Competing order in the mixed state of high-temperature superconductors

Physical Review B - Condensed Matter and Materials Physics

Volumn 66, Issue 14, 2002, Pages 1-8

  Kivelson, Steven A ^a   Lee, Dung Hai ^b   Fradkin, Eduardo ^c   Oganesyan, Vadim ^d  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b UNIVERSITY OF CALIFORNIA   (United States)

^c University of Illinois at Urbana Champaign   (United States)

^d PRINCETON UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84881142761     PISSN: 10980121     EISSN: 1550235X     Source Type: Journal    
DOI: 10.1103/PhysRevB.66.144516     Document Type: Article

References (41)

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25. In a crystal with a (formula presented) rotation axis, for instance a tetragonal crystal, the two nematic axes are related by rotations by (formula presented) For crystals with a mirror plane symmetry, so long as the preferred stripe direction is not in the mirror plane, the two nematic axes are related by reflection through this plane. In either case, the stripe orientational order parameter is Ising-like. It is also possible to imagine more complex stripe orientational order parameters in crystals with a screw symmetry, which still yield an Ising order parameter;
26. for example, in the low-temperature tetragonal phase of (formula presented) there are two planes per unit cell and a fourfold screw symmetry. It is possible to imagine more exotic possibilities, such as a crystal with a (formula presented) axis, in which case the nematic order parameter could be related to that of a three state Potts model.
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29. In 2D, or in the absence of interplane couplings, vortices are pointlike with a halo of the competing order parameter (formula presented) with a size of order (formula presented) Since an isolated vortex cannot order, it is only when the interactions between neighboring vortices are sufficiently strong that (formula presented) order occurs. Where (formula presented) is an (formula presented) rotor with N large, Demler et al. (Ref. 11) showed that, by the time interactions between vortices are sufficiently strong to trigger this order, the vortices are so strongly overlapping that (formula presented) is essentially uniform. They argued that the same is true in the (formula presented) Heisenberg case, as well. A simple argument shows that in the Ising (formula presented) case, this argument breaks down, at least for (formula presented) near (formula presented) An array of votices with spacing (formula presented) can be treated as a (formula presented)-dimensional transverse field Ising model, with one effective Ising spin per vortex halo, an effective exchange coupling which falls off exponentially as (formula presented) and a transverse field (formula presented) for which we can use the same Landau-Ginzburg estimate (formula presented) derived in Eq. (20). As a function of increasing magnetic field (formula presented) decreases, and the transition to the (formula presented) ordered state occurs when (formula presented) Neglecting any (formula presented) dependence that is less singular than that in h, we find that this transition occurs when (formula presented) This latter inequality insures the self-consistency of the approach. Consequently, the phase boundary has the very singular magnetic field dependence, (formula presented) where (formula presented) is the correlation length exponent of the (formula presented)-dimensional Ising model (formula presented) (For the (formula presented) and Heisenberg cases, the (formula presented) dependence of h is no more singular than that of J, so it is possible that, as is the case in large N, the transition occurs when (formula presented))
30. In other terms, we are assuming that B is infinitesimally larger than (formula presented) and that (formula presented)
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32. The asymptotic shape of the phase diagram in the neighborhood of (formula presented) is derived from the quantum critical renormalization of J by quantum fluctuations. For (formula presented) this leads to a singular dependence of the susceptibility of an isolated vortex, so that (formula presented) where (formula presented) is the dynamical exponent and (formula presented) is the correlation length exponent of the transverse field Ising model. Consequently, (formula presented) Similarly, for (formula presented) the susceptibility of a single vortex does not diverge as (formula presented) but rather reaches a large, asymptotic value (formula presented) where (formula presented) Consequently, the zero temperature curve (formula presented) approaches (formula presented) in an extremely singular fashion (not shown in Fig. 11): (formula presented)
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