Boundary emptiness formation probabilities in the six-vertex model at $\mathbf{\Delta }=-\frac{\mathbf{1}}{\mathbf{2}}$

Abstract

We define a new family of overlaps C N,m for the XXZ Hamiltonian on a periodic chain of length N . These are equal to the linear sums of the groundstate components, in the canonical basis, wherein m consecutive spins are fixed to the state ↑. We define the boundary emptiness formation probabilities as the ratios C N,m /C N,0 of these overlaps. In the associated six-vertex model, they correspond to correlation functions on a semi-infinite cylinder of perimeter N . At the combinatorial point ∆ = − 1 2 , we obtain closed-form expressions in terms of simple products of ratios of integers.

Sum rules for the supersymmetric eight-vertex model

“…We note that the lowest-order term also follows from a rigorous result on the six-vertex model [35].…”

“…Fourth, the results of this article suggest that an exact finite-size computation of the emptiness or boundary emptiness formation probability for the supersymmetric eight-vertex model could be possible. In the trigonometric limit, these correlation functions are known [35,37]. Finally, we mention that there is a conjecture for a simple eigenvalue of the transfer matrix of the inhomogeneous supersymmetric eight-vertex model with open boundary conditions [38].…”

On the logarithmic bipartite fidelity of the open XXZ spin chain at $Δ=-1/2$

“…Some of them display remarkable relations with the enumerative combinatorics of alternating sign matrices and plane partitions [19]. Similar relations have been thoroughly investigated for the XXZ spin chain at ∆ = − 1 2 with (twisted) periodic boundary conditions [20][21][22][23][24][25][26][27].…”

Quantum correlations in spin chains

“…where the expectation value is computed in a system consisting of N spins, and the average is taken, for example, in a ground state of some Hamiltonian. Formation probabilities in spin-chains have been studied for some time [38][39][40][41][69][70][71][72][73][74] with the focus on behaviour of E m in the thermodynamically large system. Here, instead we focus on finite systems and, even more importantly, we build upon the hierarchy introduced in the previous section to develop detailed "tomography" of the entanglement and non-locality in the ground states of experimentally relevant Hamiltonians.…”