Tomonaga-Luttinger liquid and charge-density wave in a quasi-one-dimensional material

This paper reports the discovery of the quasi-one-dimensional material Cs$_{1-\delta}$Cr$_3$S$_3$, which uniquely exhibits the simultaneous coexistence of the mutually exclusive charge-density-wave and Tomonaga-Luttinger liquid states due to subtle cesium vacancies that shift the Fermi level into a linearly dispersive band without disrupting the CDW order.

The Gist

Imagine a crowded hallway where people are trying to walk in a single file line. In most materials, people (electrons) move around each other like a fluid, bumping and flowing together. Physicists call this a "Fermi liquid."

But in very narrow, one-dimensional hallways (like a single file line of people), things get weird. Usually, you have two possible outcomes for how these people behave, and they are opposites:

1. The "Free Flow" (Tomonaga-Luttinger Liquid): The people stop acting like individuals and start moving like a synchronized wave. If one person stops, the whole line shudders. They act like a collective dance where everyone is connected.
2. The "Gridlock" (Charge-Density Wave): The people get so nervous about bumping into each other that they decide to lock arms in pairs and stop moving entirely. They form a rigid, frozen pattern (a crystal) that blocks the flow of traffic. This turns the material into an insulator (a wall).

The Big Discovery:
For a long time, scientists thought these two behaviors were mutually exclusive. You could have the free-flowing wave or the frozen gridlock, but never both at the same time. It was like thinking a river could be both a rushing torrent and a frozen block of ice simultaneously.

However, a team of researchers from China has discovered a new material, Cs₁₋δCr₃S₃, that breaks this rule. They found a "quantum hallway" where the electrons are both flowing like a synchronized wave and locked in a frozen pattern at the same time.

How did they do it? (The Analogy)

Think of the material as a series of tiny, hollow tubes made of metal atoms (Chromium) wrapped in sulfur. Inside these tubes, the electrons are the "people."

1. The Frozen Part (The CDW):
   Normally, when electrons get crowded in a tube, they get anxious and decide to pair up, causing the tube to physically squeeze and distort. This is called a Peierls distortion. It's like the hallway floor suddenly developing a series of bumps and dips. This usually stops all movement, creating a "Charge-Density Wave" (CDW). The researchers saw this: the tubes physically squeezed, creating a gap in energy that should stop electricity.

2. The Free Flow Part (The TLL):
   Here is the magic trick. The material naturally has a few missing "guests" (Cesium atoms are missing, creating a slight shortage of positive charge). In physics terms, this is "hole doping."

   Imagine the hallway floor is bumpy (frozen), but the missing guests shift the starting line of the race. The electrons are no longer stuck in the "frozen" part of the energy landscape. Instead, they find themselves on a linear ramp right next to the bumps.

   Because of this shift, the electrons can still move freely in a wave-like pattern (the Tomonaga-Luttinger Liquid) even though the rest of the structure is frozen and distorted.

The Evidence (How they knew)

The scientists used three different "flashlights" to prove this weird state exists:

• The Electrical Test: They measured how electricity flowed. Instead of acting like a normal metal or a simple insulator, the resistance followed a strange mathematical rule (a "power law"). It was like the traffic flow obeyed a secret code that only synchronized waves follow.
• The Heat Test: They measured how heat moved through the material. In normal metals, heat and electricity travel together. Here, heat traveled much faster than electricity, a signature that the "spin" and "charge" of the electrons had separated—a hallmark of this special liquid state.
• The Microscope Test (ARPES): They used a super-powerful light microscope (using synchrotron light) to take a picture of the electrons' energy. They saw the electrons moving in a perfectly straight, diagonal line (linear dispersion), which is the fingerprint of the "free-flowing" wave state, sitting right next to the energy gap caused by the "frozen" state.

Why does this matter?

This discovery is like finding a new state of matter. It proves that nature can be more creative than our textbooks suggest. We thought "frozen" and "flowing" were enemies, but in this material, they are roommates.

This opens the door to designing new quantum computers and sensors. If we can control materials where these opposing forces coexist, we might be able to create electronic devices that are incredibly fast, efficient, and capable of doing things current technology can't imagine.

In short: They found a material where the electrons are holding hands in a frozen line (CDW) but are also surfing a wave (TLL) at the exact same time. It's a quantum paradox that actually works.

Technical Summary

1. Problem Statement

In one-dimensional (1D) electron systems, two fundamental electronic instabilities typically compete and are mutually exclusive:

• Tomonaga-Luttinger Liquid (TLL) State: Arises from strong electron-electron interactions, characterized by collective bosonic excitations (spinons and holons), linear band dispersion at the Fermi level, and power-law scaling in transport and spectral properties.
• Charge-Density Wave (CDW) / Peierls Instability: Arises from electron-phonon coupling, leading to lattice dimerization, the breaking of translational symmetry, and the opening of an energy gap that drives the system into an insulating state.

Historically, real materials exhibit either TLL behavior or CDW order, but not both simultaneously. The coexistence of these antagonistic states has remained elusive, presenting a significant challenge in condensed matter physics.

2. Methodology

The authors employed a comprehensive multi-modal approach to discover and characterize a new quasi-1D material, Cs$_{1-\delta}$Cr$_3$S$_3$:

• Material Synthesis: Single crystals were grown using a flux method (CsCl flux) with high-purity precursors. The resulting crystals are needle-like and stable in air.
• Structural Characterization:
  • Single-Crystal X-ray Diffraction (SXRD): Performed at 170 K to identify crystal symmetry and detect structural distortions.
  • Transmission Electron Microscopy (TEM): High-resolution TEM and Selected-Area Electron Diffraction (SAED) were used to confirm lattice dimerization and homogeneity at the nanoscale.
  • Chemical Analysis: Energy-dispersive X-ray spectroscopy (EDS) quantified the stoichiometry, revealing a systematic Cesium deficiency ($\delta \approx 0.044$).
• Physical Property Measurements:
  • Electrical Transport: Resistivity ($\rho$) and Current-Voltage ($I-V$) characteristics were measured to probe power-law scaling and crossover from Ohmic to non-Ohmic behavior.
  • Thermodynamics: Specific heat and thermal conductivity measurements were conducted to detect the presence of gapless excitations (spinons/holons) and test the Wiedemann-Franz law.
  • Optical Spectroscopy: Optical absorption measurements were used to determine the band gap associated with structural distortion.
• Spectroscopy: Synchrotron-based Angle-Resolved Photoemission Spectroscopy (ARPES) was utilized to map the 3D electronic band structure, focusing on the dispersion near the Fermi level ($E_F$) and spectral weight suppression.
• Theoretical Modeling:
  • Density Functional Theory (DFT): Calculated phonon spectra, electronic band structures, and density of states (DOS) for both undimerized and dimerized structures. Virtual Crystal Approximation (VCA) was used to model Cs deficiency.
  • Tight-Binding (TB) Model: A minimal two-band model based on Cr-$d_{yz/xz}$ molecular orbitals was constructed to explain the origin of linear dispersion and the mechanism of the Peierls gap.

3. Key Contributions

• Discovery of Cs$_{1-\delta}$Cr$_3$S$_3$: Identification of a new quasi-1D material where the TLL state coexists with a robust CDW phase.
• Resolution of the Antagonism: The paper provides a microscopic mechanism explaining how hole doping (induced by Cs vacancies) shifts the Fermi level into a linearly dispersive valence band within a dimerized CDW phase, allowing TLL behavior to persist despite the Peierls gap.
• Intra-Unit-Cell Dimerization: The authors clarify that the Peierls distortion in this system is an intra-unit-cell bond symmetry breaking (bond disproportionation) rather than a doubling of the unit cell periodicity, which is distinct from the canonical monatomic Peierls scenario.

4. Key Results

Structural and CDW Evidence

• Lattice Dimerization: SXRD and SAED revealed the appearance of forbidden $(00l)$ reflections (where $l$ is odd), confirming a structural distortion from space group $P6_3/m$ to $P\bar{3}$. This indicates a dimerization of the Cr$_3$S$_3$ double-walled subnanotubes.
• Optical Gap: Optical absorption measurements showed a direct optical band gap of ~250 meV, consistent with the Peierls instability.
• Robustness: The dimerization is robust against significant Cs deficiency (up to $\delta \approx 0.38$) and persists well above room temperature (no magnetic anomalies up to 400 K).

TLL Evidence

• Transport Properties:
  • Resistivity: Above ~20 K, resistivity follows a power law $\rho(T) \propto T^{-\alpha}$ with $\alpha \approx 1.9$, deviating from the Arrhenius law expected for a gapped semiconductor.
  • I-V Characteristics: At low temperatures and high bias, the system exhibits non-Ohmic behavior following $I \propto V^{\phi+1}$ with $\phi \approx 1.54$. The data collapses onto a universal TLL scaling curve.
  • Thermal Transport: The thermal conductivity shows a large non-phononic contribution, and the Lorenz number is orders of magnitude larger than that of a Fermi liquid, indicating a breakdown of the Wiedemann-Franz law.
• Specific Heat: Low-temperature specific heat data yields a sizeable linear term ($\gamma = 3.43$ mJ mol$^{-1}$ K$^{-2}$), incompatible with a fully gapped insulator but consistent with a spin-charge separated TLL state.
• ARPES Signatures:
  • Linear Dispersion: The electronic bands near $E_F$ are linearly dispersive along the chain direction ($k_z$) but flat in the transverse plane, confirming 1D character.
  • Spectral Weight Suppression: The spectral intensity vanishes smoothly near $E_F$ (unlike the sharp Fermi edge in Fermi liquids), following a power-law suppression $I(\epsilon, T) \propto T^\alpha$.
  • Scaling: Temperature-dependent EDCs collapse onto a single master curve with an exponent $\alpha \approx 1.2$.

Theoretical Understanding

• Mechanism of Coexistence: DFT and TB calculations reveal that the undimerized system has a perfect Fermi surface nesting leading to a gap. However, the Cs deficiency acts as hole doping, shifting the Fermi level from the gap into the lower bonding band.
• Preserved Linearity: In this doped regime, the Fermi level resides in a region where the band dispersion remains linear (protected by molecular orbital symmetry), allowing TLL physics to emerge even though the lattice is dimerized and a gap exists elsewhere in the spectrum.
• Interaction Strength: The extracted TLL parameter $K$ ranges from 0.11 to 0.15, indicating extremely strong repulsive electron-electron interactions, among the lowest values reported for TLL systems.

5. Significance

• Paradigm Shift: This work challenges the conventional view that TLL and CDW states are mutually exclusive. It demonstrates that in specific quasi-1D systems with intra-unit-cell distortions and controlled doping, these states can intertwine.
• New Quantum Platform: Cs$_{1-\delta}$Cr$_3$S$_3$ serves as a rare, robust material platform for studying emergent quantum phenomena, specifically the interplay between strong correlations and lattice instabilities.
• Fundamental Physics: The discovery provides a "conceptual blueprint" for realizing hybrid quantum states in 1D systems, potentially guiding the search for novel topological or superconducting phases in correlated electron materials.
• Strong Correlations: The extremely low $K$ parameter highlights the role of strong electron correlations in Cr-based systems, offering new insights into the physics of 1D quantum fluids.