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arXiv:1701.00346 · Journal of Physics A Mathematical and Theoretical

Finite-size gap, magnetization, and entanglement of deformed Fredkin spin chain

Takuma Udagawa, Hosho Katsura

Published Aug 11, 2017• 23 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1751-8121/aa85b5

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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 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).
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.
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.
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.
Frustration-free (FF) spin chains have a property that their ground state minimizes all individual terms in the chain Hamiltonian. We ask how entangled the ground state of a FF quantum spin-s chain with nearest-neighbor interactions can be for small values of s. While FF spin-1/2 chains are known to have unentangled ground states, the case s=1 remains less explored. We propose the first example of a FF translation-invariant spin-1 chain that has a unique highly entangled ground state and exhibits some signatures of a critical behavior. The ground state can be viewed as the uniform superposition of balanced strings of left and right brackets separated by empty spaces. Entanglement entropy of one half of the chain scales as 1/2 log n+O(1), where n is the number of spins. We prove that the energy gap above the ground state is polynomial in 1/n. The proof relies on a new result concerning statistics of Dyck paths which might be of independent interest.