Extreme type-ii superconductors in a magnetic field: A theory of critical fluctuations

Physical Review B - Condensed Matter and Materials Physics

Volumn 59, Issue 9, 1999, Pages 6449-6474

  Tes̆anović, Zlatko ^a  

^a JOHNS HOPKINS UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0001750188     PISSN: 10980121     EISSN: 1550235X     Source Type: Journal    
DOI: 10.1103/PhysRevB.59.6449     Document Type: Article

References (106)

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48. (Formula presented) is the vector potential associated with a given configuration of s vortices, with unit fluxes attached. In the notation of Ref. 9, (Formula presented) would correspond to (Formula presented) where (Formula presented) is the phase of the gauge transformation used in that paper. An explicit form of (Formula presented) is, however, not uniquely determined by the positions of s vortices: one must also specify the positions of branch cuts across which the phase changes from 0 to 2π. The present formulation eliminates this unnecessary complication since (Formula presented) itself is uniquely determined for any given configuration of s vortices.
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53. A similar nonrelativistic system with London-Biot-Savart-type long-range interaction of variable strength also appears to have a superfluid ground state when this interaction is weak; M. V. Feigel’man, V. B. Geshkenbein, L. B. Ioffe, and A. I. Larkin, Phys. Rev. B 48, 16 641 (1993).
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58. Extreme anisotropy and potential finite “mass terms” have been considered in Ref. 9. For example, footnote 20 of that paper considers the possibility of a finite helicity modulus (Formula presented) immediately below (Formula presented)
59. As will be discussed later in the text, the Φ transition in the gauge theory (4) is governed by an isotropic critical point. However, with (Formula presented) and, potentially, finite (Formula presented) and/or Chern-Simons terms, the same transition could be governed by an anisotropic critical point, with a different set of critical exponents in (Formula presented) and (Formula presented) directions and correspondingly different details of critical thermodynamics and transport.
60. “Perturbed” comes with quotation marks since, even though the magnetic field enters Eq. (4) through a small parameter (Formula presented) this perturbation is strongly relevant and it immediately destabilizes the zero-field (“neutral”) 3D (Formula presented) critical point.
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87. Note that (Formula presented) is not excluded at higher fields.
88. Unlike the mean-field-based approach including only the field-induced London vortex lines, where the strength of this interaction is set by the mean-field amplitude squared of (Formula presented)
89. Equation (11) for (Formula presented) differs from the estimate reported in Ref. 9 in two ways: first, the previous estimate was made for the weakly coupled layered case and, second, it was based on the mean-field approximation to Eq. (2). In contrast, the present expression (11) is exact, provided (Formula presented) is specified.
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91. This fundamental anisotropy is present even for (Formula presented) (Formula presented) and is set by (Formula presented) The value of (Formula presented) along the (Formula presented) line [more precisely, for (Formula presented) is a universal ratio of the gauge theory scenario.
92. While the rescaling procedure described here is a straightforward generalization of the familiar (Formula presented) rescaling of anisotropy, it does contain one salient point: the rescaled gauge theory looks just like the original Eq. (4) but with the Φ-dependent part isotropic, (Formula presented) (Formula presented) and (Formula presented) (Formula presented) However, the gauge is different: instead of the original Coulomb gauge (Formula presented) we now have (Formula presented) This is dealt with by introducing a gauge transformation: (Formula presented) and (Formula presented) where (Formula presented) is a regular phase defined so that the new gauge potential (Formula presented) satisfies the Coulomb gauge condition, i.e., (Formula presented) Such gauge transformation is both perfectly justified and desirable. It is justified because, even though our fictitious electrodynamics (4) is always defined in a specific gauge and does not possess the local gauge freedom of its real namesake, its free energy is still invariant under gauge transformations. It is desirable since the Coulomb gauge ensures that Φ (or (Formula presented) in the anisotropic case) can be used as a true physical order parameter of our Φ transition: see T. Kennedy and C. King, Phys. Rev. Lett. 55, 776 (1985). Related issues involving gauge transformations are discussed in Ref. 9. In the isotropic case (Formula presented) in Eq. (1)] none of these niceties are necessary and we can stay in the Coulomb gauge throughout. The fundamental anisotropy of the gauge theory, Eq. (4), expressed as (Formula presented) (Formula presented) always remains.
93. One naively expects the range of fields over which (Formula presented) is universal to be smaller than the corresponding range for (Formula presented) the melting line is first order and involves a nonuniform low-temperature state.
94. The exponent p is assumed to have the same value, (Formula presented) throughout the Φ-ordered phase. We can show that this is the case in the regime where the average size of vortex loops and s vortex overhangs is (Formula presented) and we assume that it remains true everywhere below (Formula presented) This is consistent with our physical expectation that the long-wavelength limit of the Φ-ordered phase can be related to a suitably defined “line liquid,” in the sense of Nelson (Ref. 8). An alternative possibility, that of p being a continuously varying exponent, changing from (Formula presented) far below (Formula presented) to (Formula presented) as we approach (Formula presented) appears less likely in view of the overall picture painted by the gauge theory scenario. For example, the Φ-ordered state of Eq. (4) has a true long-range order and is not a critical phase with continuously varying exponents.
95. G. W. Crabtree and D. R. Nelson, Phys. Today 50(4), 38 (1997).
96. “Vortex tachyons” denote infinite vortex paths that traverse the system along x or y direction without winding along the z axis. Thus, in the boson analogy, they travel longer intervals across “space” ((Formula presented) plane) than along “time” (z axis (Formula presented)
97. In principle, however, such a phase could result in a new transition line between (Formula presented) and (Formula presented) at least in a certain window of material parameters.
98. J. Lidmar, M. Wallin, C. Wengel, S. M. Girvin, and A. P. Young, Phys. Rev. B 58, 2827 (1998), and references therein.
99. For a useful summary of various anisotropy effects, see E. Sardella, Phys. Rev. B 53, 14 506 (1996), and references therein.
100. The same would be true for the LL-based theories, if uncritically extended to the (Formula presented) limit.
101. Eventually, for extremely low fields, the effect of electromagnetic screening (i.e., finite (Formula presented) becomes important and (Formula presented) as (Formula presented) (Ref. 8).
102. A. K. Nguyen and A. Sudbø (unpublished and private communication).
103. Also, see S.-K. Chin, A. K. Nguyen, and A. Sudbø, cond-mat/9809115 (unpublished).The author is grateful to Prof. A. Sudbø and Dr. A. K. Nguyen for numerous discussions and the opportunity to see their unpublished numerical results.
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106. G. Parisi, Statistical Field Theory (Addison-Wesley, New York, 1988), Chap. 16.