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arXiv:2408.16070

Symmetries, correlation functions, and entanglement of general quantum Motzkin spin-chains

Varun Menon, Andi Gu, Ramis Movassagh

Published Aug 28, 2024• 1 citation
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Abstract

Motzkin spin-chains, which include 'colorless' (integer spin $s=1$) and 'colorful' ($s \geq 2$) 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 $\sqrt{n}$ 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 $s \geq 2$ 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.

DOI
10.48550/arxiv.2408.16070

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Motzkin spin-chains, which include 'colorless' (integer spin s=1s=1s=1) and 'colorful' (s≥2s \geq 2s≥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\sqrt{n}n​ in system size nnn), 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=1s=1s=1 chain exhibits U(1)U(1)U(1) spontaneous symmetry breaking and ferromagnetic order. Meanwhile, the s≥2s \geq 2s≥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.
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.
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 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=1s=1s=1, where the cluster property holds. Correspondingly, for s=1s=1s=1 one gets a light-cone profile in the propagation of excitations after a local quench, while the cone is absent for s=2s=2s=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=1/2s=1/2s=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.
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.
Motzkin spin chain is a well-known mathematical model with connections to symmetry-protected topological phases, such as the Haldane phase, as well as to concepts in the AdS/CFT correspondence. They exhibit highly entangled ground states that violate the area law and are exceptionally difficult to simulate with conventional numerical methods. Numerical simulations of the Motzkin ground state become further challenging at large system sizes due to their high-dimensional spin structure, rendering it a natural test bed for quantum simulation with ultra-cold systems. Here, we propose a Rydberg-atom based quantum simulation scheme that effectively realizes Motzkin spins using an experimentally accessible set of parameters. We show that the resulting effective Motzkin ground state reproduces the characteristic entanglement scaling and the block-structure properties of the reduced density matrix associated with the ideal Motzkin state. Our results establish a pathway toward a concrete experimental realization of Motzkin spins beyond purely mathematical constructions, opening avenues for exploring other similar exotic non-area-law entangled phases in programmable Rydberg simulators.