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arXiv:2507.21007 · Physical Review Letters

High-Precision Bootstrap of Multimatrix Quantum Mechanics

Henry W. Lin, Zechuan Zheng

Published Mar 31, 2026• 1 citation
arXivHTMLPDF

Abstract

We consider matrix quantum mechanics with multiple bosonic matrices, including those obtained from dimensional reduction of Yang-Mills theories. Using the matrix bootstrap, we study simple observables like ⟨tr X^{2}⟩ in the confining phase of the theory in the infinite N limit. Exploiting the symmetries of these models and applying nonlinear relaxation, we impose constraints that include traces of words of length up to 14. Our results yield rigorous bounds on the large-N ground-state dynamics, along with estimates of selected low-order observables to eight significant digits.

DOI
10.1103/cyq8-4sd7

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We consider matrix quantum mechanics with multiple bosonic matrices, including those obtained from dimensional reduction of Yang-Mills theories. Using the matrix bootstrap, we study simple observables like ⟨tr X^{2}⟩ in the confining phase of the theory in the infinite N limit. Exploiting the symmetries of these models and applying nonlinear relaxation, we impose constraints that include traces of words of length up to 14. Our results yield rigorous bounds on the large-N ground-state dynamics, along with estimates of selected low-order observables to eight significant digits.
A bstract We implement a bootstrap method that combines stationary state conditions, thermal inequalities, and semidefinite relaxations of matrix logarithm in the ungauged one-matrix quantum mechanics, at finite rank N as well as in the large N limit, and determine finite temperature observables that interpolate between available analytic results in the low and high temperature limits respectively. We also obtain bootstrap bounds on thermal phase transition as well as preliminary results in the ungauged two-matrix quantum mechanics.
A novel bootstrap method is exemplified by lattice theories and quantum mechanics. Positivity of expectation values combined with sets of non-trivial constraint equations and semidefinite optimization rigorously and precisely bound correlation functions. Starting with the example of a one-dimensional integral, the technique is built upon to extract limit cycles in a two-dimensional classical dynamical system and spin correlation functions in the Statistical Ising Model. Further positivity constraints and simplifications arising in a large rank limit are implemented to bound the energy expectation value as a function of temperature in Matrix Quantum Mechanics. Along the way, the S-matrix of Chern-Simons-matter theory is computed as an example of computational simplification at large gauge group rank.
The many-body bootstrap certifies ground-state properties by minimizing energy over correlators constrained only by positivity, without any variational wavefunction. Its cost, however, grows exponentially with the size of the operators involved, placing the long-range string correlators that diagnose topological phases out of reach. We overcome this by treating string operators as primary objects: bare and endpoint-dressed strings satisfy a closed operator algebra with one another and with local words, yielding a semidefinite program whose cost grows only polynomially with string length. A single computation then yields estimates of string correlators of all lengths. Rigorous two-sided bounds are obtained for each target string once the ground-state energy is pinned within a window. When benchmarked against density-matrix renormalization group computations on the cluster Ising and spin−1-1−1 Heisenberg chains, the method certifies the nonlocal string order of the cluster and Haldane phases directly from the Hamiltonian. Our rigorous certification of string order parameters reveal that the sharpness of the bounds is set by which operators enter the calculation, not by their length alone. By exploiting algebraic closure of strings and local words, our framework lifts spatially extended observables into polynomially tractable bootstrap variables, opening up a route to wavefunction-free certification of nonlocal order in quantum many-body systems.
Bootstrap is a numerical many-body method that provides rigorous bounds on ground-state observables by imposing a set of necessary constraints on the expectation values of operators. The quality of the resulting bounds is sensitive to the choice of operators entering the constraints. In particular, bounds on ground-state correlations are often loose in spontaneous symmetry-breaking (SSB) phases, since local operator sets cannot exclude domain-wall excitations. In this work, we introduce non-local, string-like operators into the bootstrap and show that the program can be formulated directly in thermodynamic limit. We then apply our construction to several 1D spin models. First, we obtain a significant tightening of the bounds in the SSB phase of the 1D transverse-field Ising model. Using the 1D axial next-nearest-neighbor Ising model, we further show that this tightening allows for a quantitative estimate of the locations of phase boundaries. Finally, we generalize the string operators to the 1D Z3\mathbb{Z}_3Z3​ chiral clock model and track the behavior of the bounds across the phase diagram. Our results broaden the class of constraints available to the bootstrap and open a route toward bootstrapping more general symmetry-broken and topological phases, where the relevant constraints may involve non-local or extended operators.
We develop a bootstrap approach to Euclidean two-point correlators, in the thermal or ground state of quantum mechanical systems. We formulate the problem of bounding the two-point correlator as a semidefinite programming problem, subject to the constraints of reflection positivity, the Heisenberg equations of motion, and the Kubo-Martin-Schwinger condition or ground-state positivity. In the dual formulation, the Heisenberg equations of motion become "inequalities of motion" on the Lagrange multipliers that enforce the constraints. This enables us to derive rigorous bounds on continuous-time two-point correlators using a finite-dimensional semidefinite or polynomial matrix program. We illustrate this method by bootstrapping the two-point correlators of the ungauged one-matrix quantum mechanics, from which we extract the spectrum and matrix elements of the low-lying adjoint states. Along the way, we provide a new derivation of the energy-entropy balance inequality and establish a connection between the high-temperature two-point correlator bootstrap and the matrix integral bootstrap.
The S-matrix bootstrap is extended to a 1+1d theory with O(N) symmetry and a boundary in what we call the R-matrix bootstrap since the quantity of interest is the reflection matrix (R-matrix). Given a bulk S-matrix, the space of allowed R-matrices is an infinite dimensional convex space from which we plot a two dimensional section given by a convex domain on a 2d plane. In certain cases, at the boundary of the domain, we find vertices corresponding to integrable R-matrices with no free parameters. In other cases, when there is a one-parameter family of integrable R-matrices, the whole boundary represents integrable theories. We also consider R-matrices which are analytic in an extended region beyond the physical cuts, thus forbidding poles (resonances) in that region. In certain models, this drastically reduces the allowed space of R-matrices leading to new vertices that again correspond to integrable theories. We also work out the dual problem, in particular in the case of extended analyticity, the dual function has cuts on the physical line whenever unitarity is saturated. For the periodic Yang-Baxter solution that has zero transmission, we computed the R-matrix initially using the bootstrap and then derived its previously unknown analytic form.