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arXiv:2609.11810

Interacting Tomonaga-Lüttinger liquid with impurity for interaction constant $K = 1/2$: Thermopower investigation, entropy variation and heat capacity densities associated with thermoelectric particle transport

Abdellah Touati, Redoune Zamoum, Farhi Houssam Eddine

Published Sep 10, 2026• 0 citations
arXivHTMLPDF

Abstract

We investigate thermoelectric and thermodynamic properties of a Tomonaga-Luttinger liquid with interaction parameter $K=1/2$ in the presence of a localized impurity. Using bosonization and refermionization, we derive an exact expression for the thermopower in the nonlinear regime and obtain the corresponding Seebeck coefficient in linear response. At low temperature, the Seebeck coefficient follows a Mott-like relation governed by the energy dependence of the transmission coefficient and is found to be closely related to the entropy per particle. This connection allows us to derive using Kelvin formula the thermopower contribution to the entropy density variation and the charge-carriers contribution to the heat capacity density. Both quantities exhibit Mott-like behavior in the low-temperature regime. We apply our results to a fractional quantum Hall quantum point contact and to a one-channel quantum conductor coupled to an Ohmic environment. Our results demonstrate the close connection between thermoelectric transport and thermodynamic properties in interacting one-dimensional quantum systems.

DOI
10.48550/arxiv.2609.11810

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Interacting Tomonaga-Lüttinger liquid with impurity for interaction constant K=1/2K = 1/2K=1/2: Thermopower investigation, entropy variation and heat capacity densities associated with thermoelectric particle transport
We investigate thermoelectric and thermodynamic properties of a Tomonaga-Luttinger liquid with interaction parameter K=1/2K=1/2K=1/2 in the presence of a localized impurity. Using bosonization and refermionization, we derive an exact expression for the thermopower in the nonlinear regime and obtain the corresponding Seebeck coefficient in linear response. At low temperature, the Seebeck coefficient follows a Mott-like relation governed by the energy dependence of the transmission coefficient and is found to be closely related to the entropy per particle. This connection allows us to derive using Kelvin formula the thermopower contribution to the entropy density variation and the charge-carriers contribution to the heat capacity density. Both quantities exhibit Mott-like behavior in the low-temperature regime. We apply our results to a fractional quantum Hall quantum point contact and to a one-channel quantum conductor coupled to an Ohmic environment. Our results demonstrate the close connection between thermoelectric transport and thermodynamic properties in interacting one-dimensional quantum systems.
Recently, the influence of electron-electron interactions on the thermoelectric transport in a two-channel charge Kondo circuit has been studied in [Phys. Rev. B 105, L121405 (2022)]. In this paper, we revisit the Luttinger-liquid-based model and discuss in details the limit where the spin field is noninteracting (gσ=1g_\sigma = 1gσ​=1) and the interaction in the charge sector is repulsive (0<gρ≤10< g_\rho \leq 10<gρ​≤1). The thermoelectric transport coefficients are computed nonperturbatively with respect to the reflection amplitude at the quantum point contact. At low temperatures the thermopower shows the non-Fermi liquid behavior in the vicinity of the Coulomb peaks. We also demonstrate that repulsive interaction results in the enhancement of the thermoelectrical power.
The Seebeck and Peltier coefficients of a homogeneous Luttinger liquid are calculated in the ballistic regime. Nonlinearity of the electron spectrum is taken into account. It is shown that, in the framework of the defined approximations, the thermoelectric power of a Luttinger liquid is equal to zero, in agreement with the exponentially small thermopower of a one-dimensional degenerate Fermi gas. The Peltier coefficient is controlled by a nonequilibrium state of the system. It is finite and renormalized by the interaction in the case of a convective flow of a Luttinger liquid. The zero modes of bosonic excitations and the dispersion-induced contribution to the electric current operator are taken into account in calculations.
We study the thermoelectric transport properties of fractional quantum Hall systems based on an exact diagonalization calculation. Based on the relation between the thermoelectric response and thermal entropy, we demonstrate that thermoelectric Hall conductivity ${\ensuremath{\alpha}}_{xy}$ has power-law scaling ${\ensuremath{\alpha}}_{xy}\ensuremath{\propto}{T}^{\ensuremath{\eta}}$ for gapless composite Fermi-liquid states at filling numbers $\ensuremath{\nu}=1/2$ and 1/41/41/4 at low temperatures (TTT), with an exponent $\ensuremath{\eta}\ensuremath{\sim}0.5$ distinctly different from Fermi liquids. The power-law scaling remains unchanged for different forms of interaction including Coulomb and short-range ones, demonstrating the robustness of non-Fermi-liquid behavior of these interacting systems at low TTT. In contrast, for the 1/31/31/3 fractional quantum Hall state, ${\ensuremath{\alpha}}_{xy}$ vanishes at low TTT with an activation gap associated with neutral collective modes rather than charged quasiparticles. Our results establish another manifestation of the non-Fermi-liquid nature of quantum Hall fluids at a finite temperature.
The thermopower SL(T){S}_{L}(T)SL​(T) of a finite Luttinger liquid (LL) wire containing an impurity and connected to leads of noninteracting electrons is calculated. It is shown that at low temperatures ${k}_{B}T\ensuremath{\ll}{\ensuremath{\Delta}}_{L}\ensuremath{\simeq}\ensuremath{\Elzxh}{\mathbf{v}}_{F}/gL$ (L is the length of the wire, vF{\mathbf{v}}_{F}vF​ is the Fermi velocity of the electrons, and g is the correlation parameter of the LL), SL(T){S}_{L}(T)SL​(T) is described by a Mott-type formula. However, at higher temperatures, that is, when ${\ensuremath{\Delta}}_{L}<~{k}_{B}T\ensuremath{\ll}{\ensuremath{\varepsilon}}_{F},$ the expression for SL(T){S}_{L}(T)SL​(T) contains an interaction-dependent factor. For strong interelectron interaction, the enhancement of the thermopower is large and it is much more pronounced for spinless electrons.
In one-dimensional conductors, Coulomb interactions result in correlated electronic systems called Tomonaga-Luttinger liquids (TLL). The TLL physics also applies to other many-body phenomena, providing complementary viewpoints while benefiting from the powerful TLL framework. One such phenomenon is the low-energy conductance suppression of a quantum coherent conductor embedded in a dissipative circuit, an effect called dynamical Coulomb blockade. Here we investigate the basic class of mesoscopic circuits constituted by a short single-channel quantum conductor in series with a resistance R. Remarkably, such circuits can be mapped on a TLL of interaction parameter 1/(1+Re^2/h), with an impurity. From this mapping, generalized to realistic dissipative circuits, a scaling law for the suppressed conductance is derived at R=h/e^2, and small deviations computed for R \neq h/e^2. We find that the scaling law is obeyed by our data for arbitrary quantum channels, emulated by a Ga(Al)As quantum point contact, and by the recent data of Mebrahtu et al. [Nature 488, 61 (2012)] obtained using a carbon nanotube. This demonstrates a highly tunable test-bed for TLL physics, and consolidates a recently proposed phenomenological expression for the conductance of a quantum channel in a linear circuit.
We develop a conformal field theory approach for investigation of the quantum charge, heat, and thermoelectric transport through a quantum impurity fine-tuned to a non-Fermi liquid regime. The non-Fermi-liquid operational mode is associated with the overscreened spin Kondo effect and controlled by the number of orbital channels. The universal low-temperature scaling and critical exponents for Seebeck and Peltier coefficients are investigated for the multichannel geometry. We discuss the universality of Lorenz ratio and power factor beyond the Fermi-liquid paradigm. Different methods of verifying our findings based on the recent experiments are proposed.