Abstract The Motzkin and Fredkin quantum spin chains are described by frustration-free Hamiltonians recently introduced and studied because of their anomalous behaviors in the correlation functions and in the entanglement properties. In this paper we analyze their quantum dynamical properties, focusing in particular on the time evolution of the excitations driven by a quantum quench, looking at the correlations functions of spin operators defined along different directions, and discussing the results in relation with the cluster decomposition property. We show that, for such models, one can have absence of cone-like structure in the propagation of the excitations after a quantum quench, together with a violation of the cluster decomposition property for the ground state characterized by only a logarithmic violation of the area law for the entanglement entropy.
Abstract The Motzkin and Fredkin quantum spin chains are described by frustration-free Hamiltonians recently introduced and studied because of their anomalous behaviors in the correlation functions and in the entanglement properties. In this paper we analyze their quantum dynamical properties, focusing in particular on the time evolution of the excitations driven by a quantum quench, looking at the correlations functions of spin operators defined along different directions, and discussing the results in relation with the cluster decomposition property. We show that, for such models, one can have absence of cone-like structure in the propagation of the excitations after a quantum quench, together with a violation of the cluster decomposition property for the ground state characterized by only a logarithmic violation of the area law for the entanglement entropy.
We compute the ground-state correlation functions of an exactly solvable chain of integer spins, recently introduced in [R. Movassagh and P. W. Shor, arXiv:1408.1657], whose ground state can be expressed in terms of a uniform superposition of all colored Motzkin paths. Our analytical results show that for spin $s\ensuremath{\ge}2$ there is a violation of the cluster decomposition property. This has to be contrasted with s=1, where the cluster property holds. Correspondingly, for s=1 one gets a light-cone profile in the propagation of excitations after a local quench, while the cone is absent for s=2, as shown by time dependent density-matrix renormalization group. Moreover, we introduce an original solvable model of half-integer spins, which we refer to as Fredkin spin chain, whose ground state can be expressed in terms of superposition of all Dyck paths. For this model we exactly calculate the magnetization and correlation functions, finding that for s=1/2, a conelike propagation occurs, while for higher spins, $s\ensuremath{\ge}3/2$, the colors prevent any cone formation and clustering is violated, together with square root deviation from the area law for the entanglement entropy.
The Motzkin and Fredkin chains are frustration-free spin models with exactly solvable ground states whose q q -deformations realize an exotic quantum phase transition from a disordered phase to an ordered one under domain-wall boundary conditions. Previous work has mainly focused on their entanglement scaling and spectral gaps, particularly in color-enriched variants. Here we systematically characterize the critical behavior of this transition with three main advances. First, we interpret bulk magnetic order as arising from correlations with boundary spins subject to an effective edge field, directly relating the ordered phase to the domain-wall boundary condition. Second, using the transfer matrix (TM) constructed from the exact matrix product state (MPS) representation, together with a continuum and RG analysis, we derive the algebraic decay of spin correlations and obtain the critical exponent \eta = \frac{3}{2} η = 3 2 . We further generalize the TM method to translationally invariant gapless states with generic power-law correlations, via a zero-dimensional Hamiltonian dual to the one-dimensional TM. Third, the TM spectrum reveals a duality between the ordered and disordered phases which, combined with scale invariance at criticality and the scaling dimension of the spin operator, yields \nu_± = \frac{2}{3} ν ± = 2 3 from an RG analysis of the q q -deformed ground states. Both exponents are confirmed numerically by MPS simulations and by direct diagonalization of the TM.
We derive some entanglement properties of the ground states for two classes of quantum spin chains described by the Fredkin model, for half-integer spins, and the Motzkin model, for integer ones. Since the ground states of the two models are known analytically, we can calculate exactly the entanglement entropy, the negativity and the quantum mutual information. We show, in particular, that these systems exhibit long-distance entanglement, namely two disjoint regions of the chains remain entangled even when their separation is sent to infinity, i.e. these systems are not affected by decoherence. This strongly entangled behavior, Finally, we show that this behavior involves disjoint segments located both at the edges and in the bulk of the chains.
Entanglement is one of the most intriguing features of quantum theory and a main resource in quantum information science. Ground states of quantum many-body systems with local interactions typically obey an “area law” which means that the entanglement entropy is proportional to the boundary length. It is exceptional when the system is gapless, and the area law had been believed to be violated by at most a logarithm over two decades. Recent discovery of Motzkin and Fredkin spin chain models is striking, since these models provide significant violation of the entanglement beyond the belief, growing as a square root of the volume in spite of local interactions. In this paper, we first analytically compute the Rényi entropy of the Motzkin and Fredkin models by careful treatment of asymptotic analysis. The Rényi entropy is an important quantity, since the whole spectrum of an entangled subsystem is reconstructed once the Rényi entropy is known as a function of its parameter. We find nonanalytic behavior of the Rényi entropy with respect to the parameter, which is a novel phase transition never seen in any other spin chain studied so far. Interestingly, similar behavior is seen in the Rényi entropy of Rokhsar–Kivelson states in two dimensions.
Motzkin spin-chains, which include 'colorless' (integer spin s=1) and 'colorful' (s≥2) variants, are one-dimensional (1D) local integer spin models notable for their lack of a conformal field theory (CFT) description of their low-energy physics, despite being gapless. The colorful variants are particularly unusual, as they exhibit power-law violation of the area-law of entanglement entropy (as n in system size n), rather than a logarithmic violation as seen in a CFT. In this work, we analytically discover several unique properties of these models, potentially suggesting a new universality class for their low-energy physics. We identify a complex structure of symmetries and unexpected scaling behavior in spin-spin correlations, which deviate from known 1D universality classes. Specifically, the s=1 chain exhibits U(1) spontaneous symmetry breaking and ferromagnetic order. Meanwhile, the s≥2 chains do not appear to spontaneously break any symmetries, but display quasi-long-range algebraic order with power-law decaying correlations, inconsistent with standard Berezinskii-Kosterlitz-Thouless (BKT) critical exponents. We also derive exact asymptotic scaling expressions for entanglement measures in both colorless and colorful chains, generalizing previous results of Movassagh [J. Math Phys. (2017)], while providing benchmarks for potential quantum simulation experiments. The combination of hardness of classically simulating such systems along with the analytical tractability of their ground state properties position Motzkin spin chains as intriguing candidates for exploring quantum computational advantage in simulating many-body physics.
We present exact results on the exactly solvable spin chain of Bravyi et al. [Phys. Rev. Lett. 109, 207202 (2012)]. This model is a spin one chain and has a Hamiltonian that is local and translationally invariant in the bulk. It has a unique (frustration free) ground state with an energy gap that is polynomially small in the system’s size (2n). The half-chain entanglement entropy of the ground state is 12logn+const.[Bravyi et al., Phys. Rev. Lett. 109, 207202 (2012)]. Here we first write the Hamiltonian in the standard spin-basis representation. We prove that at zero temperature, the magnetization is along the z-direction, i.e., ⟨sx⟩=⟨sy⟩=0 (everywhere on the chain). We then analytically calculate ⟨sz⟩ and the two-point correlation functions of sz. By analytically diagonalizing the reduced density matrices, we calculate the Schmidt rank, von Neumann, and Rényi entanglement entropies for the following: 1. Any partition of the chain into two pieces (not necessarily in the middle) and 2. L consecutive spins centered in the middle. Further, we identify entanglement Hamiltonians (Eqs. (49) and (59)). We prove a small lemma (Lemma (1)) on the combinatorics of lattice paths using the reflection principle to relate and calculate the Motzkin walk “height” to spin expected values. We also calculate the, closely related (scaled), correlation functions of Brownian excursions. The known features of this model are summarized in a table in Sec. I.