Finite-$N$ Bootstrap Constraints in Matrix and Tensor Models

1. Problem Statement

The paper addresses the challenge of constraining Matrix and Tensor Models at finite $N$ (where $N$ is the dimension of the matrix or tensor) using the Bootstrap approach.

• Context: While the bootstrap method has been successful in studying large-$N$ limits (where factorization of expectation values occurs) and specific symmetry groups (like SU(2)/SU(3)), its application to general finite-$N$ regimes remains under-explored.
• Gap: In the large-$N$ limit, multi-trace expectation values factorize ($\langle \text{tr} A \text{tr} B \rangle \approx \langle \text{tr} A \rangle \langle \text{tr} B \rangle$), simplifying the problem. At finite $N$, this factorization fails, and the relationship between single-trace and multi-trace observables becomes non-trivial. The authors aim to determine if bootstrap techniques can yield $N$-dependent bounds or if the bounds are inherently independent of $N$ without additional constraints.
• Models Studied:
  1. Matrix Models: A Gaussian model with a quartic interaction ($V(M) = \frac{1}{2}M^2 + \frac{g}{4}M^4$).
  2. Tensor Models: A specific subclass of tensor models (order $D$) that can be mapped to rectangular matrix models, also with a quartic interaction.

2. Methodology

The authors employ the Matrix Bootstrap framework, which relies on two main pillars: Positivity Constraints and Schwinger-Dyson (SD) Equations.

A. Positivity Constraints

The core assumption is that the expectation value of a strictly positive quantity under a positive probability measure must be positive. The authors utilize three specific constraints:

1. Single-Trace Positivity: For an arbitrary matrix $\Phi = \sum c_i O_i$, the Gram matrix $M_{ij} = \langle \text{tr}(O_i^\dagger O_j) \rangle$ must be positive semi-definite ($M \succeq 0$).
2. Double-Trace Positivity: For a scalar $P = \sum c_i \text{tr}(O_i)$, the Gram matrix $Q_{ij} = \langle \text{tr}(O_i^\dagger) \text{tr}(O_j) \rangle$ must be positive semi-definite ($Q \succeq 0$).
3. Traceless Component Positivity: A novel constraint introduced in this work. By decomposing operators into traceless parts, they impose that the matrix $M - Q \succeq 0$. This is crucial for finite $N$ where double-trace terms do not factorize.

B. Schwinger-Dyson Equations

The authors derive SD equations by invoking the invariance of the path integral under infinitesimal field redefinitions ($M \to M + \delta M$).

• Matrix Case: The SD equations relate moments of the matrix powers. At finite $N$, these equations involve both single-trace moments ($m_k = \langle \text{tr} M^k \rangle$) and double-trace moments ($m_{k,l} = \langle \text{tr} M^k \text{tr} M^l \rangle$).
• Tensor Case: The SD equations for the tensor model (mapped to a rectangular matrix $M = TT^\dagger$) involve single-trace ($G_k$) and double-trace ($G_{k,l}$) expectation values. Crucially, the tensor SD equations contain explicit dependence on the tensor order $D$ and dimension $N$ via prefactors like $1/N^{D-2}$.

C. Numerical Optimization

The problem is formulated as a Semi-Definite Programming (SDP) problem.

• Objective: Find the extrema (bounds) of specific observables (e.g., the two-point function $m_2$) subject to the SD equations and positivity constraints.
• Tools: The authors use the SDPA solver.
• Techniques:
  • Coupling Re-scaling: To handle numerical instabilities at small coupling $g$ (where matrix elements scale as $1/g$), they apply a diagonal transformation $M \to DMD$ to tame divergences.
  • Bisection Refinement: Used to precisely determine the lower bounds on the coupling constant $g$ as a function of $N$.

3. Key Contributions and Results

A. Matrix Models: The $N$-Independence of Bounds

• Finding: When applying the bootstrap to the Gaussian matrix model with quartic interaction at finite $N$, the authors found that the resulting bounds on the two-point function do not depend explicitly on $N$.
• Observation: The allowed region in the $(m_2, g)$ plane encompasses the exact solutions for both the $N=1$ limit and the large-$N$ limit simultaneously.
• Interpretation: The bounds are "universal" in the sense that they cover all possible curves between $N=1$ and $N=\infty$. The specific $N$-dependence is encoded in the properties of multi-trace expectation values (factorization properties), not in the raw positivity constraints themselves.
• Verification:
  • Imposing Large-$N$ factorization ($m_{k,l} = m_k m_l$) recovers the large-$N$ exact solution, excluding $N=1$.
  • Imposing $N=1$ factorization ($m_{k,l} = m_{k+l}$) recovers the $N=1$ exact solution, excluding large $N$.
• Limitation: Increasing the Gram matrix size beyond $6 \times 6$ did not improve the bounds because unconstrained double-trace values ($G_{n,m}$) appeared in the single-trace Gram matrix ($M$) but not in the double-trace Gram matrix ($Q$), weakening the constraints.

B. Tensor Models: Explicit $N$-Dependence

• Finding: In contrast to matrix models, the bootstrap bounds for the tensor model vary explicitly as a function of $N$.
• Mechanism: The Schwinger-Dyson equations for tensors contain terms scaling as $1/N^{D-2}$. This explicit $N$ dependence allows the SD equations to distinguish between different $N$ values without requiring ad-hoc factorization assumptions.
• Results:
  • $g > 0$: The bounds interpolate smoothly between the exact $N=1$ solution and the large-$N$ solution. As $N$ increases, the allowed region shifts and tightens toward the large-$N$ limit.
  • $g < 0$: Similar behavior is observed. For $N=1$, bounds are inconsistent with the large-$N$ limit, but as $N$ increases, the lower bound converges to the large-$N$ solution.
  • Convergence Rate: The convergence to the large-$N$ limit is faster for higher tensor orders ($D$). For $D \ge 4$, $N < 50$ is sufficient to reach the asymptotic limit, whereas $D=3$ requires $N > 100$.
• Critical Coupling: The minimum allowed coupling $g_{\text{min}}$ asymptotes to $g_c = -1/8$ in the large-$N$ limit.

4. Significance and Implications

1. Distinction between Matrix and Tensor Models: The paper highlights a fundamental difference in how bootstrap constraints behave. Matrix models require specific multi-trace factorization assumptions to isolate finite-$N$ physics, whereas tensor models naturally encode $N$-dependence in their SD equations due to their geometric structure (Gurau degree/topology).
2. Role of Multi-Trace Observables: The work reinforces that for matrix models, the "finite-$N$ problem" is essentially a problem of constraining multi-trace expectation values. The bootstrap bounds are currently too loose to distinguish $N$ without these additional constraints.
3. Path to Quantum Gravity: The authors discuss the potential of using these finite-$N$ bounds to study quantum gravity. In the random geometric approach, the Newton constant $G$ is related to $N$ and the lattice spacing $a$. Tuning to a specific finite $N$ (rather than $N \to \infty$) could theoretically match experimental values of $G$, offering a new perspective on the continuum limit of quantum gravity.
4. Methodological Advancement: The introduction of Traceless Component Positivity and the Coupling Re-scaling trick provides robust tools for handling numerical instabilities and tightening constraints in semi-definite programming applications.

Conclusion

The paper demonstrates that while matrix bootstrap techniques struggle to isolate finite-$N$ physics without assuming specific factorization properties, tensor models offer a promising avenue where $N$-dependence emerges naturally from the theory's structure. This suggests that tensor models may be better suited for exploring the non-perturbative, finite-$N$ regimes relevant to higher-dimensional quantum gravity. Future work should focus on broader classes of tensor models (e.g., with tetrahedral interactions) and exploring the continuum limit at finite $N$.