Critical Behavior of a Classical Heisenberg Ferromagnet with Many Degrees of Freedom

Brézin, E.; Wallace, D. J.

doi:10.1103/physrevb.7.1967

Cited by 198 publications

(110 citation statements)

References 14 publications

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“…(8)] diverges for h → 0 as h −1/2 . This divergence of the uniform longitudinal susceptibility of a three-dimensional Heisenberg magnet in the ordered state is not widely appreciated, although it was noticed a long time ago 2,3 and has been confirmed by renormalization group calculations for the classical Heisenberg ferromagnet 19,20 and perturbative calculations for the corresponding quantum model. 21,22 Due to this divergence, the Gibbs free energy G(M ) of the Heisenberg ferromagnet in D = 3 does not have the generic form (7).…”

Section: Linear Spin Waves At Constant Order Parametermentioning

confidence: 99%

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Spin-wave theory at constant order parameter

2003

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We derive the low-temperature properties of spin-S quantum Heisenberg magnets from the Gibbs free energy G(M ) for fixed order parameter M . Assuming that the low-lying elementary excitations of the system are renormalized spin waves, we show that a straightforward 1/S expansion of G(M ) yields qualitatively correct results for the low-temperature thermodynamics, even in the absence of long-range magnetic order. We explicitly calculate the two-loop correction to the susceptibility of the ferromagnetic Heisenberg chain and show that it quantitatively modifies the mean-field result.

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Section: Linear Spin Waves At Constant Order Parametermentioning

confidence: 99%

“…(14) is proportional to (m − m 0 ) 3 , which can be traced to the fact that the inverse susceptibility vanishes. By contrast, in D > 4 the uniform longitudinal susceptibility of the Heisenberg ferromagnet is finite, 19,20 so that in this case the Gibbs free energy has indeed the generic form (7).…”

Section: Linear Spin Waves At Constant Order Parametermentioning

confidence: 99%

Spin-wave theory at constant order parameter

2003

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“…Power counting shows that at nonzero temperature, and for dimensions d longitudinal susceptibility to diverge everywhere in the ferromagnetic phase, so χ L is fundamentally different in the ferromagnetic phase than in the paramagnetic one. 17 Ultimately this implies the superconducting transition temperature can be very different in the two phases, and it is this aspect that previous theories missed.…”

Section: 5 14mentioning

confidence: 99%

“…In a Heisenberg ferromagnet (or in any magnet with a continuous rotation symmetry in spin space), the transverse spin waves or magnons are massless and couple to the longitudinal susceptibility χ L . 17 This effect is most easily illustrated within a nonlinear sigma-model description of the ferromagnet, 18 which treats the order parameter m as a vector of fixed length m, and parametrizes it as m = m(π 1 (x), π 2 (x), σ(x)), with σ 2 + π …”

Section: 5 14mentioning

confidence: 99%

Superconductivity and Quantum Phase Transitions in Weak Itinerant Ferromagnets

Kirkpatrick

Vojta

Belitz

et al. 2002

Recent Progress in Many-Body Theories

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It is argued that the phase transition in low-Tc clean itinerant ferromagnets is generically of first order, due to correlation effects that lead to a nonanalytic term in the free energy. A tricritical point separates the line of first order transitions from Heisenberg critical behavior at higher temperatures. Sufficiently strong quenched disorder suppresses the first order transition via the appearance of a critical endpoint. A semi-quantitative discussion is given in terms of recent experiments on MnSi and UGe 2 . It is then shown that the critical temperature for spin-triplet, p-wave superconductivity mediated by spin fluctuations is generically much higher in a Heisenberg ferromagnetic phase than in a paramagnetic one, due to the coupling of magnons to the longitudinal magnetic susceptibility. This qualitatively explains the phase diagram recently observed in UGe 2 and ZrZn 2 .

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“…An interesting analogy in this context is the corresponding OP susceptibility in a classical Heisenberg ferromagnet. Due to a coupling between the longitudinal and transverse magnetization fluctuations the longitudinal magnetic susceptibility χ L (i.e., the OP susceptibility) for 2 < d < 4 diverges everywhere in the ordered phase as 1/k 4−d [31]. This results from a one-loop contribution to χ L that is a wave-number convolution of two Goldstone modes, each of which scales as an inverse wave number squared.…”

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Quantum correlations in metals that grow in time and space

Kirkpatrick

Belitz

2016

Phys. Rev. B

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We show that the correlations of electrons with a fixed energy in metals have very anomalous time and space dependences. Due to soft modes that exist in any Fermi liquid, combined with the incomplete screening of the Coulomb interaction at finite frequencies, the correlations in 2-d systems grow with time as t 2 . In the presence of disorder, the spatial correlations grow as the distance squared. Similar, but in general weaker, effects are present in 3-d systems and in the absence of quenched disorder. We propose ways to experimentally measure these anomalous correlations.PACS numbers: 05.30.Fk; 71.27.+a Equilibrium time-correlation functions are an essential concept in statistical mechanics [1]. They describe the spontaneous fluctuations of a system in equilibrium, and together with the partition function they provide a complete description of the equilibrium state. Via the fluctuation-dissipation theorem they also describe the linear response of the system to external fields, and they are directly measurable by means of scattering experiments.An old, and seemingly plausible, assumption is that microscopic correlations decay on time scales that are much faster than macroscopic observation times. Various concepts depend on this assumption, for instance, the notion that the BBGKY hierarchy of classical kinetic equations can be truncated [2]. An analogous assumption underlies the Kadanoff-Baym scheme of deriving and solving quantum kinetic equations and its generalizations [3,4]. The assumption of a separation of time scales is also important in other areas, e.g., in signal processing [5,6]. For time-correlation functions it implies that they decay exponentially for large times. Equivalently, their Laplace transform is an analytic function of the complex frequency z at z = 0. The discovery of the non-exponential decay known as long-time tails (LTTs) [7][8][9], and the related breakdown of a virial expansion for transport coefficients [10,11] thus came as a considerable surprise [12], since it showed that the assumption is in general not true. Rather, many time-correlation functions decay only algebraically, i.e., they have no intrinsic time scale. This scale invariance is reminiscent of the behavior of correlation functions at critical points; however, it occurs in entire phases, as opposed to just at isolated points in the phase diagram, and therefore is referred to as 'generic scale invariance' [13][14][15]. The underlying physical reason is either conservation laws, or Goldstone modes that lead to a slow decay of some longwavelength fluctuations and, via mode-mode-coupling effects, affect the decay of other degrees of freedom. An example is the shear stress in a classical fluid, which is not conserved, yet its time-correlation function decays algebraically as 1/t d/2 for long times t in a d-dimensional fluid since it couples to the transverse momentum, which is conserved. As a result, the Green-Kubo expressions for various transport coefficients diverge in dimensions d ≤ 2, and the hydrodynamic equations bec...

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Critical Behavior of a Classical Heisenberg Ferromagnet with Many Degrees of Freedom

Cited by 198 publications

References 14 publications

Spin-wave theory at constant order parameter

Spin-wave theory at constant order parameter

Superconductivity and Quantum Phase Transitions in Weak Itinerant Ferromagnets

Quantum correlations in metals that grow in time and space

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