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arXiv:1710.10426 · Journal of Statistical Mechanics Theory and Experiment

Area law violations and quantum phase transitions in modified Motzkin walk spin chains

Fumihiko Sugino, Pramod Padmanabhan

Published Jan 1, 2018• 13 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1742-5468/aa9dcb

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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.
While volume violation of area law has been exhibited in several quantum spin chains, the construction of a corresponding ground state in higher dimensions, entangled in more than one direction, has been an open problem. Here we construct a 2D frustration-free Hamiltonian with maximal violation of the area law. We do so by building a quantum model of random surfaces with color degree of freedom that can be viewed as a collection of colored Dyck paths. The Hamiltonian may be viewed as a 2D generalization of the Fredkin spin chain. It relates all the colored random surface configurations subject to a Dirichlet boundary condition and hard wall constraint from below to one another, and the ground state is therefore a superposition of all such classical states and non-degenerate. Its entanglement entropy between subsystems undergoes a quantum phase transition as the deformation parameter is tuned. The area- and volume-law phases are similar to the one-dimensional model, while the critical point scales with the linear size of the system L as Llog⁡L . Further it is conjectured that similar models with entanglement phase transitions can be built in higher dimensions with even softer area law violations at the critical point.
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.
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.
We present bulk tensor networks that exactly represent the ground states of a continuous family of one-dimensional frustration-free Hamiltonians. These states, which are known as area-deformed Motzkin and Fredkin states, exhibit a novel quantum phase transition. By tuning a single parameter, they go from a phase obeying an area law to a highly entangled ``rainbow'' phase, where the half-chain entropy scales with the volume. Using the representation of these ground states as superpositions of random walks, we introduce tensor networks for these ground states where local and global rules of the walker are baked into bulk tensors, thereby providing an efficient description of the ground states (some of which satisfy a volume law scaling of entanglement entropy).
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 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 .