Deep Thermal Bootstrap Methods

• Deep thermal bootstrap is a suite of techniques that employs KMS symmetry, reflection positivity, and convex optimization to extract finite-temperature correlators in quantum systems.
• It utilizes advanced numerical methods, including semidefinite programming and neural networks, to nonperturbatively bound and reconstruct Euclidean n-point functions.
• The framework bridges zero-temperature conformal bootstrap with high-precision finite-temperature methods, enabling precise spectrum extraction and observable constraints.

The deep thermal bootstrap is a suite of theoretical and computational techniques for extracting, bounding, and reconstructing finite-temperature correlation functions and operator data in quantum many-body and quantum field systems, particularly in conformal field theories (CFTs) on thermal backgrounds. It systematically leverages symmetry constraints—most crucially the Kubo-Martin-Schwinger (KMS) condition—operator algebra, reflection positivity, and advanced numerical optimization (notably semidefinite programming and neural networks) to compute Euclidean two-point (and, more generally, $n$-point) correlators and to constrain the thermal data nonperturbatively. The program builds bridges between the traditional zero-temperature conformal bootstrap, high-precision methods at finite temperature, and emergent algorithmic paradigms such as neural network–parameterized tail resummation and positivity-free dispersion techniques (Cho et al., 11 Nov 2025, Barrat et al., 2024, Niarchos et al., 13 May 2026, Iliesiu et al., 2018).

1. Foundational Principles: Symmetry, KMS, and Reflection Positivity

At the core of the deep thermal bootstrap are universal analytic and convexity constraints on Euclidean finite-temperature correlators. The KMS condition, which arises from trace cyclicity in the thermal density matrix, imposes (anti)periodicity for Euclidean correlators: $G(\tau+\beta) = G(\tau)$ for bosonic operators. This cyclic symmetry is interpreted as a crossing symmetry in the space of thermal two-point functions, analogously to the $z\leftrightarrow 1-z$ crossing in four-point functions (Iliesiu et al., 2018).

Reflection positivity further constrains the correlator matrix built from a truncated operator basis $\{O_i\}$: $$M_{ij}(\tau)=\langle\bar O_i(\tau)\,O_j(0)\rangle_\beta,\qquad M(\tau)\succeq 0\quad\forall\,\tau$$ and, in combination with the Heisenberg equations of motion and the KMS symmetry, leads to a set of convex semidefinite constraints on the space of acceptable correlators (Cho et al., 11 Nov 2025).

2. Semidefinite Programming and Dual Formulations

The deep thermal bootstrap operationalizes these constraints via semidefinite programming (SDP), casting the problem of bounding or reconstructing thermal two-point functions as an infinite-dimensional convex optimization. In the primal formulation, the correlator matrix $M(\tau)$ and an auxiliary positivity matrix $N(\tau)$ must satisfy:

$$\begin{aligned} &\mathrm{minimize}\quad \mathrm{Tr}[\widehat O\,M(T)] \ &\mathrm{subject\;to}\quad M(\tau)\succeq0,\;N(\tau)\succeq0 \quad(0\leq\tau\leq T)\ &\phantom{\mathrm{subject\;to}\quad} \partial_\tau M(\tau)+M(\tau)D=0 \ &\phantom{\mathrm{subject\;to}\quad} D^\dagger M(\tau)-M(\tau)D=0 \ &\phantom{\mathrm{subject\;to}\quad} N(\tau)-M(\tau)D=0\ &\phantom{\mathrm{subject\;to}\quad} M(\beta)=T M(0)^{T}T^\dagger\;\;(\mathrm{KMS}) \ &\phantom{\mathrm{subject\;to}\quad} M(0)\succeq0 \end{aligned}$$

Here $D_{ij}$ encodes the commutator $[H,O_j]=O_i D_{ij}$. The dual formulation, obtained by introducing Lagrange multipliers, replaces the sharp Heisenberg and symmetry equations with matrix-positivity inequalities of motion on the multipliers: $G(\tau+\beta) = G(\tau)$0 Rigorous bounds on correlators and spectral gaps follow by truncating multipliers to positive bases (e.g., B-splines, mapped polynomials) and reducing the program to a finite-dimensional linear matrix inequality (LMI) system. Weak duality ensures any feasible dual solution yields a rigorous bound on the original correlator (Cho et al., 11 Nov 2025).

3. Sum Rules, Dispersion Relations, and Numerical Algorithms

In conformal settings, the thermal OPE and associated KMS reflection constraints generate an infinite set of sum rules for the OPE and thermal one-point coefficients. Truncating the expansion at $G(\tau+\beta) = G(\tau)$1 and resumming the "tail" via Tauberian asymptotics or explicit neural-network parameterizations is central to the accuracy of the approach: $G(\tau+\beta) = G(\tau)$2 Sum rules take the form

$G(\tau+\beta) = G(\tau)$3

and are implemented numerically via nonlinear least-squares minimization or, in the neural approach, as loss functions on suitable grids (Barrat et al., 2024, Niarchos et al., 13 May 2026). Dispersion-relation-based methods use exact integral representations for the two-point function and treat the infinite towers of higher-dimension OPE coefficients "as a package" through neural networks trained to satisfy the KMS (crossing) condition to high precision (Niarchos et al., 13 May 2026).

4. Neural Network and Non-Positivity-Based Approaches

Conventional positivity-based SDP techniques become inefficient or inapplicable in settings lacking positivity (e.g., nonunitary CFTs, holographic models). Neural network methodologies treat high-dimension tails as functions $G(\tau+\beta) = G(\tau)$4 parameterized via multi-branch feed-forward multi-layer perceptrons, with smooth activation functions imposing implicit regularity via "neural spectral bias." Physical correlator smoothness singles out the correct solution up to residual degeneracy, which is eliminated using ReLU-type anchor constraints (Niarchos et al., 13 May 2026). Losses combine KMS residual minimization and soft anchoring: $G(\tau+\beta) = G(\tau)$5 with $G(\tau+\beta) = G(\tau)$6 scanned to ensure variance minimization and stability.

Key advantages of this paradigm include bypassing unitarity, exact handling of infinite-dimensional tower sums, and control over solution smoothness and degeneracies. Systematic validation against analytic results, as in Generalized Free Fields and holographic CFTs, confirms high accuracy when anchors are placed correctly (Niarchos et al., 13 May 2026).

5. Spectrum Extraction and Physical Observables

Spectroscopic information (gaps and matrix elements) is extracted from the asymptotic (large $G(\tau+\beta) = G(\tau)$7) behavior of the bootstrapped correlators: $G(\tau+\beta) = G(\tau)$8 The log-convexity inequality,

$G(\tau+\beta) = G(\tau)$9

provides a rigorous upper bound on the lowest excitation. More precise bounds on higher gaps $z\leftrightarrow 1-z$0 are determined via a generalized eigenvalue problem. The relevant eigenvector $z\leftrightarrow 1-z$1 from

$z\leftrightarrow 1-z$2

directly yields the leading spectral weights. Agreement with analytic solutions such as the Marchesini–Onofri spectrum validates the method (Cho et al., 11 Nov 2025).

In CFTs, the one-point function of the stress tensor is linked to the free-energy density: $z\leftrightarrow 1-z$3 with $z\leftrightarrow 1-z$4 the thermal coefficient at $z\leftrightarrow 1-z$5 (Barrat et al., 2024).

6. High-Temperature Limits, Loop Equations, and Matrix Integral Bootstraps

At high temperature ($z\leftrightarrow 1-z$6), the KMS condition linearizes, and the deep thermal bootstrap recovers loop equations characteristic of classical and matrix models: $z\leftrightarrow 1-z$7 For instance, setting $z\leftrightarrow 1-z$8, $z\leftrightarrow 1-z$9 produces the classical Schwinger–Dyson equation for the one-variable Boltzmann weight, while in matrix quantum mechanics it reproduces the familiar loop equations for the one-matrix model: $\{O_i\}$0 The energy-entropy balance (EEB) inequality,

$\{O_i\}$1

emerges universally from convexity and further bounds thermal observables (Cho et al., 11 Nov 2025).

7. Extensions, Current Developments, and Future Directions

Recent and ongoing developments include extension to spinning correlators, modular invariance constraints, multi-point bootstrap, application to boundary and defect CFTs, and improved handling of subleading tail corrections via large-$\{O_i\}$2 inversion theorems or neural network surrogates (Barrat et al., 2024, Niarchos et al., 13 May 2026). Hybrid pipelines—combining dispersion-relations, neural priors, and analytic sum rule constraints—are under active exploration. The program sets the stage for systematic, precision extraction of finite-temperature data in strongly coupled theories, including those relevant for holographic duality and quantum chaos.

Systematic error control (e.g., via anchor placement and variance minimization scans), rigorous bounding via duality, and broad applicability beyond positivity-based settings position the deep thermal bootstrap as a unifying and algorithmically powerful framework for finite-temperature quantum field theory and many-body systems (Cho et al., 11 Nov 2025, Barrat et al., 2024, Niarchos et al., 13 May 2026, Iliesiu et al., 2018).