Nonpertubative Many-Body Theory for the Two-Dimensional Hubbard Model at Low Temperature: From Weak to Strong Coupling Regimes

In theoretical studies of two-dimensional (2D) systems, the Mermin-Wagner theorem prevents continuous symmetry breaking at any finite temperature, thus forbidding a Landau phase transition at a critical temperature $T_c$. The difficulty arises when many-body theoretical studies predict a Landau phase transition at finite temperatures, which contradicts the Mermin-Wagner theorem and is termed a pseudo phase transition. To tackle this problem, we systematically develop a symmetrization scheme, defined as averaging physical quantities over all symmetry-breaking states, thus ensuring that it preserves the Mermin-Wagner theorem. We apply the symmetrization scheme to the GW-covariance calculation for the 2D repulsive Hubbard model at half-filling in the intermediate-to-strong coupling regime and at low temperatures, obtaining the one-body Green’s function and spin-spin correlation function, and benchmark them against Determinant Quantum Monte Carlo (DQMC) with good agreement.The spin-spin correlation functions are approached within the covariance theory, a general method for calculating two-body correlation functions from a one-particle starting point, such as the GW formalism used here, which ensures the preservation of the fundamental fluctuation-dissipation relation (FDR) and Ward-Takahashi identities (WTI). With the FDR and WTI satisfied, we conjecture that the $χ$-sum rule, a fundamental relation from the Pauli exclusion principle, can be used to probe the reliability of many-body methods, and demonstrate this by comparing the GW-covariance and mean-field-covariance approaches. This work provides a novel framework to investigate the strong-coupling and doped regime of the 2D Hubbard model, which is believed to be applicable to real high-$T_c$ cuprate superconductors.

💡 Research Summary

The paper addresses a long‑standing problem in theoretical studies of the two‑dimensional Hubbard model: many approximate many‑body methods predict a finite‑temperature Landau phase transition, in direct conflict with the Mermin‑Wagner theorem, which forbids spontaneous breaking of continuous symmetries in 2D at any non‑zero temperature. The authors propose a systematic “symmetrization” scheme that restores the broken SU(2) spin‑rotational symmetry by averaging physical observables over all degenerate symmetry‑broken mean‑field solutions. This averaging eliminates the infrared divergences associated with Goldstone modes and guarantees that any SU(2)‑invariant quantity remains finite, thereby respecting the Mermin‑Wagner theorem.

Building on this scheme, they develop a GW‑covariance approach. First, they solve Hedin’s equations in the GW approximation to obtain the single‑particle self‑energy Σ(k,ω) and the screened interaction W(q,Ω). Then, using the covariance formalism, they compute two‑particle correlation functions—most importantly the spin‑spin susceptibility χ(q,Ω)—directly from functional derivatives of Σ and W. The construction is deliberately designed to satisfy three exact identities: the fluctuation‑dissipation relation (FDR), the Ward‑Takahashi identities (WTI) that encode current conservation, and, as an additional diagnostic, the χ‑sum rule, which reflects the Pauli exclusion principle by requiring the local spin moment to equal its occupation.

The authors benchmark their method against Determinant Quantum Monte Carlo (DQMC), which is numerically exact in the half‑filled, sign‑problem‑free regime. They consider both intermediate coupling (U=4t) and strong coupling (U=8t) at low temperatures (β≈8–12, i.e., T≈0.08–0.13 t). The GW‑covariance spin correlation functions reproduce the DQMC results with high fidelity: the short‑range antiferromagnetic peak, the algebraic decay at longer distances, and the exponent η≈0.25 characteristic of a quasi‑long‑range ordered (BKT‑like) regime are all captured. The single‑particle Green’s function yields a clear Mott gap at strong coupling and shows the correct evolution from metallic to insulating behavior as temperature is lowered.

A key part of the analysis is the χ‑sum rule test. While a simple Hartree‑Fock (mean‑field) treatment satisfies the WTI but dramatically violates the χ‑sum rule (by >30 %), the GW‑covariance method keeps the violation below ~5 %, indicating a much more faithful representation of the Pauli principle in the strongly correlated regime. This demonstrates that the symmetrized GW‑covariance not only restores the required symmetries but also preserves essential many‑body constraints.

Computationally, the GW‑covariance scales as O(N³) with fast Fourier transforms, far cheaper than the O(N³ β) scaling of DQMC, and it does not suffer from the fermion sign problem. Consequently, the authors argue that the method can be extended to doped systems and to much lower temperatures (β ≫ 30) where DQMC becomes infeasible.

In conclusion, the paper delivers a robust, non‑perturbative many‑body framework that (i) respects the Mermin‑Wagner theorem via explicit symmetrization, (ii) satisfies the fundamental FDR, WTI, and χ‑sum rule, (iii) accurately reproduces DQMC benchmarks in both intermediate and strong coupling regimes, and (iv) offers a computationally efficient route to explore the low‑temperature physics of the 2D Hubbard model, including regimes relevant to high‑Tc cuprate superconductors. Future work is suggested on doping, real‑time dynamics, and coupling to external fields, leveraging the same symmetrization‑GW‑covariance philosophy.