Establishing the (non)existence of a spectral gap above the ground state in the thermodynamic limit is one of the fundamental steps for characterizing the topological phase of a quantum lattice model. This is particularly challenging when a model is expected to have low-lying edge excitations, but nevertheless a positive bulk gap. We review the bulk gap strategy introduced in [S. Warzel and A. Young, The spectral gap of a fractional quantum Hall system on a thin torus, J. Math. Phys. 63 (2022) 041901; S. Warzel and A. Young, A bulk spectral gap in the presence of edge states for a truncated pseudopotential, Ann. Henri Poincaré 24 (2023) 133–178], while studying truncated Haldane pseudopotentials. This approach is able to avoid low-lying edge modes by separating the ground states and edge states into different invariant subspaces before applying spectral gap bounding techniques. The approach is stated in a general context, and we reformulate specific spectral gap methods in an invariant subspace context to illustrate the necessary conditions for combining them with the bulk gap strategy. We then review its application to a truncation of the 1/3-filled Haldane pseudopotential in the cylinder geometry.
We study a fractional quantum Hall system with maximal filling ν = 1/3 in the thin torus limit. The corresponding Hamiltonian is a truncated version of Haldane’s pseudopotential, which upon a Jordan–Wigner transformation is equivalent to a one-dimensional quantum spin chain with periodic boundary conditions. Our main result is a lower bound on the spectral gap of this Hamiltonian, which is uniform in the system size and total particle number. The gap is also uniform with respect to small values of the coupling constant in the model. The proof adapts the strategy of individually estimating the gap in invariant subspaces used for the bosonic ν = 1/2 model to the present fermionic case.
Determining spectral gaps in the thermodynamic limit is a central challenge in quantum many-body physics. When a fixed boundary condition is imposed, this problem is undecidable: no algorithm, even with unlimited resources, can decide for every Hamiltonian whether it is gapped. Yet the thermodynamic limit is meant to capture the intrinsic properties of very large systems, which should not depend on how their boundaries are chosen. We therefore consider the bulk spectral gap, which concerns only the intrinsic excitations of the infinite system, and ask what can and cannot be computed about it. First, we give a general algorithm producing a hierarchy of certified upper bounds on the bulk gap, arbitrarily tight at the cost of more computation. Second, we prove that the spectral gap problem remains undecidable in the bulk, already for translationally invariant nearest-neighbour chains of fixed local dimension. Hence no general algorithm produces convergent lower bounds, and no convergent upper-bounding algorithm, ours included, admits a computable convergence rate. The bulk spectral gap is thus certifiable from above but uncomputable from below; more precisely, the bulk-gaplessness promise problem is RE-complete, exactly as hard as the halting problem. The same construction shows that no general algorithm can decide whether a given product state is a ground state, and that convergent bounds on ground-state expectation values admit no computable convergence rate in general. Finally, we obtain the first nontrivial certified upper bounds on the bulk gap of the spin-1/2 kagome Heisenberg antiferromagnet.
Abstract We identify the ground-state of a truncated version of Haldane’s pseudo-potential Hamiltonian in the thin cylinder geometry as being composed of exponentially many fragmented matrix product states. These states are constructed by lattice tilings and their properties are discussed. We also report on a proof of a spectral gap, which implies the incompressibility of the underlying fractional quantum Hall liquid at maximal filling ν = 1/3. Low-energy excitations and an extensive number of many-body scars at positive energy density, but nevertheless low complexity, are also identified using the concept of tilings.
We develop a bootstrap approach to deriving lower bounds on the spectral gap of the first Haldane pseudopotential Hamiltonian on the sphere with
NΦ flux quanta for arbitrary
NΦ, including in the thermodynamic limit
NΦ→∞. This method can in principle be generalized to pseudopotentials describing other model wavefunctions such as the Moore--Read state or Read--Rezayi sequence. Our approach is based on Knabe's observation that if
H is a Hamiltonian with ground state energy zero then the spectral gap is at least
ε>0 if and only if
H2⪰εH. To show this inequality, we split
H2 into two parts, one that can be analytically bounded and another that can be bounded using a finite semidefinite program (SDP). Crucially, the size of this program is determined only by cutoff parameters, and the program depends on
NΦ only parametrically, allowing us to obtain numerically exact (up to solver precision) lower bounds on the gap. In order to turn these numerical bounds into mathematically rigorous results, we introduce an algorithm that can produce nearly-optimal exact interior feasible points for large rational SDPs even if the affine constraints are highly redundant and ill-conditioned. We additionally provide an algorithm that establishes the rank stability of the SDP in the
NΦ→∞ limit, which is needed to certify the bound in the thermodynamic limit. Beyond establishing a Laughlin gap greater than
1/100 in the thermodynamic limit and revealing a potential method for deriving spectral gap lower bounds for other pseudopotentials, our work provides a general method for turning numerical SDP bounds into rigorous mathematical proofs, which may have wide applicability. In particular, this work demonstrates that bootstrap methods can be used to rigorously resolve major open problems in quantum many-body physics.