Thermodynamic integration, fermion sign problem, and real-space renormalization

TL;DR

This work reexamines real-space renormalization for the 2D Ising model by retracing Wilson’s real-space decimation using unbiased Monte Carlo sampling and thermodynamic integration. It implements a modified decimation with coupling parameter $\rho$ to avoid identifications that spoil critical scaling, and develops a Monte Carlo framework that also contends with the emergent fermion sign problem through an absolute-value reformulation and a correlated-sampling strategy. The study analyzes two-factor and fourteen-factor interaction sets on the Wilson patch, locating perturbative fixpoints and showing the sign problem remains manageable for practical parameter ranges, with significant precision gains from correlated sampling. The results illuminate the feasibility of Wilson’s program in a modern computational setting and suggest pathways to generalize real-space renormalization beyond simple models, while acknowledging open questions about uniqueness and universality.

Abstract

We reconsider real-space renormalization for the two-dimensional Ising model, following the path traced out by Wilson in Sect. VI of his 1975 Reviews of Modern Physics. In that reference, Wilson considerably extended the Kadanoff decimation procedure towards a possibly rigorous construction of a real-space scale-invariant hamiltonian. Wilson's construction has, to the best of our knowledge, never been fully understood and thus neither reproduced nor generalized. In the present work, we use Monte Carlo sampling in combination with thermodynamic integration in order to retrace Wilson's computation for a real-space renormalization with a number of terms in the hamiltonian. We elaborate on the connection of real-space renormalization with the fermion sign problem and discuss to which extent our Monte Carlo procedure actually implements Wilson's program from half a century ago.

Figures (7)

• Figure 1: The $14$ dominant factor types $T \in \{1 ,\ldots, 14\}$ identified in Ref. Wilson1975. In the Wilson patch of old sites (set $\mathcal{S}$ of eq. \ref{['equ:SDefinition']}), they appear in $9447$ factors. In the patch of new sites (set $\mathcal{N}$ of eq. \ref{['equ:NDefinition']}), they appear in $4795$ factors.
• Figure 2: Wilson patch from Ref. Wilson1975. Naively, the old-only spins---the "$\times$" in the picture---are to be decimated (summed over) (see Section \ref{['sec:OriginalDecimation']}), yielding a partition function and at the same time the Boltzmann weight of the new spins---the "$\cdot$" in the picture. At the fixpoint, the partition function over the old spins agrees with the Boltzmann weight of the new spins. The naive procedure has a flaw, which was corrected by Kadanoff (see Section \ref{['sec:ModifiedDecimation']}).
• Figure 3: Wilson patch with the modified decimation. All $s$ spins (lower-level crosses and dots) are summed over in order to compute the effective interaction between $t$ spins (upper-level dots). Spins $s_k$ and $t_k$, for $k \in \mathcal{N}$ (dots on the same sites on both levels) are coupled by $1 + \rho t_k s_k$ (vertical lines), but not identified (see eq. \ref{['equ:ModifiedDecimation']}).
• Figure 4: Average sign of Alg. \ref{['alg:alpha-rho-Metropolis']} (alpha-rho-Metropolis) for the two-factor-type model with interactions $(K, L) = (0.3,0.10736)$. (a): Average sign and thermodynamic integration path ($t_{\mathcal{B} \cap \mathcal{N}}=2$, $t_{\mathcal{A} \cap \mathcal{N}} = \{1,\ldots, 1\}$) connecting $(\alpha,\rho)=(1,1.05)$ to the known partition function at $(\alpha, \rho) = (0, 1.001)$, with exact values (red dots) from eq. \ref{['equ:AlphaZeroSign']}. (b): Average sign at $\alpha=1$ for the spin configurations $t_{\mathcal{B} \cap \mathcal{N}} = \{1,\ldots, 250\}$, for two different coupling parameters $\rho$.
• Figure 5: Plot of the variation ratio $\Psi(K,L|\{1,\ldots, 1\})$ for the two-factor-field model on a grid of interactions $(K, L)$ for correlation parameter $\rho=1.04$. The variations are smallest for $(K, L) \simeq (0.31, 0.11)$, in good agreement with second-order perturbation theory.