We report an arithmetically certified enclosure of the zero-temperature entropy per site of the six-state square-lattice Potts antiferromagnet: ln W(6) lies in [1.435107300597, 1.435107684912]. Both endpoints follow from established free-strip ratio and even-cylinder inequalities evaluated with colour and spatial symmetry quotients. Complete integer matrices and positive integer vectors support the spectral certificates. All 4,790,948 entries of eight archived quotient matrices have been independently reconstructed from the colouring compatibility rule. Four matched width budgets separate finite-root certification uncertainty from the remaining thermodynamic gap. At the largest budget the finite-root contribution is below 1.168 x 10^-13, compared with an interval width of 3.84315 x 10^-7. The record contains the eight-page preprint, LaTeX source, complete computational materials and checksums. We acknowledge Gewu Intelligence Lab for providing the collaborative research environment in which this project was developed. W.J. acknowledges support from the Strategic Priority Research Program of the Chinese Academy of Sciences under Grant No. XDB1680102.
We report an arithmetically certified enclosure of the zero-temperature entropy per site of the six-state square-lattice Potts antiferromagnet: ln W(6) lies in [1.435107300597, 1.435107684912]. Both endpoints follow from established free-strip ratio and even-cylinder inequalities evaluated with colour and spatial symmetry quotients. Complete integer matrices and positive integer vectors support the spectral certificates. All 4,790,948 entries of eight archived quotient matrices have been independently reconstructed from the colouring compatibility rule. Four matched width budgets separate finite-root certification uncertainty from the remaining thermodynamic gap. At the largest budget the finite-root contribution is below 1.168 x 10^-13, compared with an interval width of 3.84315 x 10^-7. The record contains the eight-page preprint, LaTeX source, complete computational materials and checksums. We acknowledge Gewu Intelligence Lab for providing the collaborative research environment in which this project was developed. W.J. acknowledges support from the Strategic Priority Research Program of the Chinese Academy of Sciences under Grant No. XDB1680102.
We present exact calculations of the zero-temperature partition function (chromatic polynomial) and W(q), the exponent of the ground-state entropy, for the q-state Potts antiferromagnet with next-nearest-neighbor spin-spin couplings on square lattice strips, of width L(y)=3 and L(y)=4 vertices and arbitrarily great length Lx vertices, with both free and periodic boundary conditions. The resultant values of W for a range of physical q values are compared with each other and with the values for the full two-dimensional lattice. These results give insight into the effect of such nonnearest-neighbor couplings on the ground-state entropy. We show that the q=2 (Ising) and q=4 Potts antiferromagnets have zero-temperature critical points on the Lx-->infinity limits of the strips that we study. With the generalization of q from Z+ to C, we determine the analytic structure of W(q) in the q plane for the various cases.
Proving a quantum-response inequality at every temperature requires control beyond a finite temperature grid or a truncated series. We give a sufficient finite certificate for strict growth of the pairing-source curvature 𝜅2(𝛽)=𝜕2ℎ log Tr 𝑒−𝛽(𝐻−ℎ𝑀)|ℎ=0, which equals 𝛽 times the physical source susceptibility. For finite Hermitian systems with source reversal, a positive spectral shift produces nonnegative two-insertion temperature coefficients. Ordering their ratios to the partition coefficients implies 𝜅′2(𝛽)> 0 for every 𝛽 > 0. A simple ground state coupled to the source forces these ratios toward a line with positive slope; an explicit spectral error bound proves every increment beyond a computable cutoff. The complete certificate therefore requires a finite coefficient prefix and finitely many spectral bounds. For the half-filled uniform-attraction Hubbard dimer, two ordered temperature-series quotients and the atomic formula prove curvature growth over all couplings, with connected- cluster consequences near gapped dimer products. For a connected four-site attractive chain, integer and rational arithmetic verifies 3199 increments and closes the infinite tail. A uniform perturbation bound also gives an explicit rectangle in hopping and attraction parameters. A self-contained finite-field certificate shows that the same chain’s unscaled polarization can decrease while i
We investigate the finite-temperature corrections to scaling in the three-state square-lattice Potts antiferromagnet, close to the critical point at T=0. Numerical diagonalization of the transfer matrix on semi-infinite strips of width L sites, 4 < or = L < or = 14, yields finite-size estimates of the corresponding scaled gaps, which are extrapolated to L --> infinity. Owing to the characteristics of the quantities under study, we argue that the natural variable to consider is x identical with L e(-2 beta). For the extrapolated scaled gaps we show that square-root corrections, in the variable x, are present, and provide estimates for the numerical values of the amplitudes of the first- and second-order correction terms, for both the first and second scaled gaps. We also calculate the third scaled gap of the transfer matrix spectrum at T=0, and find an extrapolated value of the decay-of-correlations exponent, eta(3)=2.00(1). This is at odds with earlier predictions, to the effect that the third relevant operator in the problem would give eta(P(stagg))=3, corresponding to the staggered polarization.
. Finite-size corrections to scaling of critical correlation lengths and free energies of Ising and three-state Potts ferromagnets are analysed by numerical methods, on strips of width N sites of square, triangular and honeycomb lattices. Strong evidence is given that the amplitudes of the `analytical' correction terms, N -2 , are identically zero for triangular and honeycomb Ising systems. For Potts spins, our results are broadly consistent with this lattice-dependent pattern of cancellations, though for correlation lengths non-vanishing (albeit rather small) amplitudes cannot be entirely ruled out. 1. Introduction The systematic study of subdominant terms in scaling provides researchers with guidelines on how to extrapolate, e.g. finite-size results to the thermodynamic limit [1--5]. Also, it may in itself bring out connections to underlying physical properties. These latter may be universal, such as the relationship between critical free-energy finite-width correction and conforma...
Corrected version. We prove an exact separator-compression theorem for the complete pre-Arf spin-structure family of zero-field Ising models on cubic-lattice strips G_{n,w}=P_n x P_w x P_w. This version corrects the internal-cut character-duality proof: lambda_a uses PD(b_i), while lambda_b uses PD(a_i); the former longitude-H3 step is withdrawn and replaced by an arbitrary-width pushed-off-meridian proof with exact triangular-transport and two-route replay firewalls. The all-sector tensor train has denominator-free bond at most 2^(w^2-1), with generic equality for independent nonuniform weights at every width. For cubic boxes the remaining carrier is still 2^(L^2-1); no thermodynamic limit, exact critical temperature, critical exponents, or exact solution of the three-dimensional Ising model is claimed. The record contains the corrected paper, standalone source, deterministic proof/replay archive, and checksums.
We study three-state Potts spins on a square lattice, in which all bonds are ferromagnetic along one of the lattice directions, and antiferromagnetic along the other. Numerical transfer-matrix methods are used on infinite strips of width L sites, 4 ⩽ L ⩽ 14. Based on the analysis of the ratio of scaled mass gaps (inverse correlation lengths) and scaled domain-wall free energies, we provide strong evidence that a critical (Kosterlitz–Thouless) phase is present, whose upper limit is, in our best estimate, T c = 0.29 ± 0.01. From an analysis of the (extremely anisotropic) nature of excitations below T c , we argue that the critical phase extends all the way down to T = 0. While domain walls parallel to the ferromagnetic direction are soft for the whole extent of the critical phase, those along the antiferromagnetic direction seem to undergo a softening transition at a finite temperature. Assuming a bulk correlation length varying, for T > T c , as ξ( T ) = a ξ exp[ b ξ ( T − T c ) −σ ], σ ≃ 1/2, we attempt finite-size scaling plots of our finite-width correlation lengths. Our best results are for T c = 0.50 ± 0.01. We propose a scenario in which such inconsistency is attributed to the extreme narrowness of the critical region.