Thermalized TLL: Collective Modes in 1D Systems

• Thermalized Tomonaga–Luttinger Liquid is a state of strongly correlated 1D systems characterized by collective plasmon-like excitations and temperature-dependent scaling.
• It transforms zero-temperature power-law singularities into temperature-dependent forms, affecting correlation, response, and transport functions.
• Universal behavior across experiments is evidenced by conductance suppression, LDOS scaling, and dynamic thermalization driven by many-body interactions.

A thermalized Tomonaga–Luttinger liquid (TLL) is a universal state of strongly correlated one-dimensional systems–of either fermions or bosons–in which low-energy excitations are collective plasmon-like modes. In a thermalized TLL, equilibrium is (locally or globally) established via either intrinsic many-body interactions, coupling to an external bath, or both, such that dynamical and statistical properties are described by finite-temperature generalizations of the TLL low-energy effective theory. Thermalization impacts correlation functions, response functions, and transport through the replacement of zero-temperature power-law singularities by temperature-dependent scaling forms, and by introducing thermal correlation lengths set by the ratio of velocity to temperature.

1. Foundational Concepts: TLL Theory and the Impact of Thermalization

The TLL framework replaces Fermi-liquid theory in one dimension, accounting for the breakdown of quasiparticle excitations due to enhanced quantum fluctuations and strong correlations. The effective low-energy Hamiltonian is quadratic in bosonic fields $\phi(x)$ and $\theta(x)$,

$$H = \int \frac{dx}{\pi} [uK (\partial_x \theta)^2 + \frac{u}{K} (\partial_x \phi)^2],$$

where $u$ is the sound velocity and $K$ is the dimensionless TLL parameter encoding the interaction (repulsive: $K<1$, attractive: $K>1$).

Thermalization is introduced through the density matrix, which at finite temperature $T$ is $e^{-H/k_BT}$, and modifies observable correlation functions such that for a generic vertex operator,

$$\langle e^{i[n\theta(x)+m\phi(x)]} e^{-i[n\theta(0)+m\phi(0)]} \rangle \propto \left| \frac{\pi \alpha}{u \sinh \left(\frac{\pi x k_BT}{u}\right)} \right|^{\eta},$$

where $\eta$ depends on $K$, $n$, and $m$. The finite $T$ correlation length $\xi_T=u/(\pi k_BT)$ manifests as a crossover from algebraic power-law to exponential decay of correlations at long distances or times.

2. Universal Scaling Laws: Conductance, Spectroscopy, and Critical Exponents

A haLLMark of thermalized TLL physics is the appearance of universal scaling functions in equilibrium and transport experiments. In electronic quantum circuits, the conductance $G(V, T)$ through a one-dimensional conductor in a dissipative environment maps onto the TLL impurity problem. Specifically, conductance data for different circuit parameters collapse onto a universal curve,

$G_\mathrm{TLL}(V_\mathrm{DS}/V_B),$

with $V_B$ a scaling bias. More generally, at low temperature and bias, the relation

$$\frac{\tau(V_{\mathrm{DS,1}})/(1-\tau(V_{\mathrm{DS,1}}))}{\tau(V_{\mathrm{DS,2}})/(1-\tau(V_{\mathrm{DS,2}}))} = \left(\frac{V_{\mathrm{DS,1}}}{V_{\mathrm{DS,2}}}\right)^{2R/R_q}$$

is satisfied, where $R$ is the environmental resistance, $R_q = h/e^2$, and $\tau = R_q G$ is the dimensionless transmission.

Spectroscopic probes such as scanning tunneling spectroscopy (STS) detect power-law suppression of the local density of states (LDOS) near the Fermi level, which is replaced at finite temperature by

$$\rho_\mathrm{TLL}(E, T) \propto T^\alpha \cosh\left(\frac{E}{2k_BT}\right) \left|\Gamma \left( \frac{1+\alpha}{2} + i\frac{E}{2\pi k_BT} \right)\right|^2,$$

with critical exponent $\alpha$ determined by the Luttinger parameter $K$ and model details (Stühler et al., 2019, Zhang et al., 2023, Lou et al., 2021).

3. Signatures Across Experimental Platforms

Thermalized TLL behavior has been demonstrated in a broad range of experimental contexts:

• Electronic quantum circuits and quantum point contacts (QPCs): Universal conductance suppression and dynamical Coulomb blockade are mapped onto TLL models with impurities, with the effective interaction parameter $K=1/(1+Re^2/h)$ set by circuit resistance (Jezouin et al., 2013, Anthore et al., 2018).
• Quantum Hall and topological insulator edge channels: Bismuthene and multilayer FeSe edge modes exhibit power-law LDOS suppression and temperature scaling indicative of thermalized helical TLLs (Stühler et al., 2019, Zhang et al., 2023).
• Quasi-1D van der Waals and transition-metal systems: Nanoscale systems such as Nb$_9$Si$_4$Te$_{18}$, CoSb$_{1-x}$ nanoribbons, and MoSe$_2$ domain boundaries were shown to display TLL scaling of DOS with strong thermal robustness (Yao et al., 2023, Lou et al., 2021, Xia et al., 2019).
• Spin ladders and antiferromagnetic chains: Magnetic compounds (e.g., (Hpip)$_2$CuCl$_4$, NaVOPO$_4$) realize thermalized TLLs, with specific heat, magnetization, and relaxation rates confirming the scaling predictions and allowing extraction of $K$ as a function of field or doping (Ward et al., 2013, Islam et al., 2023).
• Ultracold atoms in 1D traps: Matter-wave interferometry and Bragg spectroscopy directly probe TLL correlation functions, with controlled splitting and quantum quenches accessing both prethermal and global thermalized states (Cavazos-Cavazos et al., 2022, Ruggiero et al., 2020).
• Quantum simulators with Rydberg-encoded spins: Direct measurement of the power-law decay of spin-spin correlations, tunable through interaction range, establishes the TLL paradigm and enables extraction of K, with the ballistic spread of correlations observed in quenched dynamics (Emperauger et al., 14 Jan 2025).

4. Thermalization Dynamics and Nonequilibrium Regimes

A detailed understanding of how thermalization is dynamically established in TLLs comes from quantum quench studies: after a sudden change of interaction (quench), correlations initially decay algebraically with nonthermal exponents (prethermal regime), followed by the emergence of an exponentially decaying thermal regime on shorter length scales (Buchhold et al., 2015). The approach to equilibrium is governed by phonon scattering and slow energy transport via hydrodynamic modes, with the effective temperature approaching its equilibrium value algebraically in time, and the thermalization exponent tied to the KPZ universality class ($\mu = 2/3$). Crossover length and time scales separating prequench, prethermal, and thermal regions are set by light-cone propagation velocities and the strength of interactions or integrability-breaking terms.

For two coupled TLLs (e.g., split condensates), quench dynamics exhibits multiple light-cone velocities, prethermalization with distinct effective temperatures for symmetric/antisymmetric sectors, and a drift to a quasi-thermal long-time regime characterized by a unique correlation length, all with direct applicability to cold atom experiments (Ruggiero et al., 2020).

5. Boundaries, Disorder, and Environmental Coupling

Thermalized TLLs with boundaries present rich physics, especially in the local density of states near an open end. Analytic expressions for spatially resolved correlation and spectral functions reveal how boundary conditions (backscattering pinning, radiative coupling) alter scaling exponents and generate boundary resonances. Strong environmental coupling can induce spatially dependent TLL parameters $K(x)$ and, in the strong-coupling limit, plasmon-polariton hybrid excitations, which bridge the fields of TLL and nano-optics (Grigoryan et al., 20 Dec 2024).

Weak disorder can drive a transition from a thermalized TLL to a Bose glass state (gapless, compressible insulator), as explored through magnetic insulator quantum simulators where controlled substitution tunes the disorder landscape (Ward et al., 2013).

Impurity effects in finite chains manifest as Friedel oscillations of the local magnetization, with temperature and filling dependence governed by TLL scaling relations; these have been quantitatively measured in engineered Rydberg spin arrays (Emperauger et al., 14 Jan 2025).

6. Universality and Material Parameter Dependence

In all cases, the Luttinger parameter $K$ remains the central quantity, set by the interaction strength, nature of the constituent particles (fermionic/bosonic, spinful/spinless, helical/chiral), and, in low-disorder or well-tuned artificial systems, is directly extractable from temperature or energy scaling exponents. Notably, in systems with additional symmetry or multiple channels (e.g., MWNTs with valley and spin degrees of freedom), TLL parameters separate into charge ("holon," often universal due to averaging over disorder) and neutral modes (sensitive to inter-shell coupling or disorder) (Grigoryan et al., 12 Dec 2024).

The quantum-critical nature of the thermalized TLL, described by a conformal field theory with central charge $c=1$, underpins all power-law scaling forms, providing a unified description across diverse platforms. The absence of true long-range order persists at all $T>0$, with quantum criticality manifest in transport, thermodynamic, and spectral observables (Bouchoule et al., 21 Jan 2025).

7. Theoretical Extensions and Future Directions

Beyond the standard free-boson TLL paradigm, recent work introduces dynamical squeezing fields as fluctuating Bogoliubov parameters, leading to additional branches in the spectral function and nonlinear, temperature-driven modifications of excitation velocities—phenomena potentially observable as side-band peaks in spectroscopic measurements (Nagao et al., 2020).

Further research avenues include extending the mapping of TLL physics to more complex environments (frequency-dependent impedances, engineered quantum circuits), leveraging TLL universality for device functionalities (quantum switching, amplification, information processing), and exploring the effect of strong correlations, disorder, and proximity-induced phenomena (parafermion excitations at topological edges).

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In summary, the thermalized Tomonaga–Luttinger liquid constitutes a paradigmatic quantum-critical state in one dimension, exhibiting power-law correlations modified by thermal scaling laws, universal conductance curves, and collective plasmonic excitations. Its realization and characterization across materials and platforms, and under equilibrium and nonequilibrium conditions, provide a comprehensive and quantitatively predictive framework for exploring strong-correlation phenomena and criticality in low-dimensional quantum systems.