Graded Thermodynamic Bethe Ansatz

• Graded Thermodynamic Bethe Ansatz is a framework that extends conventional TBA to incorporate cyclic symmetry and fractional-spin phenomena in integrable quantum systems.
• It revises scattering matrices, Y-system relations, and free energy computations by modifying the TBA equations with graded pullbacks and generalized Gibbs ensembles.
• The approach unifies orbifold twists, superalgebraic deformations, and fractional-spin flows, providing new tools for analyzing phase transitions and integrable deformations.

The graded Thermodynamic Bethe Ansatz (TBA) generalizes the standard TBA formalism to integrable quantum field theories (IQFTs) and spin chain systems with internal grading structures—such as cyclic $\mathbb{Z}_n$ symmetries or superalgebraic gradings. In such constructions, the hierarchy of conserved charges is extended to include fractional-spin quantities, and S-matrices and associated TBA/Y-system equations must be consistently modified to account for both integer and fractional degrees of freedom. This approach unifies features from cyclic orbifolds, ODE/IM correspondences, and fractional-spin generalizations of known solvable irrelevant deformations.

1. Graded Structure in Integrable Systems

Grading in integrable systems arises from implementing internal discrete structures on the Hilbert space. In one construction, an internal $\mathbb{Z}_n$ grading is introduced by viewing the rapidity variable $\theta$ as living on an $n$-sheeted complex cover, defined via conformal maps

$$f_k(\theta) = \xi\left[\frac{\theta}{n} - \frac{2\pi i w k}{n}\right],\quad k \in \mathbb{Z}_n,$$

for integers $w$, $n$ with $\gcd(w, n) = 1$, and sign $\xi = \pm 1$. Each physical species $a$ is promoted to an $n$-component multiplet $a_k$; the two-body graded S-matrix amplitudes are

$$S_{a_k b_\ell}(\theta) = S_{ab}(f_{k-\ell}(\theta)).$$

Unitarity, crossing, and bootstrap relations lift to the graded context, with closure of the bootstrap requiring that $n = 2w h + \xi$ ($h$ the Coxeter number of the species graph $G$). These conditions generate consistent cyclic multiplets of scattering amplitudes, forming the foundation for a graded integrable theory (Brizio et al., 5 Nov 2025).

In AdS/CFT applications, the grading reflects superalgebraic content: for the AdS$_5\times$S$^5$ superstring, four species arise (bosonic $Q$- and magnonic strings, left/right wings, and fermionic $y_\pm$ roots), each with parity $F_a$ used to grade physical and auxiliary roots within the TBA formulation (Bajnok, 2010).

2. Generalized Gibbs Ensembles and Fractional-Spin Charges

Integrable QFTs admit towers of commuting local charges $Q_s^\pm$ of Lorentz spin $s$ (exponents of $G \bmod h$), acting additively on multi-particle states. In a generic generalized Gibbs ensemble (GGE), all these charges can be sourced: $$\rho \propto \exp\left(-\sum_{s \in S} \beta_s H_s(L)\right), \quad H_s(L) = \frac{Q_s^+(L)+Q_s^-(L)}{2},$$ where $\beta_1$ is the inverse temperature and higher $\beta_s$ couple to higher-spin conserved fields. The source term $\nu(\theta)$ in the TBA equations generalizes correspondingly, containing contributions from both integer and fractional spins: $$\nu(\theta) = \sum_{s \in S} \gamma_s r^s \cosh(s \theta), \qquad r = m_* \beta,\qquad \gamma_s = \frac{\beta_s}{\beta^s}.$$ In the graded construction, pulling back along $f_k$ further introduces effective fractional-spin source terms, specifically, spins $s/n$ (Brizio et al., 5 Nov 2025).

3. Graded TBA Equations: Structure and Solution

The graded TBA formalism requires extending the familiar saddle-point equations to all $(a_k)$ species: $$\epsilon_{a_k}(\theta) = \nu_{a_k}(\theta) - \sum_{b \in G}\sum_{\ell \in \mathbb{Z}_n} \int_{\mathbb{R}} \frac{dy}{2\pi} \ \varphi_{a_k b_\ell}(\theta - y) \log[1 + e^{-\epsilon_{b_\ell}(y)}],$$ with kernels

$$\varphi_{ab}(\theta) = -i \,\partial_\theta \log S_{ab}(\theta),$$

and graded pullback

$$\varphi_{a_k b_\ell}(\theta) = (\xi/n) \varphi_{ab}(f_{k-\ell}(\theta)).$$

With GGE sources, the graded driving term, when only spin-$n$ and some spin-$s$ charge are turned on, takes the form

$$\nu_{a_k}(\theta) = \sum_{s \in S} \alpha_s q_{s, a_*} r^{s/n} \cosh\left[\frac{s\theta}{n} - \frac{2\pi i w s k}{n}\right],$$

with the normalization $\alpha_n = 1$. For each $a_k$, the physical free energy and effective central charge are expressed as

$$f(\{\beta_s\}) = -\sum_{a,k} \frac{m_a}{2\pi\beta} \int d\theta \cosh\theta\,L_{a_k}(\theta), \quad c_\text{eff}(r,\{\gamma_s\}) = \frac{3r}{\pi^2} \sum_{a,k} \int d\theta \,\frac{m_a}{m_*} \cosh\theta\,L_{a_k}(\theta),$$

with $L_{a_k} = \ln(1 + e^{-\epsilon_{a_k}})$.

Numerical solution proceeds via iteration, truncation of sums where species sets are infinite, and careful contour analysis when auxiliary temperatures or coupling parameters lead to singularities or branch cuts in the TBA kernel or driving terms.

4. Functional Y-system and Closure

The fusion and bootstrap relations in the graded TBA framework culminate in a system of functional relations: $$Y_{a_{k-\xi}}(\theta + i\pi/h)\, Y_{a_{k+\xi}}(\theta - i\pi/h) = \prod_{b \in G} [1 + Y_{b_k}(\theta)]^{G_{ab}},$$ for $k \in \mathbb{Z}_n$. Periodicity properties are

$$Y_{a_k}(\theta + i\pi P) = Y_{\overline{a}, k+\xi h}(\theta), \quad P = \frac{h+2}{h}.$$

Closure of the Y-system is ensured by the consistency condition $n=2wh+\xi$ on the grading data. This structure generalizes the canonical Y-system of ungraded factorizable scattering to accommodate the effects of internal cyclicity and fractional-spin degrees (Brizio et al., 5 Nov 2025).

In the AdS/CFT mirror TBA, the grading of Y-functions manifests as a reversal of fusion rules for fermionic nodes, leading to Y-system relations with factors $1+1/Y$ at those nodes, in contrast to $1+Y$ for bosonic sites (Bajnok, 2010).

5. Special Cases: ODE/IM Correspondence and Level-Crossing Phenomena

A notable realization of the graded TBA appears in the $n=5$ Lee–Yang model, where the associated Y-system exactly matches that arising from the ODE/IM correspondence for the quantum cubic oscillator, with monodromy invariants $R_k(E, \alpha)$ obeying

$$Y_{k+1}(\theta^+) Y_{k-1}(\theta^-) = 1 + Y_k(\theta), \quad \theta^\pm = \theta \pm i \pi/3,\ k \in \mathbb{Z}_5,$$

and $$Y_k(\theta) = -R_0(e^{2\pi i k/5}\alpha, e^{6\theta/5})$$ (Brizio et al., 5 Nov 2025). This context reveals the compatibility of the graded TBA with spectral analysis from differential equations.

Another significant feature is the emergence of infinite towers of level crossings as auxiliary temperature or deformation parameters are analytically continued. These appear as discrete jumps in the effective central charge $c_\text{eff}(r = 0, \alpha)$, corresponding to residue contributions from roots of the quantization condition

$$\epsilon_{a_k}(\theta_j) = i\pi(2j + 1), \quad j \in \mathbb{N}.$$

In the graded Ising model ($n=3$), this mechanism produces an unbounded sequence of crossing points. Analogous phase transitions are found in interacting theories like graded Lee–Yang (Brizio et al., 5 Nov 2025).

6. Deformations, Fractional-Spin CDD Factors, and Hagedorn Phenomena

The graded TBA admits a class of exactly soluble deformations via CDD factors implementing flows sourced by fractional-spin charges: $$F_{a_k b_\ell}(\theta, s) = \exp\left[i\mu\, q_{s,a}\, q_{s,b}\, \sinh(s\,f_{k-\ell}(\theta))\right].$$ For $s=n$, this mechanism reduces to the $T\bar{T}$ deformation. For generic (fractional) $s$ one obtains new integrable flows, leading to exotic thermodynamic and spectral properties (Brizio et al., 5 Nov 2025). For instance, in the graded Ising GGE, the deformed ground-state energy follows

$$E^\pm(r; \mu) \sim -\frac{1}{\mu^2} \left(\sqrt{r\mu + 8\pi} \pm 2\sqrt{2\pi}\right)^2,$$

with a critical radius $r_*$ where a Hagedorn-like singularity signals a finite limiting temperature—a generalization of the Yang-Lee edge singularity at nontrivial internal symmetry and fractional-spin deformation flow.

7. Connections, Applications, and Significance

The graded TBA framework offers a unified approach encompassing:

• Cyclic orbifold twists via chemical potentials,
• GGE flows incorporating both integer and fractional spins,
• ODE/IM correspondence for nonlocal spectral theory of differential equations,
• Fractional-spin $T\bar{T}$-like deformations,
• Level crossing sequences and emergent discrete phase transitions in the ground-state spectrum.

Its domain of applicability spans both quantum field theoretic integrable models and lattice systems (e.g., AdS/CFT superstrings), with substantial consequences for analysis of finite-volume spectra, deformation-driven phase structure, and integrable deformations beyond the canonical $T\bar{T}$ line. The algebraic and analytic machinery developed for graded TBA is necessary for interpreting phenomena in models with cyclic symmetry and higher/fractional-spin dynamics, including those with connections to conformal field theory, string theory, and exactly solvable quantum mechanical problems.