape-dependent finite-size effect of the critical two-dimensional Ising model on a triangular lattice.