Fredkin Spin Chain

@article{Salberger2016FredkinSC,
  title={Fredkin Spin Chain},
  author={Olof Salberger and Vladimir E. Korepin},
  journal={arXiv: Quantum Physics},
  year={2016}
}
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 of Ramis Movassagh and Peter Shor. Our model can be solved by means of Catalan combinatorics in the form of random walks on the upper half of a square lattice [Dyck walks]. Each Dyck path… 

Figures from this paper

Can a spin chain relate combinatorics to number theory?
The Motzkin spin chain is a spin-1 frustration-free model introduced by Shor & Movassagh. The ground state is constructed by mapping of random walks on upper half of the square lattice to spin
Deformed Fredkin Spin Chain with Extensive Entanglement
We introduce a new spin chain which is a deformation of the Fredkin spin chain and has a phase transition between bounded and extensive entanglement entropy scaling. In this chain, spins have a local
Finite-size gap, magnetization, and entanglement of deformed Fredkin spin chain
We investigate ground- and excited-state properties of the deformed Fredkin spin chain proposed by Salberger, Zhang, Klich, Korepin, and the authors. This model is a one-parameter deformation of the
Non-Interacting Motzkin Chain - Periodic Boundary Conditions
The Motzkin spin chain is a spin-1 model introduced in \cite{shor} as an example of a system exhibiting a high degree of quantum fluctuations whose ground state can be mapped to Motzkin paths that
Hilbert space fragmentation and exact scars of generalized Fredkin spin chains
In this work, based on the Fredkin spin chain, we introduce a family of spin-1/2 many-body Hamiltonians with a three-site interaction featuring a fragmented Hilbert space with coexisting quantum
Area law violations and quantum phase transitions in modified Motzkin walk spin chains
Area law violations for entanglement entropy in the form of a square root have recently been studied for one-dimensional frustration-free quantum systems based on the Motzkin walks and their
Quantum phase transitions and localization in semigroup Fredkin spin chain
TLDR
An extended quantum spin chain model is constructed by introducing new degrees of freedom to the Fredkin spin chain, and it is shown that excited states due to disconnections with respect to the arrow indices are localized without disorder.
Excitations and ergodicity of critical quantum spin chains from non-equilibrium classical dynamics
We study a quantum spin-1/2 chain that is dual to the canonical problem of non-equilibrium Kawasaki dynamics of a classical Ising chain coupled to a thermal bath. The Hamiltonian is obtained for the
The pair-flip model: a very entangled translationally invariant spin chain
Investigating translationally invariant qudit spin chains with a low local dimension, we ask what is the best possible tradeoff between the scaling of the entanglement entropy of a large block and
Shor–Movassagh chain leads to unusual integrable model
The ground state of the Shor–Movassagh chain can be analytically described by the Motzkin paths. There is no analytical description of the excited states. The model is not solvable. We prove the
...
...

References

SHOWING 1-8 OF 8 REFERENCES
Criticality without frustration for quantum spin-1 chains.
TLDR
This work proposes the first example of a FF translation-invariant spin-1 chain that has a unique highly entangled ground state and exhibits some signatures of a critical behavior.
Power law violation of the area law in quantum spin chains
Entanglement between two quantum systems is a non-classical correlation between them. Entanglement is a feature of quantum mechanics which does not appear classically, and it serves as a resource for
Supercritical entanglement in local systems: Counterexample to the area law for quantum matter
TLDR
This work suggests that simple quantum matter is richer and can provide much more quantum resources than expected, and introduces a class of exactly solvable one-dimensional physical models which can prove violate the area law by a square root, i.e., exponentially more than the logarithm.
Crystallization in Ising quantum magnets
TLDR
The precise control of Rydberg many-body systems is demonstrated and a magnetization staircase is observed as a function of the system size and the emergence of crystalline states with vanishing susceptibility is shown.
Far-from-equilibrium spin transport in Heisenberg quantum magnets.
TLDR
The far-from-equilibrium dynamics in ferromagnetic Heisenberg quantum magnets realized with ultracold atoms in an optical lattice is studied and a profound dependence of the decay rate on the wave vector is found.
Experimental realization of plaquette resonating valence-bond states with ultracold atoms in optical superlattices.
TLDR
Direct experimental evidence is shown of a time-resolved valence-bond quantum resonance with ultracold bosonic atoms in an optical lattice by means of a superlattice structure that creates a three-dimensional array of independent four-site plaquettes, which the author can fully control and manipulate in parallel.
Concentrating partial entanglement by local operations.
TLDR
Any pure or mixed entangled state of two systems can be produced by two classically communicating separated observers, drawing on a supply of singlets as their sole source of entanglement.
Effective quantum spin systems with trapped ions.
TLDR
This work shows that the physical system consisting of trapped ions interacting with lasers may undergo a rich variety of quantum phase transitions, and allows for an analogue quantum simulator of spin systems with trapped ions.