Large-N One-Matrix Anharmonic Oscillator

• The paper demonstrates that bootstrap techniques yield rigorous bounds for thermal energies and excitation gaps in the large-N one-matrix anharmonic oscillator.
• It employs a combination of semiclassical methods and modern numerical approaches, including WKB and semidefinite programming, for precise spectral analysis.
• The study reveals detailed interaction coefficients, such as h1111, and validates the dual long-string description of non-singlet excitations.

The large-N one-matrix anharmonic oscillator is a paradigmatic quantum mechanical matrix model with quartic nonlinearity, widely studied for its connections with nonperturbative aspects of quantum field theory, matrix quantum mechanics, and dual gravitational descriptions. In the large-N limit, the spectral problem and thermal properties admit a precise formulation in terms of collective variables of single-trace operators, supporting both traditional analytic techniques—such as WKB in the singlet sector—and modern numerical/computational approaches, including semidefinite and conic programming-based bootstrap methods. Recent advances have enabled the extraction of finite-temperature excited spectra and interaction data with rigorous error bounds, illuminating the structure and universality of matrix model dynamics.

1. Model Definition and Large-N Structure

The one-matrix anharmonic oscillator consists of a single Hermitian $N \times N$ matrix $X$ with conjugate momentum $P$, governed by the Hamiltonian

$H = \Tr\left(\tfrac12 P^2 + \tfrac12 m^2 X^2 + g X^4\right), \qquad Z(\beta) = \Tr\left(e^{-\beta H}\right).$

For both gauged and ungauged versions, the large-N limit is obtained by rescaling $X,P \rightarrow X/\sqrt{N},\,P/\sqrt{N}$ so that all single-trace correlators remain $O(1)$.

In the strict $N \rightarrow \infty$ limit, singlet states (invariant under unitary conjugation) dominate. However, in the absence of gauge constraints, non-singlet excitations appear, which are described by a dual “long string” picture: these are collective modes propagating in an effective inverted double-well potential $v(\lambda) = \tfrac12 \lambda^2 + g \lambda^4$. The energy spectrum and couplings of these long-string states,

$\{\Delta_n\}_{n=0}^\infty,\quad h_{abcd},$

correspond to gaps above the Fermi sea (singlet sector), with $h_{abcd}$ encoding their interactions and two-loop effects in the thermal free energy (Adams, 3 Nov 2025).

2. Bootstrap and Optimization Formulations

Matrix model bootstrap refers to the program of imposing all symmetry, factorization, and positivity constraints obeyed by finite- and infinite-temperature correlators, seeking their allowed range or extremal values consistent with quantum mechanics.

Bootstrap Constraints

Key constraints at large N include:

• Moment Matrix Positivity: For all single-trace monomials $\mathcal{O}_i$, the moment matrix

$\mathcal{M}_{ij} = \langle \Tr(\mathcal{O}_i^\dagger \mathcal{O}_j) \rangle_\beta \succeq 0$

is positive semidefinite.

• Parity/Time-Reversal: Thermal averages of monomials with odd total degree vanish, with further real/imaginary conditions according to $P$ parity.
• Stationarity (Heisenberg Equations): $\langle [H,\Tr \mathcal{O}_i] \rangle_\beta = 0$.
• Trace Cyclicity and Factorization: Linear and quadratic relations among correlators.
• SU(N) Commutation Relations: $$\frac{1}{N}[X, P] = \frac{i}{N}\mathbb{I} + O(N^{-2})$$.
• Thermal (KMS) Condition: Expressed as

$$\beta C \succeq A^{1/2} \log(A^{1/2} B^{-1} A^{1/2}) A^{1/2},$$

where $A, B, C$ are covariance and commutator matrices built from operator traces.

Conic and Semidefinite Programming

The bootstrap optimization naturally takes the form of a convex program:

• Primal Problem: Minimize (or maximize) the energy $E = c^T x$ over the vector $x$ of correlator variables, subject to semidefinite and linear equality/inequality constraints (moment matrix, KMS cone, factorization, symmetry).
• Dual Problem: Maximizes a linear functional using Lagrange multipliers for the constraints, including dual cones to the semidefinite and operator-relative-entropy cones.

Earlier work (Han et al., 2020) was based on linear semidefinite programming (SDP) with rational approximations to nonlinearities (e.g., logarithms in KMS conditions). Modern implementations, such as those leveraging the Quantum Information Conic Solver (QICS), now directly implement both the semidefinite and operator-relative-entropy cones, removing log-approximations and enabling exact enforcement of the KMS thermal constraint (Adams, 3 Nov 2025).

3. Numerical Implementation and Algorithmic Advances

Matrix model bootstrapping proceeds via basis truncation in operator degree $L$, constructing all allowed monomials up to $L$ and building the moment (Gram) matrix $\mathcal{M}$ in this basis. Notably:

• The number of variables $M_k$ (moments) grows as $O(L^2)$.
• Each block $\mathcal{M}_{ij}$ is a linear or quadratic function of moments.
• Linear constraints (from symmetry and stationarity) are handled by projection or explicit elimination.
• The resulting (possibly noncommutative) SDP or conic program is passed to numerical solvers such as SDPA, MOSEK, or QICS.

QICS, implementing the Skajaa–Ye algorithm for nonsymmetric cones, can handle nonlinear cone constraints such as operator-relative-entropy exactly, yielding feasible dual/primal certificates and rigorous two-sided bounds $E_{\min} \leq E \leq E_{\max}$. This avoids systematic overestimation or loss of precision incurred by polynomial/logarithmic relaxations.

Convergence with respect to $L$ is rapid; for $L = 12$, sub-permille precision is obtained for ground and low-lying excited state energies. Standard performance on modern hardware is in the sub-minute regime for L up to 5–7, and up to several hours for L near 12, owing to rapid matrix growth.

4. Spectral Results, Long-String States, and Couplings

The central achievement of modern conic bootstrapping (notably (Adams, 3 Nov 2025)) is the extraction of not only the thermal energy but also individual excitation gaps and interaction coefficients in the large-N one-matrix model.

At $g = 2$, $L = 12$, QICS yields:

Parameter	Bootstrap Bound (±1σ)	Known Value
$e_0$	0.865457750210 ± $3\times 10^{-7}$	0.8654577
$\Delta_1$	2.1283360 ± $2\times 10^{-4}$	2.1281936
$h_{1111}$	0.32731 ± 0.07	0.3278

These quantities are defined as follows:

• $e_0$: Ground state energy density in the large-N limit.
• $\Delta_1$: First long-string excitation gap (above the Fermi sea in the singlet sector).
• $h_{1111}$: The lowest-order four-point long-string coupling, controlling two-loop thermal corrections.

The fit aligns bootstrap envelopes for $E(\beta)/N^2$ in the low-temperature regime to the ansatz

$$\frac{E(\beta)}{N^2} \approx e_0 + \Delta_1 e^{-\beta\Delta_1} + [\Delta_1 + h_{1111}(1 - 2\beta\Delta_1)] e^{-2\beta\Delta_1}.$$

For $\Delta_1$, the bootstrap deviation from the known value is less than $0.001\%$ ($$|\Delta_1^{\mathrm{bootstrap}} - \Delta_1^{\mathrm{exact}}| / \Delta_1^{\mathrm{exact}} \lesssim 10^{-5}$$), establishing the tightest nonperturbative bounds to date. Notably, $h_{1111}$ is determined entirely from symmetry and self-consistency (KMS) without explicit input from higher-loop data, supporting the predictive completeness of the bootstrap formulation.

5. Analytic Benchmarks and Hermitian/PT-Symmetric Equivalents

Traditional large-N analysis is anchored in mapping the matrix model spectral problem to a system of noninteracting fermions (singlet sector) or to multi-dimensional WKB problems in cases with PT-symmetry (0804.0778). Notable results include:

• Diagonalizing the matrix $M$ and reducing to eigenvalue integrals, with Vandermonde determinant building in Pauli exclusion for the singlet sector.
• For quartic coupling ($p=4$), isospectral mapping to a Hermitian problem is possible, with the leading “quantum anomaly” vanishing at $N \rightarrow \infty$.
• The leading large-N ground-state energy per matrix entry is given by

$E_0(\infty) = \frac{p+2}{3p+2}\, \varepsilon_F,$

where $\varepsilon_F$ is the Fermi energy determined by implicit quantization conditions involving Gamma functions and the coupling $g$.

Rapid convergence of finite-N spectra to WKB (large-N) predictions is observed, with less than 0.001 difference for $N=8$ and negligible anomaly contributions at leading order.

6. Significance, Implications, and Future Prospects

The rigorous computability achieved for the large-N one-matrix anharmonic oscillator exemplifies a new standard for nonperturbative control in strongly coupled quantum systems. The replacement of relaxed SDP approximations with direct nonlinear conic programming—via QICS—marks a substantial strengthening of the bootstrap approach, extending its reach to excited-state gaps and interaction coefficients with controlled errors (Adams, 3 Nov 2025).

These results validate the long-string effective theory—i.e., the description of non-singlet excitations as dual “string” modes—up to at least the precision of the employed bootstrap. The success in determining coupling coefficients such as $h_{1111}$ from general symmetry and KMS conditions alone suggests that higher excitations $(\Delta_2, \Delta_3, \ldots)$ and higher interaction data could be systematically constrained at larger system sizes $L$.

A plausible implication is that further algorithmic and computational improvements will allow the bootstrap to definitively resolve multi-matrix and field-theoretical analogs, where the analytic solution is unknown. This approach may also illuminate universal features of nonperturbative quantum dynamics, including holographic duality and rigorous characterization of thermalization in matrix quantum mechanics.

7. Related Models and Extensions

Extensions of the large-N bootstrap methodology to multi-matrix models, higher-degree polynomial potentials, and vector models (e.g., O(N)-invariant systems) proceed along analogous lines, although the exponential growth of the moment basis poses increasing computational challenges. The core framework—imposing all symmetry, positivity, and thermal consistency constraints, casting the problem as an SDP or conic program—remains applicable.

In summary, the large-N one-matrix anharmonic oscillator serves as an archetype for the synergy between analytic and computational methods in modern quantum many-body theory, exemplifying both the reach and precision of the bootstrap paradigm in nonperturbative spectral analysis.