Strange Luttinger liquids in a cavity-embedded one-dimensional electronic chain

Technical Summary: Strange Luttinger Liquids in a Cavity-Embedded One-Dimensional Electronic Chain

Problem Statement
This work investigates the low-energy physics of a one-dimensional (1D) electronic chain coupled to a homogeneous, quantized vacuum field (a single-mode cavity) in the presence of nearest-neighbor electron-electron interactions. While strong light-matter coupling has been shown to modify various condensed matter phenomena (e.g., quantum Hall effects, superconductivity gaps), the specific impact of cavity vacuum fluctuations on the universality class of 1D interacting electron systems remains to be fully characterized. The authors aim to determine whether the standard Luttinger liquid (LL) theory, which relies on specific velocity relations, holds when the electronic hopping is modulated by a quantized photon field, and how this "strange" regime evolves when finite electron-electron interactions are introduced.

Methodology
The study employs a unified microscopic model described by a tight-binding Hamiltonian with periodic boundary conditions, incorporating:

1. Light-Matter Coupling: Implemented via a Peierls substitution ($t \to t e^{-ig(\hat{a}+\hat{a}^\dagger)}$), where $g$ is the dimensionless coupling strength and $\hat{a}$ is the photon annihilation operator.
2. Electron-Electron Interactions: A nearest-neighbor interaction term $U$.

The analysis proceeds in two regimes:

• Non-Interacting Regime ($U=0$): The authors derive an effective low-energy Hamiltonian by projecting the full light-matter Hamiltonian onto specific photon sectors. They utilize the Hellmann-Feynman theorem to calculate expansion coefficients of the energy spectrum with respect to kinetic and current operators. This allows for an analytical derivation of renormalized velocities.
• Interacting Regime ($U \neq 0$): Since bosonization becomes complicated by photon-mediated long-range interactions, the authors rely on Exact Diagonalization (ED) for finite systems ($N=12$) to map the phase diagram. They compute charge susceptibilities to identify phase boundaries and analyze the local density of states (LDOS) to detect Majorana-like zero modes.
• Perturbative Extension: For the interacting case, the authors propose an extension of the non-interacting effective theory by incorporating the interaction term perturbatively into the cavity-renormalized Hamiltonian, assuming the decoupling of photonic sectors in the thermodynamic limit.

Key Contributions and Results

1. Identification of the "Strange Luttinger Liquid":
   In the absence of electron-electron interactions ($U=0$), the system is found to be described by a Hamiltonian that formally resembles a conventional Luttinger liquid but with cavity-renormalized velocities ($\bar{v}_S, \bar{v}_N, \bar{v}_J$). Crucially, the authors demonstrate that the fundamental relation governing Luttinger universality, $v_S = \sqrt{v_J v_N}$, is violated.

   • The cavity coupling suppresses the current velocity ($\bar{v}_J$) significantly more than the density velocity ($\bar{v}_N$) or the sound velocity ($\bar{v}_S$).
   • Despite this violation, the ground-state correlation exponents (e.g., the decay of single-particle correlations $G(x, \tau) \sim |x|^{-1}$) remain unchanged from the non-interacting case, as the ground state remains a non-interacting Fermi sea. This invariance of exponents alongside the breakdown of velocity relations defines the "strange" nature of this liquid.

2. Modification of Phase Diagrams via Exact Diagonalization:
   For finite interactions, the cavity field significantly alters the phase boundaries between the Luttinger liquid (LL), Charge Density Wave (CDW), and Phase Separation (PS) phases.

   • The cavity-induced renormalization of the hopping amplitude effectively increases the ratio $U/t$, meaning weaker interaction strengths are required to drive transitions into the CDW ($U > 2t$) or PS ($U < -2t$) phases compared to the vacuum case.
   • The phase diagram is plotted as a function of interaction strength $U/t$ and coupling $g$, showing a clear shift in transition points.

3. Fate of Majorana-like Zero Modes:
   The study examines the localized zero modes that emerge at the edges of open chains in the attractive interaction regime ($-2t < U < 0$).

   • The spectral weight of these edge modes is found to be maximized near the phase transition points.
   • The presence of the cavity field ($g > 0$) induces shifts in the spectral weight distribution similar to those observed in the bulk phase transitions, highlighting that vacuum fluctuations can tune topological edge features in number-conserving systems without invoking mean-field symmetry breaking.

4. Unified Low-Energy Description:
   The authors propose a perturbative framework extending the strange Luttinger liquid description to the interacting regime. By treating the interaction term within the cavity-renormalized basis, they derive dressed velocities ($\tilde{v}_S, \tilde{v}_N, \tilde{v}_J$) that retain the inequality $\tilde{v}_S \neq \sqrt{\tilde{v}_J \tilde{v}_N}$. This analytical approach aligns with the trends observed in the ED results, suggesting a consistent low-energy description that accounts for both cavity renormalization and electron correlations.

Significance and Claims
The paper claims to establish cavity quantum electrodynamics (cQED) as a mechanism for engineering unconventional 1D quantum liquids that lie beyond the standard Luttinger liquid paradigm. The primary significance lies in the demonstration that light-matter coupling can qualitatively modify the low-energy sector of 1D systems by breaking the standard velocity relations that define Luttinger universality, without necessarily altering the ground-state correlation exponents.

Furthermore, the work highlights the potential of cavity fields to control topological many-body features (specifically Majorana-like modes) in systems that conserve particle number, offering a new perspective on engineering topological phases without relying on the breaking of global $U(1)$ symmetry often required in superconducting models. The results suggest that vacuum fluctuations provide a tunable parameter to shift phase boundaries and modify edge state properties in correlated electronic systems.