arXiv subjects

H. W. Diehl

Publications and source records attributed to H. W. Diehl.

At least 19 recordsLinked to original sources

arXiv2020

Why boundary conditions do not generally determine the universality class for boundary critical behavior

Interacting field theories for systems with a free surface frequently exhibit distinct universality classes of boundary critical behaviors depending on gross surface properties. The boundary condition satisfied by the continuum field theory on some scale may or may not be decisive for the universality class that applies. In many recent papers on boundary field theories it is taken for granted that Dirichlet or Neumann boundary conditions decide whether the ordinary or special boundary universality class is observed. While true in a certain sense for the Dirichlet boundary condition, this is not the case for the Neumann boundary condition. Building on results that have been worked out in the 1980s, but have not always been appropriately appreciated in the literature, the subtle role of boundary conditions and their scale dependence is elucidated and the question of whether or not they determine the observed boundary universality class is discussed.

hep-th↗

arXiv2017

Fluctuation-induced forces in confined ideal and imperfect Bose gases

Fluctuation-induced forces are investigated for ideal and imperfect Bose gases confined to $d$-dimensional films of size $\infty^{d-1}\times D$ under periodic (P), antiperiodic (A), Dirichlet-Dirichlet (DD), Neumann-Neumann (NN), and Robin (R) boundary conditions (BCs). The full scaling functions $Υ^{\text{BC}}_d(x_λ=D/λ_{th},{x_ξ=D/ξ})$ of the residual reduced grand potential per area, $φ_{\text{res},d}^{\text{BC}}(T,μ,D)=D^{-(d-1)}Υ_d^{\text{BC}}(x_λ,x_ξ)$, are determined for the ideal gas case with these BCs, where $λ_{th}$ and $ξ$ are the thermal de-Broglie wavelength and the bulk correlation length, respectively. The scaling functions $Θ^{\text{BC}}_d(x_ξ)\equiv Υ_d^{\text{BC}}(\infty,x_ξ)$ describing the critical behavior at the bulk condensation transition are shown to agree with those previously determined from a massive free $O(2)$ theory for $\text{BC}=\text{P},\text{A},\text{DD},\text{DN},\text{NN}$. For $d=3$, they are expressed in closed analytical form. The analogous functions $Υ_d^{\text{BC}}(x_λ,x_ξ,c_1D,c_2D)$ and $Θ^{\text{R}}_d(x_ξ,c_1D,c_2D)$ under the RBCs $(\partial_z-c_1)ϕ|_{z=0}=(\partial_z+c_2)ϕ|_{z=D}=0$ with $c_1\ge 0$ and $c_2\ge 0$ are also determined. The functions $Υ_{\infty,d}^{\text{P}}(x_λ,x_ξ)$ and $Φ_{\infty,d}^{\text{P}}(x_ξ)$ for the imperfect Bose gas are shown to agree with those of the interacting Bose gas with $n\to\infty$ internal degrees of freedom. Hence for ${d=3}$, $Φ_{\infty,d}^{\text{P}}(x_ξ)$ is known exactly in closed analytic form. A modified imperfect Bose-gas model with free BC is introduced that corresponds to the limit $n\to\infty$ of this interacting Bose gas. Exact results for the function $Θ_{\infty,3}^{\mathbb{DD}}(x_ξ)$ therefore follow from those of the $O(2n)$ $ϕ^4$ model for $n\to\infty$.

cond-mat.quant-gas↗

arXiv2017

The three-dimensional $O(n)$ $ϕ^4$ model on a strip with free boundary conditions: exact results for a nontrivial dimensional crossover in the limit $n\to\infty$

Recent exact $n\to\infty$ results for critical Casimir forces of the $O(n)$ $ϕ^4$ model on a three-dimensional strip bounded by two planar free surfaces at a distance $L$ are surveyed. This model has long-range order below the bulk critical temperature $T_c$ if $L=\infty$, but remains disordered for all $T>0$ when $L<\infty$. A proper analysis of its scaling behavior near $T_c$ is quite challenging: Besides with bulk, boundary, and finite-size critical behaviors, one must deal with a nontrivial dimensional crossover. The model can be solved exactly in the limit $n\to\infty$ in terms of the eigenvalues and eigenenergies of a selfconsistent Schrödinger equation involving a potential $v(z)$ with the near-boundary singular behavior $v(z\to 0+)\approx -1/(4z^2)+4m/(π^{2}z)$, where $m=1/ξ_+(|t|)$ is the inverse bulk correlation length and $t\sim (T-T_c)/T_c$, and a corresponding singularity at the second boundary plane. The potential $v(z)$, the excess free energy, and the Casimir force have been determined numerically with high precision. Exact analytical results for a variety of properties such as series expansion coefficients of $v(z)$, the scattering data of $v(z)$ in the semi-infinite case $L=\infty$ for all $m\gtreqless 0$, and the low-temperature asymptotic behavior of the residual free energy and the Casimir force can be obtained by a combination of boundary-operator and short-distance expansions, proper extensions of inverse scattering theory, new trace formulae, and semiclassical expansions.

cond-mat.stat-mech↗

arXiv2016

Comment on "Nonlocal quartic interactions and universality classes in perovskite manganites"

In a recent paper [Phys. Rev. E \textbf{92}, 012123 (2015)] a modified $d$-dimensional $Φ^4$ model was investigated which differs from the standard one in that the $Φ^4$ term was replaced by a nonlocal one with a potential $u(\bm{x}-\bm{x}')$ that depends on a parameter $σ$ and decays exponentially as $|\bm{x}-\bm{x}'|\to\infty$ on a scale $|m|^{-1}<\infty$. The authors claim the upper critical dimension of this model to be $d_σ=4+2σ$. Performing a one-loop calculation they arrive at expansions in powers of $ε_σ=d_σ-d$ for critical exponents such as $η$ and related ones to $O(ε_σ)$ whose $O(ε_σ)$ coefficients depend on $σ$ and the ratio $w=m^2/Λ^2$, where $Λ$ is the UV cutoff. We show that these claims are unfounded and based on misjudgments and an ill-conceived renormalization group calculation.

cond-mat.stat-mech↗

arXiv2015

Inverse scattering theory and trace formulae for one-dimensional Schrödinger problems with singular potentials

Inverse scattering theory is extended to one-dimensional Schrödinger problems with near-boundary singularities of the form $v(z\to 0)\simeq -z^{-2}/4+v_{-1}z^{-1}$. Trace formulae relating the boundary value $v_0$ of the nonsingular part of the potential to spectral data are derived. Their potential is illustrated by applying them to a number of Schrödinger problems with singular potentials.

math-ph↗

arXiv2015

Inverse scattering-theory approach to the exact large-$n$ solutions of $O(n)$ $ϕ^4$ models on films and semi-infinite systems bounded by free surfaces

The $O(n)$ $ϕ^4$ model on a strip bounded by a pair of planar free surfaces at separation $L$ can be solved exactly in the large-$n$ limit in terms of the eigenvalues and eigenfunctions of a self-consistent one-dimensional Schrödinger equation. The scaling limit of a continuum version of this model is considered. It is shown that the self-consistent potential can be eliminated in favor of scattering data by means of appropriately extended methods of inverse scattering theory. The scattering data (Jost function) associated with the self-consistent potential are determined for the ${L=\infty}$ semi-infinite case in the scaling regime for all values of the temperature scaling field $t=(T-T_c)/T_c$ above and below the bulk critical temperature $T_c$. These results are used in conjunction with semiclassical and boundary-operator expansions and a trace formula to derive exact analytical results for a number of quantities such as two-point functions, universal amplitudes of two excess surface quantities, the universal amplitude difference associated with the thermal singularity of the surface free energy, and potential coefficients. The asymptotic behaviors of the scaled eigenenergies and eigenfunctions of the self-consistent Schrödinger equation as function of $x= t(L/ξ_+)^{1/ν}$ are determined for $x\to-\infty$. In addition, the asymptotic ${x\to -\infty}$ forms of the universal finite-size scaling functions $Θ(x)$ and $\vartheta(x)$ of the residual free energy and the Casimir force are computed exactly to order $1/x$, including their $x^{-1}\ln|x|$ anomalies.

cond-mat.stat-mech↗

arXiv2014

Comment on "Casimir force in the $O(n\to\infty)$ model with free boundary conditions"

In a recent paper [D. Dantchev, J. Bergnoff, and J. Rudnick, Phys. Rev. E 89, 042116 (2014)] the problem of the Casimir force in the $O(n)$ model on a slab with free boundary conditions, investigated earlier by us [EPL 100, 10004 (2012)], is reconsidered using a mean spherical model with separate constraints for each layer. The authors (i) question the applicability of the Ginzburg-Landau-Wilson approach to the low-temperature regime, arguing for the superiority of their model compared to the family of $ϕ^4$ models A and B whose numerically exact solutions we determined both for values of the coupling constant $0<g<\infty$ and $g=\infty$. They (ii) report consistency of their results with ours in the critical region and a strong manifestation of universality, but (iii) point out discrepancies with our results in the region below $T_{\mathrm{c}}$. We show here that (i) is unjustified and prove that our model B with $g=\infty$ is identical to their spherical model. Hence evidence for the reported universality is already contained in our work. Moreover, the results we determined for anyone of the models A and B for various thicknesses $L$ are all numerically exact. (iii) is due to their misinterpretation of our results for the scaling limit. We also show that their low-temperature expansion, which does not hold inside the scaling regime, is limited to temperatures lower than they anticipated.

cond-mat.stat-mech↗

arXiv2014

Large-$n$ approach to thermodynamic Casimir effects in slabs with free surfaces

The classical $n$-vector $ϕ^4$ model with $O(n)$ symmetrical Hamiltonian ${\cal H}$ is considered in a $\infty^2\times L$ slab geometry bounded by a pair of parallel free surface planes at separation $L$. The temperature-dependent scaling functions of the excess free energy and the thermodynamic Casimir force are computed in the large-$n$ limit for temperatures $T$ at, above, and below the bulk critical temperature $T_{\rm c}$. Their $n=\infty$ limits can be expressed exactly in terms of the eigensystem of a self-consistent one-dimensional Schrödinger equation. This equation is solved by numerical means for two distinct discretized versions of the model: in the first ("model A"), only the coordinate $z$ across the slab is discretized and the integrations over momenta conjugate to the lateral coordinates are regularized dimensionally; in the second ("model B"), a simple cubic lattice with periodic boundary conditions along the lateral directions is used. Renormalization-group ideas are invoked to show that, in addition to corrections to scaling $\propto L^{-1}$, anomalous ones $\propto L^{-1}\ln L$ should occur. They can be considerably decreased by taking an appropriate $g\to\infty$ ($T_{\rm c}\to\infty$) limit of the $ϕ^4$ interaction constant $g$. Depending on the model A or B, they can be absorbed completely or to a large extent in an effective thickness $L_{\rm eff}=L+δL$. Excellent data collapses and consistent high-precision results for both models are obtained. The approach to the low-temperature Goldstone values of the scaling functions is shown to involve logarithmic anomalies. The scaling functions exhibit all qualitative features seen in experiments on the thinning of wetting layers of ${}^4$He and Monte Carlo simulations of $XY$ models, including a pronounced minimum of the Casimir force below $T_{\rm c}$.

cond-mat.stat-mech↗

arXiv2014

The O(n) $ϕ^4$ model with free surfaces in the large-$n$ limit: Some exact results for boundary critical behaviour, fluctuation-induced forces and distant-wall corrections

The $O(n)$ $ϕ^4$ model on a slab $\mathbb{R}^{d-1}\times[0,L]$ bounded by free surfaces is studied for $2<d<4$ in the limit $n\to\infty$. The self-consistent potential $V(z)$ which the exact $n\to\infty$ solution of the model involves is analysed by means of boundary operator expansions. Building on the known exact $n\to\infty$ solution for $V(z)$ in the semi-infinite case $L=\infty$ at the bulk critical point, we exactly determine two types of corrections to this potential: (i) those linear in the temperature scaling field $t$ at $L=\infty$, and (ii) the leading $L$-dependent (distant-wall) corrections at the critical point. From (i) exact analytical results at $d=3$ are obtained for the leading temperature singularity of the excess surface free energy and the implied asymptotic behaviours of the scaling functions $Θ_3(x)$ and $\vartheta_3(x)$ of the residual free energy $f_{\rm res} =L^{1-d}\,Θ_d(tL)$ and the critical Casimir force $β\mathcal{F}_{\rm C}(T,L)=L^{-d}\,\vartheta_d(tL)$ in the limit $x\to 0\pm$. The second derivative $\vartheta_3''(0)$ is computed exactly.

cond-mat.stat-mech↗

arXiv2012

Exact thermodynamic Casimir forces for an interacting three-dimensional model system in film geometry with free surfaces

The limit n to infinity of the classical O(n) phi^4 model on a 3d film with free surfaces is studied. Its exact solution involves a self-consistent 1d Schrödinger equation, which is solved numerically for a partially discretized as well as for a fully discrete lattice model. Numerically exact results are obtained for the scaled Casimir force at all temperatures. Obtained via a single framework, they exhibit all relevant qualitative features of the thermodynamic Casimir force known from wetting experiments on Helium-4 and Monte Carlo simulations, including a pronounced minimum below the bulk critical point.

cond-mat.stat-mech↗

arXiv2012

Critical Casimir effect in films for generic non-symmetry-breaking boundary conditions

Systems described by an O(n) symmetrical $ϕ^4$ Hamiltonian are considered in a $d$-dimensional film geometry at their bulk critical points. A detailed renormalization-group (RG) study of the critical Casimir forces induced between the film's boundary planes by thermal fluctuations is presented for the case where the O(n) symmetry remains unbroken by the surfaces. The boundary planes are assumed to cause short-ranged disturbances of the interactions that can be modelled by standard surface contributions $\propto \bmϕ^2$ corresponding to subcritical or critical enhancement of the surface interactions. This translates into mesoscopic boundary conditions of the generic symmetry-preserving Robin type $\partial_n\bmϕ=\mathring{c}_j\bmϕ$. RG-improved perturbation theory and Abel-Plana techniques are used to compute the $L$-dependent part $f_{\mathrm{res}}$ of the reduced excess free energy per film area $A\to\infty $ to two-loop order. When $d<4$, it takes the scaling form $f_{\mathrm{res}}\approx D(c_1L^{Φ/ν},c_2L^{Φ/ν})/L^{d-1}$ as $L\to\infty$, where $c_i$ are scaling fields associated with the surface-enhancement variables $\mathring{c}_i$, while $Φ$ is a standard surface crossover exponent. The scaling function $D(\mathsf{c}_1,\mathsf{c}_2)$ and its analogue $\mathcal{D}(\mathsf{c}_1,\mathsf{c}_2)$ for the Casimir force are determined via expansion in $ε=4-d$ and extrapolated to $d=3$ dimensions. In the special case $\mathsf{c}_1=\mathsf{c}_2=0$, the expansion becomes fractional. Consistency with the known fractional expansions of D(0,0) and $\mathcal{D}(0,0)$ to order $ε^{3/2}$ is achieved by appropriate reorganisation of RG-improved perturbation theory. For appropriate choices of $c_1$ and $c_2$, the Casimir forces can have either sign. Furthermore, crossovers from attraction to repulsion and vice versa may occur as $L$ increases.

cond-mat.stat-mech↗

arXiv2010

On conjectured local generalizations of anisotropic scale invariance and their implications

The theory of generalized local scale invariance of strongly anisotropic scale invariant systems proposed some time ago by Henkel [Nucl. Phys. B \textbf{641}, 405 (2002)] is examined. The case of so-called type-I systems is considered. This was conjectured to be realized by systems at m-axial Lifshitz points; in support of this claim, scaling functions of two-point cumulants at the uniaxial Lifshitz point of the three-dimensional ANNNI model were predicted on the basis of this theory and found to be in excellent agreement with Monte Carlo results [Phys. Rev. Lett. \textbf{87}, 125702 (2001)]. The consequences of the conjectured invariance equations are investigated. It is shown that fewer solutions than anticipated by Henkel generally exist and contribute to the scaling functions if these equations are assumed to hold for all (positive and negative) values of the d-dimensional space (or space time) coordinates $(t,\bm{r})\in \mathbb{R}\times\mathbb{R}^{d-1}$. Specifically, a single rather than two independent solutions exists in the case relevant for the mentioned fit of Monte Carlo data for the ANNNI model. Renormalization-group improved perturbation theory in $4+m/2-ε$ dimensions is used to determine the scaling functions of the order-parameter and energy-density two-point cumulants in momentum space to two-loop order. The results are mathematically incompatible with Henkel's predictions except in free-field-theory cases. However, the scaling function of the energy-density cumulant we obtain for m=1 upon extrapolation of our two-loop RG results to d=3 differs numerically little from that of an effective free field theory.

cond-mat.stat-mech↗

arXiv2009

Critical Casimir amplitudes for $n$-component $ϕ^4$ models with O(n)-symmetry breaking quadratic boundary terms

Euclidean $n$-component $ϕ^4$ theories whose Hamiltonians are O(n) symmetric except for quadratic symmetry breaking boundary terms are studied in films of thickness $L$. The boundary terms imply the Robin boundary conditions $\partial_nϕ_α=\mathring{c}^{(j)}_αϕ_α$ at the boundary planes $\mathfrak{B}_{j=1,2}$ at $z=0$ and $z=L$. Particular attention is paid to the cases in which $m_j$ of the $n$ variables $\mathring{c}^{(j)}_α$ take the special value $\mathring{c}_{m_j\text{-sp}}$ corresponding to critical enhancement while the remaining ones are subcritically enhanced. Under these conditions, the semi-infinite system bounded by $\mathfrak{B}_j$ has a multicritical point, called $m_j$-special, at which an $O(m_j)$ symmetric critical surface phase coexists with the O(n) symmetric bulk phase, provided $d$ is sufficiently large. The $L$-dependent part of the reduced free energy per area behaves as $Δ_C/L^{d-1}$ as $L\to\infty$ at the bulk critical point. The Casimir amplitudes $Δ_C$ are determined for small $ε=4-d$ in the general case where $m_{c,c}$ components $ϕ_α$ are critically enhanced at both boundary planes, $m_{c,D} + m_{D,c}$ components are enhanced at one plane but satisfy asymptotic Dirichlet boundary conditions at the respective other, and the remaining $m_{D,D}$ components satisfy asymptotic Dirichlet boundary conditions at both $\mathfrak{B}_j$. Whenever $m_{c,c}>0$, these expansions involve integer and fractional powers $ε^{k/2}$ with $k\ge 3$ (mod logarithms). Results to $O(ε^{3/2})$ for general values of $m_{c,c}$, $m_{c,D}+m_{D,c}$, and $m_{D,D}$ are used to estimate the $Δ_C$ of 3D Heisenberg systems with surface spin anisotropies when $(m_{c,c}, m_{c,D}+ m_{D,c}) = (1,0)$, $(0,1)$, and $(1,1)$.

cond-mat.stat-mech↗

arXiv2009

Dynamic critical behavior of model A in films: Zero-mode boundary conditions and expansion near four dimensions

The critical dynamics of relaxational stochastic models with nonconserved $n$-component order parameter $\bmϕ$ and no coupling to other slow variables ("model A") is investigated in film geometries for the cases of periodic and free boundary conditions. The Hamiltonian $\mathcal{H}$ governing the stationary equilibrium distribution is taken to be O(n) symmetric and to involve, in the case of free boundary conditions, the boundary terms $\int_{\mathfrak{B}_j}\mathring{c}_j ϕ^2/2$ associated with the two confining surface planes $\mathfrak{B}_j$, $j=1,2$, at $z=0$ and $z=L$, where the enhancement variables $\mathring{c}_j$ are presumed to be subcritical or critical. A field-theoretic RG study of the dynamic critical behavior at $d=4-ε$ bulk dimensions is presented, with special attention paid to the cases where the classical theories involve zero modes at $T_{c,\infty}$. This applies when either both $\mathring{c}_j$ take the critical value $\mathring{c}_{\text{sp}}$ associated with the special surface transition, or else periodic boundary conditions are imposed. Owing to the zero modes, the $ε$ expansion becomes ill-defined at $T_{c,\infty}$. Analogously to the static case, the field theory can be reorganized to obtain a well-defined small-$ε$ expansion involving half-integer powers of $ε$, modulated by powers of $\lnε$. Explicit results for the scaling functions of $T$-dependent finite-size susceptibilities at temperatures $T\ge T_{c,\infty}$ and of layer and surface susceptibilities at the bulk critical point are given to orders $ε$ and $ε^{3/2}$, respectively. For the case of periodic boundary conditions, the consistency of the expansions to $O(ε^{3/2})$ with exact large-$n$ results is shown.

cond-mat.stat-mech↗

arXiv2008

Crossover from Attractive to Repulsive Casimir Forces and Vice Versa

Systems described by an O(n) symmetrical $ϕ^4$ Hamiltonian are considered in a $d$-dimensional film geometry at their bulk critical points. The critical Casimir forces between the film's boundary planes $\mathfrak{B}_j, j=1,2$, are investigated as functions of film thickness $L$ for generic symmetry-preserving boundary conditions $\partial_n\bmϕ=\mathring{c}_j\bmϕ$. The $L$-dependent part of the reduced excess free energy per cross-sectional area takes the scaling form $f_{\text{res}}\approx D(c_1L^{Φ/ν},c_2L^{Φ/ν})/L^{d-1}$ when $d<4$, where $c_i$ are scaling fields associated with the variables $\mathring{c}_i$, and $Φ$ is a surface crossover exponent. Explicit two-loop renormalization group results for the function $D(\mathsf{c}_1,\mathsf{c}_2)$ at $d=4-ε$ dimensions are presented. These show that (i) the Casimir force can have either sign, depending on $\mathsf{c}_1$ and $\mathsf{c}_2$, and (ii) for appropriate choices of the enhancements $\mathring{c}_j$, crossovers from attraction to repulsion and vice versa occur as $L$ increases.

cond-mat.stat-mech↗

arXiv2008

Thermodynamic Casimir effects involving interacting field theories with zero modes

Systems with an O(n) symmetrical Hamiltonian are considered in a $d$-dimensional slab geometry of macroscopic lateral extension and finite thickness $L$ that undergo a continuous bulk phase transition in the limit $L\to\infty$. The effective forces induced by thermal fluctuations at and above the bulk critical temperature $T_{c,\infty}$ (thermodynamic Casimir effect) are investigated below the upper critical dimension $d^*=4$ by means of field-theoretic renormalization group methods for the case of periodic and special-special boundary conditions, where the latter correspond to the critical enhancement of the surface interactions on both boundary planes. As shown previously [\textit{Europhys. Lett.} \textbf{75}, 241 (2006)], the zero modes that are present in Landau theory at $T_{c,\infty}$ make conventional RG-improved perturbation theory in $4-ε$ dimensions ill-defined. The revised expansion introduced there is utilized to compute the scaling functions of the excess free energy and the Casimir force for temperatures $T\geqT_{c,\infty}$ as functions of $\mathsf{L}\equiv L/ξ_\infty$, where $ξ_\infty$ is the bulk correlation length. Scaling functions of the $L$-dependent residual free energy per area are obtained whose $\mathsf{L}\to0$ limits are in conformity with previous results for the Casimir amplitudes $Δ_C$ to $O(ε^{3/2})$ and display a more reasonable small-$\mathsf{L}$ behavior inasmuch as they approach the critical value $Δ_C$ monotonically as $\mathsf{L}\to 0$.

cond-mat.stat-mech↗

arXiv2008

Compatibility of 1/n and epsilon expansions for critical exponents at m-axial Lifshitz points

The critical behaviour of d-dimensional n-vector models at m-axial Lifshitz points is considered for general values of m in the large-n limit. It is proven that the recently obtained large-N expansions [J. Phys.: Condens. Matter 17, S1947 (2005)] of the correlation exponents η_{L2}, η_{L4} and the related anisotropy exponent θare fully consistent with the dimensionality expansions to second order in ε=4+m/2-d [Phys. Rev. B 62, 12338 (2000); Nucl. Phys. B 612, 340 (2001)] inasmuch as both expansions yield the same contributions of order ε^2/n.

cond-mat.stat-mech↗

arXiv2006

Fluctuation-induced forces in periodic slabs: Breakdown of epsilon expansion at the bulk critical point and revised field theory

Systems described by $n$-component $ϕ^4$ models in a $\infty^{d-1}\times L$ slab geometry of finite thickness $L$ are considered at and above their bulk critical temperature $T_{c,\infty}$. The renormalization-group improved perturbation theory commonly employed to investigate the fluctuation-induced forces (``thermodynamic Casimir effect'') in $d=4-ε$ bulk dimensions is re-examined. It is found to be ill-defined beyond two-loop order because of infrared singularities when the boundary conditions are such that the free propagator in slab geometry involves a zero-energy mode at bulk criticality. This applies to periodic boundary conditions and the special-special ones corresponding to the critical enhancement of the surface interactions on both confining plates. The field theory is reorganized such that a small-$ε$ expansion results which remains well behaved down to $T_{c,\infty}$. The leading contributions to the critical Casimir amplitudes $Δ_{\mathrm{per}}$ and $Δ_{\mathrm{sp},\mathrm{sp}}$ beyond two-loop order are $\sim (u^*)^{(3-ε)/2}$, where $u^*=O(ε)$ is the value of the renormalized $ϕ^4$ coupling at the infrared-stable fixed point. Besides integer powers of $ε$, the small-$ε$ expansions of these amplitudes involve fractional powers $ε^{k/2}$, with $k\geq 3$, and powers of $\ln ε$. Explicit results to order $ε^{3/2}$ are presented for $Δ_{\mathrm{per}}$ and $Δ_{\mathrm{sp},\mathrm{sp}}$, which are used to estimate their values at $d=3$.

cond-mat.stat-mech↗