Criticality without Frustration for Quantum Spin-1 Chains

Bravyi, Sergey; Caha, Libor; Movassagh, Ramis; Nagaj, Daniel

doi:10.1103/physrevlett.109.207202

Cited by 95 publications

(149 citation statements)

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“…For this reason finding and studying Hamiltonians with highly entangled spins is currently one of the most challenging and intriguing fields in quantum physics. In this direction local integer frustration-free spin Hamiltonians whose ground-state can be expressed as a combination of Motzkin paths [9] have been recently introduced [10,11]. Among others interesting aspects, their importance is given by the fact that they own a level of entanglement entropy which strongly exceeds the one exhibited by other previously known local models.…”

Section: Pacs Numbersmentioning

confidence: 99%

“…The deformation induced by t = 1 keeps the model frustration-free [14], and, while for t = 1 we recover the undeformed model [10][11][12], for t > 1 (t < 1) the paths having larger (smaller) h are favored in the ground state. Notice that for t = 1 one can have analytical expressions for the magnetizatazion and the z − z correlation functions, which were tested against DMRG results in [12].…”

Section: Pacs Numbersmentioning

confidence: 99%

“…The spin model we consider has the peculiarity of having a ground state which can be expressed in terms of Motzkin paths describing all the possible 2n moves that one can make to go from a point of height h = 0 to an other point of the same h without crossing the 0 line [10,11]. As shown in Fig.…”

Section: Pacs Numbersmentioning

confidence: 99%

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Haldane topological orders in Motzkin spin chains

et al. 2017

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Motzkin spin chains are frustration-free models whose ground-state is a combination of Motzkin paths. The weight of such path contributions can be controlled by a deformation parameter t. As a function of the latter these models, beside the formation of domain wall structures, exhibit gapped Haldane topological orders with constant decay of the string order parameters for t < 1. A behavior compatible with a Berezinskii-KosterlitzThouless phase transition at t = 1 is also presented. By means of numerical calculations we show that the topological properties of the Haldane phases depend on the spin value. This allows to classify different kinds of hidden antiferromagnetic Haldane gapped regimes associated to nontrivial features like symmetry-protected topological order. Our results from one side allow to clarify the physical properties of Motzkin frustration-free chains and from the other suggest them as a new interesting and paradigmatic class of local spin Hamiltonians. [6] has been discovered for spin-1 XXZ Heisenberg chains. This has driven significant efforts to look for new kinds of models whose topological order can be described in terms of a SOP [7] motivating the discovery of the celebrated AffleckKennedy-Lieb-Tasaki (AKLT) model [8]. Although the argument of Haldane is given for integer spin chains, only integer spin XXZ-like and AKLT-like chains own topological HP and is therefore non-trivial to find and study new classes of Hamiltonians where HP emerges. Thanks to the strongest quantum "resource", namely the entanglement, spin models have also a fundamental role in the simulation of quantum logical gates for quantum computation [4]. For this reason finding and studying Hamiltonians with highly entangled spins is currently one of the most challenging and intriguing fields in quantum physics. In this direction local integer frustration-free spin Hamiltonians whose ground-state can be expressed as a combination of Motzkin paths [9] have been recently introduced [10,11]. Among others interesting aspects, their importance is given by the fact that they own a level of entanglement entropy which strongly exceeds the one exhibited by other previously known local models. Relevantly, also for half-integer spins, a similar class of Hamiltonians, the Fredkin spin chains, exhibiting the same features [12,13] has been introduced. In addition to their entanglement properties Motzkin chains own further very peculiar properties. Indeed, even if they are purely local models, for high spin values s (i.e., s ≥ 2) they behave as de facto long range Hamiltonians being able to violate cluster decomposition properties (CDP) and area law (AL) scaling DMRG local magnetization for a system of length 2n = 60 at different t deformation values S z i for c) s = 1 and d) s = 2. Lower panels: Thermodynamic limit of the gap ∆ = E1 − E0 as a function of t for e) s = 1 (red circles) and f) s = 2 (blue squares). The continuos lines are fitted with the form ∼ exp(−b/ √ tc − t) with tc = 1 and b a fitting parameter. The thermodynamic limit ...

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Section: Pacs Numbersmentioning

confidence: 99%

Section: Pacs Numbersmentioning

confidence: 99%

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Haldane topological orders in Motzkin spin chains

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“…Gapless frustration-free systems include the spin-1=2 ferromagnetic Heisenberg chain and the Rokhsar-Kivelson quantum dimer model [27]. Two recent papers construct gapless frustration-free spin chains in which the half-chain entanglement entropy diverges in the thermodynamic limit n → ∞: a spin-1 example based on parenthesized expressions [with logðnÞ divergence] [28] and a higher spin generalization ( ffiffiffi n p divergence) [29]. The classification of spin-1=2 chains in Ref.…”

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confidence: 99%

“…This may be relevant to understanding possible scaling limits of critical frustration-free systems, an issue which has been raised in Refs. [28][29][30]. The third and final reason, explained below, is its potential relevance to the area law for the entanglement entropy in onedimensional spin systems.…”

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Correlation Length versus Gap in Frustration-Free Systems

Gosset

Huang

2016

Phys. Rev. Lett.

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Hastings established exponential decay of correlations for ground states of gapped quantum many-body systems. A ground state of a (geometrically) local Hamiltonian with spectral gap ϵ has correlation length ξ upper bounded as ξ ¼ Oð1=ϵÞ. In general this bound cannot be improved. Here we study the scaling of the correlation length as a function of the spectral gap in frustration-free local Hamiltonians, and we prove a tight bound ξ ¼ Oð1= ffiffi ffi ϵ p Þ in this setting. This highlights a fundamental difference between frustration-free and frustrated systems near criticality. The result is obtained using an improved version of the combinatorial proof of correlation decay due to Aharonov, Arad, Vazirani, and Landau. DOI: 10.1103/PhysRevLett.116.097202 Exponential decay of correlations is a basic feature of the ground space in gapped quantum many-body systems. The setting is as follows. We consider a geometrically local Hamiltonian H which acts on particles of constant dimension s; i.e., the Hilbert space is ðC s Þ ⊗n , where n is the total number of particles. The particles are located at the sites of a finite lattice (of some arbitrary dimension). We write the Hamiltonian aswhere distinct terms H i ; H j are supported on distinct subsets of particles. Here the support of a term H i is the set of particles on which it acts nontrivially. We assume that H has constant range r with respect to the usual distance function d on the lattice, the shortest path metric. This means that the diameter of the support of each term H i is upper bounded by r (e.g., r ¼ 2 for nearest-neighbor interactions). Without loss of generality we assume that the smallest eigenvalue of each term H i is equal to zero, and that ∥H i ∥ ≤ 1.If H has a unique ground state jψi and spectral gap ϵ, connected correlation functions decay exponentially as a function of distance [1][2][3]. In particular, where dðA; BÞ denotes the distance between the supports of two (arbitrary) local observables A, B, and C is a positive constant which depends on r and the lattice. In the transverse field Ising chain the scaling ξ ¼ Θð1=ϵÞ is achieved [4,5], which shows that the upper bound on ξ in Eq. (2) cannot be improved. In gapped systems with (exactly) degenerate ground states, a modification of Eq. (1),holds with ξ ¼ Oð1=ϵÞ for any ground state jψi [6], where G is the projector onto the ground space. An overview of these results and the proof techniques used to obtain them is given in Ref. [7].Here we specialize to frustration-free geometrically local Hamiltonians. Frustration-freeness means that any ground state of H is also in the ground space of each term H i . Since we assume that H i has smallest eigenvalue zero, this means that any ground state jψi of H satisfies H i jψi ¼ 0 for all i. The ground energy of H is therefore zero and in general the ground space may be degenerate. The spectral gap ϵ of H is defined to be its smallest nonzero eigenvalue. Henceforth we assume (without loss of generality [8]) that each term H i in the Hamiltonian is a projector,...

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Generalized Entanglement Entropy in New Spin Chains

Sugino

Korepin

2020

Springer Proceedings in Physics

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Criticality without Frustration for Quantum Spin-1 Chains

Cited by 95 publications

References 36 publications

Haldane topological orders in Motzkin spin chains

Haldane topological orders in Motzkin spin chains

Correlation Length versus Gap in Frustration-Free Systems

Generalized Entanglement Entropy in New Spin Chains

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