Fermionic Green's Function Analysis

• Green's Function for Fermions is the time-ordered correlator of fermionic operators that captures single-particle propagation, excitations, and spectral weights in quantum systems.
• Analytical and numerical schemes, including equations-of-motion, Krylov space methods, and continued fractions, enable precise study of interacting electrons and strongly correlated lattice models.
• Fermionic Green's functions underpin the analysis of response functions, topological invariants, and emergent phenomena such as non-Fermi liquid behavior and density-wave orders.

A Green's function for fermions is a central object in quantum many-body theory and quantum field theory, encoding information about single-particle propagation, excitations, correlations, and response in systems governed by Fermi statistics. Among its broad uses, the Green's function forms the foundation for both analytical and numerical approaches to interacting electrons, strongly correlated lattice models, field-theoretical formulations of condensed matter and high-energy systems, and the analysis of experimental observables ranging from spectral functions to collective phenomena.

1. Fundamental Definitions and Formulation

The fermionic Green's function represents the time-ordered correlator of fermionic field operators, most commonly employed as the single-particle (one-body) propagator: $$G_{\alpha\beta}(t-t') = -i\langle \Psi_0| \mathcal{T} [c_\alpha(t) c_\beta^\dagger(t') ]|\Psi_0\rangle$$ where $c_\alpha$ and $c_\beta^\dagger$ are annihilation and creation operators in a basis (typically momentum or real space), $\mathcal{T}$ denotes time ordering, and $\Psi_0$ is the ground or thermal state. In frequency space, the Lehmann representation reveals its analytic structure, expressing $G_{\alpha\beta}(\omega)$ as a sum over system eigenstates, with poles and discontinuities reflecting the excitation energies and spectral weights.

The retarded Green's function in momentum and energy variables takes the typical form: $$G_k(\omega) = \langle 0| f_k \frac{1}{\omega + i\eta - H + E_0} f_k^\dagger |0\rangle$$ with $H$ the Hamiltonian, $E_0$ the ground state energy, and $\eta \to 0^+$ a convergence parameter.

2. Analytical and Numerical Schemes

a. Equations-of-Motion and Decoupling: Edwards Fermion-Boson Model

In models such as the Edwards fermion-boson Hamiltonian, analytical computation adopts an equation-of-motion method, generating a hierarchy of Green's functions due to the coupling between fermions and bosons (1007.1342). To render the hierarchy tractable, a controlled approximation (e.g., "2-boson approximation") is introduced, retaining only up to two bosonic degrees of freedom coupled to fermion hopping. Higher-order terms are closed using scheme akin to Hartree-Fock decoupling.

For general density $n$ on bipartite lattices, the resulting Green's function on each sublattice reads: $$G_k^{ll}(E) = [E - z t_0^2 (1-n_l) C(E, n_l) - \gamma_k^2 n_l \bar{D}(E, n_l)]^{-1}$$ with auxiliary functions $C, D, \bar{D}$ reflecting virtual boson processes, $\gamma_k$ the band-structure term, and $n_l$ the sublattice occupations.

Self-consistent numerical iteration then determines the chemical potential and density-wave order as functions of filling; comparison with density matrix renormalization group (DMRG) and dynamical DMRG (DDMRG) shows strong agreement for ground and spectral properties. This framework can be extended to higher dimensions by modifying only the density of states and band structure inputs.

b. Krylov Space and Lanczos Approaches

The calculation of the Green's function in moderately large Hilbert spaces relies on Krylov subspace methods, particularly variants of the Lanczos algorithm (1109.1205). The resolvent $(\omega - H)^{-1}$ is approximated in a Krylov subspace generated from the application of $H$ on a reference state, enabling the calculation of spectral functions and time-dependent correlation functions, and enabling numerically exact results on the time domain by "restarting" the homogeneous propagation. The approach is suitable for both equilibrium and nonequilibrium scenarios (e.g., implemented on the Keldysh contour), and forms the basis for cluster-embedding methods such as dynamical mean-field theory (DMFT) and cluster perturbation theory.

c. Recursion and Continued Fraction Methods

For lattice models with nearest-neighbor hopping, Green's functions for few-body problems are expressed through recurrence relations in a suitable basis of relative particle coordinates (1112.1928, Komnik et al., 2017). For two fermions, the recursion links Green's functions at neighboring distances, and solvable analytic expressions involving Chebyshev polynomials or continued fractions are obtained. This approach generalizes to more particles and higher dimensions, facilitating the paper of bound state formation, phase diagrams at low concentration, and identification of phase separation or multi-particle complexes.

3. Exact Solutions and Special Function Representations

a. Bethe-Ansatz Integrable Models

In exactly solvable models, such as one-dimensional fermions with delta interactions, determinantal representations of eigenstates permit reduction of the Green's function and related correlation functions to compact determinant or Fredholm determinant forms (1111.2972, Gamayun et al., 2014). In special limits, these determinants map onto known classes (e.g., Toeplitz determinants), and the asymptotic behavior of the Green's function is related to the solution of nonlinear Painlevé V equations. In the hardcore (impenetrable) limit, the Green's function exhibits algebraic decay with an exponent depending continuously on the momentum, bridging free fermion and hardcore boson behavior.

b. Path Integral Approaches

The relativistic Green's function of a free (or externally coupled) Dirac fermion can be formulated in path integral language, with coordinate or field-space transformations simplifying the calculation (Banerjee et al., 2023, Merdaci et al., 2017). For the Dirac equation, spherical coordinate path integrations reduce to harmonic oscillator path integrals, yielding results in terms of (modified) Bessel or spherical Bessel functions and spherical spinors. For Dirac/Pauli-Dirac particles with magnetic moment couplings, the propagator is expressed using parabolic cylindrical functions, encapsulating the response to inhomogeneous fields and permitting explicit calculation of particle creation probabilities analogous to the Schwinger mechanism.

c. Green's Functions in Gauge/Gravity Duality

In holographic dualities, fermionic Green's functions encode response of boundary (field-theory) fermions to strongly coupled gauge dynamics by solving the Dirac equation in nontrivial bulk geometries (1210.4560, Fan, 2014). For certain gravitational backgrounds, exact solutions are given in terms of Whittaker, hypergeometric, or Heun's functions, providing analytic access to the spectral function, location of Fermi surfaces (poles at $\omega=0$), and characterization of Fermi vs. non-Fermi liquid behavior as a function of parameters such as fermion charge and background temperature. Multiple Fermi surfaces and nontrivial pole structure in the complex frequency plane are captured by these analytic forms.

4. Zeros of the Green's Function and Topological Aspects

The topology and global properties of fermionic systems are encoded not only in the poles but also in the zeros of the Green's function (Lu et al., 2023, Flores-Calderón et al., 29 Oct 2024). In particular:

• Symmetric Mass Generation (SMG): Strong interactions can gap the Fermi surface without symmetry breaking, resulting in Green's function zeros encircling the same Fermi volume as the original poles—a "Luttinger surface." The Luttinger theorem, which counts particle number via the sign change of $\text{Re}\, G(0, k)$, is preserved under the pole–to–zero transition.
• Weyl–Mott Point: In models with Weyl nodes subject to strong finite-momentum interations (e.g., Hatsugai-Kohmoto at half–filling), interaction-driven zeros appear at the original Weyl points, inheriting their topological charge. The emergent phase is non-Fermi liquid, with a divergent self-energy and split pole structure, and supports a strongly correlated chiral anomaly.

Explicitly, the Green's function near such a point assumes the structure

$$G(\omega, k) = \frac{1}{\omega + i0^+ - \frac{U^2}{4(\omega + i0^+)}}$$

with the zero at $\omega \to 0$. Topological invariants calculated from $-G^{-1}(k, 0)$ confirm inherited Chern numbers and associated anomaly phenomena.

5. Computation and Simulation in the Modern Context

a. Tensor Network and Quantum Computation

Advances in tensor network methods, such as Grassmann higher–order tensor renormalization group (GHOTRG) (Yoshimura et al., 2017), enable efficient, sign-problem–free calculations of Green's functions in Hamiltonians of significant dimensionality and complexity. This approach leverages modified (impure) tensors for operator insertions, careful management of Grassmann algebra, and block-decomposed isometric projections during renormalization. Results for chiral condensates and Green's functions are quantitatively consistent with analytical solutions for free models.

Quantum computing methodologies now propose variational algorithms for simulating Green's functions directly on near-term devices (Endo et al., 2019). Two approaches are prominent: (1) time-domain variational quantum simulation (VQS) of the real-time evolution, and (2) Lehmann representation reconstruction from excited-state estimation. Both approaches require only shallow circuits and have been validated on Fermi-Hubbard model instances.

b. Path Integral Molecular Dynamics

Numerical calculation of Green's functions in many-body systems is further advanced via path integral molecular dynamics (PIMD) (Yunu et al., 2022). Here, the antisymmetry constraint among fermions is imposed via a sign-alternating configuration sum, and the Green's function is sampled using ring polymer representations that allow direct extraction of momentum distributions relevant for cold atom and Mott insulator systems.

6. Green's Function in External and Non-Abelian Fields

The calculation of fermionic propagators in external electromagnetic or non-Abelian gauge backgrounds is analytically possible in select cases of classical field configurations (Parazian, 20 Jul 2025). For example, in an SU(N) Yang–Mills field given by a plane wave on the light cone, the exact Green's function is constructed as: $$G_F(x, x') = 2N \int_C \frac{d^4 p}{(2\pi)^4} \frac{(\gamma^\mu p_\mu + m) U(p)}{p^2 - m^2 + i\epsilon} e^{-i p \cdot (x - x')}$$ where $U(p)$ is built from trigonometric functions and the SU(N) generators, encoding the full non-perturbative fermion–gauge dynamics. This propagator resums all orders of interaction with the background, aligning with theoretical developments in high-energy physics and applications ranging from heavy-ion collisions to early-universe field dynamics.

7. Physical Interpretation and Observable Consequences

Fermionic Green's functions underpin the prediction and interpretation of a wide variety of physical phenomena, including:

• Spectral Functions: The imaginary part, $-\frac{1}{\pi} \Im G(k, \omega)$, yields the single-particle spectral function, directly observed in photoemission and tunneling experiments.
• Correlations and Order: Off-diagonal elements characterize long-range order, quasiparticle fractions, and emergent orders such as charge density waves or superconductivity.
• Response Functions and Transport: The Green's function enters linear response theory, Kubo formulas, and the evaluation of transport coefficients.
• Topological and Anomalous Phenomena: Zeros and pole structures determine Fermi volumes, topological invariants, and are linked to global constraints such as the Luttinger theorem, Berry phases, and anomalies.

In summary, the Green's function for fermions serves as a unifying analytical, numerical, and conceptual framework for understanding single-particle excitations, correlations, and emergent collective phenomena in interacting quantum systems throughout contemporary theoretical physics.