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arXiv:1405.5176

Non-Linear Methods in Strongly Correlated Electron Systems

Matteo Bazzanella, Johan Nilsson

Published May 20, 2014• 2 citations
arXivHTMLPDF

Abstract

We analyze the structure of the group of (local) non-linear canonical transformations that exist in a system with n fermionic modes. To perform our study we develop an alternative framework to represent the generators of these canonical transformations; indeed we show how their definition, understanding and control is significantly improved using the Majorana fermion representation. These canonical transformations have the structure of a Lie group and we provide a representation for the elements of the Lie algebra that is very convenient both conceptually and practically (computationally): indeed our framework yields as side product an extremely effective tool to handle and work with SU(2n) Lie groups and algebras. Granting an enhanced control of the group of non-linear canonical transformation, our framework can be helpful in the study of strongly correlated electron systems, since it allows to easily identify fermionic degrees of freedom able to capture part of the correlations and thus may give a simpler representation of the Hamiltonian. Thanks to our analysis, also symmetry-based studies of the quantum Hamiltonians can be improved, since the simple representation of the generators of the canonical group permits to identify and understand otherwise hidden symmetries of difficult interpretation. The main aim of this work is to provide a comprehensive, general and scrupulous analysis of this framework, that we already applied in some circumstances. Therefore specific applications will not be presented in detail, since they can be found elsewhere, while only the formal and conceptual aspects will be developed thoroughly.

DOI
10.48550/arxiv.1405.5176

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We analyze the structure of the group of (local) non-linear canonical transformations that exist in a system with n fermionic modes. To perform our study we develop an alternative framework to represent the generators of these canonical transformations; indeed we show how their definition, understanding and control is significantly improved using the Majorana fermion representation. These canonical transformations have the structure of a Lie group and we provide a representation for the elements of the Lie algebra that is very convenient both conceptually and practically (computationally): indeed our framework yields as side product an extremely effective tool to handle and work with SU(2n) Lie groups and algebras. Granting an enhanced control of the group of non-linear canonical transformation, our framework can be helpful in the study of strongly correlated electron systems, since it allows to easily identify fermionic degrees of freedom able to capture part of the correlations and thus may give a simpler representation of the Hamiltonian. Thanks to our analysis, also symmetry-based studies of the quantum Hamiltonians can be improved, since the simple representation of the generators of the canonical group permits to identify and understand otherwise hidden symmetries of difficult interpretation. The main aim of this work is to provide a comprehensive, general and scrupulous analysis of this framework, that we already applied in some circumstances. Therefore specific applications will not be presented in detail, since they can be found elsewhere, while only the formal and conceptual aspects will be developed thoroughly.
In this thesis some new ideas to perform the analysis of Strongly Correlated Electronic Systems (SCES) are developed. In particular the use of non-linear canonical transformations is considered thoroughly. Using such transformations it is possible, in some circumstances, to simplify the quantum problem redefining the fermionic degrees of freedom used to describe the system. To understand and use effectively these non-linear transformations it is convenient to work in the Majorana fermion representation, i.e., to represent the quantum mechanical operators in terms of Majorana fermions. These objects can be imagined as algebraic constituents of the fermionic degrees of freedom. In a fermionic system, different equivalent sets of (emergent) Majorana fermions can be used to build the fermionic operators that characterize the system. The non-linear transformations can be seen as a way to mix these equivalent sets. Thanks to this insight, it becomes possible to determine the full structure of the group of canonical transformations and to identify an advantageous framework, which allows their use in the study of a generic SCES system. To test these statements the Hubbard and the Kondo lattice models were intensively studied making use of non-linear canonical transformations, obtaining interesting results in both cases. For example, in the Hubbard model a free fermion mean-field description of the paramagnetic Mott insulator was identified, while in the Kondo lattice it was possible to describe already at mean-field level the spin-selective Kondo insulating phase, consistently (from a quantitative and qualitative point of view) with the known numerical results. Moreover the method elaborated for the study of the Hubbard model is suitable for a systematic generalization to other situations and shows great room for improvement. These results prove that, thanks to the redefinition of the degrees of freedom used in the analysis of the system, it becomes possible to obtain quite non-trivial results already at mean-field level, or to consider very involved (but meaningful) correlated quantum states via simple variational trial states. This will potentially permit a more judicious and profitable choice of the fundamental degrees of freedom, allowing for an improvement of the efficiency of the analytical and numerical techniques used in the analysis of many SCES systems.
We study in this series the group theoretical structure of Fermion many-body systems arising from the canonical anticommutation relation of the annihilation-creation operators. Owing to the canonical anticommutation relation, a Fermion system with N single particle states has at least six Lie algebras of Fermion operators, a U(N), an SO(2N), an SO(2N+1), an SO(2N+2) and two U(N+1) Lie algebras. There are also two Clifford algebras of 2N and 2N+1 dimensions. The Fermion space is shown to belong to the spinor representations of the SO(2N), SO(2N+1) and SO(2N+2) groups. The canonical transformations generated by the U(N), SO(2N) and SO(2N+1) Lie algebras are characterized as the transformations to induce the linear U(N), SO(2N) and SO(2N+1) transformations for the Clifford algebras. The independent (quasi-) particle type wave functions of three kinds including the Hartree-Fock and Hartree-Bogoliubov wave functions are constructed by means of the canonical transformations and their relationship is studied. We derive three exact generator coordinate representations for state vectors in which the generator coordinates are the U(N), SO(2N) or SO(2N+1) group and the generating functions are the independent (quasi-) particle type wave functions. We characterize the structures of state vectors in the generator coordinate representations and study the relationship of the three representations.
Symmetry plays a central role in modern physics, from classifying quantum states to characterizing phases of matter through spontaneous symmetry breaking. In interacting fermionic systems with multiple internal degrees of freedom, however, determining the full continuous symmetry group and classifying possible order parameters remain challenging. In this work, we present a systematic framework for analyzing continuous symmetries and identifying candidate order parameters in such systems. By mapping the Hamiltonian to a Majorana representation, we obtain the generators of continuous symmetries from the Lie algebra of operators that commute with the Hamiltonian. We then identify the structure of this Lie algebra using the theory of semisimple Lie algebras. Building on representation theory, we further develop a systematic method for exhaustively enumerating candidate order parameters. By decomposing the exterior-power representations induced by the symmetry algebra on the Majorana space and incorporating discrete lattice symmetries, we classify these order parameters according to the symmetries they break. (Abridged. Please see the PDF manuscript for the complete abstract and specific model applications.)
We address the question of identifying degrees of freedom for quantum systems. Typically, quasi-particle descriptions of correlated matter are based upon the canonical algebras of bosons or fermions. Here we highlight that a special class of non-canonical algebras also offer useful quantum degrees of freedom, allowing for the development of quasi-particle descriptions which go beyond the weakly correlated paradigm. We give a broad overview of such algebras for spin, electron and local moment systems, and outline important test problems upon which to develop the framework.
A new many-body theory for fermions is proposed wihich is based on the SO(2N+1) Lie algebra of the fermion operators consisted of the annihilation-creation operators and the pair operators. A New cannonical transformation, which is the extension of the Bogoliubov transformation to the SO(2N+1) group, is introduced. A new bose representation for the fermion Lie operators is obtained by mapping the fermion Lie operators into the regular representation of the SO(2N+1) group. The annihilation-creation operators and the pair operators of fermions are represented by the closed first order differential operators on the SO(2N+1) group. An exact representation of fermion wavefunctions in a form similar to the wavefunction of the generator coordinate method is obtained making use of the SO(2N+1) canonical transformation. The physical fermion space is shown to be the irreducible spinor representation of the SO(2N+1) group. The dynamics of fermions in the bose representation space is shown to represent rotations of a 2N+1 dimensional rotator.
Non-canonical degrees of freedom provide one of the most promising routes towards characterising a range of important phenomena in condensed matter physics. Potential candidates include the pseudogap regime of the cuprates, heavy-fermion behaviour, and also indeed magnetically ordered systems. Nevertheless it remains an open question whether non-canonical algebras can in fact provide legitimate quantum degrees of freedom. In this paper we survey progress made on this topic, complementing distinct approaches so as to obtain a unified description. In particular we obtain a novel exact representation for a self-energy-like object for non-canonical degrees of freedom. We further make a resummation of density correlations to obtain analogues of the RPA and GW approximations commonly employed for canonical degrees of freedom. We discuss difficulties related to generating higher-order approximations which are consistent with conservation laws, which represents an outstanding issue. We also discuss how the interplay between canonical and non-canonical degrees of freedom offers a useful paradigm for organising the phase diagram of correlated electronic behaviour.