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arXiv:quant-ph/0701160 · Physical Review A

Entangled rings, matrix product states, and exact solutions of XYZ spin chains

M. Asoudeh, Vahid Karimipour, Afsaneh Sadrolashrafi

Published Jul 18, 2007• 22 citations
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Abstract

We show that the ground state of the Heisenberg spin-1/2 chain in an external magnetic field, can be exactly expressed as a matrix product state, provided that the coupling constants are constrained to be on a specific two dimensional surface. This ground state has a very interesting property: all the pairs of spins are equally entangled with each other. In this last respect, the results are of interest for engineering long-range entanglement in experimentally realizable finite arrays of qubits, where the ground state will act as the initial state of a quantum computer.

DOI
10.1103/physreva.76.012320

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We show that the ground state of the Heisenberg spin-1/2 chain in an external magnetic field, can be exactly expressed as a matrix product state, provided that the coupling constants are constrained to be on a specific two dimensional surface. This ground state has a very interesting property: all the pairs of spins are equally entangled with each other. In this last respect, the results are of interest for engineering long-range entanglement in experimentally realizable finite arrays of qubits, where the ground state will act as the initial state of a quantum computer.
We determine the conditions for the existence of a pair of degenerate parity breaking separable eigenstates in general arrays of arbitrary spins connected through XYZXYZXYZ couplings of arbitrary range and placed in a transverse field, not necessarily uniform. Sufficient conditions under which they are ground states are also provided. It is then shown that in finite chains, the associated definite parity states, which represent the actual ground state in the immediate vicinity of separability, can exhibit entanglement between any two spins regardless of the coupling range or separation, with the reduced state of any two subsystems equivalent to that of pair of qubits in an entangled mixed state. The corresponding concurrences and negativities are exactly determined. The same properties persist in the mixture of both definite parity states. These effects become specially relevant in systems close to the XXZXXZXXZ limit. The possibility of field induced alternating separable solutions with controllable entanglement side limits is also discussed. Illustrative numerical results for the negativity between the first and the jth{j}^{\text{th}}jth spin in an open spin sss chain for different values of sss and jjj are as well provided.
We examine the entanglement of cyclic spin-1/2 chains with anisotropic XYZXYZXYZ Heisenberg couplings of arbitrary range at transverse factorizing magnetic fields. At these fields the system exhibits a degenerate symmetry-breaking separable ground state (GS). It is shown, however, that the side limits of the GS pairwise entanglement at these fields are actually nonzero in finite chains, corresponding such fields to a GS spin-parity transition. These limits exhibit universal properties like being independent of the pair separation and interaction range, and are directly related to the magnetization jump. Illustrative exact results are shown for chains with (I) full range and (II) nearest-neighbor couplings. Global entanglement properties at such points are also discussed.
We consider a one-parameter family of matrix-product states of spin-1 particles on a periodic chain and study in detail the entanglement properties of such a state. In particular, we calculate exactly the entanglement of one site with the rest of the chain, and the entanglement of two distant sites with each other, and show that the derivative of both these properties diverge when the parameter ggg of the states passes through a critical point. Such a point can be called a point of quantum phase transition, since at this point the character of the matrix-product state, which is the ground state of a Hamiltonian, changes discontinuously. We also study the finite size effects and show how the entanglement depends on the size of the chain. This later part is relevant to the field of quantum computation where the problem of initial state preparation in finite arrays of qubits or qutrits is important. It is also shown that the entanglement of two sites have scaling behavior near the critical point.
Spins models, despite their simplicity, are known to describe with great precision physical properties of the materials. Of particular interest are the one-dimensional models that present an exact analytical solution, which can be obtained by the transfer matrix method and the decoration-iteration transformation technique. In this thesis, we studied a spin-electron model and two Ising-Heisenberg spin models, where all models are exactly soluble. The first model we investigate concerns a spin-electron diamond chain with Ising spins located on the local sites and mobile electrons dislocated in the interstitial sites under the external magnetic field action. We analyzed the ground states and the correlation between the spins. We show that thermal fluctuations can lead to the emergence of quantum entanglement above the unfrustrated ground states. We also analyze this same model but for different Landé g-factors. We investigated the phase diagrams at zero temperature. We also analyze the magnetization and we show that the magnetic field can cause the emergence of quantum entanglement in some conditions above the non-entanglement ground states. In this same context, we studied the Ising-XYZ diamond model of spin-1/2, with interaction between the second neighbours of the nodal Ising spins, and an external magnetic field applied to the z-axis. We investigate the phase diagram at zero temperature and the quantum entanglement. We notice that the bipartite entanglement between the pairs of Heisenberg interstitial spins in the natural mineral azurite is insensitive to the external magnetic field applied up to approximately 30 K. The third model we studied is a classical-quantum chain consisting of regularly alternating Ising and Heisenberg spins and with two different local anisotropy axes, subject to an applied external magnetic field. We studied the ground states, the magnetization and the susceptibility. It is evidenced that the investigated spin model reproduces magnetic characteristics of the heterobimetallic coordination compound Dy(NO)3(DMSO)2Cu(opba)(DMSO)2. In this thesis we show that the exact soluble spin models can describe with good precision a class of real magnetic materials, which can present interesting magnetic and thermodynamic properties, and quantum entanglement.
Several recent experiments have shown that long-range exchange interactions can determine collective magnetic ground states of nanostructures in bulk and on surfaces. The ability to generate and control entanglement in a system with long-range interaction will be of great importance for future quantum technology. An important step forward to reach this goal is the creation of entangled states for spins of distant magnetic atoms. Herein, the generation of long-distance entanglement between remote spins at large separations in bulk and on surface is studied theoretically, based on a quantum spin Hamiltonian and time-dependent Schrödinger equation for experimentally realized conditions. We demonstrate that long-distance entanglement can be generated between remote spins by using an appropriate quantum spin chain (a quantum mediator), composed by sets of antiferromagnetically coupled spin dimers. Ground state properties and quantum spin dynamics of entangled atoms are studied. We demonstrate that one can increase or suppress entanglement by adding a single spin in the mediator. The obtained result is explained by monogamy property of entanglement distribution inside a quantum spin system. We present a novel approach for non-local sensing of remote magnetic adatoms via spin entanglement.
We present a manifestly rotational invariant formulation of the matrix product method valid for spin chains and ladders. We apply it to two-legged spin ladders with spins , 1 and and different magnetic structures labelled by the exchange coupling constants, which can be ferromagnetic or antiferromagnetic along the legs and the rungs of the ladder. We compute ground-state energy densities, correlation lengths and string order parameters. We present numerical evidence of the duality properties of the three different nonferromagnetic spin ladders. We show that the long-range topological order characteristic of isolated spin 1 chains is broken by the interchain coupling. The string order correlation function decays exponentially with a finite correlation length that we compute. A physical picture of the spin 1 ladder is given in terms of a collection of resonating spin 1 chains. Finally, for ladders with spin equal to or greater than we define a class of AKLT states whose matrix product coefficients are given by 9- j symbols.