Bootstrapping Euclidean Two-point Correlators

TL;DR

We develop a convex bootstrapping framework for Euclidean two-point correlators in thermal or ground states by enforcing reflection positivity, Heisenberg equations, and KMS/ground-state positivity, casting the bounds as finite-dimensional SDPs. The dual formulation converts dynamics into inequalities on Lagrange multipliers, enabling rigorous bounds on correlators for a truncated operator basis, with practical implementation via PMP and clamped B-splines. Applied to the ungauged 1-MQM, the method yields tight bounds on $\langle \mathrm{tr} X(\tau) X(0) \rangle_\beta$, from which one can extract adjoint-spectrum data and matrix elements, and it recovers MO-equation insights in the ground state while matching or surpassing some finite-T Monte Carlo results. The work also proves an exponential-type bound and the Energy-Entropy balance inequality, and it connects high-temperature bootstrap to matrix-integral bootstrap, laying groundwork for extending to multi-point or Lorentzian correlators and more complex large-$N$ quantum systems.

Abstract

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.

Figures (17)

• Figure 1: Comparison of the allowed region (in pink) for the level 8 two-point correlator bootstrap (\ref{['dualsdp:thermalNoInit']}) using the KMS condition with the level 7 one-point function bootstrap using the EEB inequality Fawzi:2023fpgCho:2024kxn for the anharmonic oscillator $V(x) = \frac{1}{2} x^2 + \frac{1}{4} x^4$. We also show the "exact" value from Hamiltonian truncation (in black).
• Figure 2: Comparison of the integral bootstrap (in dashed blue) with the two-point correlator bootstrap (in pink, uses \ref{['dualsdp:thermalNoInit']}) for the anharmonic oscillator $V(x) = \frac{1}{2} x^2 + \frac{1}{4} x^4$. This shows good agreement at high temperature, where a bosonic quantum system reduces to classical statistical mechanics. Both bootstrap results are obtained at level 6.
• Figure 3: The ground-state Euclidean two-point correlator of the anharmonic oscillator with $H=\frac{1}{2} p^2+ \frac{1}{2} x^2 + \frac{1}{4} x^4$. The shaded regions indicate the bootstrap allowed regions at level-$\{4,6,8\}$. Note that level $L$ means $\mathcal{M}$ with level $L$ and $\mathcal{N}$ with level $L+1$. The green dashed line indicates the exponential lower bound \ref{['universal-lower-bound']}. This plot is obtained by feeding the zero-time data at $\tau=0$ from Hamiltonian truncation into the dual problem \ref{['SDP:MNbootPractical']}, discretized using the PMP formulation described in Appendix \ref{['app:PMP']} with $d=16$.
• Figure 4: The lower and upper bounds converge as the parameters of either the polynomial or spline basis increase (such as the degrees of the polynomial used in the Lagrange multipliers). On the left, we show the convergence in the polynomial basis where $\mathsf{d}$ is the degree of the polynomial. On the right, we show the convergence in the B-spline parameters, which include the number of nodes $\mathsf{N}$ and the degree $\mathsf{d}$ (see Appendix \ref{['app:b-spline']}). We show the results in the case of the ground state of an anharmonic oscillator with $V(x) = \tfrac{1}{2} x^2 + \frac{1}{4} x^4$.
• Figure 5: Bootstrap bounds on the Euclidean two-point correlator in the thermal state of anharmonic oscillator with $H=\frac{1}{2} x^2+ \frac{1}{2} p^2 + \frac{1}{4} x^4$ at $\beta=2$. Left: Bootstrap results using the zero-time input obtained from Hamiltonian truncation, as stated in \ref{['SDP:dualThermal']} and Appendix \ref{['app:withInitThermal']}. Right: Bootstrap results without the zero-time input, as stated in \ref{['dualsdp:thermalNoInit']} and Appendix \ref{['app:noInitThermal']}. The shaded regions indicate the bootstrap allowed regions at different levels. The green dashed line indicates the exponential lower bound \ref{['eqn:univExpThermal']}. Both figures are obtained with the B-spline basis.