---
title: 1D Interacting Spinless Fermion Model
url: https://www.emergentmind.com/topics/one-dimensional-interacting-spinless-fermion-model
type: topic
---

# 1D Interacting Spinless Fermion Model

The one-dimensional interacting spinless fermion model is a canonical quantum lattice system central to the study 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) [1002.1147]. 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 $k = \pi$ [1002.1147].

- **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 [1002.1147].

## 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 [1012.3003]. 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 [1407.2050]. 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 [1307.0760]:
  $$
  \epsilon_{\text{edge}}(k) = m v_F^2 / K - (k - 2 m v_F)^2 / (2m K),
  $$
  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 [2305.13374, 2406.10063]. 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 [1703.10587]:
  $$
  S_2(n=1) = \ln N - \ln[A(g, \Lambda/\rho_0)] + b(g,\Lambda/\rho_0) N^{-(4g+1)} + O(N^{-(4g+2)}),
  $$
  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 [1905.03312]. 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 [2001.01080]. 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 [1909.07399]. 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 [2508.02370].

- **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) [2406.17034]. 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.

Source: https://www.emergentmind.com/topics/one-dimensional-interacting-spinless-fermion-model