We test the Gluing Method on the square-lattice two-dimensional Ising model by reconstructing global free-energy generators from complementary local relations. The emphasis is kept explicit throughout: the inputs used as \emph{gluing material} are separated from quantities obtained \emph{after gluing} and from data reserved for validation. At zero magnetic field, Kramers--Wannier duality is absorbed into the coordinate $\tau=\log\sinh 2K$ , so that the reduced free energy decomposes as $\Phi(\tau)=\tau/2+G(|\tau|)$ . The gluing materials are then only a critical logarithmic local expansion and a low-order high-temperature outer expansion; the low-temperature sector is generated by duality rather than fitted independently. Positive flat weights preserve the prescribed critical asymptotics to every finite algebraic order. With one crossover parameter fixed from three scalar anchors, the resulting bulk free energy agrees with the exact Onsager benchmark at a maximum relative error of $1.915\times10^{-5}$ on the tested interval. Internal energy and specific heat are derived from the same glued generator. Zero-field finite-size corrections are organized separately as one-variable scaling functions multiplying $L^{-2},L^{-4},\ldots$ . At nonzero field in the thermodynamic limit, the gluing materials are the established high-temperature, critical, and low-temperature local expansions of the universal Ising magnetic scaling free energy $\Phi_{\rm mag}(\eta)$ , $\eta=m/|h|^{8/15}$ . The low-temperature input contains the $|h|$ cusp associated with spontaneous symmetry breaking. Mapping the glued scaling free energy back to the square lattice through nonlinear scaling fields yields a single generator for the magnetic free-energy increment, magnetization, and susceptibility. The construction reproduces high-temperature linear response, the critical isotherm $M\propto |H|^{1/15}$ , and the two one-sided limits $M(H\to0^\pm)=\pm M_0$ without separately fitting magnetization branches. Finally, finite size and finite field are combined in the scaling variables $x_t=Lm$ and $x_h=L^{15/8}h$ . The two protected one-dimensional axes, a central mixed-response expansion, a low-temperature spectral-rounding relation, and a massive outer description are glued into a two-variable scaling surface $\mathcal Y_{\rm glue}(x_t,x_h)$ . The resulting lattice free energy closes in the compact form $\Phi_L^{glue}(K,H)=\Phi_{lat}^{glue}(K,H)+\frac{\mathcal{Y}_{glue}(Lm,L^{15/8}h)- Y_{bulk}(Lm,L^{15/8}h)}{L^2}+O(L^{-4})$ where $m=m(r,H)$ and $h=h(r,H)$ are the nonlinear scaling fields and $Y_{\rm bulk}$ is the thermodynamic singular scaling surface expressed in the same variables. Thus all finite-size/finite-field information is carried by the excess $\mathcal Y_{\rm glue}-Y_{\rm bulk}$ , while the thermodynamic lattice free energy remains the previously constructed $\Phi_{\rm lat}^{\rm glue}$ . On eleven transfer-matrix points withheld from the construction, the blind free-energy test gives RMS absolute error $3.90\times10^{-4}$ and maximum absolute error $9.55\times10^{-4}$ in $\mathcal Y$ . Differentiation of the same free energy produces finite-size magnetization and susceptibility; magnetization is especially stable on the low-temperature side, while the second derivative is more sensitive in crossover regions.We test the Gluing Method on the square-lattice two-dimensional Ising model by reconstructing global free-energy generators from complementary local relations. The emphasis is kept explicit throughout: the inputs used as \emph{gluing material} are separated from quantities obtained \emph{after gluing} and from data reserved for validation. At zero magnetic field, Kramers--Wannier duality is absorbed into the coordinate $\tau=\log\sinh 2K$, so that the reduced free energy decomposes as $\Phi(\tau)=\tau/2+G(|\tau|)$. The gluing materials are then only a critical logarithmic local expansion and a low-order high-temperature outer expansion; the low-temperature sector is generated by duality rather than fitted independently. Positive flat weights preserve the prescribed critical asymptotics to every finite algebraic order. With one crossover parameter fixed from three scalar anchors, the resulting bulk free energy agrees with the exact Onsager benchmark at a maximum relative error of $1.915\times10^{-5}$ on the tested interval. Internal energy and specific heat are derived from the same glued generator. Zero-field finite-size corrections are organized separately as one-variable scaling functions multiplying $L^{-2},L^{-4},\ldots$. At nonzero field in the thermodynamic limit, the gluing materials are the established high-temperature, critical, and low-temperature local expansions of the universal Ising magnetic scaling free energy $\Phi_{\rm mag}(\eta)$, $\eta=m/|h|^{8/15}$. The low-temperature input contains the $|h|$ cusp associated with spontaneous symmetry breaking. Mapping the glued scaling free energy back to the square lattice through nonlinear scaling fields yields a single generator for the magnetic free-energy increment, magnetization, and susceptibility. The construction reproduces high-temperature linear response, the critical isotherm $M\propto |H|^{1/15}$, and the two one-sided limits $M(H\to0^\pm)=\pm M_0$ without separately fitting magnetization branches. Finally, finite size and finite field are combined in the scaling variables $x_t=Lm$ and $x_h=L^{15/8}h$. The two protected one-dimensional axes, a central mixed-response expansion, a low-temperature spectral-rounding relation, and a massive outer description are glued into a two-variable scaling surface $\mathcal Y_{\rm glue}(x_t,x_h)$. The resulting lattice free energy closes in the compact form $\Phi_L^{glue}(K,H)=\Phi_{lat}^{glue}(K,H)+\frac{\mathcal{Y}_{glue}(Lm,L^{15/8}h)- Y_{bulk}(Lm,L^{15/8}h)}{L^2}+O(L^{-4})$ where $m=m(r,H)$ and $h=h(r,H)$ are the nonlinear scaling fields and $Y_{\rm bulk}$ is the thermodynamic singular scaling surface expressed in the same variables. Thus all finite-size/finite-field information is carried by the excess $\mathcal Y_{\rm glue}-Y_{\rm bulk}$, while the thermodynamic lattice free energy remains the previously constructed $\Phi_{\rm lat}^{\rm glue}$. On eleven transfer-matrix points withheld from the construction, the blind free-energy test gives RMS absolute error $3.90\times10^{-4}$ and maximum absolute error $9.55\times10^{-4}$ in $\mathcal Y$. Differentiation of the same free energy produces finite-size magnetization and susceptibility; magnetization is especially stable on the low-temperature side, while the second derivative is more sensitive in crossover regions.

Sep 2, 20260 citations

Finite Size Effects in Phase Transitions

E. Brézin, Jean Zinn‐Justin

Current physics sources and commentsJan 1, 19886 citations

Thermal Casimir Effect in A Schwarzschild-like Wormhole Spacetime

Arista Romadani, Apriadi Salim Adam, Ar Rohim +2 more

We study the finite-temperature Casimir effect for a massless scalar field confined between two parallel plates in a Schwarzschild-like wormhole spacetime. Imposing Dirichlet boundary conditions, we compute the renormalized Casimir free energy in the comoving frame. We find that the thermal correction to the renormalized Casimir free energy decreases gradually with the temperature and becomes independent of the background geometry in this frame. Thermodynamic quantities derived from the Casimir free energy, namely, the renormalized Casimir entropy, internal energy, and heat capacity at constant volume, exhibit distinct temperature dependence. At low temperatures, all thermodynamic quantities recover the expected behavior, consistent with the fundamental laws of thermodynamics. These results provide a compact framework for analyzing quantum vacuum forces in gravitational backgrounds.We study the finite-temperature Casimir effect for a massless scalar field confined between two parallel plates in a Schwarzschild-like wormhole spacetime. Imposing Dirichlet boundary conditions, we compute the renormalized Casimir free energy in the comoving frame. We find that the thermal correction to the renormalized Casimir free energy decreases gradually with the temperature and becomes independent of the background geometry in this frame. Thermodynamic quantities derived from the Casimir free energy, namely, the renormalized Casimir entropy, internal energy, and heat capacity at constant volume, exhibit distinct temperature dependence. At low temperatures, all thermodynamic quantities recover the expected behavior, consistent with the fundamental laws of thermodynamics. These results provide a compact framework for analyzing quantum vacuum forces in gravitational backgrounds.

hep-thCross-listedarXiv:2605.26743May 27, 2026

Scaling of the finite-size effect of the α -Rényi entropy in disjointed intervals under dilation

Long Xiong, Shunyao Zhang, Guang‐Can Guo +1 more

The $\ensuremath{\alpha}$-R\'enyi entropy in the gapless models has been obtained by the conformal field theory, which is exact in the thermodynamic limit. However, the calculation of its finite-size effect (FSE) is challenging. So far only the FSE in a single interval in the $XX$ model has been understood and the FSE in the other models and in the other conditions is totally unknown. Here we report the FSE of this entropy in disjointed intervals $A={\ensuremath{\cup}}_{i}{A}_{i}$ under a uniform dilation $\ensuremath{\lambda}A$ in the $XY$ model, showing a universal scaling law as ${\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\lambda}A}^{\ensuremath{\alpha}}={\mathrm{\ensuremath{\Delta}}}_{A}^{\ensuremath{\alpha}}{\ensuremath{\lambda}}^{\ensuremath{-}\ensuremath{\eta}}\mathcal{B}(A,\ensuremath{\lambda})$, where $|\mathcal{B}(A,\ensuremath{\lambda})|\ensuremath{\le}1$ is a bounded function and $\ensuremath{\eta}=\text{min}(2,2/\ensuremath{\alpha})$ when $\ensuremath{\alpha}<10$. We verify this relation in the phase boundaries of the $XY$ model, in which the different central charges correspond to the physics of free fermion and free boson models. We find that in the disjointed intervals two FSEs, termed as extrinsic FSE and intrinsic FSE, are required to fully account for the FSE of the entropy. Physically, we find that only the edge modes of the correlation matrix localized at the open ends $\ensuremath{\partial}A$ have contribution to the total entropy and its FSE. Our results provide some incisive insight into the entanglement entropy in the many-body systems.The $\ensuremath{\alpha}$-R\'enyi entropy in the gapless models has been obtained by the conformal field theory, which is exact in the thermodynamic limit. However, the calculation of its finite-size effect (FSE) is challenging. So far only the FSE in a single interval in the $XX$ model has been understood and the FSE in the other models and in the other conditions is totally unknown. Here we report the FSE of this entropy in disjointed intervals $A={\ensuremath{\cup}}_{i}{A}_{i}$ under a uniform dilation $\ensuremath{\lambda}A$ in the $XY$ model, showing a universal scaling law as ${\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\lambda}A}^{\ensuremath{\alpha}}={\mathrm{\ensuremath{\Delta}}}_{A}^{\ensuremath{\alpha}}{\ensuremath{\lambda}}^{\ensuremath{-}\ensuremath{\eta}}\mathcal{B}(A,\ensuremath{\lambda})$, where $|\mathcal{B}(A,\ensuremath{\lambda})|\ensuremath{\le}1$ is a bounded function and $\ensuremath{\eta}=\text{min}(2,2/\ensuremath{\alpha})$ when $\ensuremath{\alpha}<10$. We verify this relation in the phase boundaries of the $XY$ model, in which the different central charges correspond to the physics of free fermion and free boson models. We find that in the disjointed intervals two FSEs, termed as extrinsic FSE and intrinsic FSE, are required to fully account for the FSE of the entropy. Physically, we find that only the edge modes of the correlation matrix localized at the open ends $\ensuremath{\partial}A$ have contribution to the total entropy and its FSE. Our results provide some incisive insight into the entanglement entropy in the many-body systems.

arXiv:2203.10272Physical Review AApr 21, 20222 citations

Finite size effects at phase transitions

Kurt Binder

Springer eBooksApr 6, 2008116 citations

\tau=\log\sinh 2K

, so that the reduced free energy decomposes as

\Phi(\tau)=\tau/2+G(|\tau|)

. The gluing materials are then only a critical logarithmic local expansion and a low-order high-temperature outer expansion; the low-temperature sector is generated by duality rather than fitted independently. Positive flat weights preserve the prescribed critical asymptotics to every finite algebraic order. With one crossover parameter fixed from three scalar anchors, the resulting bulk free energy agrees with the exact Onsager benchmark at a maximum relative error of

1.915\times10^{-5}

on the tested interval. Internal energy and specific heat are derived from the same glued generator. Zero-field finite-size corrections are organized separately as one-variable scaling functions multiplying

L^{-2},L^{-4},\ldots

. At nonzero field in the thermodynamic limit, the gluing materials are the established high-temperature, critical, and low-temperature local expansions of the universal Ising magnetic scaling free energy

\Phi_{\rm mag}(\eta)

\eta=m/|h|^{8/15}

. The low-temperature input contains the

|h|

cusp associated with spontaneous symmetry breaking. Mapping the glued scaling free energy back to the square lattice through nonlinear scaling fields yields a single generator for the magnetic free-energy increment, magnetization, and susceptibility. The construction reproduces high-temperature linear response, the critical isotherm

M\propto |H|^{1/15}

, and the two one-sided limits

M(H\to0^\pm)=\pm M_0

without separately fitting magnetization branches. Finally, finite size and finite field are combined in the scaling variables

x_t=Lm

and

x_h=L^{15/8}h

. The two protected one-dimensional axes, a central mixed-response expansion, a low-temperature spectral-rounding relation, and a massive outer description are glued into a two-variable scaling surface

\mathcal Y_{\rm glue}(x_t,x_h)

. The resulting lattice free energy closes in the compact form

\Phi_L^{glue}(K,H)=\Phi_{lat}^{glue}(K,H)+\frac{\mathcal{Y}_{glue}(Lm,L^{15/8}h)- Y_{bulk}(Lm,L^{15/8}h)}{L^2}+O(L^{-4})

where

m=m(r,H)

and

h=h(r,H)

are the nonlinear scaling fields and

Y_{\rm bulk}

is the thermodynamic singular scaling surface expressed in the same variables. Thus all finite-size/finite-field information is carried by the excess

\mathcal Y_{\rm glue}-Y_{\rm bulk}

, while the thermodynamic lattice free energy remains the previously constructed

\Phi_{\rm lat}^{\rm glue}

. On eleven transfer-matrix points withheld from the construction, the blind free-energy test gives RMS absolute error

3.90\times10^{-4}

and maximum absolute error

9.55\times10^{-4}

\mathcal Y

. Differentiation of the same free energy produces finite-size magnetization and susceptibility; magnetization is especially stable on the low-temperature side, while the second derivative is more sensitive in crossover regions.

XX

model has been understood and the FSE in the other models and in the other conditions is totally unknown. Here we report the FSE of this entropy in disjointed intervals $A={\ensuremath{\cup}}_{i}{A}_{i}$ under a uniform dilation $\ensuremath{\lambda}A$ in the

XY

model, showing a universal scaling law as ${\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\lambda}A}^{\ensuremath{\alpha}}={\mathrm{\ensuremath{\Delta}}}_{A}^{\ensuremath{\alpha}}{\ensuremath{\lambda}}^{\ensuremath{-}\ensuremath{\eta}}\mathcal{B}(A,\ensuremath{\lambda})$, where $|\mathcal{B}(A,\ensuremath{\lambda})|\ensuremath{\le}1$ is a bounded function and $\ensuremath{\eta}=\text{min}(2,2/\ensuremath{\alpha})$ when $\ensuremath{\alpha}<10$. We verify this relation in the phase boundaries of the

XY

model, in which the different central charges correspond to the physics of free fermion and free boson models. We find that in the disjointed intervals two FSEs, termed as extrinsic FSE and intrinsic FSE, are required to fully account for the FSE of the entropy. Physically, we find that only the edge modes of the correlation matrix localized at the open ends $\ensuremath{\partial}A$ have contribution to the total entropy and its FSE. Our results provide some incisive insight into the entanglement entropy in the many-body systems.