Abstract We prove that the energy gap of the model proposed by Zhang et al (2016 arXiv:1606.07795) is exponentially small in the square of the system size. In Movassagh and Shor (2016 Proc. Natl Acad. Sci. USA ) a class of exactly solvable quantum spin chain models was proposed that have integer spins ( s ), with a nearest neighbors Hamiltonian, and a unique ground state. The ground state can be seen as a uniform superposition of all s -colored Motzkin walks. The half-chain entanglement entropy provably violates the area law by a square root factor in the system’s size (∼ n ) for s > 1. For s = 1, the violation is logarithmic (Bravyi et al 2012 Phys. Rev. Lett . 109 207202). Moreover in Movassagh and Shor (2016 Proc. Natl Acad. Sci. USA ) it was proved that the gap vanishes polynomially and is O ( n − c ) with c ⩾ 2 . Recently, a deformation of Movassagh and Shor (2016 Proc. Natl Acad. Sci. USA ), which we call ‘weighted Motzkin quantum spin chain’ was proposed Zhang et al (2016 arXiv:1606.07795). This model has a unique ground state that is a superposition of the s -colored Motzkin walks weighted by t a r e a { M o t z k i n w a l k } with t > 1. The most surprising feature of this model is that it violates the area law by a factor of n . Here we prove that the gap of this model is upper bounded by 8 n s t − n 2 / 3 for t > 1 and s > 1.
Abstract We prove that the energy gap of the model proposed by Zhang et al (2016 arXiv:1606.07795) is exponentially small in the square of the system size. In Movassagh and Shor (2016 Proc. Natl Acad. Sci. USA ) a class of exactly solvable quantum spin chain models was proposed that have integer spins ( s ), with a nearest neighbors Hamiltonian, and a unique ground state. The ground state can be seen as a uniform superposition of all s -colored Motzkin walks. The half-chain entanglement entropy provably violates the area law by a square root factor in the system’s size (∼ n ) for s > 1. For s = 1, the violation is logarithmic (Bravyi et al 2012 Phys. Rev. Lett . 109 207202). Moreover in Movassagh and Shor (2016 Proc. Natl Acad. Sci. USA ) it was proved that the gap vanishes polynomially and is O ( n − c ) with c ⩾ 2 . Recently, a deformation of Movassagh and Shor (2016 Proc. Natl Acad. Sci. USA ), which we call ‘weighted Motzkin quantum spin chain’ was proposed Zhang et al (2016 arXiv:1606.07795). This model has a unique ground state that is a superposition of the s -colored Motzkin walks weighted by t a r e a { M o t z k i n w a l k } with t > 1. The most surprising feature of this model is that it violates the area law by a factor of n . Here we prove that the gap of this model is upper bounded by 8 n s t − n 2 / 3 for t > 1 and s > 1.
Abstract We introduce a multi-parameter deformation of the Fredkin spin 1 / 2 chain whose ground state is a weighted superposition of Dyck paths, depending on a set of parameters t i along the chain. The parameters are introduced in such a way to maintain a frustration-free system while allowing the exploration of a range of possible phases. In the case where the parameters are uniform, and a color degree of freedom is added, we establish a phase diagram with a transition between an area law and a volume law. The volume entropy obtained for half a chain is n log s where n is the half-chain length and s is the number of colors. Next, we prove an upper bound on the spectral gap of the t > 1 , s > 1 phase, scaling as Δ = O ( ( 4 s ) n t − n 2 / 2 ) , similar to a recent a result about the deformed Motzkin model (Levine and Movassagh 2017 J. Phys. A: Math. Theor . 50 255302), albeit derived in a different way. Finally, using an additional variational argument we prove an exponential lower bound on the gap of the model for t > 1 , s = 1 , which provides an example of a system with bounded entanglement entropy and a vanishing spectral gap.
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.
Abstract 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 variations. Here we consider a Motzkin walk with a different Hilbert space on each step of the walk spanned by the elements of a symmetric inverse semigroup with the direction of each step governed by its algebraic structure. This change alters the number of paths allowed in the Motzkin walk and introduces a ground state degeneracy that is sensitive to boundary perturbations. We study the frustration-free spin chains based on three symmetric inverse semigroups, S 1 3 , S 2 3 and S 1 2 . The system based on S 1 3 and S 2 3 provides examples of quantum phase transitions in one dimension, with the former exhibiting a transition between the area law and a logarithmic violation of the area law and the latter providing an example of transition from logarithmic scaling to a square root scaling in the system size, mimicking a colored S 1 3 system. The system with S 1 2 is much simpler and produces states that continue to obey the area law.
The spin-one Motzkin chain is gapped for any area weight t<1
We consider the spin-one Motzkin chain with area weight t>0. We resolve three questions from the literature about this model. We prove (i) existence of a uniform spectral gap for all t<1 as conjectured by Zhang--Ahmadein--Klich \cite{zhang2017novel} (ii) an explicit formula for the long-distance limit of the string order parameter, which implies it is non-vanishing at small t, confirming a conjecture by Barbiero et al. \cite{barbiero2017haldane}, and (iii) that gaplessness for t>1 is robust and extends to hard boundary conditions, answering a question of Zhang--Klich \cite{zhang2017entropy}. These conclusions rest on an effective approximate description of the ground states of finite open Motzkin chains in terms of height-controlled imbalanced Motzkin walks.
Abstract 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 interaction of three nearest neighbors. The Hamiltonian is frustration-free and its ground state can be described analytically as a weighted superposition of Dyck paths that depends on a deformation parameter t . In the purely spin 1/2 case, whenever t ≠ 1 , the entanglement entropy obeys an area law: it is bounded from above by a constant, when the size of the block n increases. When a local color degree of freedom is introduced and t > 1 the entanglement entropy increases linearly, while for t < 1 an area law is obeyed. The half-chain entanglement entropy is tightly bounded by n log s where 2 n is the length of the chain, and s is the number of colors. Our chain fosters a new example for a significant boost to entropy and for the existence of the associated critical rainbow phase where the entanglement entropy scales with volume that discovered by Zhang, Ahmadain and Klich ( Proc. Natl Acad. Sci 2017).