Upper bounds to the spin gap for quantum systems from coupled chains to square lattice

Wolfgang L. Spitzer, Shannon L. Starr

Colby Austin Brown

The ferromagnetic Heisenberg spin-1⁄2 quantum spin chain describes a quantum spin system of spin-1⁄2 particles on a one-dimensional lattice. In the thermodynamic limit, the spectral gap of this system closes at rate of O(n−2). We demonstrate this by proving an upper bound on the spectral gap using the variational principle and a lower bound using the martingale method.

David Gosset, Evgeny Mozgunov

We improve Knabe’s spectral gap bound for frustration-free translation-invariant local Hamiltonians in 1D. The bound is based on a relationship between global and local gaps. The global gap is the spectral gap of a size-m chain with periodic boundary conditions, while the local gap is that of a subchain of size n < m with open boundary conditions. Knabe proved that if the local gap is larger than the threshold value 1/(n − 1) for some n > 2, then the global gap is lower bounded by a positive constant in the thermodynamic limit m → ∞. Here we improve the threshold to 6n(n+1), which is better (smaller) for all n > 3 and which is asymptotically optimal. As a corollary we establish a surprising fact about 1D translation-invariant frustration-free systems that are gapless in the thermodynamic limit: for any such system the spectral gap of a size-n chain with open boundary conditions is upper bounded as O(n−2). This contrasts with gapless frustrated systems where the gap can be Θ(n−1). It also limits the extent to which the area law is violated in these frustration-free systems, since it implies that the half-chain entanglement entropy is O(1/ϵ) as a function of spectral gap ϵ. We extend our results to frustration-free systems on a 2D square lattice.

P. Lemmens, M. Fischer, M. Grove +5 more

Wojciech Dybalski

Lionel Levine, Ramis Movassagh

Abstract We prove that the energy gap of the model proposed by Zhang et al (2016 arXiv:1606.07795) is exponentially small in the square of the system size. In Movassagh and Shor (2016 Proc. Natl Acad. Sci. USA ) a class of exactly solvable quantum spin chain models was proposed that have integer spins ( s ), with a nearest neighbors Hamiltonian, and a unique ground state. The ground state can be seen as a uniform superposition of all s -colored Motzkin walks. The half-chain entanglement entropy provably violates the area law by a square root factor in the system’s size (∼ n ) for s > 1. For s = 1, the violation is logarithmic (Bravyi et al 2012 Phys. Rev. Lett . 109 207202). Moreover in Movassagh and Shor (2016 Proc. Natl Acad. Sci. USA ) it was proved that the gap vanishes polynomially and is O ( n − c ) with c ⩾ 2 . Recently, a deformation of Movassagh and Shor (2016 Proc. Natl Acad. Sci. USA ), which we call ‘weighted Motzkin quantum spin chain’ was proposed Zhang et al (2016 arXiv:1606.07795). This model has a unique ground state that is a superposition of the s -colored Motzkin walks weighted by t a r e a { M o t z k i n w a l k } with t > 1. The most surprising feature of this model is that it violates the area law by a factor of n . Here we prove that the gap of this model is upper bounded by 8 n s t − n 2 / 3 for t > 1 and s > 1.