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arXiv:cond-mat/0212204

Solution of mass/gap equations for strong velocity anisotropy in the QED3 theory of underdoped cuprates

Dominic J. Lee

Published Dec 10, 2002• 0 citations
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Abstract

The low-lying excitations at the nodes of the d-wave gap in the normal state for underdoped cuprates, close to the superconducting phase transition, may be described by an effective QED3 theory. There are three characteristic velocities: v_F, v_Delta and c. For v_Delta=v_F=c, the model reduces to N-flavour QED3. Here, in the isotropic limit, for N100. However, we argue that N_c=N_0c, when we go beyond the approximations used to derive our mass gap equations. We expect, though, that our solution for N=2 is roughly correct, at low momentum, for large enough anisotropies. Implications and possible extensions to this work are also discussed.

DOI
10.48550/arxiv.cond-mat/0212204

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The low-lying excitations at the nodes of the d-wave gap in the normal state for underdoped cuprates, close to the superconducting phase transition, may be described by an effective QED3 theory. There are three characteristic velocities: v_F, v_Delta and c. For v_Delta=v_F=c, the model reduces to N-flavour QED3. Here, in the isotropic limit, for N100. However, we argue that N_c=N_0c, when we go beyond the approximations used to derive our mass gap equations. We expect, though, that our solution for N=2 is roughly correct, at low momentum, for large enough anisotropies. Implications and possible extensions to this work are also discussed.
A version of non‐compact QED in 2+1 dimensions (QED3), with anisotropies introduced by differing Fermi and gap velocities in the spatial directions has been proposed as an effective theory of the SDW‐pseudogap phase transition in underdoped cuprates at T=0. Preliminary simulations using lattice gauge theory techniques suggest that matters are a little more complicated than previously thought; in particular, measurements of the fermion propagator in Landau gauge indicate that one should be cautious in inferring properties of this form of QED3 from those of isotropic QED3.
We develop a slave-boson theory for the $t\ensuremath{-}J$ model at finite doping that respects an SU(2) symmetry: a symmetry previously known to be important at half filling. The mean-field phase diagram is found to be consistent with the phases observed in the cuprate superconductors, which contain ddd-wave superconductor, spin-gap, strange metal, and Fermi-liquid phases. The spin-gap phase is best understood as the staggered flux phase, which is nevertheless translationally invariant for physical quantities. The physical electron spectral function shows small Fermi segments at low doping that continuously evolve into the large Fermi surface at high-doping concentrations. The close relation between the SU(2) and the U(1) slave-boson theory is discussed. The low-energy effective theory for the low-lying fluctuations is derived and additional lying modes [which were overlooked in the U(1) theory] are identified.
With the coupled Dyson-Schwinger equations in the framework of the unified QED3 theory,we study the phase transition between the antiferromagnet(AF) and the d-wave superconductor (dSC) of planar cuprates at T=0.By solving the coupled Dyson-Schwinger equations both analytically and numerically in rainbow approximation in Landau gauge and comparing the obtained results with that given in the 1/N expansion,we find that there exists a chiral symmetry breaking from dSC phase to AF phase when the quasi- fermion flavors N■4 in half-filling and the AF phase can possibly coexist with the dSC phase in the underdoped region.By comparing the pressure between the coexistent AF-dSC phase and dSC phase,we find that AF-dSC coexisting phase is the stable phase,the AF phase can then coexist with the dSC phase.
The ubiquitous phase diagram of the cuprate superconductors has a phase where the density of states still has a gap above the superconducting transition temperature. Due to the presence of a gap without any other superconducting characteristics, this phase is known as the pseudogap (PG) state. Emery and Kivelson proposed that in this state the superconducting order parameter still has a finite amplitude, however the quantal phase is disordered in such a way that it does not possess long range order. Using these ideas, Franz and Tesanovic developed a theory to explain the pseudogap phase that is identical to three dimensional quantum electrodynamics (QED3 ) apart from an intrinsic anisotropy. Amazingly, this theory can easily explain the antiferromagnetic region. The main purpose of this thesis is twofold. First, to explore some finite temperature properties of QED3 . We do this by calculating a Bosonic polarization bubble at finite temperature in the Matsubara formalism. Secondly we wish to derive some experimentally testable predictions of QED3 . To this end we calculate the modification to the Patrick Lee universal conductivity in the pseudogap state.
We study the effect of gapless quasiparticles in a d-wave superconductor on the T=0T=0T=0 end point of the Kosterlitz-Thouless transition line in underdoped high-temperature superconductors. Starting from a lattice model that has gapless fermions coupled to three-dimensional XY\mathrm{XY}XY phase fluctuations of the superconducting order parameter, we propose a continuum field theory to describe the quantum phase transition between the d-wave superconductor and the spin-density-wave insulator. Without fermions, the theory reduces to the standard Higgs scalar electrodynamics (HSE), which is known to have the critical point in the inverted XY\mathrm{XY}XY universality class. Extending the renormalization group calculation for the HSE to include the coupling to fermions, we find that the qualitative effect of fermions is to increase the portion of the space of coupling constants where the transition is discontinuous. The critical exponents at the stable fixed point vary continuously with the number of fermion fields N, and we estimate the correlation length exponent $(\ensuremath{\nu}=0.65)$ and the vortex field anomalous dimension $({\ensuremath{\eta}}_{\ensuremath{\Phi}}=\ensuremath{-}0.48)$ at the quantum critical point for the physical case N=2.N=2.N=2. The stable critical point in the theory disappears for the number of Dirac fermions N>Nc,N>{N}_{c},N>Nc​, with ${N}_{c}\ensuremath{\approx}3.4$ in our approximation. We discuss the relationship between the superconducting and the chiral (SDW) transitions, and point to some interesting parallels between our theory and the Thirring model.
We propose that the recently observed violation of the Wiedemann-Franz law in the normal state of underdoped cuprates is caused by spin-charge separation and dynamical chiral symmetry breaking in a (2+1)(2+1)(2+1)-dimensional system consisting of massless Dirac fermions, charged bosons, and a gauge field. While the d-wave spinon gap vanishes at the Fermi points, the nodal fermions acquire a finite mass due to strong gauge fluctuations. This mass provides a gap below which no free fermions can be excited. This implies that there is not a residual linear term for the thermal conductivity, in good agreement with experiments. Other physical implications of the chiral symmetry breaking are also discussed.