Exact finite-size scaling of the three-dimensional mean spherical model with Neumann–Dirichlet boundary conditions and surface fields in the regime at and below the film critical point

The three-dimensional mean spherical model confined to a film of thickness $L$ with Neumann–Dirichlet boundary conditions and surface magnetic fields acting only on the two bounding planes is one of the few settings in which finite-size scaling with mixed boundary conditions and surface fields can be computed exactly. Amin (2006) derived its scaled finite-size equation of state, specific-heat scaling function, closed-form magnetization profile, and local fluctuation profiles — but only in the parameter regime $\tfrac{\phi}{2}+\cos\varphi_{L}^{(\mathrm{ND})}(1)>1$, equivalently $y=\phi^{1/2}L>\pi/2$, i.e. on the disordered side of the film's shifted critical coupling. The complementary regime $y\le\pi/2$ — the film critical point itself and the ordered side, where with surface fields the analytically continued equation of state still admits real solutions at least generically — lies outside the domain of validity of those closed forms. The zero-field analogue of the complementary regime is settled in the adjacent literature, but no published work treats the surface-field problem there, nor provides an explicit evaluation of the implicitly defined scaling function $X^{(\mathrm{ND})}(\eta_{1},\eta_{L},\zeta)$. The problem posed here is to complete the exact analysis in $y\le\pi/2$: the continued equation of state with its existence, uniqueness, and branch structure; the magnetization-profile, specific-heat, and fluctuation scaling functions; the singular surface-field-free limits matching the known ordered-phase solution; and the explicit evaluation of $X^{(\mathrm{ND})}$ across both regimes.

Background

The mean spherical model is a lattice spin system with real-valued spins whose fixed length is enforced only on the average, $\langle\sum_{i}\sigma_{i}^{2}\rangle=N$; it is exactly solvable and coincides with the $n\to\infty$ limit of the $O(n)$ vector models, which makes it a standard benchmark for finite-size scaling theory. In three dimensions it has a bulk critical coupling $K_{c}$ (with $K=J/(k_{B}T)$ the dimensionless coupling) and critical exponents $\nu=1$, $\beta=1/2$, $\alpha=-1$. The Lagrange multiplier of the spherical constraint — the spherical field $\phi\ge0$ — acts as a mass term: it vanishes at bulk criticality from the disordered side, and its finite-size behavior encodes the rounding of the singularity.

In the film geometry $L\times\infty^{2}$ (finite thickness $L$, extended in the other two directions) the modes transverse to the film are discrete, and the boundary conditions imposed on the two bounding planes select the spectrum. Neumann–Dirichlet (ND) boundary conditions mean a free surface (zero normal derivative of the spin field) on the plane $r=1$ and a Dirichlet condition (zero spin on the ghost plane $r=L+1$, the lattice realization of the ordinary surface class in which surface order is suppressed) on the plane $r=L$. The ND eigenfunctions are $u_{L}^{(\mathrm{ND})}(r,k)=\tfrac{2}{\sqrt{2L+1}}\cos[(r-\tfrac12)\varphi_{L}^{(\mathrm{ND})}(k)]$ with $\varphi_{L}^{(\mathrm{ND})}(k)=\pi(2k-1)/(2L+1)$, so the lowest transverse eigenvalue is $2[1-\cos\varphi_{L}^{(\mathrm{ND})}(1)]\simeq(\pi/2L)^{2}$. Surface magnetic fields $h(r)=h_{1}\delta_{r,1}+h_{L}\delta_{r,L}$ act only on the two bounding planes and couple directly to these modes. Confinement shifts the film critical coupling from $K_{c}$ to $K_{c,L}^{(\mathrm{ND})}$; at $d=3$ this shift is of order $1/L$ (Dantchev and Brankov 2003), in contrast to Dirichlet–Dirichlet and Neumann–Neumann films, which exhibit anomalous $\pm\ln L/L$ shifts and nonscaling finite-size behavior (Chen and Dohm 2003). According to the Privman–Fisher finite-size scaling hypothesis (Privman and Fisher 1984), near criticality the finite-size corrections of thermodynamic quantities are governed by universal scaling functions of scaled variables; here the scaling variables are $y=\phi^{1/2}L$, the scaled surface fields $\eta_{1}=K^{-1/2}h_{1}L^{3/2}$ and $\eta_{L}=K^{-1/2}h_{L}L^{1/2}$ (their different powers of $L$ reflect the different surface critical behavior of the free and ordinary faces), and $\zeta=(K_{c,L}^{(\mathrm{ND})}-K)L$.

Danchev, Brankov, and Amin (1997) constructed the exact ND mode-sum machinery for this model with surface fields and extracted the corresponding surface critical exponents. Building on it, Amin (2006) derived, in the critical finite-size scaling regime, the scaled equation of state

$$\frac{1}{4\pi}\ln\!\left[\cosh\sqrt{y^{2}-\pi^{2}/4}\,\right]+\frac{1}{4y}\Big[\eta_{1}^{2}Y_{1}'(y)+2\eta_{1}\eta_{L}Y_{2}'(y)+\eta_{L}^{2}Y_{3}'(y)\Big]=2\zeta,$$

where $Y_{1}(y)=\tanh\sqrt{y^{2}-\pi^{2}/4}/\sqrt{y^{2}-\pi^{2}/4}$, $Y_{2}(y)=1/\cosh\sqrt{y^{2}-\pi^{2}/4}$, and $Y_{3}(y)=-\sqrt{y^{2}-\pi^{2}/4}\,\tanh\sqrt{y^{2}-\pi^{2}/4}$. Primes denote differentiation with respect to the displayed argument: $Y_{i}'=dY_{i}/dy$ in the equation of state, and $X'^{(\mathrm{ND})}=\partial X^{(\mathrm{ND})}/\partial\zeta$ in the specific-heat scaling form quoted below. It yields the scaling of the spherical field $\phi\simeq L^{-2}X^{(\mathrm{ND})}(\eta_{1},\eta_{L},\zeta)$ with $X^{(\mathrm{ND})}$ defined implicitly, the specific-heat scaling form $C_{L,3}-\pi/2\cong-K_{c}^{2}L^{-1}X'^{(\mathrm{ND})}(\eta_{1},\eta_{L},\zeta)$, the closed-form magnetization profile

$$m_{L}(K,h(r);r)=\frac{1}{2K\cosh[(L+\tfrac12)x]}\left\{\frac{h_{1}\sinh[(L-r+1)x]\cosh(x/2)}{\sinh x}+h_{L}\cosh[(r-\tfrac12)x]\right\},\quad x=\operatorname{arccosh}\!\left[\tfrac{\phi}{2}+\cos\varphi_{L}^{(\mathrm{ND})}(1)\right],$$

and the mean-square local magnetization, whose analysis confirms the exponents $\beta=1/2$ and $\alpha=-1$.

The quantity $y=\phi^{1/2}L$ orders the regimes. The condition $\tfrac{\phi}{2}+\cos\varphi_{L}^{(\mathrm{ND})}(1)>1$ is equivalent to $y>\pi/2$ and places the film on the disordered side of the zero-field film critical coupling $K_{c,L}^{(\mathrm{ND})}$; $y=\pi/2$ is the film critical point itself, and $y<\pi/2$ corresponds to $K>K_{c,L}^{(\mathrm{ND})}$, the ordered side. The analysis of Amin (2006) is restricted to $y>\pi/2$, where $x$ is real and the closed forms are hyperbolic. In the complementary regime $y\le\pi/2$ the underlying finite-$L$ mode sums remain perfectly well defined; $x$ becomes imaginary (the hyperbolic functions continue to trigonometric ones), and with surface fields present the continued equation of state still admits real solutions there for generic parameter values — so the observables are physically meaningful — yet this regime, the exact domain of $(\eta_{1},\eta_{L},\zeta)$ in which real solutions exist, the branch selection of the now possibly non-monotonic implicit equation, and any explicit (numerical) evaluation of $X^{(\mathrm{ND})}$ at nonzero $\eta_{1},\eta_{L}$ remain untreated in the published literature.

For zero surface fields the complementary regime is settled: Dantchev and Brankov (2003) derived the ND constraint in its trigonometric (arccos) form, established the absence of a logarithmic shift for ND films, obtained the below-$T_{c}$ solution $\phi\simeq L^{-2}e^{4\pi L(K_{c,3}-K)}$ and the exact zero-field susceptibility, and Chamati (2008) inverted the $d=3$ ND equation of state explicitly in closed form and evaluated the ND Casimir amplitude $\Delta^{(\mathrm{ND})}=0.01922\ldots$ (the amplitude of the fluctuation-induced force between the film surfaces at criticality); Kastening and Dohm (2008) reported the corresponding critical Casimir force scaling functions of the mean spherical model for nonperiodic boundary conditions. What no published work treats is the surface-field problem ($\eta_{1},\eta_{L}\neq0$) in $y\le\pi/2$.

Problem Statement

For the three-dimensional mean spherical model in film geometry $L\times\infty^{2}$ with Neumann–Dirichlet boundary conditions and surface magnetic fields $h(r)=h_{1}\delta_{r,1}+h_{L}\delta_{r,L}$, complete the exact finite-size scaling analysis of Amin (2006) in the complementary parameter regime $\tfrac{\phi}{2}+\cos\varphi_{L}^{(\mathrm{ND})}(1)\le1$ (equivalently $y=\phi^{1/2}L\le\pi/2$), i.e. at the film critical point $K=K_{c,L}^{(\mathrm{ND})}$ and on the ordered side $K>K_{c,L}^{(\mathrm{ND})}$, which the Amin (2006) analysis — valid only for $\tfrac{\phi}{2}+\cos\varphi_{L}^{(\mathrm{ND})}(1)>1$ — does not reach. With the model, conventions, and scaling variables $(y,\eta_{1},\eta_{L},\zeta)$ fixed exactly as in the Background above, the problem is to determine:

(1) The finite-size equation of state in $y\le\pi/2$ (the analytic continuation of the log-cosh form together with its surface-field terms), together with an analysis of for which $(\eta_{1},\eta_{L},\zeta)$ real solutions exist, whether they are unique, and which branch is physical (continuity from the disordered branch, or minimization of the free energy).

(2) The closed-form magnetization profile $m_{L}(K,h(r);r)$, the specific-heat scaling function, and the mean-square local magnetization in the same regime, with their finite-size scaling forms; the singular limits $\eta_{1},\eta_{L}\to0$ at fixed $\zeta\le0$, where the surface fields vanish and the solution must cross over to the condensation mechanism of the mean spherical model (below the transition the spherical constraint is saturated by a macroscopic occupation of the lowest transverse mode, the analogue of Bose condensation) in the ordered film, together with the subsequent deep-ordered-phase limit $L(K-K_{c,3})\to\infty$, in which the solution must match asymptotically the known zero-field below-transition solution; and the matching at $y=\pi/2$ to Amin (2006)'s $y>\pi/2$ formulas.

(3) An explicit evaluation of the scaling function $X^{(\mathrm{ND})}(\eta_{1},\eta_{L},\zeta)$ — defined implicitly by the equation of state — over a domain covering both regimes.

Together these describe the exact finite-size scaling of this model across the entire critical finite-size window, of which the Amin (2006) treatment covers only the disordered half. Two further extensions lie beyond this core — other boundary-condition classes with surface fields, and spatially distributed (non-surface-only) field profiles. The field-free versions of the other boundary-condition classes are already settled in the literature, and the surface-field versions of arbitrary other boundary conditions or arbitrary interior field profiles form open-ended families without a natural completion boundary; they are therefore recorded as context rather than posed as part of the problem.

References

1. Magdy E. Amin (2006). Surface critical behavior and scaling functions for the three-dimensional mean spherical model. Physics Letters A 358, 94–99. DOI: 10.1016/j.physleta.2006.05.005. https://doi.org/10.1016/j.physleta.2006.05.005
2. D. M. Danchev, J. G. Brankov, M. E. Amin (1997). New surface critical exponents in the spherical model. Journal of Physics A: Mathematical and General 30, 1387–1402. DOI: 10.1088/0305-4470/30/5/010. https://doi.org/10.1088/0305-4470/30/5/010
3. M. E. Amin, J. G. Brankov (1998). New effects in the magnetization profile of the spherical model in a step-like field. Journal of Physics A: Mathematical and General 31, 4821–4834. DOI: 10.1088/0305-4470/31/21/003. https://doi.org/10.1088/0305-4470/31/21/003
4. Magdy E. Amin (2001). The influence of three external fields on the mean spherical model: magnetization profile. Physica A: Statistical Mechanics and its Applications 291, 501–511. DOI: 10.1016/S0378-4371(00)00538-0. https://doi.org/10.1016/S0378-4371(00)00538-0
5. X. S. Chen, V. Dohm (2003). Scaling and nonscaling finite-size effects in the Gaussian and the mean spherical model with free boundary conditions. Physical Review E 67, 056127. DOI: 10.1103/PhysRevE.67.056127. https://doi.org/10.1103/PhysRevE.67.056127
6. Daniel M. Dantchev, Jordan G. Brankov (2003). On the finite-size behaviour of systems with asymptotically large critical shift. Journal of Physics A: Mathematical and General 36, 8915–8933. DOI: 10.1088/0305-4470/36/34/301. https://doi.org/10.1088/0305-4470/36/34/301 (preprint arXiv:cond-mat/0305470)
7. H. Chamati (2008). Finite-size effects in the spherical model of finite thickness. Journal of Physics A: Mathematical and Theoretical 41, 375002. DOI: 10.1088/1751-8113/41/37/375002. https://doi.org/10.1088/1751-8113/41/37/375002 (preprint arXiv:0805.0715)
8. B. Kastening, V. Dohm (2008). Critical Casimir force scaling functions of the mean spherical model in 2 < d ≤ 3 dimensions for nonperiodic boundary conditions. In: Path Integrals — New Trends and Perspectives (World Scientific), pp. 283–288. DOI: 10.1142/9789812837271_0039. https://doi.org/10.1142/9789812837271_0039
9. D. M. Dantchev, S. Dietrich (2023). Critical Casimir effect: Exact results. Physics Reports 1005, 1–130. DOI: 10.1016/j.physrep.2022.12.004. https://doi.org/10.1016/j.physrep.2022.12.004 (preprint arXiv:2203.15050)
10. Daniel Dantchev, Vassil Vassilev, Joseph Rudnick (2025). Gaussian Versus Mean-Field Models: Contradictory Predictions for the Casimir Force Under Dirichlet–Neumann Boundary Conditions. Entropy 27, 468. DOI: 10.3390/e27050468. https://doi.org/10.3390/e27050468
11. V. Privman, M. E. Fisher (1984). Universal critical amplitudes in finite-size scaling. Physical Review B 30, 322–327. DOI: 10.1103/PhysRevB.30.322. https://doi.org/10.1103/PhysRevB.30.322