Physica A Statistical Mechanics and its Applications

Duality-based approximation for the critical point of the square lattice Ising ferromagnet within Tsallis statistics

L. R. da Silva, Eugene Stanley

Published Dec 1, 1996• 6 citations

Abstract

Within the generalized thermostatistics of Tsallis, we propose for the spin-12 Ising ferromagnet a transmissivity variable which extends that defined by Tsallis and Levy for thermal magnetic systems. By using this generalized transmissivity as well as duality arguments, we calculate the q-dependence of the critical temperature corresponding to the square lattice, where q is the entropic index (q = 1 reproduces standard thermostatistics). Our approximate results are compared with those previously obtained using renormalization group and mean-field approximation.

DOI: 10.1016/s0378-4371(96)00270-1

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Physica A Statistical Mechanics and its Applications

Duality-based approximation for the critical point of the square lattice Ising ferromagnet within Tsallis statistics

L. R. da Silva, Eugene Stanley

Published Dec 1, 1996• 6 citations

Publisher

Abstract

DOI: 10.1016/s0378-4371(96)00270-1

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For the two-dimensional ferromagnetic Ising critical point, I show that the known values of the critical exponents imply the absence of logarithms of the reduced temperature in the leading contributions to any field derivative of the free energy at zero magnetic field. For the square-lattice Ising antiferromagnet in a weak magnetic field, I compute the critical line ${T}_{c}$ (H)= ${T}_{c}^{0}$ (1-0.038 023 259 ${H}^{2}$ ) and the leading contribution to the susceptibility \ensuremath{\chi}=0.014 718 006 6 ${H}^{2}$ ln(1/\ensuremath{\Vert}t\ensuremath{\Vert}), where t is the reduced temperature.For the two-dimensional ferromagnetic Ising critical point, I show that the known values of the critical exponents imply the absence of logarithms of the reduced temperature in the leading contributions to any field derivative of the free energy at zero magnetic field. For the square-lattice Ising antiferromagnet in a weak magnetic field, I compute the critical line ${T}_{c}$(H)=${T}_{c}^{0}$(1-0.038 023 259${H}^{2}$) and the leading contribution to the susceptibility \ensuremath{\chi}=0.014 718 006 6${H}^{2}$ln(1/\ensuremath{\Vert}t\ensuremath{\Vert}), where t is the reduced temperature.

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By using self-dual clusters (whose sizes are characterized by the numbers $b=2,3,4,5$ ), we calculate within a real-space renormalization-group framework, the longitudinal surface tension of the square-lattice first-neighbor \textonehalf{}-spin ferromagnetic Ising model. The exact critical temperature ${T}_{c}$ is recovered for any value of $b$ ; the exact asymptotic behavior of the surface tension in the limit of low temperatures is analytically recovered; the approximate correlation-length critical exponents monotonically tend towards the exact value $\ensuremath{\nu}=1$ (which, for two dimensions, coincides with the surface-tension critical exponent $\ensuremath{\mu}$) for increasingly large cells; the same behavior is noticed in the approximate values for the surface-tension amplitude in the limit $T\ensuremath{\rightarrow}{T}_{c}$. We develop four different numerical procedures for extrapolating to $b\ensuremath{\rightarrow}\ensuremath{\infty}$ the renormalization-group (RG) results for the surface tension, and quite satisfactory agreement is obtained with Onsager's exact expression (error varying from zero to a few percent on the whole temperature domain). Furthermore, we compare the set of RG surface tensions with a set of biased surface tensions (associated with appropriate misfit seams) and find only fortuitous coincidence among them.By using self-dual clusters (whose sizes are characterized by the numbers $b=2,3,4,5$), we calculate within a real-space renormalization-group framework, the longitudinal surface tension of the square-lattice first-neighbor \textonehalf{}-spin ferromagnetic Ising model. The exact critical temperature ${T}_{c}$ is recovered for any value of $b$; the exact asymptotic behavior of the surface tension in the limit of low temperatures is analytically recovered; the approximate correlation-length critical exponents monotonically tend towards the exact value $\ensuremath{\nu}=1$ (which, for two dimensions, coincides with the surface-tension critical exponent $\ensuremath{\mu}$) for increasingly large cells; the same behavior is noticed in the approximate values for the surface-tension amplitude in the limit $T\ensuremath{\rightarrow}{T}_{c}$. We develop four different numerical procedures for extrapolating to $b\ensuremath{\rightarrow}\ensuremath{\infty}$ the renormalization-group (RG) results for the surface tension, and quite satisfactory agreement is obtained with Onsager's exact expression (error varying from zero to a few percent on the whole temperature domain). Furthermore, we compare the set of RG surface tensions with a set of biased surface tensions (associated with appropriate misfit seams) and find only fortuitous coincidence among them.

Physical review. B, Condensed matterFeb 1, 198119 citations

We investigate the critical behavior and the duality property of the ferromagnetic

q

-state clock model on the square lattice based on the tensor-network formalism. From the entanglement spectra of local tensors defined in the original and dual lattices, we obtain the exact self-dual points for the model with $q \leq 5 $ and approximate self-dual points for

q \geq 6

. We calculate accurately the lower and upper critical temperatures for the six-state clock model from the fixed-point tensors determined using the higher-order tensor renormalization group method and compare with other numerical results.

{T}_{c}

. For the canonical two-dimensional square lattice near-neighbor Ising ferromagnet, it is shown that compact ``extended scaling'' expressions analogous to the high-dimensional limit form but with the appropriate exponents give accurate approximations to the true temperature dependencies, again over the entire temperature range from

{T}_{c}

to infinity. Within this approach, for near-neighbor interaction systems, there is no crossover temperature above which mean-field-like behavior sets in.

{T}_{c}

(H)=

{T}_{c}^{0}

(1-0.038 023 259

{H}^{2}

) and the leading contribution to the susceptibility \ensuremath{\chi}=0.014 718 006 6

{H}^{2}

ln(1/\ensuremath{\Vert}t\ensuremath{\Vert}), where t is the reduced temperature.

By using self-dual clusters (whose sizes are characterized by the numbers

b=2,3,4,5

), we calculate within a real-space renormalization-group framework, the longitudinal surface tension of the square-lattice first-neighbor \textonehalf{}-spin ferromagnetic Ising model. The exact critical temperature

{T}_{c}

is recovered for any value of

b

; the exact asymptotic behavior of the surface tension in the limit of low temperatures is analytically recovered; the approximate correlation-length critical exponents monotonically tend towards the exact value $\ensuremath{\nu}=1$ (which, for two dimensions, coincides with the surface-tension critical exponent $\ensuremath{\mu}$) for increasingly large cells; the same behavior is noticed in the approximate values for the surface-tension amplitude in the limit $T\ensuremath{\rightarrow}{T}_{c}$. We develop four different numerical procedures for extrapolating to $b\ensuremath{\rightarrow}\ensuremath{\infty}$ the renormalization-group (RG) results for the surface tension, and quite satisfactory agreement is obtained with Onsager's exact expression (error varying from zero to a few percent on the whole temperature domain). Furthermore, we compare the set of RG surface tensions with a set of biased surface tensions (associated with appropriate misfit seams) and find only fortuitous coincidence among them.