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arXiv:cond-mat/0404180 · Physical Review B

Transport properties of clean and disordered superconductors in matrix field theory

Lubo Zhou, T. R. Kirkpatrick

Published Mar 22, 2004• 1 citation
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Abstract

A comprehensive field theory is developed for superconductors with quenched disorder. We first show that the matrix field theory, used previously to describe a disordered Fermi liquid and a disordered itinerant ferromagnet, also has a saddle-point solution that describes a disordered superconductor. A general gap equation is obtained. We then expand about the saddle point to Gaussian order to explicitly obtain the physical correlation functions. The ultrasonic attenuation, number density susceptibility, spin-density susceptibility, and the electrical conductivity are used as examples. Results in the clean limit and in the disordered case are discussed, respectively. This formalism is expected to be a powerful tool to study the quantum phase transitions between the normal-metal state and the superconductor state.

DOI
10.1103/physrevb.69.104516

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A comprehensive field theory is developed for superconductors with quenched disorder. We first show that the matrix field theory, used previously to describe a disordered Fermi liquid and a disordered itinerant ferromagnet, also has a saddle-point solution that describes a disordered superconductor. A general gap equation is obtained. We then expand about the saddle point to Gaussian order to explicitly obtain the physical correlation functions. The ultrasonic attenuation, number density susceptibility, spin-density susceptibility, and the electrical conductivity are used as examples. Results in the clean limit and in the disordered case are discussed, respectively. This formalism is expected to be a powerful tool to study the quantum phase transitions between the normal-metal state and the superconductor state.
In this thesis a systematic, functional matrix field theory is developed to describe both clean and disordered s-wave and d-wave superconductors and the quantum phase transitions associated with them. The thesis can be divided into three parts. The first part includes chapters 1 to 3. In chapter one a general physical introduction is given. In chapters two and three the theory is developed and used to compute the equation of state as well as the number-density susceptibility, spin-density susceptibility, the sound attenuation coefficient, and the electrical conductivity in both clean and disordered s-wave superconductors. The second part includes chapter four. In this chapter we use the theory to describe the disorder-induced metal - superconductor quantum phase transition. The key physical idea here is that in addition to the superconducting order-parameter fluctuations, there are also additional soft fermionic fluctuations that are important at the transition. We develop a local field theory for the coupled fields describing superconducting and soft fermionic fluctuations. Using simple renormalization group and scaling ideas, we exactly determine the critical behavior at this quantum phase transition. Our theory justifies previous approaches. The third part includes chapter five. In this chapter we study the analogous quantum phase transition in disordered d-wave superconductors. This theory should be related to high TcT_cTc​ superconductors. Surprisingly, we show that in both the underdoped and overdoped regions, the coupling of superconducting fluctuations to the soft disordered fermionic fluctuations is much weaker than that in the s-wave case. The net result is that the disordered quantum phase transition in this case is a strong coupling, or described by an infinite disordered fixed point, transition and cannot be described by the perturbative RG description that works so well in the s-wave case. The transition appears to be related to the one that occurs in disordered quantum antiferromagnets.
xi The transport properties of a one dimensional disordered system of interacting spinpolarized electrons is studied using a bosonized field theory. In contrast to other approaches in that field [3], [2], a fully bosonized action on the Keldysh contour is used to calculate disorder averaged observables. After a brief introduction to Fermi and Luttinger liquids, we review bosonization, the Keldysh technique and disorder effects in mesoscopic systems. We calculate the first order correction to the clean conductivity in the disorder strength in order to develop a diagrammatic technique for a bosonized system on the Keldysh contour. Furthermore, the dissipative part of the self energy of the retarded plasmonic Green’s function is calculated via equations of motion in the high temperature regime. Corrections to the Drude conductivity of order = 1−K (weak interactions) and ωβ (high temperature) are calculated representing the terms beyond the saddle point approximation.
We develop an analytical theory, based on the quantum cavity method, describing the quantum phase transitions in low-temperature, strongly disordered ferromagnets and superconductors. At variance with the usual quantum critical points, we find a phase diagram with two critical points separating three phases. When the disorder increases, the systems goes from the ordered phase to an intermediate disordered phase characterized by activated transport and then to a second disordered phase where no transport is possible. Both the ordered and disordered phases exhibit strong inhomogeneity of their basic properties typical of glassy physics.
The effects of quenched disorder on the critical properties of itinerant quantum antiferromagnets and ferromagnets are considered. Particular attention is paid to locally ordered spatial regions that are formed in the presence of quenched disorder even when the bulk system is still in the paramagnetic phase. These rare regions or local moments are reflected in the existence of spatially inhomogeneous saddle points of the Landau-Ginzburg-Wilson functional. We derive an effective theory that takes into account small fluctuations around all of these saddle points. The resulting free energy functional contains a new term in addition to those obtained within the conventional perturbative approach, and it comprises what would be considered nonperturbative effects within the latter. A renormalization group analysis shows that in the case of antiferromagnets, the previously found critical fixed point is unstable with respect to this new term, and that no stable critical fixed point exists at one-loop order. This is contrasted with the case of itinerant ferromagnets, where we find that the previously found critical behavior is unaffected by the rare regions due to an effective long-ranged interaction between the order parameter fluctuations.
Zero-temperature or quantum phase transitions in itinerant electronic systems both with and without quenched disordered are discussed. Phase transitions considered include, the ferromagnetic transition, the antiferromagnetic transition, the superconductor-metal transition, and various metal-insulator transitions. Emphasis is placed on how to determine the universal properties that characterize these quantum phase transitions. For the first three of the phase transitions listed above, one of the main physical ideas established is that in zero-temperature systems there are soft or slow modes that exist in addition to the soft order parameter fluctuations, and that these modes can couple to the critical modes. These extra soft modes are shown to have a profound effect on the quantum critical properties. For quantum phase transitions involving zero wavenumber order parameters, i.e., the ferromagnetic and superconductor-metal transitions, these extra modes effectively lead to long-ranged effective interactions between order parameter fluctuations, which in turn lead to exactly soluble critical behaviors. For the antiferromagnetic case, we argue that while in low enough dimensions disorder fluctuation effects tend to destroy long-range order, quantum fluctuations counteract this effect and in some parameter regions manage to re-establish antiferromagnetic long-range order. For the metal-insulator transition, some recent new ideas are reviewed. In particular, it is pointed out that for interacting disordered electrons, one expects that in high dimensions the metal-insulator transition is related to the phase transition that occurs in random-field magnets in high dimensions. If the analogy also holds in three dimensions this suggests that the metal-insulator transition might have glassy characteristics.
We calculate the dynamical conductivity σðωÞ and the bosonic (pair) spectral function PðωÞ from quantum Monte Carlo simulations across clean and disorder-driven superconductor-insulator transitions (SITs).We identify characteristic energy scales in the superconducting and insulating phases that vanish at the transition due to enhanced quantum fluctuations, despite the persistence of a robust fermionic gap across the SIT.Disorder leads to enhanced absorption in σðωÞ at low frequencies compared to the SIT in a clean system.Disorder also expands the quantum critical region, due to a change in the universality class, with an underlying T ¼ 0 critical point with a universal low-frequency conductivity σ Ã ≃ 0.5ð4e 2 =hÞ.