A Hierarchy of Spectral Gap Certificates for Frustration-Free Spin Systems

Abstract

Estimating spectral gaps of quantum many-body Hamiltonians is a highly challenging computational task, even under assumptions of locality and translation-invariance. Yet, the quest for rigorous gap certificates is motivated by their broad applicability, ranging from many-body physics to quantum computing and classical sampling techniques. Here we present a general method for obtaining lower bounds on the spectral gap of frustration-free quantum Hamiltonians in the thermodynamic limit. We formulate the gap certification problem as a hierarchy of optimization problems (semidefinite programs) in which the certificate -- a proof of a lower bound on the gap -- is improved with increasing levels. Our approach encompasses existing finite-size methods, such as Knabe's bound and its subsequent improvements, as those appear as particular possible solutions in our optimization, which is thus guaranteed to either match or surpass them. We demonstrate the power of the method on one-dimensional spin-chain models where we observe an improvement by several orders of magnitude over existing finite size criteria in both the accuracy of the lower bound on the gap, as well as the range of parameters in which a gap is detected.

Figures (4)

• Figure 1: Illustration of the two relaxation steps. (a) Geometrical locality condition: The terms discarded from $H^2$ in \ref{['eq:H_2_n']} are positive semi-definite since $h_i$ and $h_j$ have disjoint support. (b) Freedom in choosing a local generating term: Two local terms $g_n$ and $q_n$ generating the same global operator can differ as in \ref{['eq:Y_condition']} as this difference cancels telescopically. Inside the $n=5$-site support of each local term (orange rectangle), the operators $Y_{n-1}$ are depicted as rectangles of different colors and the identity operator is depicted by a gray circle. When summed over all $i$, the terms colored identically cancel out as indicated by the crosses. The purple and gray terms are canceled by the preceding and the following terms in the sum respectively.
• Figure 2: The gap of the $S=1$ AKLT chain. Lower bounds on the energy gap above the ground state of the infinite AKLT chain obtained with different methods are plotted as a function of the size $n$ of the finite system: The Knabe bound \ref{['eq:knabe_bound']}, the Gosset--Mozgunov bound \ref{['eq:gosset_bound']}, and our LTI SDP method \ref{['eq:primal_TI_Y']}. In addition, gaps of finite systems with periodic boundary conditions (PBC) obtained with exact diagonalization (ED) are plotted for rings of sizes $n=8,12$, and $16$ by dotted, dashed, and solid horizontal black lines. Note that those lines lie essentially on top of each other in the plot (the value for $n=12$ and $n=16$ differs in the 6th digit). The LTI SDP significantly outperforms the other two methods and gives a lower bound of $0.34976$ for the AKLT gap.
• Figure 3: Gaps of the $\mathbb{Z}_3$ family of deformed Potts clock models. For a family of spin chain models parameterized by two parameters $r,s$ (\ref{['eq:deformed_potts']}) lower bounds on the gap were computed at different values of the parameters using different methods: the Knabe and Gosset--Mozgunov finite size criteria (\ref{['eq:knabe_bound']} and \ref{['eq:gosset_bound']} respectively), and our LTI SDP method \ref{['eq:primal_TI_Y']}. The shaded regions indicate the parameter values for which the finite-size methods detected a gap using exact diagonalization results for systems up to size $n=10$ (green region with dashed border for Knabe and blue region with dash-dotted border for Gosset-Mozgunov). The different markers indicate parameter values for which the LTI SDP detected a gap, with different markers corresponding to different values of $n$ (see legend). For each coordinate $(r,s)$, we only plot the marker corresponding to the lowest value $n$ for which the LTI SDP detected a gap. Darker marker colors indicate larger gaps (see color bar). Each marker corresponds to the larger of the two gaps resulting from using either the original Hamiltonian or the modified Hamiltonian---in which the interaction term is a projection---as the input for the LTI SDP.
• Figure 4: Gaps of the 1D Glauber Hamiltonian family. For a family of spin chain models parameterized by two parameters $\gamma,\delta$ (\ref{['eq:glauber_model']}) lower bounds on the gap were computed at different values of the parameters using different methods: the Knabe and Gosset--Mozgunov finite size criteria (\ref{['eq:knabe_bound']} and \ref{['eq:gosset_bound']} respectively) computed for the coarse-grained Hamiltonian $\tilde{h}$ (\ref{['eq:glauber_blocked_hamiltonian']}), and our LTI SDP method \ref{['eq:primal_TI_Y']} computed for the original Hamiltonian. The green shaded region (dashed border) marks the region in which the Knabe method detected a gap ($\gamma\lesssim 0.92$), the blue shaded region (dash-dotted border) marks the region where the Gosset--Mozgunov bound did ($\gamma\lesssim 0.96$). The gaps detected by our LTI method are plotted as individual data points with marker colors corresponding to the size of the gap (see color bar).