🔬 mesoscale physics

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

This paper investigates the thermoelectric and thermodynamic properties of a Tomonaga-Luttinger liquid with interaction constant $K=1/2$ and a localized impurity, deriving exact expressions for thermopower and demonstrating a close connection between transport coefficients and entropy or heat capacity densities in the low-temperature regime.

Original authors: Abdellah Touati, Redoune Zamoum, Farhi Houssam Eddine

Published 2026-09-11

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CC BY 4.0

Original authors: Abdellah Touati, Redoune Zamoum, Farhi Houssam Eddine

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the microscopic world of quantum physics, electricity is not just a flow of charge; it is also a flow of heat and information. When scientists study how electrons move through tiny, one-dimensional wires, they often look at a property called thermopower. This is the ability of a material to generate a voltage simply because one end is hotter than the other. It is a sensitive probe, revealing not just how easily electrons travel, but how their energy is distributed and how they interact with one another. In many materials, these interactions are weak, but in certain exotic quantum states, electrons behave like a tightly knit crowd, influencing each other's movement in complex ways. Understanding how this collective behavior affects the transport of heat and charge is crucial for developing future technologies that could harvest waste heat or manipulate energy at the nanoscale.

A team of researchers has recently explored this phenomenon in a specific, highly controlled quantum system. They focused on a theoretical model known as a Tomonaga-Luttinger liquid, which describes how electrons behave in a one-dimensional wire when they strongly repel one another. To make the problem solvable with exact mathematical precision, they set the strength of this interaction to a specific value and introduced a single, localized impurity—a tiny obstacle in the path of the flowing electrons. By using advanced theoretical techniques to map the behavior of these interacting electrons onto a simpler system of non-interacting particles, the team calculated exactly how the thermopower would behave under different conditions, from very low temperatures to higher ones, and under various applied voltages.

The researchers discovered that the presence of the impurity acts as a powerful filter for energy. When the electrons encounter this obstacle, the system becomes highly selective about which electrons can pass through. Electrons with specific energies are blocked, while others are allowed to flow, creating a sharp asymmetry between those with higher energy and those with lower energy. This energy filtering is the key driver of the thermopower. The team found that as the temperature rises from absolute zero, the thermopower does not simply increase steadily. Instead, it climbs to a distinct peak before falling off again. This peak occurs when the thermal energy of the electrons matches the specific energy scale of the impurity's effect, allowing the system to most effectively distinguish between different energy levels.

A particularly striking finding was the relationship between this electrical voltage and the concept of entropy, which measures the disorder or information content of a system. The researchers showed that in the low-temperature limit, the thermopower is directly proportional to the entropy carried by each individual electron. This connection allowed them to use a fundamental thermodynamic rule to calculate how the total disorder of the system changes as electrons are transported. They found that the entropy variation associated with this transport also exhibits a peak at low temperatures, mirroring the behavior of the thermopower. Furthermore, they calculated the heat capacity density, a measure of how much energy is required to change the temperature of the system due to these transported charges. This quantity also showed a sharp peak, indicating a rapid change in the system's thermodynamic state as the temperature increased.

The study applied these general findings to two specific physical setups that can be described by the same mathematical model. The first was a quantum point contact, a narrow bridge connecting a normal metal to a special state of matter found in the fractional quantum Hall effect. The second was a one-dimensional conductor coupled to a resistive environment. In both cases, the results were consistent: strong scattering from the impurity, which reduces the overall electrical conductance, dramatically enhances the thermopower. This creates a trade-off where blocking the flow of electricity actually makes the system better at generating voltage from heat. The researchers observed that while the thermopower and entropy variation remain stable and predictable at very low temperatures, they become unstable and change behavior as the temperature rises, eventually smoothing out as the thermal energy overwhelms the specific quantum effects of the impurity.

These findings provide a unified picture of how non-equilibrium quantum transport is linked to thermodynamics in one-dimensional systems. The work demonstrates that the same quantum interactions that hinder the flow of electricity can be harnessed to create strong thermoelectric responses. By understanding how a single impurity can act as an energy filter, scientists gain insight into the fundamental connection between the movement of charge and the flow of entropy. This knowledge could be valuable for designing future quantum devices that rely on precise control of heat and charge, offering a clearer path toward manipulating energy at the smallest possible scales. The study confirms that in the quantum realm, the obstacles in a system are not just barriers to be overcome, but essential features that define how energy and information are transported.

Technical Summary: Thermoelectric and Thermodynamic Properties of an Interacting Tomonaga-Luttinger Liquid with Impurity ( $K=1/2$ )

Problem Statement
The paper investigates the thermoelectric and thermodynamic properties of an interacting one-dimensional quantum system, specifically a Tomonaga-Luttinger liquid (TLL) with an interaction parameter $K=1/2$ in the presence of a localized impurity. The study focuses on calculating the thermopower ( $Q$ ) in the nonlinear regime and the Seebeck coefficient ( $S_{Seebeck}$ ) in the linear-response regime. A central goal is to explore the connection between these transport coefficients and thermodynamic quantities, specifically the entropy per particle and heat capacity density, utilizing the Kelvin formula. The authors aim to apply these exact theoretical results to two specific physical systems that map onto this TLL model: a quantum point contact (QPC) between fractional quantum Hall (FQH) edge states and a one-channel quantum conductor coupled to an Ohmic environment.

Methodology
The authors employ the techniques of bosonization and refermionization to solve the system exactly.

Model: The system is described by a bosonized Hamiltonian density containing a free chiral mode and a tunneling term representing the impurity at $x=0$ . The specific value $K=1/2$ is crucial as it allows the bosonic fields to be "refermionized" into new fermionic fields ( $\gamma_\pm$ ), rendering the Hamiltonian exactly solvable without perturbation theory.
Derivation: Using the exact wave functions of the charge carriers, the authors derive exact expressions for the electric current and differential conductance. These are then used to calculate the thermopower $Q$ in the nonlinear regime (finite voltage $V$ and temperature $T$ ).
Linear Response: The Seebeck coefficient is derived by taking the linear-response limit ( $V \to 0$ ) of the thermopower.
Thermodynamic Connection: The authors utilize the Kelvin formula, which relates the Seebeck coefficient to the entropy per particle ( $s = e S_{Seebeck}$ ), to derive expressions for the entropy density variation ( $\Delta S$ ) and the heat capacity density ( $C$ ) associated with thermoelectric particle transport.
Applications: The general results are applied to two specific mappings:
1. A QPC connecting a $\nu=1/3$ FQH edge state to a $\nu=1$ normal metal.
2. A one-channel coherent conductor (OCC) coupled to a dissipative Ohmic environment.

Key Contributions and Results

Exact Thermopower Expressions: The paper provides exact analytical expressions for the thermopower $Q$ valid for all voltage and temperature regimes. In the low-temperature limit ( $k_B T \ll \hbar\omega$ ), the thermopower is shown to obey a Mott-like relation:
$Q \approx \frac{\pi^2 k_B^2 T}{3e} \frac{\partial \ln T(\mu)}{\partial \mu}$
where $T(\mu)$ is the transmission coefficient. This confirms that $Q$ is governed by the energy dependence of the transmission around the chemical potential.
Linear Response and Mott Formula: In the linear-response regime, the Seebeck coefficient $S_{Seebeck}$ is derived. At low temperatures, it also follows a Mott-like formula dependent on the logarithmic derivative of the transmission coefficient at the Fermi energy. The authors note that $S_{Seebeck}$ is generally smaller in magnitude than the nonlinear thermopower $Q$ because $Q$ integrates transmission over a broader energy window defined by the applied voltage.
Entropy and Heat Capacity: By establishing the close relationship between $S_{Seebeck}$ and the entropy per particle ( $s$ ) at low temperatures, the authors derive the entropy density variation ( $\Delta S$ ) and the heat capacity density ( $C$ ) contributed by the transported charges.
- Both $\Delta S$ and $C$ exhibit Mott-like behavior in the low-temperature limit.
- The heat capacity density $C$ can exhibit both positive and negative values. A positive peak occurs when the entropy variation $\Delta S$ is activated most rapidly at low temperatures, interpreted as a Schottky peak. Conversely, a negative $C$ indicates that $\Delta S$ is decreasing with temperature, characterizing a depletion of the transport-related entropy contribution as temperature increases.
- The analysis reveals that strong backscattering (low effective transmission) enhances the energy selectivity of the transport channel, thereby increasing the thermopower, entropy variation, and the magnitude of heat capacity peaks (both positive and negative), despite suppressing electrical conductance.
System-Specific Behaviors:
- QPC System: Higher backscattering amplitude ( $\Gamma_B$ ) leads to stronger energy dependence of transmission, enhancing thermopower and entropy variation.
- OCC System: Lower effective transmission ( $\tau$ ) corresponds to stronger backscattering, similarly enhancing thermoelectric effects.
- Temperature Regimes: At low temperatures, the system exhibits robust Mott-like power-law scaling. At high temperatures, thermal broadening smears out the energy asymmetry, causing thermopower and related quantities to decrease.

Significance and Claims
The paper claims to demonstrate a "close connection between thermoelectric transport and thermodynamic properties in interacting one-dimensional quantum systems." Specifically, it shows that:

The Seebeck coefficient in the linear regime is directly proportional to the entropy per particle at low temperatures and low Fermi energies.
This proportionality allows for the derivation of entropy density variations and heat capacity contributions directly from transport measurements (via the Kelvin formula).
The interplay between backscattering and thermoelectricity reveals a trade-off: while strong backscattering reduces electrical conductance, it significantly enhances thermopower and local thermodynamic signatures (entropy and heat capacity) due to increased energy filtering.
The results provide a unified theoretical picture linking non-equilibrium quantum transport to mesoscopic thermodynamics, offering a framework to probe energy-filtering mechanisms in quantum devices like FQH QPCs and Ohmic-coupled conductors.

The authors conclude that their work offers valuable insights for understanding how many-body correlations and impurities influence thermoelectric efficiency and thermodynamic state functions in 1D quantum systems, without proposing new experimental setups or future applications beyond the systems explicitly analyzed.

Technical Summary: Thermoelectric and Thermodynamic Properties of an Interacting Tomonaga-Luttinger Liquid with Impurity (K=1/2K=1/2K=1/2)

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Technical Summary: Thermoelectric and Thermodynamic Properties of an Interacting Tomonaga-Luttinger Liquid with Impurity ( $K=1/2$ )