We study the crossover from low- to high-temperature fluctuations including critical fluctuations in confined isotropic O$(n)$-symmetric systems on the basis of a finite-size renormalization-group approach at fixed dimension $d$ introduced previously [V. Dohm, Phys. Rev. Lett. {\bf 110}, 107207 (2013)]. Our theory is formulated within the $φ^4$ lattice model in a $d$-dimensional block geometry with periodic boundary conditions. We derive the finite-size scaling functions $F^{\text ex}$ and $X$ of the excess free energy density and of the thermodynamic Casimir force, respectively, for $1\leq n \leq \infty$, $2<d<4$. Applications are given for $ L_\parallel^{d-1} \times L$ slab geometries with a finite aspect ratio $ρ=L/L_\parallel$ as well as for the film limit $ρ\to 0$ at fixed $L$. For $n=1$ and $ρ=0$ the low-temperature limits of $F^{\text ex}$ and $X$ vanish whereas they are finite for $n\geq 2$ and $ρ= 0$ due to the effect of the Goldstone modes. For $n=1$ and $ρ>0$ we find a finite low-temperature limit of $F^{\text ex}$ which deviates from that of the the Ising model. We attribute this deviation to the nonuniversal difference between the $φ^4$ model with continuous variables $φ$ and the Ising model with discrete spin variables $s=\pm1$. For $n\geq 2$ and $ρ>0$, a logarithmic divergence of $F^{\text ex}$ in the low-temperature limit is predicted, in excellent agreement with Monte Carlo (MC) data for the $d=3$ $XY$ model. For $2\leq n \leq \infty$ and $0\leq ρ<ρ_0=0.8567$ the Goldstone modes generate a negative (attractive) low-temperature Casimir force that vanishes for $ρ= ρ_0$ and becomes positive (repulsive) for $ρ> ρ_0$. Our predictions are compared with MC data for Ising, $XY$, and Heisenberg models in slab geometries with $0.01\leqρ\leq1$. Good overall agreement is found.
In confined systems near a continuous phase transition the long-ranged fluctuations of the corresponding order parameter are subject to boundary conditions. These constraints result in so-called critical Casimir forces acting as effective forces on the confining surfaces. For systems belonging to the Ising bulk universality class corresponding to a scalar order parameter the critical Casimir force is studied for the film geometry in the crossover regime characterized by different surface fields at the two surfaces. The scaling function of the critical Casimir force is calculated within mean fi…
We study the crossover between thermodynamic Casimir forces arising from long-range fluctuations due to Goldstone modes and those arising from critical fluctuations. Both types of forces exist in the low-temperature phase of O$(n)$ symmetric systems for $n>1$ in a $d$-dimensional ${L_\parallel^{d-1} \times L}$ slab geometry with a finite aspect ratio $ρ= L/L_\parallel$. Our finite-size renormalization-group treatment for periodic boundary conditions describes the entire crossover from the Goldstone regime with a nonvanishing constant tail of the finite-size scaling function far below $T_c$ up …
We study the crossover from the ordinary to the normal surface universality class in the three-dimensional Ising universality class. This crossover is relevant for the behavior of films of binary mixtures near the demixing point and a weak adsorption at one or both surfaces. We perform Monte Carlo simulations of the improved Blume-Capel model on the simple cubic lattice. We consider systems with film geometry, where various boundary conditions are applied. We discuss corrections to scaling that are caused by the surfaces and their relation with the so called extrapolation length. To this end w…
To study the critical Casimir force between chemically structured boundaries immersed in a binary mixture at its demixing transition, we consider a strip of Ising spins subject to alternating fixed spin boundary conditions. The system exhibits a boundary induced phase transition as function of the relative amount of up and down boundary spins. This transition is associated with a sign change of the asymptotic force and a diverging correlation length that sets the scale for the crossover between different universal force amplitudes. Using conformal field theory and a mapping to Majorana fermion…
The finite-size renormalization-group approach for isotropic O$(n)$-symmetric systems introduced previously [V. Dohm, Phys. Rev. Lett. {\bf 110}, 107207 (2013)] is extended to weakly anisotropic O$(n)$-symmetric systems. Our theory is formulated within the $φ^4$ model with lattice anisotropy in a $d$-dimensional block geometry with periodic boundary conditions. It describes the crossover from low- to high-temperature fluctuations including Goldstone-dominated and critical fluctuations for $1\leq n \leq \infty$ in $2<d<4$ dimensions. An exact representation is derived for the large-distance beh…
We consider the Bose gas on a $d$-dimensional anisotropic lattice employing the imperfect (mean-field) gas as a prototype example. We study the dimensional crossover arising as a result of varying the dispersion relation at finite temperature $T$. We analyze in particular situations where one of the relevant effective dimensionalities is located at or below the lower critical dimension, so that the Bose-Einstein condensate becomes expelled from the system by anisotropically modifying the lattice parameters controlling the kinetic term in the Hamiltonian. We clarify the mechanism governing this…
We study fluctuation-induced interaction in confined fluids above the isotropic-lamellar transition. At an ideal continuous transition, the disjoining pressure has the asymptotic form $Π(d\to\infty)\approx -C k_BT q_0^2/d$, where $d$ is the interwall distance, $q_0$ is the wavenumber of the scattering peak, and $C= 1/(4π)$ in the strong anchoring limit. The long-rangedness is enhanced due to continuous distribution of soft modes in the ${\bf q}$-space. An unconventionally strong Casimir force with a range of several lamella thicknesses is realistic above the transition. We also find an oscilla…
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 variab…
Universal aspects of the thermodynamic Casimir effect in wetting films of 3He-4He mixtures near their bulk tricritical point are studied within suitable models serving as representatives of the corresponding universality class. The effective forces between the boundaries of such films arising from the confinement are calculated along isotherms at several fixed concentrations of 3He. Nonsymmetric boundary conditions impose nontrivial concentration profiles leading to repulsive Casimir forces which exhibit a rich behavior of the crossover between the tricritical point and the line of critical po…
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 eig…
By employing the methods of renormalized field theory we show that the percolation behavior of random resistor-diode networks near the multicritical line belongs to the universality class of isotropic percolation. We construct a mesoscopic model from the general epidemic process by including a relevant isotropy-breaking perturbation. We present a two-loop calculation of the crossover exponent $ϕ$. Upon blending the $ε$-expansion result with the exact value $ϕ=1$ for one dimension by a rational approximation, we obtain for two dimensions $ϕ= 1.29\pm 0.05$. This value is in agreement with the re…
The elastic interactions between objects embedded in a nematic liquid crystal are usually caused by the average distorsion-rather than by the fluctuations-of the nematic orientational field. We argue that for sufficiently small particles, the nematic-mediated interaction originates purely from the fluctuations of the nematic director. This Casimir interaction decays as d^(-6), d being the distance between the particles, and it dominates van der Waals interactions close to the isotropic-to-nematic transition. Considering the nematic as a polymer solvent, we show that the onset of this Casimir i…
We consider the d-dimensional imperfect (mean-field) Bose gas confined in a slit-like geometry and subject to periodic boundary conditions. Within an exact analytical treatment we first extract the bulk critical properties of the system at Bose-Einstein condensation and identify the bulk universality class to be the one of the classical d-dimensional spherical model. Subsequently we consider finite slit width D and analyze the excess surface free energy and the related Casimir force acting between the slit boundaries. Above the bulk condensation temperature (T>T_c) the Casimir force decays exp…
We generalize the directed percolation (DP) model by relaxing the strict directionality of DP such that propagation can occur in either direction but with anisotropic probabilities. We denote the probabilities as $p_{\downarrow}= p \cdot p_d$ and $p_{\uparrow}=p \cdot (1-p_d)$, with $p $ representing the average occupation probability and $p_d$ controlling the anisotropy. The Leath-Alexandrowicz method is used to grow a cluster from an active seed site. We call this model with two main growth directions {\em biased directed percolation} (BDP). Standard isotropic percolation (IP) and DP are the…
We study the critical behavior of the free energy and the thermodynamic Casimir force in a $L_\parallel^{d-1} \times L$ block geometry in $2<d<4$ dimensions with aspect ratio $ρ=L/L_\parallel$ above, at, and below $T_c$ on the basis of the O$(n)$ symmetric $ϕ^4$ lattice model with periodic boundary conditions (b.c.). We consider a simple-cubic lattice with isotropic short-range interactions. Exact results are derived in the large - $n$ limit describing the geometric crossover from film ($ρ=0$) over cubic $ρ=1$ to cylindrical ($ρ= \infty$) geometries. For $n=1$, three perturbation approaches ar…
On the example of the spherical model we study, as a function of the temperature $T$, the behavior of the Casimir force in O(n) systems with a diffuse interface and slab geometry $\infty^{d-1}\times L$, where $2<d<4$ is the dimensionality of the system. We consider a system with nearest-neighbor anisotropic interaction constants $J_\parallel$ parallel to the film and $J_\perp$ across it. The model represents the $n\to\infty$ limit of O(n) models with antiperiodic boundary conditions applied across the finite dimension $L$ of the film. We observe that the Casimir amplitude $Δ_{\rm Casimir}(d|J_…
A two-loop renormalization group analysis of the critical behaviour at an isotropic Lifshitz point is presented. Using dimensional regularization and minimal subtraction of poles, we obtain the expansions of the critical exponents $ν$ and $η$, the crossover exponent $ϕ$, as well as the (related) wave-vector exponent $β_q$, and the correction-to-scaling exponent $ω$ to second order in $ε_8=8-d$. These are compared with the authors' recent $ε$-expansion results [{\it Phys. Rev. B} {\bf 62} (2000) 12338; {\it Nucl. Phys. B} {\bf 612} (2001) 340] for the general case of an $m$-axial Lifshitz point…
The Casimir force between macroscopic bodies depends strongly on their shape and orientation. To study this geometry dependence in the case of two deformed metal plates, we use a path integral quantization of the electromagnetic field which properly treats the many-body nature of the interaction, going beyond the commonly used pairwise summation (PWS) of van der Waals forces. For arbitrary deformations we provide an analytical result for the deformation induced change in Casimir energy, which is exact to second order in the deformation amplitude. For the specific case of sinusoidally corrugate…
The exact critical Casimir amplitude is derived for anisotropic systems within the $d=2$ Ising universality class by combining conformal field theory (CFT) with anisotropic $φ^4$ theory. Explicit results are presented for the general anisotropic scalar $φ^4$ model and for the fully anisotropic triangular-lattice Ising model in finite rectangular and infinite strip geometries with periodic boundary conditions (PBC). These results demonstrate the validity of multiparameter universality for confined anisotropic systems and the nonuniversality of the critical Casimir amplitude. We find an unexpect…
The proximity force approximation (PFA) relates the interaction between closely spaced, smoothly curved objects to the force between parallel plates. Precision experiments on Casimir forces necessitate, and spur research on, corrections to the PFA. We use a derivative expansion for gently curved surfaces to derive the leading curvature modifications to the PFA. Our methods apply to any homogeneous and isotropic materials; here we present results for Dirichlet and Neumann boundary conditions and for perfect conductors. A Padé extrapolation constrained by a multipole expansion at large distance …
The proceeding comment raises a few points concerning our paper Dantchev \textit{et al.}, Phys. Rev. E. {\bf 89}, 042116 (2014). In this reply we stress that while Refs. Diehl \textit{et al.} EPL {\bf 100}, 10004 (2012) and Phys. Rev. E. {\bf 89}, 062123 (2014) use three different models to study the the Casimir force for the $O(n \rightarrow \infty)$ model with free boundary conditions we study a single model over the entire range of temperatures, from above the bulk critical temperature, $T_c$, to absolute temperatures down to $T=0$. The use of a single model renders more transparent the cro…
The surface scaling behavior of classical isotropic Heisenberg magnets is investigated by Monte - Carlo methods in d=3 dimensions for various values of the surface - to - bulk coupling ratio J_1/J. For J_1/J <= 1.0 critical behavior according to the ordinary surface universality class is found. New estimates for magnetic surface exponents are presented and compared to older estimates and their theoretical counterparts. For J_1/J >= 2.0 scaling is still valid with effective exponents which depend on J_1/J. The surface structure factor S_1(p,L) is investigated at bulk criticality as function of …
We consider the fluctuation-induced interaction of two thin, rod-like particles or "needles" immersed in a two-dimensional critical fluid of Ising symmetry right at the critical point. Conformally mapping the plane containing the needles onto a simpler geometry in which the stress tensor is known, we analyze the force and torque between needles of arbitrary length, separation, and orientation. For infinite and semi-infinite needles we utilize the mapping of the plane bounded by the needles onto the half plane, and for two needles of finite length the mapping onto an annulus. For semi-infinite …
We study the geometry dependence of the Casimir energy for deformed metal plates by a path integral quantization of the electromagnetic field. For the first time, we give a complete analytical result for the deformation induced change in Casimir energy δ\cal E in an experimentally testable, nontrivial geometry, consisting of a flat and a corrugated plate. Our results show an interesting crossover for δ\cal E as a function of the ratio of the mean plate distance H, to the corrugation length λ: For λ\ll H we find a {\em slower} decay \sim H^{-4}, compared to the H^{-5} behavior predicted by the …
Finite-size effects are investigated in the Gaussian model with isotropic and anisotropic short-range interactions in film geometry with nonperiodic boundary conditions (b.c). We have obtained exact results for the free energy and the Casimir force for antiperiodic, Neumann, Dirichlet, and Neumann-Dirichlet mixed b.c. in 1<d<4 dimensions. For the Casimir force, finite-size scaling is found to be valid for all b.c.. For the free energy, finite-size scaling is valid in 1<d<3 and 3<d<4 dimensions for antiperiodic, Neumann, and Dirichlet b.c., but logarithmic deviations from finite-size scaling ex…
We study the critical Casimir force for a film geometry in the Ising universality class. We employ a homogeneous adsorption preference on one of the confining surfaces, while the opposing surface exhibits quenched random disorder, leading to a random local adsorption preference. Disorder is characterized by a parameter p, which measures, on average, the portion of the surface that prefers one component, so that p=0, 1 correspond to homogeneous adsorption preference. By means of Monte Carlo simulations of an improved Hamiltonian and finite-size scaling analysis, we determine the critical Casimi…
We analyze the thermodynamic Casimir effect occurring in a gas of non-interacting bosons confined by two parallel walls with a strongly anisotropic dispersion inherited from an underlying lattice. In the direction perpendicular to the confining walls the standard quadratic dispersion is replaced by the term $|{\bf p}|^α$ with $α\geq 2$ treated as a parameter. We derive a closed, analytical expression for the Casimir force depending on the dimensionality $d$ and the exponent $α$, and analyze it for thermodynamic states in which the Bose-Einstein condensate is present. For $α\in\{4,6,8,\dots\}$ …
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 interac…
We present exact results on the behavior of the thermodynamic Casimir force and the excess free energy in the framework of the $d$-dimensional spherical model with a power law long-range interaction decaying at large distances $r$ as $r^{-d-σ}$, where $σ<d<2σ$ and $0<σ\leq2$. For a film geometry and under periodic boundary conditions we consider the behavior of these quantities near the bulk critical temperature $T_c$, as well as for $T>T_c$ and $T<T_c$. The universal finite-size scaling function governing the behavior of the force in the critical region is derived and its asymptotics are inve…
Equilibrium properties of the three-dimensional isotropic Heisenberg spin glass are studied by extensive Monte Carlo simulations, with particular attention to the nature of its phase transition. A finite-size-scaling analysis is performed both for the spin-glass (SG) and the chiral-glass (CG) orders. Our results suggest that the model exhibits the CG long-range order at finite temperatures without accompanying the conventional SG long-range order, in contrast to some of the recent works claiming the simultaneous SG and CG transition. Typical length and time scales which represent a crossover f…
We study the effective interaction between two ellipsoidal particles at the interface of two fluid phases which are mediated by thermal fluctuations of the interface. In this system the restriction of the long--ranged interface fluctuations by particles gives rise to fluctuation--induced forces which are equivalent to interactions of Casimir type and which are anisotropic in the interface plane. Since the position and the orientation of the colloids with respect to the interface normal may also fluctuate, this system is an example for the Casimir effect with fluctuating boundary conditions. In…
Casimir Sign Reversal from Finite-Range Bogoliubov Fluctuations
Universidad San Sebastián（圣塞巴斯蒂安大学）；Centro de Estudios Científicos (CECs)（科学研究中心）；Instituto de Ciencias Físicas y Matemáticas, Universidad Austral de Chile（智利南方大学物理与数学科学研究所）；IPNP - Faculty of Mathematics and Physics, Charles University（查理大学数学与物理学院）；INFN, Sezione di Napoli（意大利国家核物理研究院那不勒斯分部）；Dipartimento di Matematica e Applicazioni “R. Caccioppoli,” Universitá di Napoli Federico II（那不勒斯费德里科二世大学应用数学系）
The scaling limit of the Heisenberg XXZ spin chain at zero magnetic field is studied in the gapped antiferromagnetic phase. For a spin-chain ring having $N_x$ sites, the universal Casimir scaling function, which characterises the leading finite-size correction term in the large-$N_x$ expansion of the ground state energy, is calculated by numerical solution of the nonlinear integral equation of the convolution type. It is shown, that the same scaling function describes the temperature dependence of the free energy of the infinite XXZ chain at low enough temperatures in the gapped scaling regime…
We study dimensional crossover in Ising systems at complex temperatures by comparing three types of system: the infinite isotropic 2D Ising model; the infinite anisotropic 2D Ising model; and Ising ladders with a finite number of legs. In particular we present evidence, from both tensor-network calculations and numerical evaluations based on the exact solution of the model, that the infinite anisotropic 2D Ising model exhibits long-range spatially modulated magnetization in certain regions of the complex-temperature plane. We discuss the physics of the special unitary points that exists in the…
High-resolution ac-calorimetry has been carried out on dispersions of aerosils in the liquid crystal octyloxycyanobiphenyl (8OCB) as a function of aerosil concentration and temperature spanning the crystal to isotropic phases. The liquid-crystal 8OCB is elastically stiffer than the previously well studied octylcyanobiphenyl (8CB)+aerosil system and so, general quenched random disorder effects and liquid-crystal specific effects can be distinguished. A double heat capacity feature is observed at the isotropic to nematic phase transition with an aerosil independent overlap of the heat capacity w…
We have experimentally quantified the temporal structural diversity from the coordinate fluctuations of a laser beam propagating through isotropic optical turbulence. The main focus here is on the characterization of the long-range correlations in the wandering of a thin Gaussian laser beam over a screen after propagating through a turbulent medium. To fulfill this goal, a laboratory-controlled experiment was conducted in which coordinate fluctuations of the laser beam were recorded at a sufficiently high sampling rate for a wide range of turbulent conditions. Horizontal and vertical displacem…
The van der Waals force established between two surfaces plays a central role in many phenomena, such as adhesion or friction. However, the dependence of this forces on the distance of separation between plates is very complex. Two widely different non-retarded and retarded regimes are well known, but these have been traditionally studied separately. Much less is known about the important experimentally accesible cross-over regime. In this study, we provide analytical approximations for the van der Waals forces between two plates that interpolates exactly between the short distance and long di…
We consider the Heisenberg antiferromagnet on the square lattice with S=1/2 and very weak easy-plane exchange anisotropy; by means of the quantum Monte Carlo method, based on the continuous-time loop algorithm, we find that the thermodynamics of the model is highly sensitive to the presence of tiny anisotropies and is characterized by a crossover between isotropic and planar behaviour. We discuss the mechanism underlying the crossover phenomenon and show that it occurs at a temperature which is characteristic of the model. The expected Berezinskii-Kosterlitz-Thouless transition is observed bel…
