1D Interacting Spinless Fermion Model

• The model is a canonical quantum lattice system that encapsulates particle interactions and emergent phenomena in one-dimensional conductors.
• It employs the Bethe ansatz to uncover detailed spectral, topological, and entanglement properties, distinguishing it from traditional spin chain models.
• Extensions introduce Luttinger liquid physics and quasi-Fermi liquid behaviors, offering pathways for quantum simulation and novel material designs.

The one-dimensional interacting spinless fermion model is a canonical quantum lattice system central to the paper of strongly correlated physics in reduced dimensions. It serves as a minimal paradigm for understanding quantum statistics, particle interactions, spectral properties, entanglement, topological order, and emergent phenomena in one-dimensional conductors. Recent investigations have extended its theoretical foundation, revealed exact solutions, clarified dynamical, topological, and entanglement features, and established its role as a parent system for new quantum phases such as the quasi-Fermi liquid.

1. Model Definition and Bethe Ansatz Solution

The basic spinless fermion (SF) model on a chain of length $L$ with $M$ fermions is characterized by the Hamiltonian

$$H = -t \sum_{j}(c_{j}^\dagger c_{j+1} + h.c.) + V \sum_{j} n_j n_{j+1} - \mu \sum_j n_j,$$

where $t$ is the nearest-neighbor hopping amplitude, $V$ is the interaction strength (repulsive for $V > 0$), $c_j^\dagger$ creates a fermion at site $j$, $n_j = c_j^\dagger c_j$, and $\mu$ is the chemical potential. For certain values of $V/t$ (notably $V/t=2$), the model is exactly solvable via the Bethe ansatz. The spectrum is encoded in sets of rapidities $\{\Lambda_j\}$ satisfying

$$L \arctan \Lambda_j = \pi I_j + \sum_{l=1}^M \arctan \frac{\Lambda_j - \Lambda_l}{2},$$

with quantum numbers $I_j$ (integers/half-odd-integers, depending on $M$ parity) (Kohno et al., 2010). Ground states arise from a contiguous, symmetric distribution of $I_j$ ("Fermi sea"), while elementary excitations are generated by introducing "holes" (psinons $\psi$) or "particles" (antipsinons $\psi^*$) into this distribution. Physical observables, including total momentum and energy, are functions of these quantum numbers and rapidities.

2. Spectral Properties, Edge Features, and High-Energy Continuum

Bethe ansatz analysis uncovers distinctive statistical features affecting the spectral properties:

• Gapless Points: In the SF model, the gapless excitations occur at $k = \pi \pm k_F$, with $k_F = \pi M/L$ specifying the Fermi momentum. This is a direct result of the asymmetry in label distributions after single particle/hole addition, contrasting the spin-1/2 XXZ chain where gapless points remain at $k = \pi$ (Kohno et al., 2010).
• Spectral Function Shape: The one-particle addition/removal spectral functions $A^\pm(k,\omega)$ exhibit almost delta-function–like peaks at these gapless points, sharply localized in energy and momentum. In bosonic or spin chains (e.g., XXZ), spectral weight is broadly distributed over a continuum.
• High-Energy Branches and Two-String Solutions: High-energy continua arise from complex "two-string" rapidity solutions, representing bound pairs of complex conjugate rapidities. In some momentum regions (notably near $k=0$), the two-string branch can account for over 60% of the spectral weight. This feature is necessary to recover sum-rule completeness and is observed in DMRG and dynamical simulations.
• Comparison to XXZ Chain:

| Model | Gapless Points | Spectral Line Shape | High-Energy Features | |--------------|------------------------|--------------------------------------|------------------------------| | SF (fermion) | $k = \pi \pm k_F$ | $\delta$-like at gapless points | Shifted two-string continuum | | XXZ chain | $k = \pi$ (fixed) | Broad, continuum with tails | Two-string at $k = \pi$ |

The shift in the two-string branch by $k_F$ in the SF model, compared to the XXZ chain, reflects the different underlying quantum statistics (Kohno et al., 2010).

3. Thermodynamic and Phase Structure: Diamond Chain and Particle-Hole Symmetry

The interplay of hopping and interaction can be studied on decorated lattices such as the diamond chain (Rojas et al., 2010). Using the decoration/iteration transformation, the SF Hamiltonian can be mapped onto an effective atomic limit model with exact solution by the transfer matrix method. The phase diagram at $T=0$ reveals four phases with discrete densities ($\rho = 0,1,2,3$ per unit cell), corresponding to various combinations of occupation on the "decorated" and "nodal" sites. The exact analytic treatment allows determination of explicit correlation functions and the observation of interaction-induced plateaus and particle–hole symmetric points.

Particle–hole symmetry plays a fundamental role (Thomaz et al., 2014). The model satisfies:

$$W(t,V,\mu;\beta) = W(t,V,-\mu+2V;\beta) - (\mu - V),$$

ensuring thermodynamic equivalence under $\mu \rightarrow -\mu + 2V$ (up to a shift). This symmetry is reflected in the entropy and specific heat but not the internal energy or two-point correlators. The mapping to the XXZ chain via the Jordan–Wigner transformation demonstrates that this symmetry is equivalent to field-reversal symmetry in the spin model.

4. Luttinger Liquid Physics, Beyond-Low-Energy Regimes, and Quasi-Fermi Liquids

• Luttinger Parameter at All Energies: Although the Tomonaga-Luttinger liquid (TLL) description is strictly applicable at low energies, the Luttinger parameter $K$ continues to control observables such as the parabolic edge of the spectral function $\epsilon_{\text{edge}}(k)$ at arbitrary energies (Tsyplyatyev et al., 2013):

$$\epsilon_{\text{edge}}(k) = m v_F^2 / K - (k - 2 m v_F)^2 / (2m K),$$

where $m = (2t)^{-1}$.

• Quasi-Fermi Liquid (qFL): Fine-tuning nearest- and next-nearest-neighbor (and correlated hopping) interactions can nullify all marginal operators, making only irrelevant interactions dominant (Baktay et al., 2023, Baktay et al., 14 Jun 2024). This produces states where:
  • $n(k)$ displays a finite, FL-like discontinuity at $k_F$.
  • The spectral function $A(k,\omega)$ reveals both FL features (Lorentzian peaks, well-defined Landau quasiparticles for $\omega > \mu$) and LL-like edge singularities (power-law, non-quasiparticle) for $\omega < \mu$.
  • The dynamic structure factor $S(q,\omega)$ exhibits features of both high-energy bound states (attractive interactions) or spectral weight concentration in the continuum (repulsive). These properties are robust across multiple microscopic Hamiltonians in this class and for a range of energy scales.

5. Entanglement, Accessible Entropy, and Quantum Information Diagnostics

• Particle Partition Entanglement: The Rényi entanglement entropy for a particle bipartition in the TLL regime follows the universal scaling (Barghathi et al., 2017):

$$S_2(n=1) = \ln N - \ln[A(g, \Lambda/\rho_0)] + b(g,\Lambda/\rho_0) N^{-(4g+1)} + O(N^{-(4g+2)}),$$

with $g = (K + K^{-1} - 2)/4$. The leading term arises from fermion antisymmetry; subleading, $K$-dependent power-law corrections quantify interaction effects. The sensitivity of the entanglement entropy to boundary conditions and degeneracies renders it a useful probe of quantum liquids.

• Operationally Accessible Entanglement: For systems with fixed global particle number, only entanglement within fixed particle-number sectors is operationally available for quantum protocols (Barghathi et al., 2019). The accessible part $S_1^{\text{acc}}$ is:

$$S_1^{\text{acc}}(\rho_A) = \sum_n P_n S_1(\rho_{A,n}),$$

with $P_n$ the probability to find $n$ particles in subregion $A$. In the TLL phase, most spatial entanglement is accessible; in classical charge-density wave (CDW) or phase-separated states, the accessible part vanishes or is subleading, indicating the physical distinction between classical and quantum entanglement content.

6. Extensions: Topological Order, Gauge Fields, Embedding, and Non-Abelian Couplings

• Topological Phases: Modulation of interactions as $V_j = V[1 + \lambda \cos(2\pi \alpha j + \delta)]$ induces Kosterlitz–Thouless transitions to insulating charge-density-wave (CDW) states with nontrivial topological invariants (Zuo et al., 2020). These include many-body Berry phases $\gamma$, Chern numbers $C$, and fractionalized edge charges $\pm e/2$.
• Coupling to $\mathbb{Z}_2$ or Dynamical Axion Fields: Coupling to $\mathbb{Z}_2$ gauge fields induces linear confinement, producing bosonic dimers as effective low-energy degrees of freedom—a phenomenon accompanied by doubled Friedel oscillation periods and the possibility of Luttinger or Mott liquid behavior depending on filling and coupling (Borla et al., 2019). Coupling to dynamical link spins, as in emerging axion field models, realizes a $1+1$D axion-electrodynamics system where the spin orientation (axion angle $\theta$) couples to an electric field $E$ via $\theta E$, with classical and fully quantum dynamics accessible via tensor network methods (Hosogi et al., 4 Aug 2025).
• Higher-Dimensional Embedding: Grid structures can embed stacks of decoupled 1D SF liquids or solids (including Majorana sectors) in higher-dimensional, exactly solvable quantum spin Hamiltonians via bond-dependent coupling patterns (XY or X-Ising along "chains," ZZ-Ising off-chain) (Pujari, 24 Jun 2024). The conserved nature of the off-chain spins enables coexistence of classical order and quantum criticality, offering a platform for quantum engineering with site-local qubit control.

7. Physical Significance and Applications

The one-dimensional interacting spinless fermion model unifies themes across quantum statistical mechanics, condensed matter, and quantum information science:

• It serves as a reference system for the verification of exotic non-Fermi liquid phases, including the quasi-Fermi liquid, with direct experimental consequences for ARPES and cold-atom tunneling/transport.
• It encapsulates the essential role of statistics, boundary effects, and interactions in spectral and entanglement phenomena.
• It is central in describing systems exhibiting confinement, emergent gauge structures, and fractionalization.
• Recent advances in tensor network techniques (infinite MPS, DMRG) and exact diagonalization have enabled precision characterization of static and dynamic responses.
• Its embedding in higher dimensions via bond engineering may enable new architectures for quantum simulation and information processing with controllable, exactly known excitations.

Collectively, these features make the one-dimensional interacting spinless fermion model a touchstone for theory and experiment in the physics of strongly correlated, low-dimensional quantum matter.