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The Philosophical Magazine A Journal of Theoretical Experimental and Applied Physics

The emergence domain of an exact ground state in a non-integrable system: the case of the polyphenylene type of chains

Réka Trencsényi, Zsolt Gulácsi

Published Sep 3, 2012• 5 citations
Publisher

Abstract

A technique based on the transformation in positive semidefinite form of the Hamiltonian (Ĥ) is developed to allow the study of the enlargement possibilities of the emergence domain of a given state. For this reason the calculation of exact ground states for the polyphenylene type of non-integrable chains is performed. Using different exact transcriptions of Ĥ in positive semidefinite form and different solutions of the matching equations, the same type of ground state is deduced in different regions of the parameter space, hence a more global view of its emergence possibilities is obtained. With this procedure an enlarged view of the appearance in the phase diagram of different studied phases can be achieved.

DOI
10.1080/14786435.2012.716527

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A technique based on the transformation in positive semidefinite form of the Hamiltonian (Ĥ) is developed to allow the study of the enlargement possibilities of the emergence domain of a given state. For this reason the calculation of exact ground states for the polyphenylene type of non-integrable chains is performed. Using different exact transcriptions of Ĥ in positive semidefinite form and different solutions of the matching equations, the same type of ground state is deduced in different regions of the parameter space, hence a more global view of its emergence possibilities is obtained. With this procedure an enlarged view of the appearance in the phase diagram of different studied phases can be achieved.
Quantum mechanical many-body systems described by an arbitrary Hamiltonian ( ) are analyzed. It is shown how positive semidefinite operator ( ) properties are able to lead in this case to exact results related to the ground state and the low lying part of the excitation spectrum. This is done independent on dimensionality and integrability. The technique first casts the Hamiltonian in a positive semidefinite form in exact terms ( = + C , where C is a scalar). Second, the ground state is deduced by constructing the most general Hilbert space vector, on which applying , one obtains zero as a result. It is underlined, that the procedure, if applied for variable total number of particles N, allows to obtain information also related to the low lying part of the excitation spectrum. The uniqueness of the ground states can be demonstrated via the study of the kernel of ' = − C . The physical properties of the obtained phases are deduced based on ground state expectation values calculated in terms of the constructed ground states. Since for a fixed structure of , usually the transformation = + C is exact only in a restricted parameter space domain ( ), the deduced ground states are present only in . A global view on the phase diagram is obtained by different transformations of in positive semidefinite form.
Abstract A technique based on positive semidefinite operator properties has been used in deducing exact ground states for hexagon chains of polyphenylene type (hexagons interconnected by bonds build up the 1D periodic structure), placed in a constant external magnetic field B perpendicular to the surface containing the system. The used Hubbard type of model takes into account B by Peierls phase factors, and has the peculiar property to provide only flat bands in the bare band structure for elevated external field values independent on the parameters entering in the kinetic part of the Hamiltonian. The deduced ground states are non‐magnetic or ferromagnetic in character, localized in the thermodynamic limit, and besides the fact that B is able to switch the system between different phases, emphasize that the localization length and connectivity conditions can be tuned by external magnetic fields.
The transformation of a Hamiltonian (connected to a quantum mechanical many-body system) in positive semidefinite form represents an important procedure in a powerful tool used in deducing exact results for nonintegrable systems.Motivated by this fact, we report in this paper the exact transformation of a Hubbard type of Hamiltonian (describing in the presented case an itinerant fermionic quadrilateral chain) in positive semidefinite form, in a such a way that the kinetic and interaction energy terms of the Hamiltonian are treated similarly together, and are provided by the same positive semidefinite expression.For this reason, non-linear fermionic contributions are used in block operators from which the positive semidefinite forms are constructed.The transformation is possible to be done when a system of nonlinear equations, called matching equations, are satisfied.The procedure of solving the matching equations is also indicated.
We model conducting pentagon chains with a multi-orbital Hubbard model and prove that well below half filling, exact ferromagnetic ground states appear. The rigorous method we use is based on the transformation of original hamiltonian into positive semidefinite form. This technique is independent of the spatial dimension and does not require integrability of the model. The obtained ferromagnetism is connected to dispersionless bands but in a much broader sense than flat-band ferromagnetism requires, where on every site a Hubbard term is present. In our case, only a small percentage of, even randomly distributed, sites are only interacting.