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arXiv:1604.00668 · Europhysics Letters (EPL)

Quantum-classical equivalence and ground-state factorization

Jahanfar Abouie, Reza Sepehrinia

Published Feb 1, 2016• 0 citations
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Abstract

We have performed an analytical study of quantum-classical equivalence for quantum XY -spin chains with arbitrary interactions to explore the classical counterpart of the factorizing magnetic fields that drive the system into a separable ground state. We demonstrate that the factorizing line in the parameter space of a quantum model is equivalent to the so-called natural boundary that emerges in mapping the quantum XY -model onto the two-dimensional classical Ising model. As a result, we show that the quantum systems with the non-factorizable ground state could not be mapped onto the classical Ising model. Based on the presented correspondence we suggest a promising method for obtaining the factorizing field of quantum systems through the commutation of the quantum Hamiltonian and the transfer matrix of the classical model.

DOI
10.1209/0295-5075/113/37005

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We have performed an analytical study of quantum-classical equivalence for quantum XY -spin chains with arbitrary interactions to explore the classical counterpart of the factorizing magnetic fields that drive the system into a separable ground state. We demonstrate that the factorizing line in the parameter space of a quantum model is equivalent to the so-called natural boundary that emerges in mapping the quantum XY -model onto the two-dimensional classical Ising model. As a result, we show that the quantum systems with the non-factorizable ground state could not be mapped onto the classical Ising model. Based on the presented correspondence we suggest a promising method for obtaining the factorizing field of quantum systems through the commutation of the quantum Hamiltonian and the transfer matrix of the classical model.
By relating a transfer matrix associated with a classical system to the Hamiltonian of its corresponding quantum mechanical system, it is proved that the two-dimensional Ising model in the absence of a magnetic field is equivalent to the ground state of the linear XY-model in the presence of a magnetic field, under appropriate relations among coupling parameters appearing in the two Hamiltonians. Simple relations are established for spin correlations in both systems. Consequently, the Ising system and the ground state of the XY-model are shown to exhibit similar singularities with respect to temperature and a magnetic field, respectively. It is also proved that the two-dimensional dimer problem is equivalent to the ground state of a generalized XY-model, which is solved exactly in general.
We investigate the existence and the properties of fully separable (fully factorized) ground states in quantum spin systems. Exploiting techniques of quantum information and entanglement theory, we extend a recently introduced method and construct a general, self-contained theory of ground state factorization in frustration-free quantum spin models defined on lattices in any spatial dimension and for interactions of arbitrary range. We show that, quite generally, nonexactly solvable translationally invariant models in presence of an external uniform magnetic field can admit exact, fully factorized ground state solutions. Unentangled ground states occur at finite values of the Hamiltonian parameters satisfying well-defined balancing conditions between the applied field and the interaction strengths. These conditions are analytically determined together with the type of magnetic orderings compatible with factorization and the corresponding values of the fundamental observables such as energy and magnetization. The method is applied to a series of examples of increasing complexity, including translationally invariant models with short, long, and infinite ranges of interaction, as well as systems with spatial anisotropies, in lower and higher dimensions. We also illustrate how the general method, besides yielding a large series of exact results for complex models in any dimension, recovers, as particular cases, the results previously achieved on simple models in low dimensions exploiting direct methods based on factorized mean-field ansatz.
We exploit mappings between quantum and classical systems in order to obtain a class of two-dimensional classical systems characterised by long-range interactions and with critical properties equivalent to those of the class of one-dimensional quantum systems treated by the authors in a previous publication. In particular, we use three approaches: the Trotter-Suzuki mapping, the method of coherent states, and a calculation based on commuting the quantum Hamiltonian with the transfer matrix of a classical system. This enables us to establish universality of certain critical phenomena by extension from the results in the companion paper for the classical systems identified.
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.