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arXiv:2412.07837 · Journal of High Energy Physics

Coarse-grained bootstrap of quantum many-body systems

Minjae Cho, Colin Oscar Nancarrow, Petar Tadić, Yuan Xin, Zechuan Zheng

Published Feb 23, 2026• 3 citations
arXivHTMLPDF

Abstract

A bstract We present a new computational framework combining coarse-graining techniques with bootstrap methods to study quantum many-body systems. The method efficiently computes rigorous upper and lower bounds on both zero- and finite-temperature expectation values of any local observables of infinite quantum spin chains. This is achieved by using tensor networks to coarse-grain bootstrap constraints, including positivity, translation invariance, equations of motion, and energy-entropy balance inequalities. Coarse-graining allows access to constraints from significantly larger subsystems than previously possible, yielding tighter bounds compared to those obtained without coarse-graining.

DOI
10.1007/jhep02(2026)222

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A bstract We present a new computational framework combining coarse-graining techniques with bootstrap methods to study quantum many-body systems. The method efficiently computes rigorous upper and lower bounds on both zero- and finite-temperature expectation values of any local observables of infinite quantum spin chains. This is achieved by using tensor networks to coarse-grain bootstrap constraints, including positivity, translation invariance, equations of motion, and energy-entropy balance inequalities. Coarse-graining allows access to constraints from significantly larger subsystems than previously possible, yielding tighter bounds compared to those obtained without coarse-graining.
A numerical bootstrap method is proposed to provide rigorous and nontrivial bounds in general quantum many-body systems with locality. In particular, lower bounds on ground state energies of local lattice systems are obtained by imposing positivity constraints on certain operator expectation values. Complemented with variational upper bounds, ground state observables are constrained to be within a narrow range. The method is demonstrated with the Hubbard model in one and two dimensions, and bounds on ground state double occupancy and magnetization are discussed.
Accurate contraction of tensor networks beyond one dimension is essential in various fields including quantum many-body physics. Existing approaches typically rely on approximate contraction schemes and do not provide certified error bars. We introduce a numerical bootstrap framework which casts the problem of tensor-network contractions into a convex optimization problem, thereby yielding certified lower and upper bounds on expectation values of physical observables. As a proof-of-principle, we construct such constraints explicitly for translationally invariant matrix product states and demonstrate that, assuming a canonical form, second-order-cone relaxation can provide tight bounds on the contraction result. We further demonstrate that when the requirement on canonical form is lifted, a more general semidefinite-programming approach could yield similar tight bounds at higher but still polynomial computational cost. Our work suggests numerical bootstrap could be a possible way forward for the rigorous contractions of tensor networks.
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.
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.
Given a renormalization scheme, we show how to formulate a tractable convex relaxation of the set of feasible local density matrices of a many-body quantum system. The relaxation is obtained by introducing a hierarchy of constraints between the reduced states of ever-growing sets of lattice sites. The coarse-graining maps of the underlying renormalization procedure serve to eliminate a vast number of those constraints, such that the remaining ones can be enforced with reasonable computational means. This process can be used to obtain rigorous lower bounds on the ground-state energy of arbitrary local Hamiltonians by performing a linear optimization over the resulting convex relaxation of reduced quantum states. The quality of the bounds crucially depends on the particular renormalization scheme, which must be tailored to the target Hamiltonian. We apply our method to 1D translation-invariant spin models, obtaining energy bounds comparable to those attained by optimizing over locally translation-invariant states of n≳100 spins. Beyond this demonstration, the general method can be applied to a wide range of other problems, such as spin systems in higher spatial dimensions, electronic structure problems, and various other many-body optimization problems, such as entanglement and nonlocality detection. Published by the American Physical Society 2024
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.