Isometric Fluctuation Relations for Equilibrium States with Broken Symmetry

We derive a set of isometric fluctuation relations, which constrain the order parameter fluctuations in finite-size systems at equilibrium and in the presence of a broken symmetry. These relations are exact and should apply generally to many condensed-matter physics systems. Here, we establish these relations for magnetic systems and nematic liquid crystals in a symmetry-breaking external field, and we illustrate them on the Curie-Weiss and the $XY$ models. Our relations also have implications for spontaneous symmetry breaking, which are discussed.

Figure 1

Probability density $p_{\mathbf{B}}(\mathbf{m})=N^3{P}_{\mathbf{B}}(N\mathbf{m})$ of the magnetization per spin $\mathbf{m}=\mathbf{M}/N=(m_x,m_y,m_z=0)$ for the three-dimensional Curie-Weiss model in the magnetic field $\mathbf{B}=(B,0,0)$ with $B=0.01$, $J=1$, and $N=100$ at the rescaled inverse temperatures (a) $\beta J=2.7$ in the paramagnetic phase and (b) $\beta J=3.3$ in the ferromagnetic phase. The lines depict the contours of $\parallel \mathbf{m}\parallel =0.1,0.2,\dots ,1.0$ where the isometric fluctuation relation (3) holds [13].

Figure 2

Cumulant generating function (9) of the magnetization in the three-dimensional Curie-Weiss model for the magnetic field $\mathbf{B}=(B,0,0)$ with $B=0.001$, $J=1$, and different temperatures across criticality. The generating function is rescaled by the average magnetization $⟨m{⟩}_B$ in the direction of the external field and plotted versus the rescaled parameter $\lambda /(2\beta B)$ in the same direction $\mathbf{\lambda }=(\lambda ,0,0)$. The generating function is computed by taking the Legendre-Fenchel transform of the large-deviation function ${\mathrm{\Phi }}_{\mathbf{B}}(\mathbf{m})$ introduced in Eq. (7) [13]. The isometric fluctuation relation (3) implies the symmetry $\lambda \to 2\beta B-\lambda$ of the generating function according to Eq. (10).

Figure 3

$XY$ model of two-dimensional magnetism on a square lattice with $L=10$ and $J=1$ in the external magnetic field $\mathbf{B}=(0.1,0)$ at the rescaled inverse temperature $\beta J=0.8$ above the critical temperature $T_{\mathrm{KT}}$. (a) The distribution $Q_{\mathbf{B}}(M,\theta )\equiv 2\pi M{P}_{\mathbf{B}}(\mathbf{M})$ versus the modulus of the magnetization per spin $M/N=\parallel \mathbf{M}\parallel /N$ for different values of the angle $\theta$ separated by $\mathrm{\Delta }\theta =\pi /6$, with the top curve corresponding to the direction along the magnetic field. (b) The distribution compensated by Boltzmann weights $f_{\mathbf{B}}(M,\theta )\equiv Q_{\mathbf{B}}(M,\theta )\exp (-\beta BM\cos \theta )$ confirming the isometric fluctuation relation. Inset: Check of the equality between the left- and right-hand sides of Eq. (17). After a transitory run of $10^4$ spin flips, the statistics is carried out over $10^8$ values of the magnetization, each one separated by $10^3$ spin flips.