Some aspects of Affleck-Kennedy-Lieb-Tasaki models: tensor network, physical properties, spectral gap, deformation, and quantum computation
Tzu-Chieh Wei, Robert Raussendorf, Ian Affleck
备注：
47 pages, single column, 12 figures, to appear in a Chapter of the Springer Volume on Entanglement in Spin Chains - Theory and Quantum Technology Applications
发表信息：
In Bayat, A., Bose, S., Johannesson, H. (eds) Entanglement in Spin Chains. Quantum Science and Technology. Springer (2022)
Affleck, Kennedy, Lieb, and Tasaki constructed a spin-1 model that is isotropic in spins and possesses a provable finite gap above the ground state more than three decades ago. They also constructed models in two dimensions. Their construction has impacted subsequent research that is still active. In this review article, we review some selected the progresses, such as magnetic ordering of the AKLT models, emerging phases under deforming the AKLT Hamiltonians, symmetry-protected topological order in several AKLT models, their spectral gap, and applications for quantum computation.
We study the dynamical properties of both bulk and edge spins of a two-dimensional Affleck-Kennedy-Lieb-Tasaki (AKLT) model mainly by using the stochastic series expansion quantum Monte Carlo method with stochastic analytic continuation. In the deep AKLT phase, we obtain a spin spectrum with flat band, which is a strong evidence for a localized state. Through the spectrum analysis, we see a clear continuous phase transition from the AKLT phase to the Néel phase in the model, and the energy gap becomes closed at the corresponding momentum point. In comparison with linear spin-wave theory, the d…
A symmetry-protected topological phase has nontrivial surface states in the presence of certain symmetries, which can either be gapless or be degenerate. In this work, we study the physical consequence of such gapless surface states at the bulk quantum phase transition (QPT) that spontaneously breaks these symmetries. The two-dimensional Affleck-Kennedy-Lieb-Tasaki phase on a square lattice and its QPTs to Néel ordered phases are realized with the spin-$1/2$ Heisenberg model on a decorated square lattice. With large-scale quantum Monte Carlo simulations, we show that even though the bulk QPTs …
Quantum entanglement under an extensive bipartition can reveal the critical boundary theory of a topological phase in the parameter space. In this study we demonstrate that the infinite-randomness fixed point for spin-1/2 degrees of freedom can emerge from an extensive random bipartition of the spin-1 Affleck-Kennedy-Lieb-Tasaki chain. The nested entanglement entropy of the ground state of the reduced density matrix exhibits a logarithmic scaling with an effective central charge $\tilde{c} = 0.72 \pm 0.02 \approx \ln 2$. We further discuss, in the language of bulk quantum entanglement, how to …
Applying a symmetric bulk bipartition to one-dimensional Affleck-Kennedy-Lieb-Tasaki valence bond solid (VBS) states for the integer spin-S Haldane gapped phase, we can create an array of fractionalized spin-S/2 edge states with the super unit cell $l$ in the reduced bulk system, and the topological properties encoded in the VBS wave functions can be revealed. The entanglement Hamiltonian (EH) with $l=even$ corresponds to the quantum antiferromagnetic Heisenberg spin-S/2 model. For the even integer spins, the EH still describes the Haldane gapped phase. For the odd integer spins, however, the …
The elementary excitation in the antiferromagnetic spin-1 model known as the Affleck-Kennedy-Lieb-Tasaki (AKLT) Hamiltonian has been described alternatively as magnons or kink-like solitons (triplons). The latter, which we call the triplon throughout this paper, has been proven equivalent descriptions of the same magnon excitation and not an independent branch of excited states. On the other hand, no careful examination of multi-magnon and multi-triplon equivalence was made in the past. In this paper we prove that two-magnon and two-triplon states are also identical descriptions of the same ex…
We exploit a natural Projected Entangled-Pair State (PEPS) representation for the resonating Affleck-Kennedy-Lieb-Tasaki loop (RAL) state. By taking advantage of PEPS-based analytical and numerical methods, we characterize the RAL states on various two-dimensional lattices. On square and honeycomb lattices, these states are critical since the dimer-dimer correlations decay as a power law. On kagome lattice, the RAL state has exponentially decaying correlation functions, supporting the scenario of a gapped spin liquid. We provide further evidence that the RAL state on the kagome lattice is a $\…
The $S=1$ Affleck-Kennedy-Lieb-Tasaki (AKLT) quantum spin chain was the first rigorous example of an isotropic spin system in the Haldane phase. The conjecture that the $S=3/2$ AKLT model on the hexagonal lattice is also in a gapped phase has remained open, despite being a fundamental problem of ongoing relevance to condensed-matter physics and quantum information theory. Here we confirm this conjecture by demonstrating the size-independent lower bound $Δ>0.006$ on the spectral gap of the hexagonal model with periodic boundary conditions in the thermodynamic limit. Our approach consists of two…
We argue that the $q$-deformed spin-1 AKLT Hamiltonian should be regarded as a representative of a symmetry protected topological phase. Even though it fails to exhibit any of the standard symmetries known to protect the Haldane phase it still displays all characteristics of this phase: Fractionalized spin-$\frac{1}{2}$ boundary spins, non-trivial string order and - when using an appropriate definition - a two-fold degeneracy in the entanglement spectrum. We trace these properties back to the existence of an $SO_q(3)$ quantum group symmetry and speculate about potential links to discrete duali…
We construct a Hamiltonian whose ground state encodes a time-independent emulation of quan- tum teleportation. We calculate properties of the Hamiltonian, using exact diagonalization and a mean-field theory, and argue that it has a gap. The system exhibits an illuminating relationship to the well-known AKLT (Affleck, Lieb, Kennedy and Tasaki) model.
On the basis of a Berry-phase analysis, we study the ground state of the $J_1$-$J_2$ Heisenberg chain for $S=2,3,4$. We find that changes of the Berry phase occur $S$ times for spin-$S$ systems, indicating the sequential phase transitions. The sequential phase transitions are associated with the change of valence-bond configurations of the ground states. To demonstrate this numerically, we connect the $J_1$-$J_2$ Heisenberg Hamiltonian with the generalized Affleck-Kennedy-Lieb-Tasaki model, and show that the different phases are connected to the different valence-bond-solid states.
We show that a local Hamiltonian of spin-3/2 particles with only two-body nearest-neighbor Affleck-Kennedy-Lieb-Tasaki and exchange-type interactions has an unique ground state, which can be used to implement universal quantum computation merely with single-spin measurements. We prove that the Hamiltonian is gapped, independent of the system size. Our result provides a further step towards utilizing systems with condensed matter-type interactions for measurement-based quantum computation.
A class of exact spin ground states with nonzero averages of vector spin chirality, $<Š_i \times Š_j \cdot \hat{z}>$, is presented. It is obtained by applying non-uniform O(2) rotations of spin operators in the XY plane on the SU(2)-invariant Affleck-Kennedy-Lieb-Tasaki (AKLT) states and their parent Hamiltonians. Excitation energies of the new ground states are studied with the use of single-mode approximation in one dimension for S=1. The excitation gap remains robust. Construction of chiral AKLT states is shown to be possible in higher dimensions. We also present a general idea to produce v…
Frustrated one-dimensional (1D) magnets are known as ideal playgrounds for new exotic quantum phenomena to emerge. We consider an elementary frustrated 1D system: the spin-$\frac{1}{2}$ ferromagnetic ($J_1$) Heisenberg chain with next-nearest-neighbor antiferromagnetic ($J_2$) interactions. On the basis of density-matrix renormalization group calculations we show the existence of a finite spin gap at $J_2/|J_1|>1/4$ and we find the ground state in this region to be a valence bond solid (VBS) with spin-singlet dimerization between third-neighbor sites. The VBS is the consequence of spontaneous …
The Haldane state is a typical quantum and topological state of matter, which exhibits an edge state corresponding to symmetry-protected topological order in a one-dimensional integer spin chain. This edge state can be utilized for a processing unit of quantum computation. Its realization, however, has difficulties with synthesis of integer spin compounds. In contrast, one-half spin systems are more designable due to recent progress on intended synthesis of organic materials, quantum dots, and optical lattices. Here we propose a concept to design the Haldane state with one-half spins by making…
A new method of writing down the path integral for spin-1 Heisenberg antiferromagnetic chain is introduced. In place of the conventional coherent state basis that leads to the non-linear sigma-model, we use a new basis called the fluctuating matrix product states (fMPS) which embodies inter-site entanglement from the outset. It forms an overcomplete set spanning the entire Hilbert space of the spin-1 chain. Saddle-point analysis performed for the bilinear-biquadratic spin model predicts Affeck-Kennedy-Lieb-Tasaki (AKLT) state as the ground state in the vicinity of the AKLT Hamiltonian. Quadrat…
The peculiar features of quantum magnetism sometimes forbid the existence of gapped `featureless' paramagnets which are fully symmetric and unfractionalized. The Lieb-Schultz-Mattis theorem is an example of such a constraint, but it is not known what the most general restriction might be. We focus on the existence of featureless paramagnets on the spin-1 square lattice and the spin-1 and spin-1/2 honeycomb lattice with spin rotation and space group symmetries in 2+1D. Although featureless paramagnet phases are not ruled out by any existing theorem, field theoretic arguments disfavor their exis…
Optimizing translationally invariant infinite-Projected Entangled Pair States (iPEPS), we investigate the spin-2 Affleck-Kennedy-Lieb-Tasaki (AKLT) and spin-1 Heisenberg models on the Kagome lattice as a function of magnetic field. We found that the magnetization curves offer a wide variety of compressible and incompressible phases. Incompressible nematic phases breaking the lattice $C_3$ rotation -- for which we propose simple qualitative pictures -- give rise to magnetization plateaux at reduced magnetization $m_z=5/6$ and $m_z=1/3$ for spin-2 and spin-1, respectively, in addition to the $m_…
We introduce a simple representation for irreducible spherical tensor operators of the rotation group of arbitrary integer or half integer rank and use these tensor operators to construct matrix product states corresponding to all the variety of valence-bond states proposed in the Affleck-Kennedy-Lieb-Tasaki (AKLT) construction. These include the fully dimerized states of arbitrary spins, with uniform or alternating patterns of spins, which are ground states of Hamiltonians with nearest and next-nearest neighbor interactions, and the partially dimerized or AKLT/VBS (Valence Bond Solid) states,…
Two-dimensional (spin-$2$) Affleck-Kennedy-Lieb-Tasaki (AKLT) type valence bond solids on the square lattice are known to be symmetry protected topological (SPT) gapped spin liquids [Shintaro Takayoshi, Pierre Pujol, and Akihiro Tanaka Phys. Rev. B ${\bf 94}$, 235159 (2016)]. Using the projected entangled pair state (PEPS) framework, we extend the construction of the AKLT state to the case of $SU(3)$, relevant for cold atom systems. The entanglement spectrum is shown to be described by an alternating $SU(3)$ chain of "quarks" and "antiquarks", subject to exponentially decaying (with distance) …
Single-spin measurements on the ground state of an interacting spin lattice can be used to perform a quantum computation. We show how such measurements can mimic renormalization group transformations and remove the short-ranged variations of the state that can reduce the fidelity of a computation. This suggests that the quantum computational ability of a spin lattice could be a robust property of a quantum phase. We illustrate our idea with the ground state of a spin-1 chain, which can serve as a quantum computational wire not only at the Affleck-Kennedy-Lieb-Tasaki point, but within the rotat…
We propose that a valence-bond-solid (VBS) order can be stabilized in certain two-dimensional antiferromagnets due to spin-lattice coupling. In contrast to the VBS state of the Affleck-Kennedy-Lieb-Tesaki (AKLT) type in which the spin $2S$ and the lattice coordination $z$ must be commensurate, the spin-lattice coupling-induced VBS state can occur when $2S$ is not an integer multiple of $z$. As a concrete example, S=2 spins on the triangular network with $z=6$ is discussed. Within the Schwinger boson mean-field theory it is shown that the ground state is given by the $\sqrt{3}\times\sqrt{3}$ mo…
We present a spin-1 chain with a Hamiltonian which has three exactly solvable ground states. Two of these are fully dimerized, analogous to the Majumdar-Ghosh (MG) states of a spin-1/2 chain, while the third is of the Affleck-Kennedy-Lieb-Tasaki (AKLT) type. We use variational and numerical methods to study the low-energy excitations which interpolate between these ground states in different ways. In particular, there is a spin-1/2 excitation which interpolates between the MG and AKLT ground states; this is the lowest excitation of the system and it has a surprisingly small gap. We discuss gen…
Motivated by the magnetic properties of the spin-chain compounds LiCuSbO$_4$$\equiv$LiSbCuO$_4$ and Rb$_2$Cu$_2$Mo$_3$O$_{12}$, we study the ground state of the Heisenberg chain with dimerized nearest-neighbor ferromagnetic (FM)($J_1, J_1^\prime<0$) and next-nearest-neighbor antiferromagnetic ($J_2>0$) couplings. Using the density-matrix renormalization group technique and spin-wave theory we find a first-order transition between a fully-polarized FM and an incommensurate spiral state at $2α=β/(1+β)$, where $α$ is the frustration ratio $J_2/|J_1|$ and $β$ the degree of dimerization $J_1^\prime…
We investigate the entanglement properties of the valence-bond-solid (VBS) state defined on two-dimensional lattices, which is the exact ground state of the Affleck-Kennedy-Lieb-Tasaki model. It is shown that the entanglement entropy obeys an area law and the non-universal prefactor of the leading term is strictly less than $\ln 2$. The analysis of entanglement spectra for various lattices reveals that the reduced density matrix associated with the VBS state is closely related to a thermal density matrix of a {\it holographic} spin chain, whose spectrum is reminiscent of that of the spin-1/2 H…
Recently it has been shown that the non-local correlations needed for measurement based quantum computation (MBQC) can be revealed in the ground state of the Affleck-Kennedy-Lieb-Tasaki (AKLT) model involving nearest neighbor spin-3/2 interactions on a honeycomb lattice. This state is not singular but resides in the disordered phase of ground states of a large family of Hamiltonians characterized by short-range-correlated valence bond solid states. By applying local filtering and adaptive single particle measurements we show that most states in the disordered phase can be reduced to a graph of…
The models constructed by Affleck, Kennedy, Lieb, and Tasaki describe a family of quantum antiferromagnets on arbitrary lattices, where the local spin S is an integer multiple M of half the lattice coordination number. The equal time quantum correlations in their ground states may be computed as finite temperature correlations of a classical O(3) model on the same lattice, where the temperature is given by T=1/M. In dimensions d=1 and d=2 this mapping implies that all AKLT states are quantum disordered. We consider AKLT states in d=3 where the nature of the AKLT states is now a question of det…
We study the spectral properties of a family of quantum antiferromagnets on two-dimensional (2D) lattices. This family of models is obtained by a deformation of the well-studied 2D quantum antiferromagnetic model of Affleck, Kennedy, Lieb and Tasaki (AKLT); they are described by two-body, frustration-free Hamiltonians on a three-colourable lattice of spins. Although the existence of a spectral gap in the 2D AKLT model remains an open question, we rigorously prove the existence of a gap for a subset of this family of quantum antiferromagnets. Along with providing new progress for the gap proble…
The single-copy entanglement of a given many-body system is defined [J. Eisert and M. Cramer, Phys. Rev. A. 72, 042112 (2005)] as the maximal entanglement deterministically distillable from a bipartition of a single specimen of that system. For critical (gapless) spin chains, it was recently shown that this is exactly half the von Neumann entropy [R. Orus, J. I. Latorre, J. Eisert, and M. Cramer, Phys. Rev. A 73, 060303(R) (2006)], itself defined as the entanglement distillable in the asymptotic limit: i.e. given an infinite number of copies of the system. It is an open question as to what the…
The AKLT spin chain is the prototypical example of a frustration-free quantum spin system with a spectral gap above its ground state. Affleck, Kennedy, Lieb, and Tasaki also conjectured that the two-dimensional version of their model on the hexagonal lattice exhibits a spectral gap. In this paper, we introduce a family of variants of the two-dimensional AKLT model depending on a positive integer $n$, which is defined by decorating the edges of the hexagonal lattice with one-dimensional AKLT spin chains of length $n$. We prove that these decorated models are gapped for all $n \geq 3$.
One-way quantum computation proceeds by sequentially measuring individual spins (qubits) in an entangled many-spin resource state. It remains a challenge, however, to efficiently produce such resource states. Is it possible to reduce the task of generating these states to simply cooling a quantum many-body system to its ground state? Cluster states, the canonical resource for one-way quantum computing, do not naturally occur as ground states of physical systems. This led to a significant effort to identify alternative resource states that appear as ground states in spin lattices. An appealing …
We study the structure of quantum ground states of simplex solid models, which are generalizations of the valence bond construction for quantum antiferromagnets originally proposed by Affleck, Kennedy, Lieb, and Tasaki (AKLT) [Phys. Rev. Lett. 59, 799 (1987)]. Whereas the AKLT states are created by application of bond singlet operators for SU(2) spins, the simplex solid construction is based on N-simplex singlet operators for SU(N) spins. In both cases, a discrete one-parameter family of translationally-invariant models with exactly solvable ground states is defined on any regular lattice, and…
Harvard University（哈佛大学）；Boston University（波士顿大学）；Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences（中国科学院物理研究所）
The strange correlator [Phys. Rev. Lett. 112, 247202 (2014)] has been proposed as a measure of symmetry protected topological order in one- and two-dimensional systems. It takes the form of a spin-spin correlation function, computed as a mixed overlap between the state of interest and a trivial local product state. We demonstrate that it can be computed exactly (asymptotically, in the Monte Carlo sense) for various Affleck-Kennedy-Lieb-Tasaki states by direct evaluation of the wave function within the valence bond loop gas framework. We present results for lattices with chain, square, honeycom…
I define a set of wavefunctions for SU(N) lattice antiferromagnets, analogous to the valence bond solid states of Affleck, Kennedy, Lieb, and Tasaki (AKLT), in which the singlets are extended over N-site simplices. As with the valence bond solids, the new simplex solid (SS) states are extinguished by certain local projection operators, allowing us to construct Hamiltonians with local interactions which render the SS states exact ground states. Using a coherent state representation, we show that the quantum correlations in each SS state are calculable as the finite temperature correlations of a…
Motivated by recent experiments on Bi$_3$Mn$_4$O$_{12}$(NO$_3$), and a broader interest arising from numerical work on the honeycomb lattice Hubbard model, we have studied the effect of a magnetic field on honeycomb lattice spin models with quantum paramagnetic ground states. For a model with frustrating second-neighbor exchange, $J_2$, we use a Lindemann-like criterion within spin wave theory to show that Néel order melts beyond a critical $J_2$. The critical $J_2$ increases with a magnetic field, implying the existence of a field-induced paramagnet-Néel transition over a range of $J_2$. We a…
Engineering Biquadratic Interactions in Spin-1 Chains by Spin-1/2 Spacers
TU Dortmund（多特蒙德工业大学）；Julius-Maximilians-Universität Würzburg（维尔茨堡大学）；Nicolaus Copernicus University（哥白尼大学）；University of Ottawa（渥太华大学）；National Research Council（国家研究委员会）
The discovery of Quantum Many-Body Scars (QMBS) both in Rydberg atom simulators and in the Affleck-Kennedy-Lieb-Tasaki (AKLT) spin-1 chain model, have shown that a weak violation of ergodicity can still lead to rich experimental and theoretical physics. In this review, we provide a pedagogical introduction to and an overview of the exact results on weak ergodicity breaking via QMBS in isolated quantum systems with the help of simple examples such as the fermionic Hubbard model. We also discuss various mechanisms and unifying formalisms that have been proposed to encompass the plethora of syste…
Quantum phases of naturally-occurring systems exhibit distinctive collective phenomena as manifestation of their many-body correlations, in contrast to our persistent technological challenge to engineer at will such strong correlations artificially. Here we show theoretically that quantum correlations exhibited in the two-dimensional valence bond solid phase of a quantum antiferromagnet, modeled by Affleck, Kennedy, Lieb, and Tasaki as a precursor of spin liquids and topological orders, are sufficiently complex yet structured enough to simulate universal quantum computation when every single s…
Using the Matrix Product State framework, we generalize the Affleck-Kennedy-Lieb-Tasaki (AKLT) construction to one-dimensional spin liquids with global color ${\rm SU}(N)$ symmetry, finite correlation lengths, and edge states that can belong to any self-conjugate irreducible representation (irrep) of ${\rm SU}(N)$. In particular, ${\rm SU}(2)$ spin-$1$ AKLT states with edge states of arbitrary spin $s=1/2,1,3/2,\cdots$ are constructed, and a general formula for their correlation length is given. Furthermore, we show how to construct local parent Hamiltonians for which these AKLT states are uni…
In quantum spin-1 chains, there is a nonlocal unitary transformation known as the Kennedy-Tasaki transformation $U_{\text{KT}}$, which defines a duality between the Haldane phase and the $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetry-breaking phase. In this paper, we find that $U_{\text{KT}}$ also defines a duality between a topological Ising critical phase and a trivial Ising critical phase, which provides a "hidden symmetry breaking" interpretation for the topological criticality. Moreover, since the duality relates different phases of matter, we argue that a model with self-duality (i.e., inva…
University of California, Irvine（加州大学尔湾分校）；Budapest University of Technology and Economics（布达佩斯技术与经济大学）；Institute of Physics, Budapest University of Technology and Economics（布达佩斯技术与经济大学物理研究所）；Institute for Theoretical Solid State Physics, IFW Dresden（德累斯顿IFW理论固体物理研究所）；Würzburg-Dresden Cluster of Excellence（维尔茨堡-德累斯顿卓越集群）；ELTE Eötvös Loránd University（厄特沃什·罗兰大学）；HUN-REN Wigner Research Centre for Physics（匈牙利研究与创新网络维格纳物理研究中心）；Eötvös Loránd University（厄特沃什·罗兰大学）
Neural network quantum states (NQS) have been widely applied to spin-1/2 systems where they have proven to be highly effective. The application to systems with larger on-site dimension, such as spin-1 or bosonic systems, has been explored less and predominantly using spin-1/2 Restricted Boltzmann Machines (RBMs) with a one-hot/unary encoding. Here we propose a more direct generalisation of RBMs for spin-1 that retains the key properties of the standard spin-1/2 RBM, specifically trivial product states representations, labelling freedom for the visible variables and gauge equivalence to the ten…
