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Multipole and Berezinskii-Kosterlitz-Thouless Transitions in the Two-component Plasma

Jeanne Boursier, Sylvia Serfaty

Abstract

We study the two-dimensional two-component Coulomb gas in the canonical ensemble and at inverse temperature $β>2$. In this regime, the partition function diverges and the interaction needs to be cut off at a length scale $λ\in (0,1)$. Particles of opposite charges tend to pair into dipoles of length scale comparable to $λ$, which themselves can aggregate into multipoles. Despite the slow decay of dipole--dipole interactions, we construct a convergent cluster expansion around a hierarchical reference model that retains only intra-multipole interactions. This yields a large deviations result for the number of $2p$-poles as well as a sharp free energy expansion as $N\to\infty$ and $λ\to0$ with three contributions: (i) the free energy of $N$ independent dipoles, (ii) a perturbative correction, and (iii) the contribution of a non-dilute subsystem. The perturbative term has two equivalent characterizations: (a) a convergent Mayer series obtained by expanding around an i.i.d.\ dipole model; and (b) a variational formula as the minimum of a large-deviation rate function for the empirical counts of $2p$-poles. The Mayer coefficients exhibit transitions at $β_p=4-\tfrac{2}{p}$, that accumulate at $β=4$, which corresponds to the Berezinskii-Kosterlitz-Thouless transition in the low-dipole-density limit. At $β=β_p$ the $p$-dipole cluster integrals switch from non-integrable to integrable tails. The non-dilute system corresponds to the contribution of large dipoles: we exhibit a new critical length scale $R_{β, λ}$ which transitions from $λ^{-(β-2)/(4-β)}$ to $+\infty$ as $β$ crosses the critical inverse temperature $β=4$, and which can be interpreted as the maximal scale such that the dipoles of that scale form a dilute set.

Multipole and Berezinskii-Kosterlitz-Thouless Transitions in the Two-component Plasma

Abstract

We study the two-dimensional two-component Coulomb gas in the canonical ensemble and at inverse temperature . In this regime, the partition function diverges and the interaction needs to be cut off at a length scale . Particles of opposite charges tend to pair into dipoles of length scale comparable to , which themselves can aggregate into multipoles. Despite the slow decay of dipole--dipole interactions, we construct a convergent cluster expansion around a hierarchical reference model that retains only intra-multipole interactions. This yields a large deviations result for the number of -poles as well as a sharp free energy expansion as and with three contributions: (i) the free energy of independent dipoles, (ii) a perturbative correction, and (iii) the contribution of a non-dilute subsystem. The perturbative term has two equivalent characterizations: (a) a convergent Mayer series obtained by expanding around an i.i.d.\ dipole model; and (b) a variational formula as the minimum of a large-deviation rate function for the empirical counts of -poles. The Mayer coefficients exhibit transitions at , that accumulate at , which corresponds to the Berezinskii-Kosterlitz-Thouless transition in the low-dipole-density limit. At the -dipole cluster integrals switch from non-integrable to integrable tails. The non-dilute system corresponds to the contribution of large dipoles: we exhibit a new critical length scale which transitions from to as crosses the critical inverse temperature , and which can be interpreted as the maximal scale such that the dipoles of that scale form a dilute set.

Paper Structure

This paper contains 147 sections, 72 theorems, 1127 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Setting of the problem and objectives
  3. Physical background
  4. The BKT transition
  5. The renormalization group picture
  6. Multipole transitions
  7. Rigorous approaches to the transition
  8. Existence of two phases
  9. Duality with integer-valued height functions
  10. Implementing the renormalization group
  11. Comparison with our model
  12. Prior results on our model
  13. Definitions, method of proof and main results
  14. Model
  15. Dipoles and multipoles
  16. ...and 132 more sections

Key Result

Theorem 1

Let $\beta\in (2,\infty)$ and let $p(\beta)$ be as in Definition def:pbeta. Let $\lambda\in (0,1)$. Let $\mathcal{Z}_\beta$ be as in def:Zbeta and $\mathsf{K}_{\beta,\lambda}^\mathrm{dip}$ be as in Definition def:dipole activity. Recall the Ursell function $\mathrm{I}$ from Definition def:graphU.

Figures (12)

  • Figure 1: Kosterlitz--Thouless approximate RG flow in the $(K,z)$-plane. The bold curve is the separatrix of the ODE \ref{['eq:ODE']}.
  • Figure 2: A stable matching. The positive charges are represented as red dots and negative charges as blue dots.
  • Figure 3: Three multipoles: $\{1,2\}$, $\{4\}$, $\{3,5\}$
  • Figure 4: Left: the partition $X$ with blocks $A,B,C,D,E$ and $F$. Let $X_1=\{A,B\}$ and $X_2=\{C,D\}$. Right: the coarsening $\mathrm{Coarse}_X(X_1,X_2)$ with blocks $A\cup B$, $C\cup D$, $E$, and $F$.
  • Figure 5: A tree relative to $X\coloneqq \{S_1,S_2,S_3,S_4,S_5\}$. Notice that $T\in \mathsf{T}^X$ implies that the quotient graph $(V_X,T)/X$ is a tree on $X$, but the converse is not true.
  • ...and 7 more figures

Theorems & Definitions (212)

  • Definition 2.1
  • Definition 2.2: Multipoles
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5: Dipole activity
  • Definition 2.6: Hypertrees
  • Definition 2.7: Truncated Mayer series
  • Definition 2.8: Series truncation parameter
  • Definition 2.9: $\lambda$-optimal error rate
  • Theorem 1: Free energy expansion
  • ...and 202 more