Lee-Yang Zero Theory: Critical Phenomena Analysis

• Lee-Yang zero theory is a framework that studies phase transitions by analyzing the zeros of partition functions in the complex plane.
• The theory emphasizes finite-size scaling and the ratio method to precisely locate critical points and suppress subleading corrections.
• It is applied in models like the Ising and 3D Potts, offering deep insights into universality, scaling, and many-body physics.

Lee-Yang zero theory provides a rigorous framework for analyzing phase transitions via the paper of the zeros of partition functions in the complex plane of a control parameter. The geometric and scaling properties of these zeros—termed Lee-Yang (or more generally, Yang-Lee) zeros—are intimately connected to the analytic structure of the free energy and, in the thermodynamic limit, to the emergence of singularities defining critical phenomena. Originally formulated in the context of the ferromagnetic Ising model, Lee-Yang zero theory underpins the universal understanding of equilibrium and, increasingly, nonequilibrium phase transitions.

1. Fundamental Principles of Lee-Yang Zeros

For a finite statistical system characterized by a control parameter $z$ (e.g., external field $h$ or fugacity), the canonical partition function can be expressed as a product over its $V$ complex roots (Lee-Yang zeros): $Z_V(z) = \prod_n (z - z_n(V)).$ A singularity in the free energy, and hence a phase transition, arises in the thermodynamic limit ($V \rightarrow \infty$) when the locus of zeros approaches and pinches the real-$z$ axis. In the classical ferromagnetic Ising model, Lee and Yang proved that all zeros in the complex magnetic field $h$ lie on the imaginary axis (the “circle theorem”). More generally, the distribution and density of these zeros encode the full analytic structure of the free energy; the pinching condition is both necessary and sufficient for the manifestation of critical points or first-order transitions.

2. Finite-Size Scaling Structure of Lee-Yang Zeros

Finite-size scaling (FSS) provides the bridge between the distribution of Lee-Yang zeros in finite systems and the universal properties at criticality. Near a critical point, the scaling hypothesis asserts that the partition function is described by the scaling variables: $$\tilde{t} = L^{y_t} t, \qquad \tilde{h} = L^{y_h} h,$$ with $t = (T-T_c)/T_c$ (reduced temperature), $h$ (external field), $L$ (system linear size), and $(y_t, y_h)$ the thermal and magnetic renormalization exponents. The scaling form reads: $Z(t, h, L^{-1}) = \tilde Z(L^{y_t} t, L^{y_h} h).$ The Lee-Yang zeros $h_{\mathrm{LY}}^{(n)}(t, L)$ obey: $$L^{y_h} h_{\mathrm{LY}}^{(n)}(t, L) = \tilde h^{(n)}_{\mathrm{LY}}(L^{y_t} t).$$ At the critical point ($t=0$), the imaginary part of the leading zero scales as: $$r_n(L) \equiv \Im h_{\mathrm{LY}}^{(n)}(0, L) \propto L^{-y_h} \sim V^{-1/\nu},$$ where $\nu=1/y_t$ is the correlation-length exponent, connecting the distribution of zeros directly to universal critical properties.

3. Ratio Method for Critical Point Location

The ratio method leverages the finite-size scaling of Lee-Yang zeros to extract the critical point in general systems, even in the presence of nontrivial mixing between scaling fields. Define the ratio,

$R_{n,m}(V; T) = \frac{r_n(V; T)}{r_m(V; T)},$

where $r_k(V; T) = \Im h_{\mathrm{LY}}^{(k)}(T, V)$. A Taylor expansion of the scaling function near criticality yields: $$R_{n,m}(V;T) = r_{n,m} + c_{n,m} L^{y_t} t + \cdots,$$ where $r_{n,m}$ is universal and volume-independent. For any pair of volumes $(V_1, V_2)$, the crossing point of the two ratio curves pinpoints the critical point: $R_{n,m}(V_1;T) = R_{n,m}(V_2;T) \implies T = T_c.$ This intrinsic cancellation of non-universal amplitudes ensures that the ratio method is robust against certain classes of systematic errors and is particularly powerful in suppressing the finite-size corrections induced by irrelevant operator mixing.

4. Suppression of Mixing-Variable Corrections

A significant advantage of the ratio method over classical cumulant-based approaches (e.g., the Binder cumulant) is its enhanced suppression of subleading finite-size corrections arising from the mixing of scaling fields. In a system with two relevant scaling fields $(\tau, \xi)$ that mix nontrivially into Ising variables $(t, h)$ via a nondiagonal transformation, the ratio expansion is: $$\mathcal{R}_{n,m}(\tau, L) = \left(r_{n,m} + C_{n,m} L^{y_t} \delta\tau + \cdots\right)\left(1 + D_{n,m} L^{2(y_t-y_h)} + \cdots\right),$$ whereas the Binder cumulant admits leading corrections $\sim L^{y_t-y_h}$. Since $2(y_t-y_h) < (y_t-y_h)$ for standard universality classes, the ratio method exhibits faster decay of finite-size corrections, leading to less distortion in the critical parameter extraction. This robust suppression of mixing-induced scaling violations is critical in multicomponent systems and models with strong operator overlap.

5. Application to the Three-Dimensional Three-State Potts Model

The ratio method has been validated by application to the three-state Potts model in three dimensions, characterized by the Hamiltonian

$$\mathcal{H}(\tau, \xi) = -\tau \sum_{\langle ij \rangle} \delta_{\sigma_i, \sigma_j} - \xi \sum_i \delta_{\sigma_i, k_0}.$$

Monte Carlo simulations (heat-bath updates) spanning system sizes $L=24,30,40,50,60,70$ allow for extraction of Lee-Yang zeros via multiparameter reweighting: $$\mathcal{Z}(\tau, \xi) / \mathcal{Z}(\tau, \Re \xi) = \langle e^{-[\mathcal{H}(\tau, \xi) - \mathcal{H}(\tau_{\mathrm{sim}}, \xi_{\mathrm{sim}})]} \rangle.$$ The ratio crossing analysis, plotting

$$\mathcal{R}_{n,1}(\tau, L) = \frac{\Im \xi_{\mathrm{LY}}^{(n)}(\tau, L)}{\Im \xi_{\mathrm{LY}}^{(1)}(\tau, L)}$$

against $\tau$ for $n=2,3,4$, yields a common intersection near $\tau_c \approx 0.54938$. Combined fits for $L \geq 40$ return

$\tau_c = 0.549379(14), \quad y_t = 1.70(16),$

in agreement with the Ising universality class. The statistical uncertainties are comparable to those of Binder-cumulant methods, but with improved control over finite-size effects.

6. Generalizations, Practical Implications, and Limitations

The ratio-of-zeros approach is adaptable to a broad class of critical phenomena:

• Universality class independence: By substituting the thermal and magnetic exponents $(y_t, y_h)$ for the appropriate universality class, the technique extends to O($N$), XY, Heisenberg, and percolation transitions.
• Higher-order zeros: Including larger $n$ in $R_{n,m}$ increases statistical precision and allows detailed scrutiny of universality.
• Mixing matrix extraction: Joint fits to the real and imaginary parts of Lee-Yang zeros enable estimation of both the critical field/location and the mixing parameters $a_{ij}$, characterizing variable correlation.
• Limitations: The method relies on accurate determination of zeros, which in reweighting approaches may be impeded by overlap problems that limit the accessible parameter window. The assumption is that no competing singularities (e.g., noncritical Yang-Lee edge singularities) exist near the critical region.

Experimentally, the method can be applied wherever partition function zeros are accessible (via fluctuation statistics, quantum probe measurements, or numerical analytic continuation). Practical deployment in QCD critical point searches, complex spin glasses, and heavy-ion fluctuation data is anticipated.

7. Comparison with Other Critical Point Locators

In practical and theoretical terms, the ratio method offers several key advantages relative to traditional cumulant/crossing methods:

• Suppression of subleading corrections: Finite-size effects from variable mixing decay more rapidly.
• Automatic cancellation of non-universal amplitudes: Only the universal structure of zeros determines the intersection.
• Robustness in complex systems: Directly deals with cases of operator entanglement or nontrivial variable transformation.
• Direct universality checks: The universal ratios $r_{n,m}$ are system-independent and can be used to benchmark universality class membership.

Nonetheless, the accuracy of the intersection approach hinges on the identification and scaling of Lee-Yang zeros, motivating ongoing efforts in efficient numerical solvers and analytic continuation algorithms.

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Lee-Yang zero theory, and in particular the ratio method for critical point location, constitute essential tools for quantitative analysis of phase transitions, offering both principled and practically advantageous frameworks for addressing universality, scaling, and finite-size effects across a broad spectrum of classical and quantum many-body systems (Wada et al., 25 Oct 2024).