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arXiv:2108.04149 · Physical review. B./Physical review. B

Multiparticle scattering and breakdown of the Wiedemann-Franz law at a junction of N interacting quantum wires

Domenico Giuliano, Andrea Kazemi Nava, Reinhold Egger, Pasquale Sodano, Francesco Buccheri

Published Jan 18, 2022• 22 citations
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Abstract

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, independent of the junction dynamics and arising from the breakdown of the Fermi-liquid picture in the wire, and a junction-related counterpart, arising from multiparticle scattering processes at the junction.

DOI
10.1103/physrevb.105.035419

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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, independent of the junction dynamics and arising from the breakdown of the Fermi-liquid picture in the wire, and a junction-related counterpart, arising from multiparticle scattering processes at the junction.
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 Lorenz number.
We study a junction of three quantum wires enclosing a magnetic flux. The wires are modelled as single-channel spinless Tomonaga–Luttinger liquids. This is the simplest problem of a quantum junction between Tomonaga–Luttinger liquids in which Fermi statistics enter in a non-trivial way. We study the problem using a mapping onto the dissipative Hofstadter model, describing a single particle moving on a plane in a magnetic field and a periodic potential coupled to a harmonic oscillator bath. Alternatively we study the problem by identifying boundary conditions corresponding to the low energy fixed points. We obtain a rich phase diagram including a chiral fixed point in which the asymmetric current flow is highly sensitive to the sign of the flux and a phase in which electron pair tunnelling dominates. We also study the effects on the conductance tensor of the junction of contacting the three quantum wires to Fermi liquid reservoirs.
We study the effect that resistive regions have on the conductance of a quantum wire with interacting electrons which is connected to Fermi liquid leads. Using the bosonization formalism and a Rayleigh dissipation function to model the power dissipation, we use both scattering theory and Green's function techniques to derive the DC conductance. The resistive regions are generally found to lead to incoherent transport. For a single wire, we find that the resistance adds in series to the contact resistance of h / e 2 for spinless electrons, and the total resistance is independent of the Luttinger parameter K W of the wire. We numerically solve the bosonic equations to illustrate what happens when a charge density pulse is incident on the wire; the results depend on the parameters of the resistive and interacting regions in interesting ways. For a junction of Tomonaga-Luttinger liquid wires, we use a dissipationless current splitting matrix to model the junction. For a junction of three wires connected to Fermi liquid leads, there are two families of such matrices; we find that the conductance matrix generally depends on K W for one family but is independent of K W for the other family, regardless of the resistances present in the system.
We study the effect that resistive regions have on the conductance of a quantum wire with interacting electrons which is connected to Fermi liquid leads. Using the bosonization formalism and a Rayleigh dissipation function to model the power dissipation, we use both scattering theory and Green's function techniques to derive the DC conductance. The resistive regions are generally found to lead to incoherent transport. For a single wire, we find that the resistance adds in series to the contact resistance of h/e(2) for spinless electrons, and the total resistance is independent of the Luttinger parameter K-W of the wire. We numerically solve the bosonic equations to illustrate what happens when a charge density pulse is incident on the wire; the results depend on the parameters of the resistive and interacting regions in interesting ways. For a junction of Tomonaga-Luttinger liquid wires, we use a dissipationless current splitting matrix to model the junction. For a junction of three wires connected to Fermi liquid leads, there are two families of such matrices; we find that the conductance matrix generally depends on K-W for one family but is independent of K-W for the other family, regardless of the resistances present in the system. Copyright (c) EPLA, 2011
We theoretically investigate the thermal-transport properties of a unitary Fermi gas between two reservoirs connected by a quantum point contact. We find several distinguished properties that are qualitatively different from those of weak or noninteracting gas systems. The particle transport figure of merit is drastically enhanced in the unitary regime. The Lorentz number violates the Wiedemann-Franz law, demonstrating the breakdown of Fermi liquid. These transport properties are the hallmarks of the unitary Fermi gas and are attributed to the existence of preformed Cooper pairs. The variation of the thermal-transport coefficients due to the changing of the transmission coefficient is also studied.
We show that in weakly disordered Luttinger liquids close to a commensurate filling the ratio of thermal conductivity $\ensuremath{\kappa}$ and electrical conductivity $\ensuremath{\sigma}$ can deviate strongly from the Wiedemann-Franz law valid for Fermi liquids scattering from impurities. In the regime where the umklapp scattering rate ${\ensuremath{\Gamma}}_{U}$ is much larger than the impurity scattering rate ${\ensuremath{\Gamma}}_{\mathrm{imp}}$, the Lorenz number $L=\ensuremath{\kappa}/(\ensuremath{\sigma}T)$ rapidly changes from very large values $L\ensuremath{\sim}{\ensuremath{\Gamma}}_{U}/{\ensuremath{\Gamma}}_{\mathrm{imp}}\ensuremath{\gg}1$ at the commensurate point to very small values $L\ensuremath{\sim}{\ensuremath{\Gamma}}_{\mathrm{imp}}/{\ensuremath{\Gamma}}_{U}\ensuremath{\ll}1$ for a slightly doped system. This surprising behavior is a consequence of approximate symmetries existing even in the presence of strong umklapp scattering.