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H. W. Diehl

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

At least 37 records · Page 2Linked to original sources
arXiv2006

Excess free energy and Casimir forces in systems with long-range interactions of van-der-Waals type: General considerations and exact spherical-model results

We consider systems confined to a $d$-dimensional slab of macroscopic lateral extension and finite thickness $L$ that undergo a continuous bulk phase transition in the limit $L\to\infty$ and are describable by an O(n) symmetrical Hamiltonian. Periodic boundary conditions are applied across the slab. We study the effects of long-range pair interactions whose potential decays as $b x^{-(d+σ)}$ as $x\to\infty$, with $2<σ<4$ and $2<d+σ\leq 6$, on the Casimir effect at and near the bulk critical temperature $T_{c,\infty}$, for $2<d<4$. For the scaled reduced Casimir force per unit cross-sectional area, we obtain the form $L^{d} {\mathcal F}_C/k_BT \approx Ξ_0(L/ξ_\infty) + g_ωL^{-ω}Ξω(L/ξ_\infty) + g_σL^{-ω_\sigm a} Ξ_σ(L ξ_\infty)$. The contribution $\propto g_σ$ decays for $T\neq T_{c,\infty}$ algebraically in $L$ rather than exponentially, and hence becomes dominant in an appropriate regime of temperatures and $L$. We derive exact results for spherical and Gaussian models which confirm these findings. In the case $d+σ=6$, which includes that of nonretarded van-der-Waals interactions in $d=3$ dimensions, the power laws of the corrections to scaling $\propto b$ of the spherical model are found to get modified by logarithms. Using general RG ideas, we show that these logarithmic singularities originate from the degeneracy $ω=ω_σ=4-d$ that occurs for the spherical model when $d+σ=6$, in conjunction with the $b$ dependence of $g_ω$.

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arXiv2005

Boundary critical behaviour at m-axial Lifshitz points of semi-infinite systems with a surface plane perpendicular to a modulation axis

Semi-infinite $d$-dimensional systems with an $m$-axial bulk Lifshitz point are considered whose ($d-1$)-dimensional surface hyper-plane is oriented perpendicular to one of the $m$ modulation axes. An $n$-component $ϕ^4$ field theory describing the bulk and boundary critical behaviour when (i) the Hamiltonian can be taken to have O(n) symmetry and (ii) spatial anisotropies breaking its Euclidean symmetry in the $m$-dimensional coordinate subspace of potential modulation directions may be ignored is investigated. The long-distance behaviour at the ordinary surface transition is mapped onto a field theory with the boundary conditions that both the order parameter $\bmϕ$ and its normal derivative $\partial_n\bmϕ$ vanish at the surface plane. The boundary-operator expansion is utilized to study the short-distance behaviour of $\bmϕ$ near the surface. Its leading contribution is found to be controlled by the boundary operator $\partial_n^2\bmϕ$. The field theory is renormalized for dimensions $d$ below the upper critical dimension $d^*(m)=4+m/2$, with a corresponding surface source term $\propto \partial_n^2\bmϕ$ added. The anomalous dimension of this boundary operator is computed to first order in $ε=d^*-d$. The result is used in conjunction with scaling laws to estimate the value of the single independent surface critical exponent $β_{\mathrm{L}1}^{(\mathrm{ord},\perp)}$ for $d=3$. Our estimate for the case $m=n=1$ of a uniaxial Lifshitz point in Ising systems is in reasonable agreement with published Monte Carlo results.

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arXiv2005

Large-n expansion for m-axial Lifshitz points

The large-n expansion is developed for the study of critical behaviour of d-dimensional systems at m-axial Lifshitz points with an arbitrary number m of modulation axes. The leading non-trivial contributions of O(1/n) are derived for the two independent correlation exponents η_{L2} and η_{L4}, and the related anisotropy index θ. The series coefficients of these 1/n corrections are given for general values of m and d with 0<m<d and 2+m/2<d<4+m/2 in the form of integrals. For special values of m and d such as (m,d)=(1,4), they can be computed analytically, but in general their evaluation requires numerical means. The 1/n corrections are shown to reduce in the appropriate limits to those of known large-n expansions for the case of d-dimensional isotropic Lifshitz points and critical points, respectively, and to be in conformity with available dimensionality expansions about the upper and lower critical dimensions. Numerical results for the 1/n coefficients of η_{L2}, η_{L4} and θare presented for the physically interesting case of a uniaxial Lifshitz point in three dimensions, as well as for some other choices of m and d. A universal coefficient associated with the energy-density pair correlation function is calculated to leading order in 1/n for general values of m and d.

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arXiv2004

Boundary critical behaviour at $m$-axial Lifshitz points: the special transition for the case of a surface plane parallel to the modulation axes

The critical behaviour of $d$-dimensional semi-infinite systems with $n$-component order parameter $\bmϕ$ is studied at an $m$-axial bulk Lifshitz point whose wave-vector instability is isotropic in an $m$-dimensional subspace of $\mathbb{R}^d$. Field-theoretic renormalization group methods are utilised to examine the special surface transition in the case where the $m$ potential modulation axes, with $0\leq m\leq d-1$, are parallel to the surface. The resulting scaling laws for the surface critical indices are given. The surface critical exponent $η_\|^{\rm sp}$, the surface crossover exponent $Φ$ and related ones are determined to first order in $ε=4+\case{m}{2}-d$. Unlike the bulk critical exponents and the surface critical exponents of the ordinary transition, $Φ$ is $m$-dependent already at first order in $ε$. The $\Or(ε)$ term of $η_\|^{\rm sp}$ is found to vanish, which implies that the difference of $β_1^{\rm sp}$ and the bulk exponent $β$ is of order $ε^2$.

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arXiv2004

Bulk and Boundary Critical Behavior at Lifshitz Points

Lifshitz points are multicritical points at which a disordered phase, a homogeneous ordered phase, and a modulated ordered phase meet. Their bulk universality classes are described by natural generalizations of the standard $ϕ^4$ model. Analyzing these models systematically via modern field-theoretic renormalization group methods has been a long-standing challenge ever since their introduction in the middle of the 1970s. We survey the recent progress made in this direction, discussing results obtained via dimensionality expansions, how they compare with Monte Carlo results, and open problems. These advances opened the way towards systematic studies of boundary critical behavior at $m$-axial Lifshitz points. The possible boundary critical behavior depends on whether the surface plane is perpendicular to one of the $m$ modulation axes or parallel to all of them. We show that the semi-infinite field theories representing the corresponding surface universality classes in these two cases of perpendicular and parallel surface orientation differ crucially in their Hamiltonian's boundary terms and the implied boundary conditions, and explain recent results along with our current understanding of this matter.

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arXiv2003

Relevance of space anisotropy in the critical behavior of m-axial Lifshitz points

The critical behavior of $d$-dimensional systems with $n$-component order parameter $\bmϕ$ is studied at an $m$-axial Lifshitz point where a wave-vector instability occurs in an $m$-dimensional subspace ${\mathbb R}^m$ ($m{>}1$). Field theoretic renormalization group techniques are exploited to examine the effects of terms in the Hamiltonian that break the rotational symmetry of the Euclidean group ${\mathbb E}(m)$. The framework for considering general operators of second order in $\bmϕ$ and fourth order in the derivatives $\partial_α$ with respect to the Cartesian coordinates $x_α$ of ${\mathbb R}^m$ is presented. For the specific case of systems with cubic anisotropy, the effects of having an additional term, $\sum_{α=1}^m(\partial_α^2\bmϕ)^2$, are investigated in an $ε$ expansion about the upper critical dimension $d^{*}(m)=4+m/2$. Its associated crossover exponent is computed to order $ε^2$ and found to be positive, so that it is a \emph{relevant} perturbation on a model isotropic in ${\mathbb R}^m$.

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arXiv2003

Boundary critical behavior at m-axial Lifshitz points for a boundary plane parallel to the modulation axes

The critical behavior of semi-infinite $d$-dimensional systems with $n$-component order parameter $\bmϕ$ and short-range interactions is investigated at an $m$-axial bulk Lifshitz point whose wave-vector instability is isotropic in an $m$-dimensional subspace of $\mathbb{R}^d$. The associated $m$ modulation axes are presumed to be parallel to the surface, where $0\le m\le d-1$. An appropriate semi-infinite $|\bmϕ|^4$ model representing the corresponding universality classes of surface critical behavior is introduced. It is shown that the usual O(n) symmetric boundary term $\propto \bmϕ^2$ of the Hamiltonian must be supplemented by one of the form $\mathringλ \sum_{α=1}^m(\partial\bmϕ/\partial x_α)^2$ involving a dimensionless (renormalized) coupling constant $λ$. The implied boundary conditions are given, and the general form of the field-theoretic renormalization of the model below the upper critical dimension $d^*(m)=4+{m}/{2}$ is clarified. Fixed points describing the ordinary, special, and extraordinary transitions are identified and shown to be located at a nontrivial value $λ^*$ if $ε\equiv d^*(m)-d>0$. The surface critical exponents of the ordinary transition are determined to second order in $ε$. Extrapolations of these $ε$ expansions yield values of these exponents for $d=3$ in good agreement with recent Monte Carlo results for the case of a uniaxial ($m=1$) Lifshitz point. The scaling dimension of the surface energy density is shown to be given exactly by $d+m (θ-1)$, where $θ=ν_{l4}/ν_{l2}$ is the anisotropy exponent.

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arXiv2003

Comment on ``Renormalization-group picture of the Lifshitz critical behavior''

We show that the recent renormalization-group analysis of Lifshitz critical behavior presented by Leite [Phys. Rev. B {\bf 67}, 104415 (2003)] suffers from a number of severe deficiencies. In particular, we show that his approach does not give an ultraviolet finite renormalized theory, is plagued by inconsistencies, misses the existence of a nontrivial anisotropy exponent $θ\ne 1/2$, and therefore yields incorrect hyperscaling relations. His $ε$-expansion results to order $ε^2$ for the critical exponents of $m$-axial Lifshitz points are incorrect both in the anisotropic ($0<m<d$) and the isotropic cases ($m=d$). The inherent inconsistencies and the lack of a sound basis of the approach makes its results unacceptable even if they are interpreted in the sense of approximations.

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arXiv2003

Surface critical behaviour at m-axial Lifshitz points: continuum models, boundary conditions and two-loop renormalization group results

The critical behaviour of semi-infinite $d$-dimensional systems with short-range interactions and an O(n) invariant Hamiltonian is investigated at an $m$-axial Lifshitz point with an isotropic wave-vector instability in an $m$-dimensional subspace of $\mathbb{R}^d$ parallel to the surface. Continuum $|\bphi|^4$ models representing the associated universality classes of surface critical behaviour are constructed. In the boundary parts of their Hamiltonians quadratic derivative terms (involving a dimensionless coupling constant $λ$) must be included in addition to the familiar ones $\proptoϕ^2$. Beyond one-loop order the infrared-stable fixed points describing the ordinary, special and extraordinary transitions in $d=4+\frac{m}{2}-ε$ dimensions (with $ε>0$) are located at $λ=λ^*=\Or(ε)$. At second order in $ε$, the surface critical exponents of both the ordinary and the special transitions start to deviate from their $m=0$ analogues. Results to order $ε^2$ are presented for the surface critical exponent $β_1^{\rm ord}$ of the ordinary transition. The scaling dimension of the surface energy density is shown to be given exactly by $d+m (θ-1)$, where $θ=ν_{l4}/ν_{l2}$ is the bulk anisotropy exponent.

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arXiv2002

Critical Behavior at M-Axial Lifshitz Points

An introduction to the theory of critical behavior at Lifshitz points is given, and the recent progress made in applying the field-theoretic renormalization group (RG) approach to $ϕ^4$ $n$-vector models representing universality classes of $m$-axial Lifshitz points is surveyed. The origins of the difficulties that had hindered a full two-loop RG analysis near the upper critical dimension for more than 20 years and produced long-standing contradictory $ε$-expansion results are discussed. It is outlined how to cope with them. The pivotal role the considered class of continuum models might play in a systematic investigation of anisotropic scale invariance within the context of thermal equilibrium systems is emphasized. This could shed light on the question of whether anisotropic scale invariance implies an even larger invariance, as recently claimed in the literature.

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arXiv2002

Dynamic surface critical behavior of isotropic Heisenberg ferromagnets: boundary conditions, renormalized field theory, and computer simulation results

The dynamic critical behavior of isotropic Heisenberg ferromagnets with a planar free surface is investigated by means of field-theoretic renormalization group techniques and high-precision computer simulations. An appropriate semi-infinite extension of the stochastic model J is constructed. The relevant boundary terms of the action of the associated dynamic field theory are identified, the implied boundary conditions are derived, and the renormalization of the model in $d<6$ bulk dimensions is clarified. Two distinct renormalization schemes are utilized. The first is a massless one based on minimal subtraction of dimensional poles and the dimensionality expansion about $d=6$. To overcome its problems in going below $d=4$ dimensions, a massive one for fixed dimensions $d\le 4$ is constructed. The resulting renormalization group (or Callan Symanzik) equations are exploited to obtain the scaling forms of surface quantities like the dynamic structure factor. In conjunction with boundary operator expansions scaling relations follow that relate the critical indices of the dynamic and static infrared singularities of surface quantities to familiar \emph{static} bulk and surface exponents. To test the predicted scaling forms and scaling-law expressions for the critical exponents involved, accurate computer-simulation data are presented for the dynamic surface structure factor. These are in conformity with our predictions.

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arXiv2002

Critical, crossover, and correction-to-scaling exponents for isotropic Lifshitz points to order $\boldsymbol{(8-d)^2}$

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. It is shown that the expansions obtained here by a direct calculation for the isotropic ($m=d$) Lifshitz point all follow from the latter upon setting $m=8-ε_8$. This is so despite recent claims to the contrary by de Albuquerque and Leite [{\it J. Phys. A} {\bf 35} (2002) 1807].

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arXiv2001

Two-loop renormalization-group analysis of critical behavior at m-axial Lifshitz points

We investigate the critical behavior that d-dimensional systems with short-range forces and a n-component order parameter exhibit at Lifshitz points whose wave-vector instability occurs in a m-dimensional isotropic subspace of ${\mathbb R}^d$. Utilizing dimensional regularization and minimal subtraction of poles in $d=4+{m\over 2}-ε$ dimensions, we carry out a two-loop renormalization-group (RG) analysis of the field-theory models representing the corresponding universality classes. This gives the beta function $β_u(u)$ to third order, and the required renormalization factors as well as the associated RG exponent functions to second order, in u. The coefficients of these series are reduced to m-dependent expressions involving single integrals, which for general (not necessarily integer) values of $m\in (0,8)$ can be computed numerically, and for special values of m analytically. The $ε$ expansions of the critical exponents $η_{l2}$, $η_{l4}$, $ν_{l2}$, $ν_{l4}$, the wave-vector exponent $β_q$, and the correction-to-scaling exponent are obtained to order $ε^2$. These are used to estimate their values for d=3. The obtained series expansions are shown to encompass both isotropic limits m=0 and m=d.

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arXiv2001

Bulk singularities at critical end points: a field-theory analysis

A class of continuum models with a critical end point is considered whose Hamiltonian ${\mathcal{H}}[ϕ,ψ]$ involves two densities: a primary order-parameter field, $ϕ$, and a secondary (noncritical) one, $ψ$. Field-theoretic methods (renormalization group results in conjunction with functional methods) are used to give a systematic derivation of singularities occurring at critical end points. Specifically, the thermal singularity $\sim|{t}|^{2-α}$ of the first-order line on which the disordered or ordered phase coexists with the noncritical spectator phase, and the coexistence singularity $\sim |{t}|^{1-α}$ or $\sim|{t}|^β$ of the secondary density $<ψ>$ are derived. It is clarified how the renormalization group (RG) scenario found in position-space RG calculations, in which the critical end point and the critical line are mapped onto two separate fixed points ${\mathcal P}_{\mathrm{CEP}}^*$ and ${\mathcal P}_λ^*$ translates into field theory. The critical RG eigenexponents of ${\mathcal P}_{\mathrm{CEP}}^*$ and ${\mathcal P}_λ^*$ are shown to match. ${\mathcal P}_{\mathrm{CEP}}^*$ is demonstrated to have a discontinuity eigenperturbation (with eigenvalue $y=d$), tangent to the unstable trajectory that emanates from ${\mathcal P}_{\mathrm{CEP}}^*$ and leads to ${\mathcal P}_λ^*$. The nature and origin of this eigenperturbation as well as the role redundant operators play are elucidated. The results validate that the critical behavior at the end point is the same as on the critical line.

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arXiv2001

Lifshitz-point critical behaviour to ${\boldsymbol{O(ε^2)}}$

We comment on a recent letter by L. C. de Albuquerque and M. M. Leite (J. Phys. A: Math. Gen. 34 (2001) L327-L332), in which results to second order in $ε=4-d+\frac{m}{2}$ were presented for the critical exponents $ν_{\mathrm{L}2}$, $η_{\mathrm{L}2}$ and $γ_{\mathrm{L}2}$ of d-dimensional systems at m-axial Lifshitz points. We point out that their results are at variance with ours. The discrepancy is due to their incorrect computation of momentum-space integrals. Their speculation that the field-theoretic renormalization group approach, if performed in position space, might give results different from when it is performed in momentum space is refuted.

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arXiv2001

Dynamic surface scaling behavior of isotropic Heisenberg ferromagnets

The effects of free surfaces on the dynamic critical behavior of isotropic Heisenberg ferromagnets are studied via phenomenological scaling theory, field-theoretic renormalization group tools, and high-precision computer simulations. An appropriate semi-infinite extension of the stochastic model J is constructed, the boundary terms of the associated dynamic field theory are identified, its renormalization in d <= 6 dimensions is clarified, and the boundary conditions it satisfies are given. Scaling laws are derived which relate the critical indices of the dynamic and static infrared singularities of surface quantities to familiar static bulk and surface exponents. Accurate computer-simulation data are presented for the dynamic surface structure factor; these are in conformity with the predicted scaling behavior and could be checked by appropriate scattering experiments.

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arXiv2000

Critical behavior at m-axial Lifshitz points: field-theory analysis and $ε$-expansion results

The critical behavior of d-dimensional systems with an n-component order parameter is reconsidered at (m,d,n)-Lifshitz points, where a wave-vector instability occurs in an m-dimensional subspace of ${\mathbb R}^d$. Our aim is to sort out which ones of the previously published partly contradictory $ε$-expansion results to second order in $ε=4+\frac{m}{2}-d$ are correct. To this end, a field-theory calculation is performed directly in the position space of $d=4+\frac{m}{2}-ε$ dimensions, using dimensional regularization and minimal subtraction of ultraviolet poles. The residua of the dimensionally regularized integrals that are required to determine the series expansions of the correlation exponents $η_{l2}$ and $η_{l4}$ and of the wave-vector exponent $β_q$ to order $ε^2$ are reduced to single integrals, which for general m=1,...,d-1 can be computed numerically, and for special values of m, analytically. Our results are at variance with the original predictions for general m. For m=2 and m=6, we confirm the results of Sak and Grest [Phys. Rev. B {\bf 17}, 3602 (1978)] and Mergulh{ã}o and Carneiro's recent field-theory analysis [Phys. Rev. B {\bf 59},13954 (1999)].

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