Deforming the Fredkin spin chain away from its frustration-free point

Abstract

The Fredkin model describes a spin-half chain segment subject to three-body, correlated-exchange interactions and twisted boundary conditions. The model is frustration-free, and its ground state wave function is known exactly. Its low-energy physics is that of a strong xy ferromagnet with gapless excitations and an unusually large dynamical exponent. We study a generalized spin chain model that includes the Fredkin model as a special tuning point and otherwise interpolates between the conventional ferromagnetic and antiferromagnetic quantum Heisenberg models. We solve for the low-lying states, using exact diagonalization and density-matrix renormalization group calculations, in order to track the properties of the system as it is tuned away from the Fredkin point; we also present exact analytical results that hold right at the Fredkin point. We identify a zero-temperature phase diagram with multiple transitions and unexpected ordered phases. The Fredkin ground state turns out to be particularly brittle, unstable to even infinitesimal antiferromagnetic frustration. We remark on the existence of an "anti-Fredkin" point at which all the contributing spin configurations have a spin structure exactly opposite to those in the Fredkin ground state. arXiv:1805.00532v3 [cond-mat.str-el]

Slow dynamics of the Fredkin spin chain

“…The dynamical exponent has been estimated, both analytically and with DMRG [5][6][7][8][9], and the most recent values are in the range 3.0 𝑧 3.2 [9]. Chen and coworkers have offered arguments for the Fredkin model's slow dynamics based on an analogy with classical spin chains [7].…”

“…Since the Fredkin spin chain cannot be mapped to a conformal field theory, there is no straightforward analytical estimate for 𝑧, and what we have are numerical estimates determined by level spectroscopy [5][6][7][8][9], i.e., from the scaling of the lowest-lying energy spacing as a function of system size. In a system of interacting particles, the correlation length 𝜉 defines a length scale at which differences between microscopic and macroscopic properties are distinguishable.…”

“…Third, it is subject to the limitations of the numerical technique used to obtain the energy spectrum. The Fredkin spin chain, for instance, sits in model-parameter-space at a special tuning point with a locally maximum entanglement entropy [7,9,[11][12][13][14], where the density matrix renormalization group (DMRG) algorithm has the greatest difficulty converging.…”

Tunable quantum spin chain with three-body interactions

“…Frustration-free quantum spin chains with local three-body interactions are relatively recent discoveries, but they have generated great excitement and have already been studied extensively [35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54]. The ground states of the original Motzkin and Fredkin spin chain models are known exactly.…”

“…This is possible because the interactions are in delicate balance. Various models [41,43,47,48,55] have been proposed that continuously deform the Fredkin model away from this specially tuned point. In Refs.…”