We study both the electrical and thermal transport near the heavy-fermion quantum critical point (QCP), identified with the breakdown of the Kondo effect as an orbital selective Mott transition. We show that the contribution to the electrical conductivity comes mainly from conduction electrons while the thermal conductivity is given by both conduction electrons and localized fermions (spinons), scattered with dynamical exponent $z = 3$. This scattering mechanism gives rise to a quasi-linear temperature dependence of the electrical and thermal resistivity. The characteristic feature of the Kondo breakdown scenario turns out to be emergence of additional entropy carriers, that is, spinon excitations. As a result, we find that the Wiedemann-Franz ratio should be larger than the standard value, a fact which enables to differentiate the Kondo breakdown scenario from the Hertz-Moriya-Millis framework.
We study the electrical and thermal transport through the SU(N) Wolff model with the use of bosonization. The Wilson ratio reaches unity as N grows to infinity. The electric conductance is dominated by the charge channel, and decreases monotonically with increasing interaction. The thermal conductivity enhances in the presence of local Hubbard U. The Wiedemann-Franz law is violated, the Lorentz number depends strongly on the interaction parameter, which can be regarded as a manifestation of spin-charge separation.
The renormalization of the Wiedemann-Franz (WF) ratio in strongly correlated electron systems is analyzed within the Landau quasiparticle picture. We demonstrate that the WF law is violated: (i) at the quantum critical point, where the effective mass diverges, and (ii) beyond a point of fermion condensation, where the single-particle spectrum $ε(p)$ becomes flat. Results of the analysis are compared with available experimental data.
The thermal conductivity measurements have been performed on the heavy-fermion compound YbRh2Si2 down to 0.04 K and under magnetic fields through a quantum critical point (QCP) at Bc = 0.66 T || c-axis. In the limit as T -> 0, we find that the Wiedemann-Franz law is satisfied within experimental error at the QCP despite the destruction of the standard signature of Fermi liquid. Our results place strong constraints on models that attempt to describe the nature of unconventional quantum criticality of YbRh2Si2.
We consider heat transport in a Luttinger liquid (LL) with weak disorder and study the Lorenz number for this system. We start at a high-$T$ regime, and calculate both the electrical and thermal conductivities using a memory function approach. The resulting Lorenz number $L$ is independent of $T$ but depends explicitly on the LL exponents. Lowering $T$, however, allows for a renormalization of the LL exponents from their bare values by disorder, causing a violation of the Wiedemann-Franz law. Finally, we extend the discussion to quantum wire systems and study the wire size dependence of the Lo…
We show that in weakly disordered Luttinger liquids close to a commensurate filling the ratio of thermal conductivity kappa and electrical conductivity sigma can deviate strongly from the Wiedemann Franz (WF) law valid for Fermi liquids scattering from impurities. In the regime where the Umklapp scattering rate Gamma_U is much larger than the impurity scattering rate Gamma_imp, the Lorenz number L = (kappa/sigma T) rapidly changes from very large values, L ~ (Gamma_U/Gamma_imp) >> 1 at the commensurate point to very small values, L ~ (Gamma_imp/Gamma_U) << 1 for a slightly doped system. This s…
The temperature dependence of the interlayer electrical and thermal resistivity in a layered metal are calculated for Fermi liquid quasiparticles which are scattered inelastically by two-dimensional antiferromagnetic spin fluctuations. Both resistivities have a linear temperature dependence over a broad temperature range. Extrapolations to zero temperature made from this linear-$T$ range give values that appear to violate the Wiedemann-Franz law. However, below a low-temperature scale, which becomes small close to the critical point, a recovery of this law occurs. Our results describe recent m…
We introduce and analyze two different scenarios for violation of the Wiedemann-Franz law in strongly correlated electron systems of solids, close to a topological quantum critical point (TQCP) where the density of states $N(0)$ diverges. The first, applicable to the Fermi-liquid (FL) side of the TQCP, involves a transverse zero-sound collective mode that opens a new channel for the thermal conductivity, thereby enhancing the Lorenz number $L(0)$ relative to the value $L_0=π^2k^2_B/3e^2$ dictated by conventional FL theory. The second mechanism for violation of the WF law, relevant to the non-F…
Quantum dot devices allow one to access the critical point of the two-channel Kondo model. The effective critical theory involves a free Majorana fermion quasiparticle localized on the dot. As a consequence, this critical point shows both the phenomenon of non-Fermi liquid physics and fractionalization. Although a violation of the Wiedemann-Franz law is considered a defining feature of non-Fermi liquid systems, we show by exact calculations that it holds at the critical point, providing a counterexample to this lore. Furthermore, we show that the fractionalized Majorana character of the critic…
The thermal conductivity (k) of the Weyl semimetal NbP was studied with the thermal gradient and magnetic field applied parallel to [0 0 1] direction. At low temperatures k(B) exhibits large quantum oscillations with frequencies matching two of several determined from the Shubnikov - de Haas effect measured on the same sample with analogous electrical current and magnetic field orientation. Both frequencies found in k(B) originate from the electron pocket enclosing a pair of Weyl nodes. The amplitude of the oscillatory component of the thermal conductivity turns out to be two orders of magnitu…
ErTe$_3$ is studied as a model system to explore thermal transport in a layered charge density wave (CDW) material. We present data from thermal diffusivity, resistivity, and specific heat measurements: There is a sharp decrease in thermal conductivity both parallel and perpendicular to the primary CDW at the CDW transition temperature. At the same time, the resistivity changes more gradually. Correspondingly, while well above and below $T_c$, a consistent description of the thermal transport applies with essentially independent electron and phonon contributions (estimated using the Wiedemann …
We show that the Wiedemann-Franz law, which holds for Landau Fermi liquids, breaks down in a large-n treatment of the t-J model. The calculated ratio of the in-plane thermal and electrical conductivities agrees quantitatively with experiments on the normal state of the electron-doped Pr_{2-x}Ce_xCuO_4 (x = 0.15) cuprate superconductor. The violation of the Wiedemann-Franz law in the uniform phase contrasts with other properties of the phase that are Fermi liquid like.
The Wiedemann-Franz (WF) law establishes a link between heat and charge transport due to electrons in solids. The extent of its validity in presence of inelastic scattering is a question raised in different contexts. We report on a study of the electrical, $σ$, and thermal, $κ$, conductivities in WP$_2$ single crystals. The WF holds at 2 K, but a downward deviation rapidly emerges upon warming. At 13 K, there is an exceptionally large mismatch between Lorenz number and the Sommerfeld value. We show that this is driven by a fivefold discrepancy between the $T$-square prefactors of electrical an…
We study the thermal transport through a Majorana island connected to multiple external quantum wires. In the presence of a large charging energy, we find that the Wiedemann-Franz law is nontrivially violated at low temperature, contrarily to what happens for the overscreened Kondo effect and for nontopological junctions. For three wires, we find that the Lorenz ratio is rescaled by a universal factor 2/3 and we show that this behavior is due to the presence of localized Majorana modes on the island.
We analyze the charge and thermal transport at a junction of interacting quantum wires close to equilibrium. Within the framework of Tomonaga-Luttinger liquids, we compute the thermal conductance for a wide class of boundary conditions and detail the physical processes leading to the breakdown of the Wiedemann-Franz law at the junction. We show how connecting external reservoirs to the quantum wires affects the conductance tensors close to the various fixed points of the phase diagram of the junction. We therefore distinguish two types of violation of the Wiedemann-Franz law: a "trivial" one, …
We study thermal transport in the disordered Fermi liquid at low temperatures. Gravitational potentials are used as sources for finding the heat density and its correlation function. For a comprehensive study, we extend the renormalization group (RG) analysis developed for electric transport by including the gravitational potentials into the RG scheme. Our analysis reveals that the Wiedemann-Franz law remains valid even in the presence of quantum corrections caused by the interplay of diffusion modes and the electron electron interaction. In the present scheme this fundamental relation is clos…
Recently, enhancement of the longitudinal magneto-electrical conductivity (LMEC) has been observed in Bi$_{1-x}$Sb$_{x}$ around $x \sim 3\%$ under $\bm{E} \parallel \bm{B}$ ($\bm{E}$: external electric field and $\bm{B}$: external magnetic field) [Phys. Rev. Lett. {\bf 111}, 246603 (2013)], where an enhancement factor proportional to $B^{2}$ is suggested to result from the $\bm{E} \cdot \bm{B}$ term. In the present study, we show that this $B^{2}$ enhancement is not limited on the LMEC, where both the Seebeck and thermal conductivities in the longitudinal setup ($\bm{E} \parallel \bm{B}$) are …
We present the theory of thermoelectric transport in metals with long-lived quasiparticles, carefully addressing the interplay of electron-electron scattering as well as electron-impurity scattering, but neglecting electron-phonon scattering. In Fermi liquids with a large Fermi surface and weak electron-impurity scattering, we provide universal and simple formulas for the behavior of the thermoelectric conductivities across the ballistic-to-hydrodynamic crossover. In this regime, the electrical conductivity is relatively unchanged by hydrodynamic effects. In contrast, the thermal conductivity …
Strain tuning Sr$_{2}$RuO$_{4}$ through the Lifshitz point, where the Van Hove singularity of the electronic spectrum crosses the Fermi energy, is expected to cause a change in the temperature dependence of the electrical resistivity from its Fermi liquid behavior $ρ\sim T^{2}$ to $ρ\sim T^{2}{\rm log}\left(1/T\right)$, a behavior consistent with experiments by Barber et al. [Phys. Rev. Lett. 120, 076601 (2018)]. This expectation originates from the same multi-band scattering processes with large momentum transfer that were recently shown to account for the linear in $T$ resistivity of the str…
We present a study of heat and charge transport in Bi$_{2+x}$Sr$_{2-x}$CuO$_{6+δ}$ focused on the size of the low-temperature linear term of the thermal conductivity at optimal doping level. In the superconducting state, the magnitude of this term implies a d-wave gap with an amplitude close to what has been reported. In the normal state, recovered by the application of a magnetic field, measurement of this term and residual resistivity yields a Lorenz number $L = κ_{N}ρ_{0}/T = 1.3 \pm 0.2 L_{0}$. The departure from the value expected by the Wiedemann-Franz law is thus slightly larger than ou…
We derive the electrical and thermal conductivities of the $d$-density-wave state in an external magnetic field $B$ in the low-temperature regime and in the presence of impurities. We show that in the zero-temperature limit, $T \to 0$, the Wiedemann-Franz (WF) law remains intact irrespectively of the value of the applied field and chemical potential $μ$ as soon as there is a scattering from impurities. For finite $T \lesssim |μ|$ the WF law violation is possible and it is enhanced by the external field. Depending on the relative values of $B$ and $T$ the electrical conduction may dominate the …
Interactions between particles in quantum many-body systems can lead to collective behavior described by hydrodynamics. One such system is the electron-hole plasma in graphene near the charge neutrality point which can form a strongly coupled Dirac fluid. This charge neutral plasma of quasi-relativistic fermions is expected to exhibit a substantial enhancement of the thermal conductivity, due to decoupling of charge and heat currents within hydrodynamics. Employing high sensitivity Johnson noise thermometry, we report the breakdown of the Wiedemann-Franz law in graphene, with a thermal conduct…
With the superconducting cuprates in mind, a set of unitary transformations was used to decouple electrons and phonons in the strong-coupling limit. While phonons remain almost unrenormalised, electrons are transformed into itinerent singlet and triplet bipolarons and thermally excited polarons. The triplet/singlet exchange energy and the binding energy of the bipolarons are thought to account for the spin and charge pseudogaps in the cuprates, respectively. We calculated the Hall Lorenz number of the system to show that the Wiedemann-Franz law breaks down due to the interference of the polaro…
A general discussion of the ratio of thermal and electrical conductivities in non-Fermi liquid metals is given. In metals with sharp Drude peaks, the relevant physics is correctly organized around the slow relaxation of almost-conserved momenta. While in Fermi liquids both currents and momenta relax slowly, due to the weakness of interactions among low energy excitations, in strongly interacting non-Fermi liquids typically only momenta relax slowly. It follows that the conductivities of such non-Fermi liquids are obtained within a fundamentally different kinematics to Fermi liquids. Among thes…
To investigate the validity of the Wiedemann-Franz (WF) law in disordered but metallic cuprates, the low-temperature charge and heat transport properties are carefully studied for a series of impurity-substituted and carrier-overdoped La_{1.8}Sr_{0.2}Cu_{1-z}M_zO_4 (M = Zn or Mg) single crystals. With moderate impurity substitution concentrations of z = 0.049 and 0.082 (M = Zn), the resistivity shows a clear metallic behavior at low temperature and the WF law is confirmed to be valid. With increasing impurity concentration to z = 0.13 (M = Zn) or 0.15 (M = Mg), the resistivity shows a low-T up…
We consider the thermoelectric properties of the mixed-dimensional quantum electrodynamics of the relativistic Dirac fermion and Wilson-Fisher boson. These models are self-dual, and can form non-trivial many-body phases depending on the values of chemical potential, background magnetic field and the electromagnetic fine-structure constant. Using particle-vortex duality, we derive a variety of thermoelectric relations for strongly-interacting phases with classic paradigms such as the Wiedemann-Franz law and the Mott's relation in the dual weakly interacting regimes. Besides, at the self-dual po…
The journey through the nontrivial band topology beyond the conventional band structure has resulted in the recent discovery of three-dimensional Dirac semimetal phase in Na$_{3}$Bi and Cd$_{3}$As$_{2}$. The bulk state of which is semi-metallic obeying linear energy dispersion, while the surface state is topology protected Fermi arc. Due to the unique band topology, they show different exotic electronic properties of both fundamental and technological interest. From electrical and thermal transport measurements, we have demonstrated a remarkable violation of Wiedemann-Franz law (WFL) under app…
The Wiedemann-Franz (WF) law links the ratio of electronic charge and heat conductivity to fundamental constants. It has been tested in numerous solids, but the extent of its relevance to the anomalous transverse transport, which represents the topological nature of the wave function, remains an open question. Here we present a study of anomalous transverse response in the noncollinear antiferromagnet Mn$_{3}$Ge extended from room temperature down to sub-Kelvin temperature and find that the anomalous Lorenz ratio remains close to the Sommerfeld value up to 100 K, but not above. The finite-temp…
The celebrated Wiedemann-Franz (WF) law which governs the relation between charge and heat transport traces back to the experimental discovery in 1853 by Wiedemann and Franz. Despite the fundamental difference of the quantum-statistical properties between fermions and bosons, the linear-in-$T$ behavior of the WF law at low temperatures has recently been found to be the universal property by the discovery of the WF law for magnon transport. However, the WF law is for the linear response, and whether or not the universal law is valid even in the nonlinear regime of Bose systems remains an open i…
We present a study of heat transport in the cuprate superconductor Bi_{2+x}Sr_{2-x}CuO_{6+δ} at subkelvin temperatures and in magnetic fields as high as 25T. In several samples with different doping levels close to optimal, the linear-temperature term of thermal conductivity was measured both at zero-field and in presence of a magnetic field strong enough to quench superconductivity. The zero-field data yields a superconducting gap of reasonable magnitude displaying a doping dependence similar to the one reported in other families of cuprate. The normal-state data together with the results of …
We present a quantum Hall effect of magnons in two-dimensional clean insulating magnets at finite temperature. Through the Aharonov-Casher effect, a magnon moving in an electric field acquires a geometric phase and forms Landau levels in an electric field gradient of sawtooth form. At low temperatures, the lowest energy band being almost flat carries a Chern number associated with a Berry curvature. Appropriately defining the thermal conductance for bosons, we find that the magnon Hall conductances get quantized and show a universal thermomagnetic behavior, i.e., are independent of materials, …
The in-plane thermal conductivity kappa(T) and electrical resistivity rho(T) of the heavy-fermion metal YbRh2Si2 were measured down to 50 mK for magnetic fields H parallel and perpendicular to the tetragonal c axis, through the field-tuned quantum critical point, Hc, at which antiferromagnetic order ends. The thermal and electrical resistivities, w(T) and rho(T), show a linear temperature dependence below 1 K, typical of the non-Fermi liquid behavior found near antiferromagnetic quantum critical points, but this dependence does not persist down to T = 0. Below a characteristic temperature T* ~…
We theoretically establish mutual relations among magnetic momentum, heat, and fluctuations of propagating magnons in a ferromagnetic insulating junction in terms of noise and the bosonic Wiedemann-Franz (WF) law. Using the Schwinger-Keldysh formalism, we calculate all transport coefficients of a noise spectrum for both magnonic spin and heat currents, and establish Onsager relations between the thermomagnetic currents and the zero-frequency noise. Making use of the magnonic WF law and the Seebeck coefficient in the low-temperature limit, we theoretically discover universal relations, i.e. bei…
We introduce an unconventional regularization of condensed matter effective field theory by a non-local bulk topological regulator in one higher dimension, which is equivalent to a bulk-interface-bulk formulation. We show its necessity motivated by a (0+1)-dimensional system. Although the system can be strictly defined on the lattice in its own dimensions, the bulk regulator is important to derive its correct physical observables, which cannot be captured by any local regulator. We explicitly obtain a constraint on the thermal and electric integer Hall conductances of half-filled translation i…
One of the main goals of spintronics is to improve transport of information carriers and to achieve new functionalities with ultra-low dissipation. A most promising strategy for this holy grail is to use pure magnon currents created and transported in insulating magnets, in the complete absence of any conducting metallic elements. Here we propose a realistic solution to this fundamental challenge by analyzing magnon and heat transport in insulating ferromagnetic junctions. We calculate all transport coefficients for magnon transport and establish Onsager relations between them. We theoreticall…
The thermal conductivity $κ$ of the cuprate superconductor La$_{1.6-x}$Nd$_{0.4}$Sr$_x$CuO$_4$ was measured down to 50 mK in seven crystals with doping from $p=0.12$ to $p=0.24$, both in the superconducting state and in the magnetic field-induced normal state. We obtain the electronic residual linear term $κ_0/T$ as $T \to 0$ across the pseudogap critical point $p^{\star}= 0.23$. In the normal state, we observe an abrupt drop in $κ_0/T$ upon crossing below $p^{\star}$, consistent with a drop in carrier density $n$ from $1 + p$ to $p$, the signature of the pseudogap phase inferred from the Hall…
A violation of the Wiedemann-Franz law in a metal can be quantified by comparing the Lorentz ratio, $L=κρ/T$, where $κ$ is the thermal conductivity and $ρ$ is the electrical resistivity, with the universal Sommerfeld constant, $L_0=(π^2/3) (k_B/e)^2$. We obtain the Lorentz ratio of a clean compensated metal with intercarrier interaction as the dominant scattering mechanism by solving exactly the system of coupled integral Boltzmann equations. The Lorentz ratio is shown to assume a particular simple form in the forward-scattering limit: $L/L_0=\overline{Θ^2}/2$, where $Θ$ is the scattering angl…
Physicists are engaged in vigorous debate on the nature of the quantum critical points (QCP) governing the low-temperature properties of heavy-fermion (HF) metals. Recent experimental observations of the much-studied compound YbRh2Si2 in the regime of vanishing temperature incisively probe the nature of its magnetic-field-tuned QCP. The jumps revealed both in the residual resistivity rho_0 and the Hall resistivity R_H, along with violation of the Wiedemann-Franz law, provide vital clues to the origin of such non-Fermi-liquid behavior. The empirical facts point unambiguously to association of t…
We consider the behavior of strongly correlated electron liquid in high-temperature superconductors within the framework of the fermion condensation model. We show that at low temperatures the normal state recovered by the application of a magnetic field larger than the critical field can be viewed as the Landau Fermi liquid induced by the magnetic field. In this state, the Wiedemann-Franz law and the Korringa law are held and the elementary excitations are the Landau Fermi Liquid quasiparticles. Contrary to what might be expected from the Landau theory, the effective mass of quasiparticles de…
The behaviour of electrons in solids is remarkably well described by Landau's Fermi-liquid theory, which says that even though electrons in a metal interact they can still be treated as well-defined fermions, called ``quasiparticles''. At low temperature, the ability of quasiparticles to transport heat is strictly given by their ability to transport charge, via a universal relation known as the Wiedemann-Franz law, which no material in nature has been known to violate. High-temperature superconductors have long been thought to fall outside the realm of Fermi-liquid theory, as suggested by seve…
Motived by time-reversal symmetry breaking and giant anomalous Hall effect in kagome superconductor \textit{A}V$_3$Sb$_5$ (\textit{A} = Cs, K, Rb), we carried out the thermal transport measurements on CsV$_3$Sb$_5$. In addition to the anomalous Hall effect, the anomalous Nernst effect and the anomalous thermal Hall effect emerge. Interestingly, the longitudinal thermal conductivity $κ_{xx}$ largely deviates from the electronic contribution obtained from the longitudinal conductivity $σ_{xx}$ by the Wiedemann-Franz law. In contrast, the thermal Hall conductivity $κ_{xy}$ is roughly consistent w…
We report on the magnetic field (0T$ \le B \le 9$T) dependence of the longitudinal thermal conductivity $κ(T,B)$ of highly oriented pyrolytic graphite in the temperature range 5 K $\le T\le$ 20 K for fields parallel to the $c-$axis. We show that $κ(T,B)$ shows large oscillations in the high-field region (B > 2 T) where clear signs of the Quantum-Hall effect are observed in the Hall resistance. With the measured longitudinal electrical resistivity we show that the Wiedemann-Franz law is violated in the high-field regime.
We study the electric and thermal transport of the Dirac carriers in monolayer graphene using the Boltzmann-equation approach. Motivated by recent thermopower measurements [F. Ghahari, H.-Y.~Xie, T. Taniguchi, K. Watanabe, M.~S.~Foster, and P.~Kim, Phys.\ Rev.\ Lett.\ {\bf 116}, 136802 (2016)], we consider the effects of quenched disorder, Coulomb interactions, and electron--optical-phonon scattering. Via an unbiased numerical solution to the Boltzmann equation we calculate the electrical conductivity, thermopower, and electronic component of the thermal conductivity, and discuss the validity …
In CeCoIn$_5$, the thermal conductivity $κ_{xx}$ and Hall conductivity $κ_{xy}$ below $T_c$ display large anomalies below $T_c$. The strong suppression of the anomalies in weak fields implies the existence of long-lived quasiparticles. We also discuss briefly the Wiedemann-Franz ratio and the existence of a strongly field-dependent spin-fluctuation heat current.
The heavy-fermion metal YbRh$_{2}$Si$_{2}$ is a weak antiferromagnet below $T_{N} = 0.07$ K. Application of a low magnetic field $B_{c} = 0.06$ T ($\perp c$) is sufficient to continuously suppress the antiferromagnetic (AF) order. Below $T \approx 10$ K, the Sommerfeld coefficient of the electronic specific heat $γ(T)$ exhibits a logarithmic divergence. At $T < 0.3$ K, $γ(T) \sim T^{-ε}$ ($ε: 0.3 - 0.4$), while the electrical resistivity $ρ(T) = ρ_{0} + aT$ ($ρ_{0}$: residual resistivity). Upon extrapolating finite-$T$ data of transport and thermodynamic quantities to $T = 0$, one observes (i)…
In this paper we study the thermopower of a quantum dot connected to two leads in the presence of Kondo correlation by employing a modified second-order perturbation scheme at nonequilibrium. A simple scheme, Ng's ansatz [Phys. Rev. Lett. {\bf 76}, 487 (1996)], is adopted to calculate nonequilibrium distribution Green's function and its validity is further checked with regard to the Onsager relation. Numerical results demonstrate that the sign of the thermopower can be changed by tuning the energy level of the quantum dot, leading to a oscillatory behavior with a suppressed magnitude due to th…
We present a study of the electrical and thermal conductivities of the $d$-density wave (DDW) state in an external magnetic field $B$ in the low temperature regime and in the presence of impurities. We show that in the zero temperature limit, $T \to 0$, the Wiedemann-Franz (WF) law remains intact. For finite $T$ the WF law violation is possible and it is enhanced by the external field.
We study electric, thermal, and thermoelectric conductivities in the vicinity of a z=2 superconductor-diffusive metal transition in two dimensions, both in the high and low frequency limits. We find violation of the Wiedemann-Franz law and a dc thermoelectric conductivity $α$ that does not vanish at low temperatures, in contrast to Fermi liquids. We introduce a Langevin equation formalism to study critical dynamics over a broad region surrounding the quantum critical point.
We investigate theoretically the thermoelectric transport properties of a T-shaped double quantum dot side-coupled to a topological superconducting nanowire hosting Majorana zero-energy modes. The calculations are performed using the numerical renormalization group method focusing on the transport regime, where the system exhibits the two-stage Kondo effect. It is shown that the leakage of Majorana quasiparticles into the double dot system results in a half-suppression of the second stage of the Kondo effect, which is revealed through fractional values of the charge and heat conductances and g…
We describe the thermal and electrical conductivities of quasi-one dimensional wires, across a quantum phase transition from a superconductor to a metal induced by pairbreaking perturbations. Fluctuation corrections to BCS theory motivate a field theory for quantum criticality. We describe deviations in the Wiedemann-Franz ratio from the Lorenz number, which can act as sensitive tests of the theory. We also describe the crossovers out of the quantum critical region into the metallic and superconducting phases.
