Coarse-grained Bootstrap of Quantum Many-body Systems

TL;DR

Coarse-grained Bootstrap of Quantum Many-body Systems tackles the challenge of obtaining rigorous bounds on local observables in infinite quantum spin chains by marrying coarse-graining with bootstrap constraints. The authors develop a coarse-grained equilibrium bootstrap that leverages tensor-network maps (uMPS) to access substantially larger subsystems, enabling two-sided zero- and finite-temperature bounds on arbitrary local observables. Their framework extends previous ground-state-focused approaches by incorporating EOM and energy–entropy balance inequalities at the coarse-grained level, and they demonstrate markedly tighter bounds for TFIM and XXZ models compared with prior work. The work advances rigorous, scalable bounding techniques for quantum many-body systems and opens avenues for symmetry-resolved analyses and MPO-based thermal treatments in future studies.

Abstract

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.

Figures (13)

• Figure 1: Left Panel: Upper and lower bounds on magnetization in TFIM (\ref{['eq:ising-hamiltonian']}) as functions of coupling $g$. Right Panel: Upper and lower bounds on correlator $\langle Z_0 Z_1 \rangle$ in TFIM as functions of coupling $g$.
• Figure 2: Upper and lower bounds on the energy density of TFIM (\ref{['eq:ising-hamiltonian']}) as functions of coupling $g$. The exact result $\bar{E}_{\text{theory}}$ is very close to the lower bounds.
• Figure 3: We show the difference between bounds of the energy density and exact values as the function of the number of sites in the sublattice $N$ for different values of $g$ in TFIM (\ref{['eq:ising-hamiltonian']}). We also show the upper bound (UB) obtained with the variational uniform MPS (VUMPS) algorithm at bond dimension $m=2$.
• Figure 4: Bounds on the ground state energy density for TFIM (\ref{['eq:ising-hamiltonian']}) with a small magnetic field $\lambda\geq0$. In the $\lambda = 0$ limit the ground state is degenerate and the upper bound is far from the ground state energy. As soon as we turn on a tiny magnetic field the upper bound converges very rapidly to the false vacuum energy, even for small subsystem size $N=5$. The value of the true and false vacuum energies are approximated by the exact diagonalization result of the 16-site periodic spin chain.
• Figure 5: Upper and lower bounds on energy density in detuned TFIM (\ref{['eq:ising-hamiltonian-detuned']}) at $g = 0.5$, i.e. TFIM in the symmetry broken phase perturbed by a small magnetic field. The dashed lines show the approximate energy of the true ground state and false vacuum. The blue and red points show the upper and lower bound of the energy, respectively. The lower bound converges fast to the true vacuum. The upper bound first saturates at the false vacuum energy, but as $N$ grows it eventually converges to the true vacuum energy.