Fermionic Multiscale RG and Fermi-Surface Problems

For a weakly interacting two-dimensional lattice Fermi system with a smooth strictly convex Fermi surface away from half filling and van Hove points, angular sector decomposition plus Grassmann determinant bounds gives a convergent finite-temperature multiscale expansion down to an exponentially small temperature. The theorem controls the two-point Schwinger function and self-energy in that window; it does not determine the zero-temperature phase.

Required background. Rigorous RG as a Dynamical System supplies scale maps and domain control. Polymer Activities and Normed RG Coordinates supplies localization and normed remainders, adapted here to Grassmann kernels.

Helpful background. Constructive Fermionic and Yukawa Models supplies Grassmann Gaussian integration. Scattering Analyticity, Crossing, and Rigorous Bounds helps distinguish Euclidean finite-temperature correlations from an S-matrix claim.

Scales and sectors near a regular Fermi surface

For the square-lattice Hubbard dispersion, write

$$e(\mathbf k)=-2t(\cos k_1+\cos k_2)-\mu,$$

and assume the Fermi curve $\mathcal F=\{\mathbf k:e(\mathbf k)=0\}$ is smooth, has nonzero curvature, and stays away from saddle points of $e$. At inverse temperature $\beta_T$, fermionic Matsubara frequencies are $k_0=(2n+1)\pi/\beta_T$, and the free covariance is

$$C(k_0,\mathbf k)=\frac{1}{-ik_0+e(\mathbf k)}.$$

Choose smooth scale functions $f_h$ supported where $|{-}ik_0+e(\mathbf k)|\asymp\gamma^h$, $h\leq0$, so $C=\sum_{h=h_T}^{0}C_h$ with $\gamma^{h_T}\asymp\beta_T^{-1}$. A scale-$h$ annulus is further cut into angular sectors of tangential width approximately $\gamma^{h/2}$. Strict curvature makes momentum conservation restrict which sector labels can occur at a vertex; this improves naive power counting.

Grassmann integration expands truncated expectations as determinants rather than sums over all pairings. A Gram representation bounds an $r\times r$ determinant by a product of vector norms, avoiding a factorial $r!$. Sector counting and determinant bounds together make the effective kernels summable over scales.

The Banach coordinate at scale $h$ is a sequence of antisymmetric kernels $W_{2m}^{(h)}$ multiplying Grassmann monomials of degree $2m$. Its norm combines an $L^1$ position-space norm (or the corresponding Fourier derivatives), sums over sector labels, and scale weights chosen from fermionic power counting. The exact integration of $C_h$ maps one such sequence to the next. Localization subtracts the Taylor jet of the two-point kernel on $\mathcal F$ and the marginal part of the four-point kernel; the remainder gains either a normal momentum, a frequency, or an angular-sector improvement. Thus the proof controls an actual map on kernels, rather than assigning a running number to each Feynman graph.

There is no bosonic large-field problem because Grassmann polynomials terminate, but this does not make convergence automatic. The number of graphs and sector assignments still grows, and a bound obtained by taking absolute values before forming determinants would reintroduce factorial growth. The Gram bound must therefore be applied before summing the tree expansion, with constants uniform in the volume and the infrared cutoff. This ordering is one of the decisive construction mechanisms in Benfatto, Giuliani, and Mastropietro 2006, §§ 2–3, pp. 823–843.

The finite-temperature theorem

For sufficiently small Hubbard coupling $U$ and chemical potential in the regular convex regime, Benfatto, Giuliani, and Mastropietro construct a convergent, non-power-series expansion for the two-point Schwinger function at temperatures satisfying a condition of the form

$$|U|\,|\log T|\leq c,$$

equivalently $T\geq e^{-c/|U|}$ after changing constants. They prove that the interacting Fermi surface remains a regular convex curve, the wave-function renormalization stays of order one, and the self-energy has the regularity stated in their theorem Benfatto, Giuliani, and Mastropietro 2006, Theorem 1.1 and § 1.4, pp. 813–822. Counterterms fix the interacting Fermi surface rather than letting it drift through the multiscale construction.

The proof localizes relevant two-point kernels into chemical-potential, Fermi-velocity, and wave-function renormalizations. Four-point kernels are marginal by power counting; antisymmetry and sector constraints supply improvements. Higher kernels are irrelevant. The running kernels remain in a normed domain because each scale estimate is uniform in volume and determinant bounds replace combinatorial pairing growth.

This is the rigorous application of Fermi-Surface Patch Theory and Low-Energy Scaling: for weak two-dimensional Hubbard coupling away from the excluded loci, decompose into angular sectors and verify the self-energy bounds in the proved temperature window. The patch picture is therefore accompanied by a stop scale $h_T$ and a convergence condition, not extrapolated indefinitely.

Independent checks

First compute $\nabla e=(2t\sin k_1,2t\sin k_2)$ on $\mathcal F$; it must not vanish. Second, check curvature in the chosen chemical-potential interval. Third, count dimensions: a normal momentum and Matsubara frequency scale as $\gamma^h$, while tangential width scales as $\gamma^{h/2}$. Treating a whole annulus as one patch loses the curvature gain.

At $U=0$, all counterterms and non-Gaussian kernels vanish and the construction must return $C$. At finite $U$, differentiating the renormalized two-point denominator at the Fermi surface checks the claimed order-one field-strength factor.

One should also verify the Ward identities actually available for the lattice model. Charge conservation constrains the density vertex, but it does not by itself cancel every marginal four-point channel. Treating a symmetry identity as a uniform kernel bound would bypass the determinant and sector estimates on which the temperature window depends.

Van Hove failure and conclusion ceiling

Move $\mu$ so that $\mathcal F$ passes through $(0,\pi)$ or $(\pi,0)$. There $\nabla e=0$, curvature coordinates become singular, the density of states has a logarithmic enhancement, and the sector-counting lemma used above fails. The model may still be analyzed by a different RG, but the regular-Fermi-surface theorem cannot simply be reused.

Likewise, the bound $T\geq e^{-c/|U|}$ leaves open what happens below the prospective instability scale and at $T=0$. Convergence of finite-temperature Schwinger functions does not prove superconductivity, rule it out, construct a Lorentzian continuum theory, or establish a material-specific phase diagram.

Exercise

Why does the lowest Matsubara frequency stop the infrared integration at a finite scale?

Solution

Fermionic frequencies satisfy $|k_0|\geq\pi T$. Therefore $|{-}ik_0+e(\mathbf k)|$ cannot be smaller than order $T$ even on the Fermi surface. Slices with $\gamma^h\ll T$ have empty support, so the last scale obeys $\gamma^{h_T}\asymp T$.

References

• Benfatto, Giuseppe, Alessandro Giuliani, and Vieri Mastropietro. “Fermi Liquid Behavior in the 2D Hubbard Model at Low Temperatures.” Annales Henri Poincaré 7 (2006): 809–898. DOI; Open PDF.
• Brydges, David C., and Gordon Slade. “A Renormalisation Group Method. I. Gaussian Integration and Normed Algebras.” Journal of Statistical Physics 159 (2015): 421–460. DOI; Open PDF.