Widom Line Structure

• Widom line is a thermodynamic boundary marking the locus of maximal response functions and correlation lengths in supercritical and quantum systems.
• It is mathematically defined through peak measurements in properties such as heat capacity, compressibility, and Ruppeiner scalar curvature.
• The concept unifies observable crossovers in fluids, magnetism, and electronic materials, guiding both theoretical analysis and experimental approaches.

The Widom line is a structural and thermodynamic concept central to the understanding of supercritical fluids, magnetism, liquid–liquid transitions, and correlated electronic and quantum systems. It describes the locus in phase space where the system exhibits maximal correlation length, pronounced maxima in thermodynamic response functions, or sharp crossovers in dynamic observables—representing the analytical continuation of a first-order coexistence boundary into the supercritical or non-ordered regime. The Widom line divides the one-phase region into subdomains with distinct physical properties, often governing observable crossovers and organizing thermodynamic and dynamical anomalies.

1. Mathematical Definitions and Geometric Context

The Widom line can be defined through multiple, closely related formulations. Its classical version in fluids is the locus in the $(T,P)$ or $(T,\rho)$ plane at which a response function (e.g., isobaric heat capacity $C_P$, isothermal compressibility $\kappa_T$, thermal expansion $\alpha_P$) achieves a maximum at fixed pressure or temperature (Han et al., 2011). In information and thermodynamic geometry, this locus is coincident with the maxima of the scalar curvature $R$ of the equilibrium thermodynamic metric, which is proportional to the correlation volume: $|R|\propto\xi^d$, with $\xi$ the correlation length and $d$ the spatial dimension (Ruppeiner et al., 2011, Leon et al., 2021, Dey et al., 2011).

In statistical mechanics, particularly for the Fisher–Widom line, the crossover is formulated via changes in the asymptotic decay of the total correlation function: $h(r) = g(r) - 1$, whose analytic structure (real and complex poles) switches between monotonic and oscillatory decay at the Widom (or Fisher–Widom) line (Montero et al., 3 Apr 2024). For quantum systems (e.g., Mott transitions), the quantum Widom line is the locus in $(U, T)$ space where the free energy curvature is minimal or response-function extrema occur, such as the inflection point in double occupancy or resistivity (Vucicevic et al., 2012, Downey et al., 2022).

2. Thermodynamic and Fluctuation Criteria

Fundamental thermodynamic relations tie the Widom line to maxima of susceptibilities:

• For supercritical fluids, $C_P$ peaks along the Widom line, which is tracked in simulations by identifying $C_P(T,P)$ maxima at fixed $P$ (Han et al., 2011).
• Fluctuation–dissipation theorems link $C_P$ to entropy fluctuations via $\langle (\delta S)^2 \rangle = N k_B C_P$.
• In models such as the square-well fluid, multiple ridges of maxima in $\beta_T, C_P, \zeta_T, \alpha_P, \xi$ are observed, with the "true" Widom line identified as the ridge of maximal $\xi$ but only very close to the critical point do these maxima collapse onto a unique line (Brazhkin et al., 2014).

Beyond classical fluids, in quantum systems (Hubbard models and doped Mott insulators), the Widom line's thermodynamic analogues include:

• Maxima or inflections of compressibility, spin susceptibility, and spectral functions, e.g., $\partial^2 n / \partial \mu^2 = 0$ or $\partial^2 A(\omega=0)/\partial T^2 = 0$ (Sordi et al., 2011, Sordi et al., 2012).
• The quantum Widom line extends this principle to free-energy curvature minima in DMFT functional formulations (Vucicevic et al., 2012, Downey et al., 2022).

3. Microscopic and Structural Interpretations

Recent work has contributed microscopic perspectives utilizing structural order parameters, machine learning, and geometrical constructions:

• In core-softened and two-scale potentials, the Widom line can be structurally defined as the locus where occupations in competing coordination shells are equal: $g(r_1)=g(r_2)$, marking maximal local fluctuations between different environments (Salcedo et al., 2012).
• In supercritical fluids, machine learning analysis reveals the "Widom delta," a deltoid-shaped region in $(T,P)$ space where fractions of liquid-like and gas-like microstates are balanced ($\pi_\mathrm{gas} = \pi_\mathrm{liq} = 0.5$), and this line coincides closely with traditional thermodynamic Widom lines near criticality (Ha et al., 2018).
• The mean-field dividing interface, obtained via the third root (Maxwell construction) in equation-of-state analysis, can act as the analytic continuation of the Widom line into the two-phase region, unifying the notion of the dividing surface in phase coexistence and the supercritical crossover in all $(T,P,V)$ planes (Liu, 2021).

4. Ruppeiner Geometry and Phase Boundaries

Thermodynamic geometry, particularly via the Ruppeiner metric, formalizes the fluctuation landscape and the partitioning of phase space:

• The Ricci scalar $R$ derived from the Ruppeiner metric is a model-independent marker of the region of maximal correlations—the Widom line—when its maxima are tracked along isobars (Ruppeiner et al., 2011, Dey et al., 2011).
• The "R-crossing method" equates $R$ in coexisting phases to determine the coexistence curve (Leon et al., 2021).
• For the van der Waals fluid, the Ruppeiner-$N$ metric (holding $N$ fixed, not $V$) provides an exact geometric Widom line coinciding with the standard thermodynamic definition $P_W(T) = 4T - 3$ (Leon et al., 2021). This defines a slope-matching between geometric and thermodynamic lines at the critical point and exact agreement for the idealized system.
• In general analytic fluid models, the geometric and thermodynamic Widom lines often deviate except close to criticality, suggesting non-universality in the matching beyond special cases.

5. Extensions to Quantum, Magnetic, and Field-Theoretic Systems

The Widom line framework has been extended to systems exhibiting quantum phase transitions and statistical field theories:

• In the half-filled Hubbard model, the quantum Widom line divides bad metal and Mott insulator regimes and governs fan-shaped quantum critical scaling in transport properties (e.g., $$\rho(T, \delta U) = \rho_c(T) f(\delta U T^{-1/z\nu})$$) (Vucicevic et al., 2012).
• In layered doped Mott insulators, the Widom line organizes entire phase diagrams: the pseudogap onset temperature $T^*$ coincides with compressibility, spin susceptibility, and c-axis conductivity crossovers, all controlled by the hidden first-order boundary (Sordi et al., 2012).
• Mean-field and information-geometric approaches in magnetic systems (Curie-Weiss, Ising model) reveal that Widom lines correspond to loci where scalar curvature or correlation lengths attain maxima, yielding universal crossovers in the supercritical regime (Dey et al., 2011).
• In field-theoretic models of QCD, the Widom line emerges naturally from the critical end point (CEP) in the $(T, \mu)$ phase diagram, marking peaks in susceptibilities and offering a predictive tool for tracking the CEP in baryon density, entropy, and specific heat curves (Sordi et al., 2023).

6. Practical Implications and Experimental Signatures

The Widom line carries significant practical and experimental utility:

• In fluids, noise power spectral analysis demonstrates that the transition across the Widom line triggers changes in the relaxation time distribution, and low-frequency fluctuation peaks provide independent dynamical indicators of its location (Han et al., 2011).
• In supercritical fluids—including water, argon, and hydrogen—experimental and simulated data for $C_P$ maxima, compressibility, entropy, and scalar curvature $R$ yield Widom lines in close agreement with predictions of thermodynamic geometry and equation-of-state theory (Ruppeiner et al., 2011, Leon et al., 2021).
• In correlated electronic systems, the Widom line chains together static and dynamic anomalies, unifying disparate transport, magnetic, and spectral phenomena via a single organizing principle (Sordi et al., 2011, Sordi et al., 2012, Downey et al., 2022).

7. Limitations, Generalizations, and Open Problems

While the Widom line offers a unifying perspective, certain caveats and open questions remain:

• In analytic fluid models beyond van der Waals and Lennard–Jones, the geometric (Ricci scalar) Widom line and the thermodynamic (response maxima) Widom line do not universally coincide, especially away from the critical point (Leon et al., 2021).
• The divergence of the "ridge bundle" of response-function maxima away from $(T_c, P_c)$ leads to the loss of a unique Widom line and its replacement by a broad crossover region, necessitating dynamic criteria (e.g., the Frenkel line) for demarcating liquid-like and gas-like behavior at high pressures and temperatures (Brazhkin et al., 2014).
• The physical significance of the choice of fixed extensive variable in thermodynamic geometry (Ruppeiner-$N$ vs. Ruppeiner-$V$) is understudied, with indications that non-equivalence in geometric constructions may impact the detection of supercritical crossovers and phase boundaries (Leon et al., 2021).
• In lower-dimensional systems and for non-generic intermolecular potentials, criteria based on vanishing compressibility or pressure (Seno, Zeno lines) may not track the structural transitions marked by the Widom (Fisher–Widom) line, pointing to the essential role of dimensionality and interaction topology (Montero et al., 3 Apr 2024).

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In summary, the Widom line structure is a multi-faceted organizing concept for crossovers in a wide range of thermodynamic, structural, and quantum critical systems. It is rigorously defined by maxima of response functions, correlation length, or scalar curvature in the appropriate thermodynamic geometry, and provides a quantitative and predictive framework for interpreting supercritical phenomena and related crossovers. Its geometric and microscopic realizations highlight both universal features and fluid-specific idiosyncrasies, underpinning both theoretical analysis and experimental identification of fundamental phase boundaries.