Platforms for the realization and characterization of Tomonaga-Luttinger liquids

TL;DR

This work surveys Tomonaga-Luttinger liquids as the cornerstone framework for understanding one-dimensional quantum systems, illustrating how collective density excitations, spin-charge separation, and power-law correlations emerge from bosonization with a single parameter $K$. It spans itinerant electronic platforms, spin chains and ladders, topological edge modes, and cold-atom realizations, detailing how experimental probes—ranging from tunneling spectroscopy to Bragg scattering—extract $u$, $K$, and related observables, and how nonlinearities, interchain coupling, and generalized hydrodynamics extend the LL picture. Key results include precise LL descriptions of Josephson junction chains, field-tuned LL behavior in spin ladders, and universal scaling of edge-state tunneling exponents in 2D topological insulators, together with prospects for Majorana/parafermion edge states and beyond-Luttinger-liquid phenomena. Collectively, the paper demonstrates the universality and versatility of TLL physics and charts a path toward exploring quantum-critical dynamics, higher-dimensional boundary realizations, and cold-atom quantum simulations with tunable interactions and geometry.

Abstract

The concept of a Tomonaga-Luttinger liquid (TLL) has been established as a fundamental theory for the understanding of one-dimensional quantum systems. Originally formulated as a replacement for Landau's Fermi-liquid theory, which accurately predicts the behaviour of most 3D metals but fails dramatically in 1D, the TLL description applies to a even broader class of 1D systems,including bosons and anyons. After a certain number of theoretical breakthroughs, its descriptive power has now been confirmed experimentally in different experimental platforms. They extend from organic conductors, carbon nanotubes, quantum wires, topological edge states of quantum spin Hall insulators to cold atoms, Josephson junctions, Bose liquids confined within 1D nanocapillaries and spin chains. In the ground state of such systems, quantum fluctuations become correlated on all length scales, but, counter-intuitively, no long-range order exists. In this respect, this review will illustrate the validity of conformal field theory for describing real-world systems, establishing the boundaries for its application and, on the other side will discuss the spectacular demonstration of how the quantum-critical TLL state governs the properties of many-body systems in one dimension.

Figures (6)

• Figure 1: Josephson junction chains as pinned TLL’s.a, SEM micrograph showing a family of chains with nominal junction size 300$\times$400 nm. The specific capacitance is 54 fF/$\mu$m2. The precise junction area is modulated by the EBL exposure dose. b, (main plot) Experimental determination of the critical voltage for a chain with 250 junctions. The blue data is obtained upon stepping up from zero voltage, and the red when stepping back down from non-zero current. (lower inset) Close up of the small voltage region where the critical voltage is extracted. A very small hysteresis region is present for this device. The critical voltage is taken to be the voltage having maximum $dI/dV$ upon stepping up from zero voltage. (Upper inset) Linear dependence of $V_c$ on chain length, $N$. c, Scaled critical voltage $v$, versus scaled Bloch bandwidth $w$. Symbols represent different 'families' distinguished by plasma frequency $\omega_p$, length and presence of ground plane. The solid red line is theory for independent QPS across each junction and no disorder and has slope = 1. Solid lines are the quantum theory of a disordered T-L mode with fitted values of screening length $\Lambda$ = 13.1 (blue), and $\Lambda$ = 4.0 (black), respectively. These exhibit a $w$-dependent slope, 4/(3 - 2$K$), where $K(w)$ is the Tomonaga-Luttinger parameter. For small $w$, $K\propto \Lambda ^{ -1} \ln w$: for the same range of $w$, $K$ is enhanced by the decreased screening length of devices with ground planes, resulting in a stronger departure from the classical result. The dotted lines show the classical depinning result, slope = 4/3, for $\Lambda$ = 13.1 (dotted blue) and $\Lambda$ = 4.0 (dotted black). Also plotted for comparison are the quantum results for $\Lambda$ = 7.7 (blue dashed), and $\Lambda$ = 3.2 (black dashed) using screening lengths inferred from the gate-dependent periodicity of $dI/dV$ at large $w$ and biases $V>V_c$. The figure is reproduced from Ref. Cedergren_etal-2017.
• Figure 2: TLL in spin chains and ladders.a, Spin chain with exchange coupling $J$, its mapping to the system of spinless fermions (spin up/down maps to the presence/absence of fermion), and spin ladder with exchange couplings $J_\perp$ along the legs and $J_\parallel$ along the rungs. b, Singlet $S=0$ and triplet $S=1$ energy levels in a spin ladder, and its normalized magnetization $m_z$ per spin as a function of the magnetic field $B$. c, TLL parameters $u$ (divided by $J_\parallel$) and $K$ as a function of $m_z$ in strong-rung or repulsive ($J_\perp/J_\parallel/=3.6$, dashed red line) and strong-leg or attractive ($J_\parallel/J_\perp=1.7$, solid blue line) regimes, together with the data for BPCB (squares) and DIMPY (circles), respectively (the figure is taken from Ref. Jeong_2016). d, Schematic intensity of the transverse magnetic excitations in a spin chain as a function of energy $\varepsilon$ and wave number $qa$. e, Phase diagram of the spin ladder material (C$_5$H$_{12}$N)$_2$CuBr$_4$ (BPCB) as a function of the magnetic field and temperature exhibits a non-magnetic quantum disordered (QD) phase of spin singlets, a TLL phase and the region of quantum criticality (QC) in between (the figure is taken from Ref. Ruegg_2008). The inset shows the crystal structure of the material. f, The boundary of the low-temperature magnetically ordered phase indicated in e measured by NMR (circles) and described in the model of weakly coupled TLLs (blue line). The figure is reproduced from Ref. Klanjsek_2008.
• Figure 3: TLLs in the edge states of 2D topological insulators (TIs). Realizations of 1D electronic structure and Luttinger liquids at the boundaries of different members of the 2D TI family. Different from trivial insulators in which conduction (CB) and valence (VB) bands are separated by the band gap, 1D metallic edge states persist at the crystalline boundaries of different TIs whose properties (chiral vs. helical, spinful vs. spinless) depend on the property of the 2D bulk they surround. The different electronic dispersions governing bulk and edge and scattering processes therein (and their suppression) are indicated.
• Figure 4: Experimental demonstrations of TLLs in 2D TIs. Luttinger liquids in the helical edges states 2D topological (quantum spin Hall) insulators as probed by scanning tunneling microscopy and spectroscopy (STM) at cryogenic temperature. a, Atomic structure of hexagonal bismuthene monolayer, visualized by STM. b, Spatial dependence of tunneling spectra of the 2D bulk at varying distance from the 1D edge as indicated in a. A suppression of the tunneling density of states (DOS) into the edge state at at $E_{\rm F}$ (ZBA) is clearly visible. c, Universal scaling of the tunneling DOS in temperature and bias voltage used to extract the Luttinger parameter $K$. d, Tunability of the Luttinger parameter as a function of the effective dielectric constant of the substrate material in the 2D TI tungsten ditelluride (WTe$_2$). Panels a-c were taken from Ref. Stuehler2020TomonagaLuttinger. Panel d was taken from Ref. jia_2022.
• Figure 5: Realizations of inter-edge tunneling devices to probe TLLs at the edges of quantum Hall and quantum spin Hall insulators. a-b, Tunneling spectroscopy of a graphene point contact connecting integer ($\nu = 1$) and fractional ($\nu = 1/3$) quantum Hall states. Panel b shows the finite bias spectroscopy of the QPC conductance and its characteristic zero-boas suppression. c-d, Point contact constriction, lithographically defined in the quantum spin Hall regime of an inverted HgTe/CdTe heterostructure. Panels a-b were taken from Ref. Cohen2022. Panels c-d were taken from Ref. Strunz2019.