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Dohm V. Multiparameter universality and intrinsic diversity of critical phenomena in weakly anisotropic systems. Phys Rev E 2023; 108:044149. [PMID: 37978693 DOI: 10.1103/physreve.108.044149] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/26/2023] [Accepted: 09/19/2023] [Indexed: 11/19/2023]
Abstract
Recently a unified hypothesis of multiparameter universality for the critical behavior of bulk and confined anisotropic systems has been formulated [V. Dohm, Phys. Rev. E 97, 062128 (2018)2470-004510.1103/PhysRevE.97.062128]. We prove the validity of this hypothesis in d≥2 dimensions on the basis of the principle of two-scale-factor universality for isotropic systems at vanishing external field. We introduce an angular-dependent correlation vector and a generalized shear transformation that transforms weakly anisotropic systems to isotropic systems. As examples we consider the O(n)-symmetric φ^{4} model, Gaussian model, and n-vector model. By means of the inverse of the shear transformation we determine the general structure of the bulk order-parameter correlation function, of the singular bulk part of the critical free energy, and of critical bulk amplitude relations of anisotropic systems at and away from T_{c}. It is shown that weakly anisotropic systems exhibit a high degree of intrinsic diversity due to d(d+1)/2-1 independent parameters that cannot be determined by thermodynamic measurements. Exact results are derived for the d=2 Ising universality class and for the spherical and Gaussian universality classes in d≥2 dimensions. For the d=3 Ising universality class we identify the universal scaling function of the isotropic bulk correlation function from the nonuniversal result of the functional renormalization group. A proof is presented for the validity of multiparameter universality of the exact critical free energy and critical Casimir amplitude in a finite rectangular geometry of weakly anisotropic systems with periodic boundary conditions in the Ising universality class. This confirms the validity of recent predictions of self-similar structures of finite-size effects in the (d=2,n=1) universality class at T=T_{c} derived from conformal field theory [V. Dohm and S. Wessel, Phys. Rev. Lett. 126, 060601 (2021)PRLTAO0031-900710.1103/PhysRevLett.126.060601]. This also substantiates the previous notion of an effective shear transformation for anisotropic two-dimensional Ising models. Our theory paves the way for a quantitative theory of nonuniversal critical Casimir forces in anisotropic superconductors for which experiments have been proposed by G. A. Williams [Phys. Rev. Lett. 92, 197003 (2004)PRLTAO0031-900710.1103/PhysRevLett.92.197003].
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Affiliation(s)
- Volker Dohm
- Institute for Theoretical Physics, RWTH Aachen University, 52056 Aachen, Germany
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Dohm V. Crossover from low-temperature to high-temperature fluctuations: Universal and nonuniversal Casimir forces of isotropic and anisotropic systems. Phys Rev E 2018; 97:062128. [PMID: 30011477 DOI: 10.1103/physreve.97.062128] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/17/2017] [Indexed: 11/07/2022]
Abstract
We study the crossover from low-temperature to high-temperature fluctuations including Goldstone-dominated and critical fluctuations in confined isotropic and weakly anisotropic 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. 110, 107207 (2013)PRLTAO0031-900710.1103/PhysRevLett.110.107207]. Our theory is formulated within the φ^{4} lattice model in a d-dimensional block geometry with periodic boundary conditions. We calculate the finite-size scaling functions F^{ex} and X of the excess free-energy density and the thermodynamic Casimir force, respectively, for 1≤n≤∞, 2<d<4. Exact results are derived for n→∞. Applications are given for L_{∥}^{d-1}×L slab geometry with an aspect ratio ρ=L/L_{∥}>0 and for film geometry (ρ=0). Good overall agreement is found with Monte Carlo (MC) data for isotropic spin models with n=1,2,3. For ρ=0, the low-temperature limits of F^{ex} and X vanish for n=1, whereas they are finite for n≥2. For ρ>0 and n=1, we find a finite low-temperature limit of F^{ex}, which deviates from that of the Ising model. We attribute this deviation to the nonuniversal difference between the φ^{4} model with continuous variables and the Ising model with discrete variables. For n≥2 and ρ>0, a logarithmic divergence of F^{ex} in the low-temperature limit is predicted, in excellent agreement with MC data. For 2≤n≤∞ and ρ<ρ_{0}=0.8567 the Goldstone modes generate a negative low-temperature Casimir force that vanishes for ρ=ρ_{0} and becomes positive for ρ>ρ_{0}. For anisotropic systems a unified hypothesis of multiparameter universality is introduced for both bulk and confined systems. The dependence of their scaling functions on d(d+1)/2-1 microscopic anisotropy parameters implies a substantial reduction of the predictive power of the theory for anisotropic systems as compared to isotropic systems. An exact representation is derived for the nonuniversal large-distance behavior of the bulk correlation function of anisotropic systems and quantitative predictions are made. The validity of multiparameter universality is proven analytically for the d=2,n=1 universality class. A nonuniversal anisotropy-dependent minimum of the Casimir force scaling function X is found. Both the sign and magnitude of X and the shift of the film critical temperature are affected by the lattice anisotropy.
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Affiliation(s)
- Volker Dohm
- Institute for Theoretical Physics, RWTH Aachen University, D-52056 Aachen, Germany
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Pelissetto A, Tripodo A, Vicari E. Criticality of O(N) symmetric models in the presence of discrete gauge symmetries. Phys Rev E 2018; 97:012123. [PMID: 29448441 DOI: 10.1103/physreve.97.012123] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/16/2017] [Indexed: 06/08/2023]
Abstract
We investigate the critical properties of the three-dimensional antiferromagnetic RP^{N-1} model, which is characterized by a global O(N) symmetry and a discrete Z_{2} gauge symmetry. We perform a field-theoretical analysis using the Landau-Ginzburg-Wilson (LGW) approach and a numerical Monte Carlo study. The LGW field-theoretical results are obtained by high-order perturbative analyses of the renormalization-group flow of the most general Φ^{4} theory with the same global symmetry as the model, assuming a gauge-invariant order-parameter field. For N=4 no stable fixed point is found, implying that any transition must necessarily be of first order. This is contradicted by the numerical results that provide strong evidence for a continuous transition. This suggests that gauge modes are not always irrelevant, as assumed by the LGW approach, but they may play an important role to determine the actual critical dynamics at the phase transition of O(N) symmetric models with a discrete Z_{2} gauge symmetry.
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Affiliation(s)
- Andrea Pelissetto
- Dipartimento di Fisica dell'Università di Roma "La Sapienza" and INFN, Sezione di Roma I, I-00185 Rome, Italy
| | - Antonio Tripodo
- Dipartimento di Fisica dell'Università di Pisa and INFN, Sezione di Pisa, I-56127 Pisa, Italy
| | - Ettore Vicari
- Dipartimento di Fisica dell'Università di Pisa and INFN, Sezione di Pisa, I-56127 Pisa, Italy
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Delfino F, Pelissetto A, Vicari E. Three-dimensional antiferromagnetic CP(N-1) models. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:052109. [PMID: 26066121 DOI: 10.1103/physreve.91.052109] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/03/2015] [Indexed: 06/04/2023]
Abstract
We investigate the critical behavior of three-dimensional antiferromagnetic CP(N-1) (ACP(N-1)) models in cubic lattices, which are characterized by a global U(N) symmetry and a local U(1) gauge symmetry. Assuming that critical fluctuations are associated with a staggered gauge-invariant (Hermitian traceless matrix) order parameter, we determine the corresponding Landau-Ginzburg-Wilson (LGW) model. For N=3 this mapping allows us to conclude that the three-component ACP(2) model undergoes a continuous transition that belongs to the O(8) vector universality class, with an effective enlargement of the symmetry at the critical point. This prediction is confirmed by numerical analyses of the finite-size scaling behaviors of the ACP(2) and the O(8) vector models, which show the same universal features at their transitions. We also present a renormalization-group (RG) analysis of the LGW theories for N≥4. We compute perturbative series in two different renormalization schemes and analyze the corresponding RG flow. We do not find stable fixed points that can be associated with continuous transitions.
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Affiliation(s)
- Francesco Delfino
- Dipartimento di Fisica dell'Università di Pisa and INFN, Largo Pontecorvo 3, I-56127 Pisa, Italy
| | - Andrea Pelissetto
- Dipartimento di Fisica dell'Università di Roma "La Sapienza" and INFN, Sezione di Roma I, I-00185 Roma, Italy
| | - Ettore Vicari
- Dipartimento di Fisica dell'Università di Pisa and INFN, Largo Pontecorvo 3, I-56127 Pisa, Italy
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Dohm V. Pronounced minimum of the thermodynamic Casimir forces of O(n) symmetric film systems: analytic theory. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:030101. [PMID: 25314378 DOI: 10.1103/physreve.90.030101] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/07/2014] [Indexed: 06/04/2023]
Abstract
Thermodynamic Casimir forces of film systems in the O(n) universality classes with Dirichlet boundary conditions are studied below bulk criticality. Substantial progress is achieved in resolving the long-standing problem of describing analytically the pronounced minimum of the scaling function observed experimentally in ^{4}He films (n=2) by Garcia and Chan [Phys. Rev. Lett. 83, 1187 (1999)] and in Monte Carlo simulations for the three-dimensional Ising model (n=1) by O. Vasilyev et al. [Europhys. Lett. 80, 60009 (2007)]. Our finite-size renormalization-group approach describes the film systems as the limit of finite-slab systems with vanishing aspect ratio. This yields excellent agreement with the depth and the position of the minimum for n=1 and semiquantitative agreement with the minimum for n=2. Our theory also predicts a pronounced minimum for the n=3 Heisenberg universality class.
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Affiliation(s)
- Volker Dohm
- Institute for Theoretical Physics, RWTH Aachen University, D-52056 Aachen, Germany
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Dohm V. Crossover from Goldstone to critical fluctuations: Casimir forces in confined O(n)-symmetric systems. PHYSICAL REVIEW LETTERS 2013; 110:107207. [PMID: 23521294 DOI: 10.1103/physrevlett.110.107207] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/11/2013] [Indexed: 06/01/2023]
Abstract
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(||)(d-1) × L slab geometry with a finite aspect ratio ρ = L/L(||). 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 to the region far above T(c) including the critical regime with a minimum of the scaling function slightly below T(c). Our analytic result for ρ << 1 agrees well with Monte Carlo data for the three-dimensional XY model. A quantitative prediction is given for the crossover of systems in the Heisenberg universality class.
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Affiliation(s)
- Volker Dohm
- Institute for Theoretical Physics, RWTH Aachen University, D-52056 Aachen, Germany
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Folk R, Holovatch Y, Moser G. Field theory of bicritical and tetracritical points. IV. Critical dynamics including reversible terms. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:021143. [PMID: 22463188 DOI: 10.1103/physreve.85.021143] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/08/2011] [Indexed: 05/31/2023]
Abstract
This article concludes a series of papers [Folk, Holovatch, and Moser, Phys. Rev. E 78, 041124 (2008); 78, 041125 (2008); 79, 031109 (2009)] where the tools of the field theoretical renormalization group were employed to explain and quantitatively describe different types of static and dynamic behavior in the vicinity of multicritical points. Here we give the complete two-loop calculation and analysis of the dynamic renormalization-group flow equations at the multicritical point in anisotropic antiferromagnets in an external magnetic field. We find that the time scales of the order parameters characterizing the parallel and perpendicular ordering with respect to the external field scale in the same way. This holds independent whether the Heisenberg fixed point or the biconical fixed point in statics is the stable one. The nonasymptotic analysis of the dynamic flow equations shows that due to cancellation effects the critical behavior is described, in distances from the critical point accessible to experiments, by the critical behavior qualitatively found in one-loop order. Although one may conclude from the effective dynamic exponents (taking almost their one-loop values) that weak scaling for the order parameter components is valid, the flow of the time-scale ratios is quite different, and they do not reach their asymptotic values.
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Affiliation(s)
- R Folk
- Institute for Theoretical Physics, Johannes Kepler University Linz, Altenbergerstrasse 69, A-4040 Linz, Austria.
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Dohm V. Critical free energy and Casimir forces in rectangular geometries. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:021108. [PMID: 21928950 DOI: 10.1103/physreve.84.021108] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/30/2010] [Indexed: 05/31/2023]
Abstract
We study the critical behavior of the free energy and the thermodynamic Casimir force in a L(∥)(d-1) × L block geometry in 2<d<4 dimensions with aspect ratio ρ=L/L(∥) on the basis of the O(n) symmetric ϕ4 lattice model with periodic boundary conditions and 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=∞) geometries. For n=1, three perturbation approaches in the minimal renormalization scheme at fixed d are presented that cover both the central finite-size regime near T(c) for 1/4≲ρ≲3 and the region well above and below T(c). At bulk T(c), we predict the critical Casimir force in the vertical (L) direction to be negative (attractive) for a slab (ρ<1), positive (repulsive) for a rod (ρ>1), and zero for a cube (ρ=1). Our results for finite-size scaling functions agree well with Monte Carlo data for the three-dimensional Ising model by Hasenbusch for ρ=1 and by Vasilyev et al. for ρ=1/6 above, at, and below T(c).
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Affiliation(s)
- Volker Dohm
- Institute for Theoretical Physics, RWTH Aachen University, D-52056 Aachen, Germany
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Kastening B, Dohm V. Finite-size effects in film geometry with nonperiodic boundary conditions: Gaussian model and renormalization-group theory at fixed dimension. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:061106. [PMID: 20866377 DOI: 10.1103/physreve.81.061106] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/08/2009] [Revised: 04/06/2010] [Indexed: 05/29/2023]
Abstract
Finite-size effects are investigated in the Gaussian model with isotropic and anisotropic short-range interactions in film geometry with nonperiodic boundary conditions (bc) above, at, and below the bulk critical temperature Tc. We have obtained exact results for the free energy and the Casimir force for antiperiodic, Neumann, Dirichlet, and Neumann-Dirichlet mixed bc in 1<d<4 dimensions. For the Casimir force, finite-size scaling is found to be valid for all bc. For the free energy, finite-size scaling is valid in 1<d<3 and 3<d<4 dimensions for antiperiodic, Neumann, and Dirichlet bc, but logarithmic deviations from finite-size scaling exist in d=3 dimensions for Neumann and Dirichlet bc. This is explained in terms of the borderline dimension d*=3 , where the critical exponent 1-α-ν=(d-3)∕2 of the Gaussian surface energy density vanishes. For Neumann-Dirichlet bc, finite-size scaling is strongly violated above Tc for 1<d<4 because of a cancelation of the leading scaling terms. For antiperiodic, Dirichlet, and Neumann-Dirichlet bc, a finite film critical temperature Tc,film(L)<Tc exists at finite film thickness L . Our results include an exact description of the dimensional crossover between the d -dimensional finite-size critical behavior near bulk Tc and the (d-1) -dimensional critical behavior near Tc,film(L). This dimensional crossover is illustrated for the critical behavior of the specific heat. Particular attention is paid to an appropriate representation of the free energy in the region Tc,film(L)≤T≤Tc. For 2<d<4 , the Gaussian results are renormalized and reformulated as one-loop contributions of the φ4 field theory at fixed dimension d and are then compared with the ε=4-d expansion results at ε=1 as well as with d=3 Monte Carlo data. For d=2 , the Gaussian results for the Casimir force scaling function are compared with those for the Ising model with periodic, antiperiodic, and free bc; unexpected exact relations are found between the Gaussian and Ising scaling functions. For both the d -dimensional Gaussian model and the two-dimensional Ising model it is shown that anisotropic couplings imply nonuniversal scaling functions of the Casimir force that depend explicitly on microscopic couplings. Our Gaussian results provide the basis for the investigation of finite-size effects of the mean spherical model in film geometry with nonperiodic bc above, at, and below the bulk critical temperature.
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Affiliation(s)
- Boris Kastening
- Institute for Theoretical Physics, RWTH Aachen University, 52056 Aachen, Germany.
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Folk R, Holovatch Y, Moser G. Field theory of bicritical and tetracritical points. III. Relaxational dynamics including conservation of magnetization (model C). PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:031109. [PMID: 19391904 DOI: 10.1103/physreve.79.031109] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/01/2008] [Indexed: 05/27/2023]
Abstract
We calculate the relaxational dynamical critical behavior of systems of O(n_{ parallel}) plus sign in circleO(n_{ perpendicular}) symmetry including conservation of magnetization by renormalization group theory within the minimal subtraction scheme in two-loop order. Within the stability region of the Heisenberg fixed point and the biconical fixed point, strong dynamical scaling holds, with the asymptotic dynamical critical exponent z=2varphinu-1 , where varphi is the crossover exponent and nu the exponent of the correlation length. The critical dynamics at n_{ parallel}=1 and n_{ perpendicular}=2 is governed by a small dynamical transient exponent leading to nonuniversal nonasymptotic dynamical behavior. This may be seen, e.g., in the temperature dependence of the magnetic transport coefficients.
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Affiliation(s)
- R Folk
- Institute for Theoretical Physics, Johannes Kepler University Linz, Altenbergerstrasse 69, A-4040, Linz, Austria.
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Folk R, Holovatch Y, Moser G. Field theory of bicritical and tetracritical points. I. Statics. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:041124. [PMID: 18999396 DOI: 10.1103/physreve.78.041124] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/06/2008] [Indexed: 05/27/2023]
Abstract
We calculate the static critical behavior of systems of O(n_||)(plus sign in circle)O(n_perpendicular) symmetry by the renormalization group method within the minimal subtraction scheme in two-loop order. Summation methods lead to fixed points describing multicritical behavior. Their stability border lines in the space of the order parameter components n_|| and n_perpendicular and spatial dimension d are calculated. The essential features obtained already in two-loop order for the interesting case of an antiferromagnet in a magnetic field ( n_|| =1, n_perpendicular =2 ) are the stability of the biconical fixed point and the neighborhood of the stability border lines to the other fixed points, leading to very small transient exponents. We are also able to calculate the flow of static couplings, which allows us to consider the attraction region. Depending on the nonuniversal background parameters, the existence of different multicritical behavior (bicritical or tetracritical) is possible, including a triple point.
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Affiliation(s)
- R Folk
- Institute for Theoretical Physics, Johannes Kepler University Linz, Altenbergerstrasse 69, Linz, Austria.
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Folk R, Holovatch Y, Moser G. Field theory of bicritical and tetracritical points. II. Relaxational dynamics. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:041125. [PMID: 18999397 DOI: 10.1103/physreve.78.041125] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/12/2008] [Indexed: 05/27/2023]
Abstract
We calculate the relaxational dynamical critical behavior of systems of O(n_||)(plus sign in circle)O(n_perpendicular) symmetry by renormalization group method within the minimal subtraction scheme in two-loop order. The three different bicritical static universality classes previously found for such systems correspond to three different dynamical universality classes within the static borderlines. The Heisenberg and the biconical fixed point lead to strong dynamic scaling whereas in the region of stability of the decoupled fixed point weak dynamic scaling holds. Due to the neighborhood of the stability border between the strong and the weak scaling dynamic fixed point to the dynamical stable fixed point a very small dynamic transient exponent of omega(Beta)_(v) =0.0044 is present in the dynamics for the physically important case n_|| =1 and n_perpendicular =2 in d=3 .
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Affiliation(s)
- R Folk
- Institute for Theoretical Physics, Johannes Kepler University Linz, Altenbergerstrasse 69, Linz, Austria.
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Dohm V. Diversity of critical behavior within a universality class. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 77:061128. [PMID: 18643238 DOI: 10.1103/physreve.77.061128] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/25/2008] [Indexed: 05/26/2023]
Abstract
We study spatial anisotropy effects on the bulk and finite-size critical behavior of the O(n) symmetric anisotropic phi;{4} lattice model with periodic boundary conditions in a d -dimensional hypercubic geometry above, at, and below Tc. The absence of two-scale factor universality is discussed for the bulk order-parameter correlation function, the bulk scattering intensity, and for several universal bulk amplitude relations. The anisotropy parameters are observable by scattering experiments at Tc. For the confined system, renormalization-group theory within the minimal subtraction scheme at fixed dimension d for 2<d<4 is employed. In contrast to the epsilon=4-d expansion, the fixed- d finite-size approach keeps the exponential form of the order-parameter distribution function unexpanded. For the case of cubic symmetry and for n=1 , our perturbation approach yields excellent agreement with the Monte Carlo (MC) data for the finite-size amplitude of the free energy of the three-dimensional Ising model at Tc by Mon [Phys. Rev. Lett. 54, 2671 (1985)]. The epsilon expansion result is in less good agreement. Below Tc, a minimum of the scaling function of the excess free energy is found. We predict a measurable dependence of this minimum on the anisotropy parameters. The relative anisotropy effect on the free energy is predicted to be significantly larger than that on the Binder cumulant. Our theory agrees quantitatively with the nonmonotonic dependence of the Binder cumulant on the ferromagnetic next-nearest-neighbor (NNN) coupling of the two-dimensional Ising model found by MC simulations of Selke and Shchur [J. Phys. A 38, L739 (2005)]. Our theory also predicts a nonmonotonic dependence for small values of the antiferromagnetic NNN coupling and the existence of a Lifshitz point at a larger value of this coupling. The nonuniversal anisotropy effects in the finite-size scaling regime are predicted to satisfy a kind of restricted universality. The tails of the large- L behavior at T++Tc violate both finite-size scaling and universality even for isotropic systems as they depend on the bare four-point coupling of the phi4 theory, on the cutoff procedure, and on subleading long-range interactions.
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Affiliation(s)
- Volker Dohm
- Institute of Theoretical Physics, RWTH Aachen University, Aachen, Germany
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Garrabos Y, Lecoutre C, Palencia F, Le Neindre B, Erkey C. Master crossover functions for one-component fluids. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 77:021116. [PMID: 18351996 DOI: 10.1103/physreve.77.021116] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/06/2007] [Revised: 10/16/2007] [Indexed: 05/26/2023]
Abstract
By introducing three well-defined dimensionless numbers, we establish the link between the scale dilatation method able to estimate master (i.e., unique) singular behaviors of the one-component fluid subclass and the universal crossover functions recently estimated [Garrabos and Bervillier, Phys. Rev. E 74, 021113 (2006)] from the bounded results of the massive renormalization scheme applied to the Phi(d)(4)(n) model of scalar order parameter (n=1) and three dimensions (d=3), representative of the Ising-like universality class. The master (i.e., rescaled) crossover functions are then able to fit the singular behaviors of any one-component fluid without adjustable parameter, using only one critical energy scale factor, one critical length scale factor, and two dimensionless asymptotic scale factors, which characterize the fluid critical interaction cell at its liquid-gas critical point. An additional adjustable parameter accounts for quantum effects in light fluids at the critical temperature. The effective extension of the thermal field range along the critical isochore where the master crossover functions seems to be valid corresponds to a correlation length greater than three times the effective range of the microscopic short-range molecular interaction.
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Affiliation(s)
- Yves Garrabos
- Equipe du Supercritique pour l'Environnement, les Matériaux et l'Espace, Institut de Chimie de la Matière Condensée de Bordeaux, UPR 9048, Centre National de la Recherche Scientifique, Université Bordeaux I, Pessac Cedex, France
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Garrabos Y. Universality and quantum effects in one-component critical fluids. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 73:056110. [PMID: 16803001 DOI: 10.1103/physreve.73.056110] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/20/2006] [Indexed: 05/10/2023]
Abstract
Nonuniversal scale transformations of the physical fields are extended to pure quantum fluids and used to calculate the susceptibility, the specific heat, and the order parameter density along the critical isochore of near its liquid-vapor critical point. Within the so-called preasymptotic domain, where the Wegner expansion restricted to the first term of confluent corrections to scaling is expected to be valid, the results are in agreement with the experimental measurements and recent predictions, either based on the minimal-substraction renormalization and the massive renormalization schemes within phi (4)(d = 3) (n = 1)the model, or based on the crossover parametric equation of state for Ising-like systems.
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Affiliation(s)
- Yves Garrabos
- Equipe du Supercritique pour l'Environnement, les Matériaux et l'Espace - Institut de Chimie de la Matière Condensée de Bordeaux - Centre National de la Recherche Scientifique - Université Bordeaux I - 87, PESSAC, France.
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Folk R, Moser G. Two-loop field theory and nonasymptotic properties of the dynamical model for the lambda transition in 3He-4He mixtures. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 73:016141. [PMID: 16486250 DOI: 10.1103/physreve.73.016141] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/16/2005] [Indexed: 05/06/2023]
Abstract
Model F' introduced by Siggia and Nelson [Phys. Rev. B 15, 1427 (1977)] describes the critical dynamics of 3He-4He mixtures near the superfluid transition. Using the minimal subtraction scheme this model is renormalized within dynamical field theory. The dynamic zeta functions needed for the nonasymptotic flow properties are presented in two-loop order. The fixed points are discussed and the stable fixed points are identified. The transition to limiting models contained in model F' is shown analytically by performing the corresponding limits and numerically by calculating the nonlinear flow. These results are the basis for further experimental comparison of the transport coefficents in 3He-4He mixtures at higher concentrations including the tricritical point.
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Affiliation(s)
- R Folk
- Institute for Theoretical Physics, University of Linz, Linz, Austria
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18
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Zhong F, Barmatz M. Comparison of theoretical models of crossover behavior near the 3He liquid-vapor critical point. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 70:066105. [PMID: 15697432 DOI: 10.1103/physreve.70.066105] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/18/2004] [Indexed: 05/24/2023]
Abstract
Parametric expressions, based on three different theoretical models, are used to calculate the isothermal susceptibility, specific heat, and order parameter along the critical isochore and coexistence curve from the asymptotic region to the crossover region. These models are (i) the minimal-subtraction renormalization scheme, (ii) the massive renormalization scheme within the phi(4) model, and (iii) the crossover parametric model based on the crossover Landau model. We fit these theories globally to experimental measurements of the isothermal susceptibility and specific heat along the critical isochore and coexistence curve, and to the coexistence curve of 3He near its liquid-vapor critical point. All of the theories provide good agreement with the experimental measurements within the reduced temperature range |t| < or =2 x 10(-2) . The differences in the fits between the theories and the correlations between the adjustable model parameters are discussed.
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Affiliation(s)
- Fang Zhong
- Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109-8099, USA
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19
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Chen XS, Dohm V. Nonuniversal finite-size scaling in anisotropic systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 70:056136. [PMID: 15600721 DOI: 10.1103/physreve.70.056136] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/22/2004] [Indexed: 05/24/2023]
Abstract
We study the bulk and finite-size critical behavior of the O(n) symmetric phi4 theory with spatially anisotropic interactions of noncubic symmetry in d<4 dimensions. In such systems of a given (d,n) universality class, two-scale factor universality is absent in bulk correlation functions, and finite-size scaling functions including the Privman-Fisher scaling form of the free energy, the Binder cumulant ratio, and the Casimir amplitude are shown to be nonuniversal. In particular it is shown that, for anisotropic confined systems, isotropy cannot be restored by an anisotropic scale transformation.
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Affiliation(s)
- X S Chen
- Institute of Theoretical Physics, Chinese Academy of Sciences, P. O. Box 2735, Beijing 100080, China
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20
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Folk R, Moser G. Critical dynamics of stochastic models with energy conservation (model C). PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 69:036101. [PMID: 15089355 DOI: 10.1103/physreve.69.036101] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/19/2003] [Indexed: 05/24/2023]
Abstract
We calculate the field-theoretic functions of the generalized dynamical model C*, in which a conserved secondary density is coupled to a nonconserved complex order parameter, in two-loop order. We show that the fixed points in this extended model are equal to the fixed points obtained in model C with a real order parameter, which has been introduced by Halperin, Hohenberg, and Ma. Our results correct long-standing errors in the field-theoretic functions in model C published by several authors leading also to different fixed point values w* for the ratio of the two time scales involved. The stability regions of the fixed points, which remained partially unclear, considered in a "phase diagram"--whose axes are the spatial dimension d and number of order parameter components n--are now clarified. Especially an anomalous region found by previous authors, in which the scaling properties remained unsolved, does not exist. There are only two regions: one with a finite fixed point w* where the dynamical exponent z of the order parameter is z=2+alpha/nu and another region where w*=0 and z is equal to the model A value.
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Affiliation(s)
- R Folk
- Institute for Theoretical Physics, University of Linz, Linz, Austria
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21
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Strösser M, Dohm V. Minimal renormalization without epsilon expansion: four-loop free energy in three dimensions for general n above and below Tc. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 67:056115. [PMID: 12786228 DOI: 10.1103/physreve.67.056115] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/10/2002] [Indexed: 05/24/2023]
Abstract
We present an analytic four-loop calculation of the free energy in three dimensions within the O(n) symmetric phi(4) theory at infinite cutoff for general n above and below T(c). It is shown that Goldstone singularities arising at intermediate stages of the calculation cancel among themselves. The correlation length above T(c) and an appropriately defined pseudocorrelation length below T(c) are calculated analytically up to four-loop order for general n. The method of minimal renormalization at fixed dimension d=3 is used to determine the analytic expressions for the four-loop series of the amplitude functions of the free energy, correlation length, and specific heat above and below T(c) in terms of the renormalized coupling. These expressions provide the basis for future accurate Borel resummations of universal amplitude ratios characterizing the asymptotic critical behavior and of crossover functions describing the nonasymptotic critical behavior. A brief application is given by a variational calculation of the universal specific-heat amplitude ratios A+/A- and P=alpha(-1)(1-A+/A-) and of the universal quantity R+(xi)=xi+0(A+)(1/d) for general n.
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Affiliation(s)
- M Strösser
- Institut für Theoretische Physik, Technische Hochschule Aachen, D-52056 Aachen, Germany
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22
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Bagnuls C, Bervillier C. Classical-to-critical crossovers from field theory. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 65:066132. [PMID: 12188808 DOI: 10.1103/physreve.65.066132] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/11/2002] [Indexed: 05/23/2023]
Abstract
We extend our previous determinations of nonasymptotic critical behavior of Phys. Rev. B 32, 7209 (1985) and 35, 3585 (1987) to accurate expressions of the complete classical-to-critical crossover (in three-dimensional field theory) in terms of the temperaturelike scaling field (i.e., along the critical isochore) for (1) the correlation length, the susceptibility, and the specific heat in the homogeneous phase for the n-vector model (n=1 to 3) and (2) the spontaneous magnetization (coexistence curve), the susceptibility, and the specific heat in the inhomogeneous phase for the Ising model (n=1). The present calculations include the seventh-loop order of Murray and Nickel and closely account for the up-to-date estimates of universal asymptotic critical quantities (exponents and amplitude combinations) provided by Guida and Zinn-Justin [J. Phys. A 31, 8103 (1998)].
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Affiliation(s)
- C Bagnuls
- Service de Physique de l'Etat Condensé, CEA/DSM/SPEC, CEA/Saclay, 91191 Gif-sur-Yvette cédex, France.
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23
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Caracciolo S, Causo MS, Pelissetto A, Rossi P, Vicari E. Crossover phenomena in spin models with medium-range interactions and self-avoiding walks with medium-range jumps. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 64:046130. [PMID: 11690113 DOI: 10.1103/physreve.64.046130] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/14/2001] [Indexed: 05/23/2023]
Abstract
We study crossover phenomena in a model of self-avoiding walks with medium-range jumps, which corresponds to the limit N-->0 of an N-vector spin system with medium-range interactions. In particular, we consider the critical crossover limit that interpolates between the Gaussian and the Wilson-Fisher fixed point. The corresponding crossover functions are computed by using field-theoretical methods and an appropriate mean-field expansion. The critical crossover limit is accurately studied by numerical Monte Carlo simulations, which are much more efficient for walk models than for spin systems. Monte Carlo data are compared with the field-theoretical predictions for the critical crossover functions, finding good agreement. We also verify the predictions for the scaling behavior of the leading nonuniversal corrections. We determine phenomenological parametrizations that are exact in the critical crossover limit, have the correct scaling behavior for the leading correction, and describe the nonuniversal crossover behavior of our data for any finite range.
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Affiliation(s)
- S Caracciolo
- Scuola Normale Superiore, INFN and INFM, I-56100 Pisa, Italy.
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24
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Kleinert H, Van den Bossche B. Three-loop critical exponents, amplitude functions, and amplitude ratios from variational perturbation theory. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 63:056113. [PMID: 11414967 DOI: 10.1103/physreve.63.056113] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/21/2000] [Indexed: 05/23/2023]
Abstract
We use variational perturbation theory to calculate various universal amplitude ratios above and below Tc in minimally subtracted straight phi4 theory with N components in three dimensions. In order to best exhibit the method as a powerful alternative to Borel resummation techniques, we consider only two- and three-loops expressions where our results are analytic expressions. For the critical exponents, we also extend existing analytic expressions for two loops to three loops.
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Affiliation(s)
- H Kleinert
- Institut für Theoretische Physik, Arnimallee 14 D-14195 Berlin, Germany
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25
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Chen XS, Dohm V, Schultka N. Order-Parameter Distribution Function of Finite O(n) Symmetric Systems. PHYSICAL REVIEW LETTERS 1996; 77:3641-3644. [PMID: 10062271 DOI: 10.1103/physrevlett.77.3641] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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26
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Koch W, Dohm V, Stauffer D. Order-Parameter Relaxation Times of Finite Three-Dimensional Ising-like Systems. PHYSICAL REVIEW LETTERS 1996; 77:1789-1792. [PMID: 10063172 DOI: 10.1103/physrevlett.77.1789] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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27
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Esser A, Dohm V, Hermes M, Wang JS. Field theory of finite-size effects in Ising-like systems. ACTA ACUST UNITED AC 1995. [DOI: 10.1007/bf01307471] [Citation(s) in RCA: 20] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
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28
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Haussmann R, Dohm V. Heat-flow induced anomalies in superfluid 4He near T lambda. PHYSICAL REVIEW LETTERS 1994; 72:3060-3063. [PMID: 10056057 DOI: 10.1103/physrevlett.72.3060] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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29
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Haussmann R, Dohm V. Depression of the superfluid transition in 4He: Renormalization-group theory. PHYSICAL REVIEW. B, CONDENSED MATTER 1992; 46:6361-6373. [PMID: 10002322 DOI: 10.1103/physrevb.46.6361] [Citation(s) in RCA: 19] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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31
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Haussmann R, Dohm V. Nonlinear heat transport near the superfluid transition of 4He. PHYSICAL REVIEW LETTERS 1991; 67:3404-3407. [PMID: 10044725 DOI: 10.1103/physrevlett.67.3404] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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32
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Dohm V. Renormalization-group flow equations of model F. PHYSICAL REVIEW. B, CONDENSED MATTER 1991; 44:2697-2712. [PMID: 9999839 DOI: 10.1103/physrevb.44.2697] [Citation(s) in RCA: 33] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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33
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Moser G, Folk R. Thermal conductivity near the superfluid transition in 3He-4He mixtures below T lambda. PHYSICAL REVIEW. B, CONDENSED MATTER 1991; 44:819-825. [PMID: 9999188 DOI: 10.1103/physrevb.44.819] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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34
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Duncan RV, Ahlers G. Singularity and nonlinearity in the Kapitza resistance between gold and superfluid 4He near T lambda. PHYSICAL REVIEW. B, CONDENSED MATTER 1991; 43:7707-7717. [PMID: 9996390 DOI: 10.1103/physrevb.43.7707] [Citation(s) in RCA: 43] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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35
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Schloms R, Dohm V. Minimal renormalization without epsilon expansion: Critical behavior above and below Tc. PHYSICAL REVIEW. B, CONDENSED MATTER 1990; 42:6142-6152. [PMID: 9994690 DOI: 10.1103/physrevb.42.6142] [Citation(s) in RCA: 43] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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36
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Krause HJ, Schloms R, Dohm V. Minimal renormalization without ?-expansion: Amplitude functions in three dimensions. ACTA ACUST UNITED AC 1990. [DOI: 10.1007/bf01406597] [Citation(s) in RCA: 38] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
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37
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Chen ZY, Albright PC, Sengers JV. Crossover from singular critical to regular classical thermodynamic behavior of fluids. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1990; 41:3161-3177. [PMID: 9903471 DOI: 10.1103/physreva.41.3161] [Citation(s) in RCA: 52] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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38
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Kolnberger S, Folk R. Critical fluctuations in superconductors. PHYSICAL REVIEW. B, CONDENSED MATTER 1990; 41:4083-4088. [PMID: 9994229 DOI: 10.1103/physrevb.41.4083] [Citation(s) in RCA: 23] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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39
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Pankert J, Dohm V. Theory of critical first sound near the lambda transition of 4He. II. Attenuation and dispersion for T >= T lambda. PHYSICAL REVIEW. B, CONDENSED MATTER 1989; 40:10856-10872. [PMID: 9991646 DOI: 10.1103/physrevb.40.10856] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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40
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Pankert J, Dohm V. Theory of critical first sound near the lambda transition of 4He. I. Model and correlation functions. PHYSICAL REVIEW. B, CONDENSED MATTER 1989; 40:10842-10855. [PMID: 9991645 DOI: 10.1103/physrevb.40.10842] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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41
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Frank D, Dohm V. Critical thermal boundary resistance of 4He near T lambda. PHYSICAL REVIEW LETTERS 1989; 62:1864-1867. [PMID: 10039790 DOI: 10.1103/physrevlett.62.1864] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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42
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Dohm V. Diagrammatic perturbation approach to finite-size and surface critical behavior for Dirichlet boundary conditions. ACTA ACUST UNITED AC 1989. [DOI: 10.1007/bf01313573] [Citation(s) in RCA: 23] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/25/2022]
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43
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Huhn W, Dohm V. Possible resolution of the finite-size scaling problem in 4He. PHYSICAL REVIEW LETTERS 1988; 61:1368-1371. [PMID: 10038776 DOI: 10.1103/physrevlett.61.1368] [Citation(s) in RCA: 17] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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44
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Tam WY, Ahlers G. Erratum: Thermal conductivity of 4He I near T lambda from vapor pressure to 28 bars: Comparison of experiment and theory. PHYSICAL REVIEW. B, CONDENSED MATTER 1988; 37:7898-7900. [PMID: 9944105 DOI: 10.1103/physrevb.37.7898] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/11/2023]
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45
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Nemirovsky AM, Wang ZG, Freed KF. Crossover renormalization-group approach to semi-infinite inhomogeneous critical behavior. PHYSICAL REVIEW. B, CONDENSED MATTER 1987; 36:3755-3762. [PMID: 9943311 DOI: 10.1103/physrevb.36.3755] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/11/2023]
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46
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Eisenriegler E, Tomaschitz R. Helmholtz free energy of finite spin systems near criticality. PHYSICAL REVIEW. B, CONDENSED MATTER 1987; 35:4876-4887. [PMID: 9940665 DOI: 10.1103/physrevb.35.4876] [Citation(s) in RCA: 28] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/11/2023]
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47
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48
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Vidal F, Maza J. Gravity effects on first-sound velocity near T lambda (P) in liquid 4He. PHYSICAL REVIEW. B, CONDENSED MATTER 1986; 34:4604-4613. [PMID: 9940252 DOI: 10.1103/physrevb.34.4604] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/11/2023]
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49
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Tam WY, Ahlers G. Thermal conductivity of 4He I near T lambda from vapor pressure to 28 bars: Comparison of experiment and theory. PHYSICAL REVIEW. B, CONDENSED MATTER 1986; 33:183-196. [PMID: 9937897 DOI: 10.1103/physrevb.33.183] [Citation(s) in RCA: 29] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/11/2023]
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50
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