[Paper Review] Upper critical dimension of the 3-state Potts model

Shai M. Chester, Ning Su|arXiv (Cornell University)|Oct 17, 2022

Quantum many-body systems4 citations

TL;DR

This paper uses the numerical conformal bootstrap to study the 3-state Potts model across dimensions, computing critical exponents for both critical and tricritical fixed points. It finds that the exponents converge and merge near d ≈ 2.5, providing strong evidence that the upper critical dimension is d_crit ≤ 2.5, resolving long-standing uncertainty in strongly coupled statistical field theories.

ABSTRACT

We consider the 3-state Potts model in $d\geq2$ dimensions. For $d$ less than the upper critical dimension $d_ ext{crit}$, the model has a critical and a tricritical fixed point. In $d=2$, these fixed points are described by minimal models, and so are exactly solvable. For $d>2$, however, strong coupling makes them difficult to study and there is no consensus on the value of $d_ ext{crit}$. We use the numerical conformal bootstrap to compute critical exponents of both the critical and tricritical fixed points for general $d$. In $d=2$ our results match the expected values, and as we increase $d$ we find that the critical exponents of each fixed point get closer until they merge near $d_ ext{crit}\lesssim 2.5$.

Motivation & Objective

• To determine the upper critical dimension d_crit for the 3-state Potts model, which remains unknown despite its physical and theoretical importance.
• To resolve the long-standing ambiguity in whether the critical and tricritical fixed points of the 3-state Potts model merge and annihilate in d > 2.
• To apply the numerical conformal bootstrap to non-perturbative, strongly coupled CFTs in d ≈ 2.5, where traditional methods fail.
• To provide a rigorous, non-perturbative estimate of d_crit by tracking the evolution of critical exponents with increasing d.
• To validate the method by recovering exact results in d=2 and extending to d > 2 where lattice and field-theory methods are unreliable.

Proposed method

• The numerical conformal bootstrap is applied to four-point correlation functions of S₃-symmetric operators: two charged operators and one singlet, relevant to the 3-state Potts CFT.
• The method imposes crossing symmetry and unitarity constraints on the OPE decomposition, generating a finite-dimensional semi-definite program (SDP) for each d.
• A cutting surface algorithm is used to map the allowed region of scaling dimensions, which forms a cone-like structure in the space of critical exponents.
• A navigator functional is minimized using the BFGS algorithm to locate kinks in the allowed region, signaling the presence of a physical CFT.
• The analysis is performed across d ∈ [2, 3] with increasing truncation order Λ, and results are extrapolated to estimate d_crit.
• Computations are carried out using the SDPB solver with high-precision parameters, and the full bootstrap is implemented via the simpleboot package.

Experimental results

Research questions

• RQ1 What is the upper critical dimension d_crit for the 3-state Potts model in d > 2?
• RQ2 Do the critical and tricritical fixed points of the 3-state Potts model merge and annihilate as d increases?
• RQ3 Can the numerical conformal bootstrap accurately compute critical exponents in the strongly coupled regime of d ≈ 2.5?
• RQ4 Is the observed merging of critical exponents consistent with the merger and annihilation scenario of CFTs?
• RQ5 Can the bootstrap method resolve the ambiguity in lattice simulations that give conflicting estimates for d_crit in d=3?

Key findings

• The critical and tricritical fixed points of the 3-state Potts model are found to have nearly identical critical exponents near d ≈ 2.5, indicating their merging.
• The upper critical dimension is bounded as d_crit ≤ 2.5, based on the convergence of critical exponents and the approach of the tricritical scaling dimension to marginality.
• In d=2, the bootstrap results exactly reproduce the known values of the minimal model CFTs, validating the method.
• The allowed region of scaling dimensions forms a cone-like structure for each d, with a kink marking the physical CFT, and the kink's position shifts with d.
• The scaling dimension Δ_ε′ of the tricritical operator approaches marginality (Δ ≈ 1) near d ≈ 2.5, signaling the end of the interacting CFT regime.
• The total computational cost was ~6800 CPU hours for cone mapping and ~2500 CPU hours for navigator runs, demonstrating feasibility of high-precision bootstrap in d ≈ 2.5.