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arXiv:1702.03581 · Journal of Physics A Mathematical and Theoretical

Entropy, gap and a multi-parameter deformation of the Fredkin spin chain

Zhao Zhang, Israel Klich

Published Aug 16, 2017• 34 citations
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Abstract

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.

DOI
10.1088/1751-8121/aa866e

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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.
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).
Abstract 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 Fredkin spin chain, whose Hamiltonian is 3-local and translationally invariant in the bulk. The model is frustration-free and its unique ground state can be expressed as a weighted superposition of colored Dyck paths. We focus on the case where the deformation parameter t > 1 . By using a variational method, we prove that the finite-size gap decays at least exponentially with increasing the system size. We prove that the magnetization in the ground state is along the z -direction, namely 〈 s x 〉 = 〈 s y 〉 = 0 , and show that the z -component 〈 s z 〉 exhibits a domain-wall structure. We then study the entanglement properties of the chain. In particular, we derive upper and lower bounds for the von Neumann and Rényi entropies, and entanglement spectrum for any bipartition of the chain.
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.
We generalize the area-law violating models of Fredkin and Motzkin spin chains into two dimensions by building quantum six- and nineteen-vertex models with correlated interactions. The Hamiltonian is frustration free, and its projectors generate ergodic dynamics within the subspace of height configuration that are non negative. The ground state is a volume- and color-weighted superposition of classical bi-color vertex configurations with non-negative heights in the bulk and zero height on the boundary. The entanglement entropy between subsystems has a phase transition as the qqq-deformation parameter is tuned, which is shown to be robust in the presence of an external field acting on the color degree of freedom. The ground state undergoes a quantum phase transition between area- and volume-law entanglement phases with a critical point where entanglement entropy scales as a function Llog⁡LL\log LLlogL of the linear system size LLL. Intermediate power law scalings between Llog⁡LL\log LLlogL and L2L^2L2 can be achieved with an inhomogeneous deformation parameter that approaches 1 at different rates in the thermodynamic limit. For the q>1q>1q>1 phase, we construct a variational wave function that establishes an upper bound on the spectral gap that scales as q−L3/8q^{-L^3/8}q−L3/8.
The stabilizer Rényi entropy is a quantitative measure of non-stabilizerness, or magic, and has typically been found to scale extensively with system size NNN (i.e., Θ(N)Θ(N)Θ(N)) for a variety of many-body quantum states. In this note, we study the stabilizer Rényi entropy of the ground state of the spin-12\frac{1}{2}21​ Dyck-Fredkin chain and its ttt-deformation, a local frustration-free model with unusual spectral-gap scaling. Exploiting the combinatorial structure, we carry out numerically exact finite-size calculations, which indicate asymptotic behavior depending on ttt: Θ(N)Θ(N)Θ(N) for t<1t<1t<1, Θ(log⁡N)Θ(\log N)Θ(logN) at t=1t=1t=1, and Θ(1)Θ(1)Θ(1) for t>1t>1t>1. The scaling at t=1t=1t=1 could be another manifestation of the unconventional criticality of the model, while the contrast with the behavior of the entanglement entropy suggests that non-stabilizerness might provide a new window into quantum many-body systems.
We generalize the area-law violating models of Fredkin and Motzkin spin chains into two dimensions by building quantum six- and nineteen-vertex models with correlated interactions. The Hamiltonian is frustration free, and its projectors generate ergodic dynamics within the subspace of height configuration that are non negative. The ground state is a volume- and color-weighted superposition of classical bi-color vertex configurations with non-negative heights in the bulk and zero height on the boundary. The entanglement entropy between subsystems has a phase transition as the q q -deformation parameter is tuned, which is shown to be robust in the presence of an external field acting on the color degree of freedom. The ground state undergoes a quantum phase transition between area- and volume-law entanglement phases with a critical point where entanglement entropy scales as a function L\log L L log L of the linear system size L L . Intermediate power law scalings between L\log L L log L and L^2 L 2 can be achieved with an inhomogeneous deformation parameter that approaches 1 at different rates in the thermodynamic limit. For the q>1 q > 1 phase, we construct a variational wave function that establishes an upper bound on the spectral gap that scales as q^{-L^3/8} q − L 3 / 8 .