Lower Bounds on Spectral Gaps of Parent Hamiltonians via Tensor Networks

Technical Summary: Lower Bounds on Spectral Gaps of Parent Hamiltonians via Tensor Networks

Problem Statement
The existence of a spectral gap—a uniform lower bound on the energy difference between the ground state manifold and the first excited state, independent of system size—is a fundamental property in quantum many-body physics. It dictates the exponential decay of correlations, the stability of topological order, and the feasibility of quasi-adiabatic evolution. However, rigorously proving the existence of a gap and, more critically, obtaining explicit quantitative lower bounds for interacting Hamiltonians is notoriously difficult. While methods exist, such as finite-size criteria (Knabe, Gosset-Mozgunov) and the martingale method (Fannes, Nachtergaele, Werner - FNW), they often yield weak bounds or fail to provide bounds in certain regimes. Specifically, the FNW method, while proving the existence of gaps for Matrix Product State (MPS) parent Hamiltonians, relies on estimating the overlap of local ground spaces in a way that yields suboptimal quantitative bounds.

Methodology
The authors revisit and refine the martingale method to create a tool for computing strong, rigorous lower bounds on spectral gaps. The approach consists of two primary steps:

1. Hamiltonian Blocking and Reshuffling:
   The original translation-invariant Hamiltonian $H_N$ is transformed into a "blocked" Hamiltonian $\tilde{H}_M$ by grouping sites into super-sites of size $p$ and $q$. The terms of the original Hamiltonian are redistributed with specific weights ($\lambda_\omega$) to form new local terms $\tilde{h}_{\tilde{\imath}}$ that act on overlapping blocks of size $2p+q$.

   • The authors introduce refined blocking schemes (Lemma 1 and Lemma 2) that allow for alternating block sizes and optimized weights.
   • This step relates the gap of the original system to the gap of the blocked system, $\Delta(H_N) \geq \Delta(\tilde{H}_M)$, while ensuring the blocked system remains frustration-free and its ground space is parametrized by the same MPS.

2. Local Gap Estimation via Tensor Networks:
   The gap of the blocked Hamiltonian is bounded using a local condition involving the relative alignment of the ground spaces of adjacent terms. This condition depends on a parameter $\gamma_{p,q}$, defined by the inequality:
   $$QR + RQ + \gamma_{p,q}(Q + R) \geq 0$$
   where $Q$ and $R$ are projectors onto the orthogonal complements of the ground spaces of adjacent blocks.

   • Key Innovation: The authors develop a technique (Theorem 4) to compute the optimal $\gamma_{p,q}$ exactly and efficiently. Instead of diagonalizing an operator on a Hilbert space that grows exponentially with the block size, they map the problem to finding the second largest eigenvalue of a matrix $\Xi$ of fixed dimension $D^4$ (where $D$ is the MPS bond dimension).
   • The matrix $\Xi$ is constructed from the transfer matrices of the MPS tensor, specifically involving terms like $T_p$, $T_{p+q}$, and the pseudoinverse $T_{2p+q}^{-1}$.
   • The final lower bound is given by $\Delta(H_N) \geq (1 - 2\gamma_{p,q})\Delta(\tilde{h}_{\tilde{\imath}})$.

Key Contributions

• Exact and Efficient Computation of Overlaps: The paper provides a method to compute the key quantity in the martingale method (the overlap of local ground spaces) exactly. This is achieved by reducing the problem to an eigenvalue problem on a space of fixed dimension $D^4$, scaling linearly with the block size and polynomially with the bond dimension. This contrasts with previous approaches that either provided loose bounds or required exponential resources.
• Refined Blocking Schemes: The authors generalize the blocking procedure to include alternating block sizes ($p$ and $q$) and optimized weights, offering greater control to tighten the bounds.
• Unified Proof of Gappedness: The method yields a significantly simplified and unified proof that parent Hamiltonians of any "well-behaved" (normal or block-normal) MPS are gapped. Because the relevant operator $\Xi$ lives in a fixed finite-dimensional space, the convergence of the bound to a non-zero value as block sizes increase can be analyzed rigorously.
• Applicability to Degenerate Ground States: The framework extends naturally to block-normal MPS, handling Hamiltonians with degenerate ground spaces (e.g., topological phases or symmetry-breaking phases) without modification.

Results
The method was benchmarked against the original FNW method, Nachtergaele's martingale method, and finite-size criteria (Knabe, Gosset-Mozgunov) on several models:

• AKLT Model: The method produced significantly tighter lower bounds than the FNW method for both Open (OBC) and Periodic (PBC) boundary conditions. For PBC, a bound of $\Delta \geq 0.2827$ was obtained, and for OBC, $\Delta \geq 0.2497$. While finite-size criteria outperformed the method at very large block sizes for AKLT, the tensor network approach provided superior bounds at smaller computational costs and succeeded in regimes where other methods failed to return any bound.
• $Z_3$-XY Model: The method provided quantitative bounds in parameter regimes where the Knabe bound failed to detect a gap (specifically for small deformation parameters $t$). It outperformed both the standard martingale method and the Knabe bound across a wide range of parameters.
• SU(3) Valence Bond Solids: The method was applied to SU(3)-symmetric models, proving they are gapped and providing explicit lower bounds.

Significance
The paper claims that its numerically motivated approach achieves two main goals:

1. Quantitative Improvement: It provides substantially stronger lower bounds on spectral gaps compared to existing techniques for MPS parent Hamiltonians, often outperforming finite-size criteria in specific regimes.
2. Conceptual Simplification: It offers a streamlined, rigorous proof of the gapped nature of parent Hamiltonians for normal and block-normal MPS. By reducing the gap analysis to a fixed-dimension eigenvalue problem, the proof avoids the complexities of previous arguments and applies uniformly to both unique and degenerate ground states.

The authors emphasize that the method is self-contained and applicable to general frustration-free Hamiltonians where the ground space can be parametrized by an MPS, making it a versatile tool for analyzing spectral gaps in quantum many-body systems.