Fredkin Spin Chain

Abstract

We introduce a new model of interacting spin 1/2. It describes interaction of three nearest neighbors. The Hamiltonian can be expressed in terms of Fredkin gates. The Fredkin gate (also known as the CSWAP gate) is a computational circuit suitable for reversible computing. Our construction generalizes the work [4]. Our model can be solved by means of Catalan combinatorics in the form of random walks on upper half of a square lattice [Dyck walks]. Each Dyck path can be mapped on a wave function of spins. The ground state is an equally weighted superposition of Dyck walks [instead of Motzkin walks]. We can also express it as a matrix product state. We further construct the model of interacting spins 3/2 and greater half-integer spins. The models with higher spins require coloring of Dyck walks. We construct SU(k) symmetric model [here k is the number of colors]. The leading term of the entanglement entropy is then proportional to the square root of the length of the lattice [like in Shor-Movassagh model]. The gap closes as a high power of the length of the lattice.

Sublogarithmic behaviour of the entanglement entropy in fermionic chains

“…Notice that the Fourier coefficients of G ν , 1/(l(log A|l|) ν ), are similar to the leading term in the expansion (20) for the Fourier coefficients s (ν) k of the function log f (ν) introduced in (19). Hence the behaviour of G ν (θ) in a neighbourhood of θ = 0 is equal to that violates the well-behaviour condition (11) of the Strong Szegő Theorem when 0 < ν ≤ 1/2.…”

“…One easily sees that the conjecture predicts a sublogarithmic growth of log D N [f (ν) ] with the dimension N . In fact, if we consider the asymptotic behaviour (20)…”

The Ground-State Vector of the XY Heisenberg Chain and the Gauss Decomposition

“…For instance, random lattice walks and boxed plane partitions, as subjects of enumerative combinatorics [6], are related to the correlation functions of the XX model [7][8][9][10][11]. Various spin lattice models [12], including the XY Heisenberg chain model, as well as its isotropic limit, the XX model, provide a base for such actively developing subjects in the theory of quantum information [13] as random lattice walks [14] and entanglement entropy [15].…”

Novel Quantum Phases in Low Dimensions

“…Following the method in the previous chapter, we introduce a parameter t that deforms the Fredkin Hamiltonian of [24,104] while remaining frustration free. The Hamiltonian is given by:…”

“…When t = 1, the entanglement entropy of the ground state scales as log n for s = 1 and as √ n for s > 1 [24,104]. The reason our deformation with the extra parameter can further increase the scaling of entropy is because when a spin is moved around its neighboring ↑↓ pair, it is separated from its own partner paired in the same color, which is the first unpaired down spin to its right (or up spin to its left).…”