Adiabatic transport of neural network quantum states

TL;DR

This work introduces a first-principles method for building neural network representations of many-body excited states by adiabatically continuing eigenstates of simple Hamiltonians into the strongly correlated regime, and obtains accurate estimates of critical exponents.

Abstract

Variational methods have offered controllable and powerful tools for capturing many-body quantum physics for decades. The recent introduction of expressive neural network quantum states has enabled the accurate representation of a broad class of complex wavefunctions for many Hamiltonians of interest. We introduce a first-principles method for building neural network representations of many-body excited states by adiabatically continuing eigenstates of simple Hamiltonians into the strongly correlated regime. With controlled access to the full many-body gap, we obtain accurate estimates of critical exponents. Successive eigenstate estimates can be run entirely in parallel, enabling precise targeting of excited-state properties without reference to the rest of the spectrum, opening the door to large-scale numerical investigations of universal properties of entire phases of matter.

Figures (7)

• Figure 1: An overview of the adiabatic transport of an NQS ground state with traditional VMC ground-state optimization. Successive eigenstate approximations trace a curve in the space of parameters $\theta$ of the trial state $\Psi _\theta$.
• Figure 2: Adiabatic transport of ground and excited states of the 2D TFIM to the critical point. Panel (a) shows the $4 \times 4$ lattice, with energies compared to exact diagonalization (solid lines) and an inset displaying the average infidelity for one representative eigenstate per degenerate manifold. Panel (b) shows the $8 \times 8$ lattice, with V-scores reported in the inset, confirming accurate wavefunctions across $\lambda$.
• Figure 3: Energy gap $\Delta$ versus $\lambda$ for 1D (a) and 2D (b) systems. The scaling $\Delta \sim L^{-z}$ at the critical point is used to extract the dynamical exponent $z$, while the insets display the finite-size data collapse used to determine the correlation-length exponent $\nu$. The extracted critical exponents are listed in Table \ref{['tab:exponents']}. Panels (c) and (d) show the corresponding ground state fidelity susceptibility for 1D and 2D systems, with the scaling collapse illustrated in the insets. In one dimension (c), the exact solution obtained via Jordan–Wigner transformation is shown as solid lines, demonstrating excellent agreement with the NQS results.
• Figure 4: The sublayer internal connectivity of the residual RBM NQS architecture used in the adiabatic transport simulations. This diagram shows the NQS amplitude factor of the full wavefunction in Eq. \ref{['eq:init']}.
• Figure S1: The dampening (learning rate) schedule, showing the cosine decay in Eq. \ref{['eq:lr-schedule']} (a) and the comparison between the Moore-Penrose weighting of inverse singular values and the soft pseudoinverse used in our simulations (b).