Strong-Coupling Corrections in Quantum Systems

• Strong-coupling corrections are systematic expansions that address non-perturbative phenomena and quantum fluctuations in high-coupling regimes.
• They utilize methods from lattice models, holography, and DMFT to bridge solvable limits with experimentally relevant intermediate-coupling behavior.
• These corrections yield improved predictions for observables such as chiral susceptibility, phase boundaries, and excitation spectra in quantum and condensed matter systems.

Strong-coupling corrections refer to systematic expansions, methodologies, or quantitative modifications that account for physical effects beyond the leading strong-coupling limit (i.e., as the coupling constant becomes large) in quantum field theory, statistical mechanics, condensed matter systems, and string/gauge dualities. These corrections are essential for understanding regimes where perturbative expansions in weak coupling fail or are qualitatively inadequate, and where non-perturbative phenomena (ordering, collective excitations, emergent scales, quantum fluctuations) dictate the physics. Strong-coupling corrections also provide the bridge between exactly solvable limiting cases and experimentally relevant intermediate-coupling regimes.

1. General Framework and Definitions

In any interacting many-body or quantum field system, the strong-coupling limit is defined as the regime where the relevant coupling constant $g$ (or $U$, $\lambda$, etc.) is large compared to all microscopic energy scales. The leading strong-coupling physics follows from truncating to $O(g^0)$ or $O(1)$, e.g., by neglecting kinetic energy relative to interaction, or by taking the classical limit of an action. Strong-coupling corrections then systematically expand observable quantities, effective actions, or partition functions in inverse powers of $g$, $1/U$, or $1/\lambda$:

• For lattice models (e.g., Hubbard, QCD), this is typically a series in $t/U$ or $\beta=2N_c/g^2$.
• In holography or AdS/CFT, corrections are organized in $1/\lambda$ or $1/N$.
• In condensed matter/quantum impurity problems, expansions in hybridization or retarded interactions are used.

The relevance of these corrections is twofold: They encode quantum/thermal fluctuations absent in the strict strong-coupling limit and allow controlled interpolation to weaker-coupling or experimentally accessible regimes.

2. Methods for Computing Strong-Coupling Corrections

Lattice Models:

• Staggered Lattice QCD: Gauge corrections enter as an expansion in the lattice gauge coupling $\beta$. To first order, the partition function is expanded as $$Z = \int e^{-S_F - \beta S_G} \simeq \int Z_F \exp(-\langle \beta S_G \rangle_U)$$, where the evaluation entails the enumeration of new fermion-plaquette diagrams (19 types for $N_c = 3$) and computation of their combinatorial weights. The explicit O($\beta$) effective action and associated algorithmic weights remain incompletely tabulated in (Forcrand et al., 2011).
• 3D Polyakov-line Actions (Finite-$T$ Yang-Mills): A resummed strong-coupling expansion yields local and non-local effective couplings (e.g., nearest-neighbor $\lambda_1 \sim u^{N_\tau}$, with $u = \beta/18$). These are non-perturbatively improved by matching to Polyakov-loop correlators measured in the full 4D theory, leading to quantitative corrections (often $\sim$20–50%) needed to accurately recover the critical $\beta_c$ for a given $N_\tau$ (Bergner et al., 2015).

Dual Variable and Anisotropy Approaches:

On anisotropic lattices, the physical anisotropy is determined via non-perturbative calibration, ensuring that the critical endpoints and first-order lines of the chiral phase boundary are accurately located and remain essentially $\beta$-independent for $\beta \leq 1$. Gauge corrections ($O(\beta)$) are encoded as additional plaquette-occupation weights in a dual representation (Kim et al., 2020).

Quantum Field Theory / Holography:

• Higher-Derivative Corrections: In AdS/CFT, higher curvature (e.g., $R^4$) terms in the bulk action generate corrections scaling as $\lambda^{-3/2}$ to quantities such as screening masses: $$M_i = \pi T \left[c_i^{(0)} + \gamma c_i^{(1)}\right] + \cdots, \quad \gamma \sim \lambda^{-3/2}.$$ The computation involves perturbatively solving for fluctuations in corrected backgrounds and extracting eigenvalues for spectral gaps (Singh et al., 2012).
• Hagedorn Temperature Expansion: Systematic strong-coupling expansion of the Hagedorn temperature in confining string backgrounds combines the sigma-model zero-point energy and quantum mechanical perturbation theory. Explicit expressions are derived up to NNLO in $1/\sqrt{g}$ for a broad class of backgrounds (Bigazzi et al., 2023).
• Flux Tube Profiles: Finite-size and $1/\lambda$ corrections to holographic flux-tube profiles and glueball masses are derived via analytic expansion of the probe string/dilaton solutions and inhomogeneous fluctuation equations, yielding quantifiable shifts to intrinsic widths and mass spectra (Canneti, 2 Oct 2025).

Impurity and Correlated Electron Systems:

• Anderson/Holstein Models: Strong-coupling corrections to the X-ray absorption spectrum are handled by an expansion in the retarded interaction, with higher order diagrams efficiently evaluated via a complex-exponential decomposition. Each additional order (NCA, OCA, etc.) captures further dynamical screening and broadening effects (2501.05825).
• Vertex Decomposition in DMFT: The local vertex is parametrized via single-boson-exchange diagrams, with strong-coupling corrections encoded in the dynamical Hedin three-leg vertex $\lambda(\nu, \omega)$. Only by including the full frequency dependence is the correct exchange scale $J = 4t^2/U$ recovered, while simplified parametrizations qualitatively fail in this regime (Harkov et al., 2021).

3. Physical Manifestations in Key Systems

A. Lattice QCD and Strong-Coupling Expansions

Gauge corrections in staggered fermion QCD shift observables such as the chiral susceptibility and the $\mu$–$T$ phase boundary, decrease the critical temperature $T_c$, shift (or reduce the curvature of) the critical line, and modify the location of the critical endpoint. The leading effect is a linear (in $\beta$) suppression of chiral order and decrease of $T_c(\mu)$ (Forcrand et al., 2013, Kim et al., 2020). Enhanced accuracy in reproducing critical values is achieved by including higher-order couplings in the effective Polyakov-line action (Bergner et al., 2015).

B. Correlated Fermion Systems

In the BCS-BEC crossover of ultracold Fermi gases, strong-coupling approaches such as the extended T-matrix approximation (ETMA) systematically incorporate superfluid fluctuations, yielding quantitatively accurate predictions for the chemical potential, compressibility, sound velocity, and other thermodynamics, especially near unitarity (Tajima et al., 2017, Kashimura et al., 2012). Similarly, shear viscosity $\eta$ acquires a nontrivial temperature and coupling dependence, including a pronounced minimum and pseudogap-related peaks, only once vertex corrections beyond mean-field are included (Kagamihara et al., 2019).

C. Holographic and String-Theoretic Observables

Corrections to operator scaling dimensions (e.g., Konishi multiplet in $AdS_5\times S^5$) emerge at the first subleading order in $1/\lambda$, computed via semiclassical fluctuation analysis. For Konishi, the anomalous dimension expands as:

$$\Delta(\lambda) = 2\lambda^{1/4} + 2\lambda^{-1/4} + O(\lambda^{-3/4}),$$

with the subleading coefficient matching integrability-based Y-system predictions (Roiban et al., 2011).

Universal $1/\lambda$ corrections to the Hagedorn temperature, flux-tube widths, and glueball masses in confining gauge theories/holographic duals are established via analytic sigma-model and quantum-mechanical quantization. The first subleading terms are directly associated with the structure of the worldsheet spectrum and the effective dilaton coupling (Bigazzi et al., 2023, Canneti, 2 Oct 2025).

D. Non-Planar and Finite-$N$ Effects

Non-planar strong-coupling corrections (as $1/N$ effects) to observables such as Wilson loop expectation values and CPO correlators admit resummations in the string dual variables. The series in $g_s^2/T$ or $g_s^2/T^2$ sum to exponential or radical (square-root) closed forms, e.g.:

$$\frac{\langle{\cal W}{\cal O}_J\rangle}{\langle{\cal W}\rangle}\bigg|_{\rm leading} = \left(1 + \frac{g_s^2}{T^2} \right)^{J/2},$$

with $T = \sqrt{\lambda}$, $g_s = \lambda/N$, corresponding to universal strong-coupling structure across gauge theories with string-theory duals (Beccaria et al., 2020).

4. Failure of Weak-Coupling Approximations and Physical Consequences

Naive or static mean-field treatments fail to capture strong-coupling physics:

• In lattice QCD, failing to include gauge corrections yields quantitatively and sometimes qualitatively incorrect phase boundaries and susceptibilities (Forcrand et al., 2013).
• In correlated fermion models, neglecting dynamical vertex corrections undermines the correct superexchange scale ($J \sim t^2/U$), the emergence of local moments, and the enhancement of spin fluctuations. Approximations such as the $w$-approximation or static RPA fail in the strong-coupling regime, instead requiring proper vertex decomposition (e.g., inclusion of full Hedin vertex) (Harkov et al., 2021).
• In pseudogap phenomena and transport, extended T-matrix approximations are necessary to avoid thermodynamic inconsistencies such as negative spin susceptibility or unphysical density of states (Kashimura et al., 2012).

5. Analytical Structures and Classification of Corrections

A distinguishing feature of strong-coupling corrections is the explicit power-law or logarithmic structure in inverse coupling, sometimes with resummation properties:

• Leading corrections often scale as $O(1/g)$, $O(1/\lambda)$, $O(1/U)$, or $O(1/N)$, with quantitatively controlled coefficients.
• In electron-phonon systems, strong-coupling BCS-ratio corrections bifurcate into distinct universality classes (double-valued branches) for conventional and near-room-temperature superconductors when plotted versus $k_BT_c/(\hbar\omega_{\ln})$ (Talantsev, 2020).
• Holographic models yield polynomial expansions in $1/\sqrt{g}$ and $1/g$, with coefficients set by worldsheet zero-point energies and effective quantum corrections (Bigazzi et al., 2023).
• In the strong-coupling expansions for impurity models, diagrammatic corrections at each order yield bona fide, physically meaningful shifts and broadenings of observables, with convergence controlled either analytically or by acceleration schemes such as exponential decomposition (2501.05825).

6. Open Problems and Numerical Considerations

While leading strong-coupling corrections provide a controlled expansion in most regimes, explicit extraction of all coefficients, especially for complex systems (e.g., lattice QCD with dynamical fermions, multi-flavor models, and higher-point or nonlocal operators), remains computationally challenging. For many applications, only partial diagrammatic classes (e.g., leading mesonic, or up to a finite range in effective actions) are accessible (Forcrand et al., 2011, Bergner et al., 2015). Empirical or "numerical matching" to fully non-perturbative correlators is often necessary for high-precision determination of criticality or thermodynamic response (Bergner et al., 2015).

7. Physical and Experimental Significance

Strong-coupling corrections are not merely technicalities, but encode the essential physical effects—quantum fluctuations, emergent scales, non-perturbative phenomena—needed to describe real correlated systems. They enable:

• Accurate mapping of phase diagrams of QCD-like theories, including modification of critical lines and endpoints.
• Quantitative prediction of thermodynamic observables, excitation spectra, and transport coefficients in ultracold atomic gases, superconductors, and quantum magnets.
• Systematic interpolation between solvable limiting cases and the regimes probed by experiments, bridging the gap between models and material reality.

The accurate inclusion and understanding of strong-coupling corrections thus remains a central objective across high-energy theory, many-body physics, and quantum materials research.

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Selected References:

• (Forcrand et al., 2011): Gauge corrections and partial fermion-plaquette diagrams in staggered lattice QCD.
• (Bergner et al., 2015): Quantitative improvement of Polyakov line actions via numerically matched strong-coupling corrections.
• (Harkov et al., 2021): SBE parametrization and Hedin vertex in DMFT, failures of simplified vertex parametrizations.
• (Tajima et al., 2017, Kagamihara et al., 2019, Kashimura et al., 2012): Strong-coupling treatments in Fermi gases and their effects on both thermodynamics and transport.
• (Bigazzi et al., 2023, Canneti, 2 Oct 2025, Singh et al., 2012, Roiban et al., 2011, Beccaria et al., 2020): Holographic strong-coupling expansions and analytic corrections in string/gauge duality.
• (Talantsev, 2020): Double-valued BCS strong-coupling correction formulas for NRT hydrides.