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Fermions in two dimensions and exactly solvable models
KTH, School of Engineering Sciences (SCI), Theoretical Physics, Mathematical Physics.
2011 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This Ph.D. thesis in mathematical physics concerns systems of interacting fermions with strong correlations. For these systems the physical properties can only be described in terms of the collective behavior of the fermions. Moreover, they are often characterized by a close competition between fermion localization versus delocalization, which can result in complex and exotic physical phenomena.

Strongly correlated fermion systems are usually modelled by many-body Hamiltonians for which the kinetic- and interaction energy have the same order of magnitude. This makes them challenging to study as the application of conventional computational methods, like mean field- or perturbation theory, often gives unreliable results. Of particular interest are Hubbard-type models, which provide minimal descriptions of strongly correlated fermions. The research of this thesis focuses on such models defined on two-dimensional square lattices. One motivation for this is the so-called high-Tc problem of the cuprate superconductors.

A main hypothesis is that there exists an underlying Fermi surface with nearly flat parts, i.e. regions where the surface is straight. It is shown that a particular continuum limit of the lattice system leads to an effective model amenable to computations. This limit is partial in that it only involves fermion degrees of freedom near the flat parts. The result is an effective quantum field theory that is analyzed using constructive bosonization methods. Various exactly solvable models of interacting fermions in two spatial dimensions are also derived and studied.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2011. , viii, 96 p.
Series
Trita-FYS, ISSN 0280-316X ; 2011:56
Keyword [en]
Bosonization, Exactly solvable models, Hubbard model, Mean field theory, Quantum field theory, Strongly correlated systems
National Category
Condensed Matter Physics
Identifiers
URN: urn:nbn:se:kth:diva-50471ISBN: 978-91-7501-174-5 (print)OAI: oai:DiVA.org:kth-50471DiVA: diva2:462009
Public defence
2011-12-16, Svedbergssalen, Roslagstullsbacken 21, AlbaNova, Stockholm, 10:00 (English)
Opponent
Supervisors
Note
QC 20111207Available from: 2011-12-07 Created: 2011-12-06 Last updated: 2011-12-07Bibliographically approved
List of papers
1. Partially Gapped Fermions in 2D
Open this publication in new window or tab >>Partially Gapped Fermions in 2D
2010 (English)In: Journal of statistical physics, ISSN 0022-4715, E-ISSN 1572-9613, Vol. 139, no 6, 1033-1065 p.Article in journal (Refereed) Published
Abstract [en]

We compute mean field phase diagrams of two closely related interacting fermion models in two spatial dimensions (2D). The first is the so-called 2D t-t'-V model describing spinless fermions on a square lattice with local hopping and density-density interactions. The second is the so-called 2D Luttinger model that provides an effective description of the 2D t-t'-V model and in which parts of the fermion degrees of freedom are treated exactly by bosonization. In mean field theory, both models have a charge-density-wave (CDW) instability making them gapped at half-filling. The 2D t-t'-V model has a significant parameter regime away from half-filling where neither the CDW nor the normal state are thermodynamically stable. We show that the 2D Luttinger model allows to obtain more detailed information about this mixed region. In particular, we find in the 2D Luttinger model a partially gapped phase that, as we argue, can be described by an exactly solvable model.

Keyword
2D lattice fermion systems, Phase diagrams, Mean field theory
National Category
Other Physics Topics
Identifiers
urn:nbn:se:kth:diva-27875 (URN)10.1007/s10955-010-9971-2 (DOI)000277715600007 ()2-s2.0-77952412411 (Scopus ID)
Note
QC 20110111Available from: 2011-01-11 Created: 2011-01-03 Last updated: 2017-12-11Bibliographically approved
2. Exact Solution of a 2D Interacting Fermion Model
Open this publication in new window or tab >>Exact Solution of a 2D Interacting Fermion Model
2012 (English)In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 314, no 1, 1-56 p.Article in journal (Refereed) Published
Abstract [en]

We study an exactly solvable quantum field theory (QFT) model describing interacting fermions in 2+1 dimensions. This model is motivated by physical arguments suggesting that it provides an effective description of spinless fermions on a square lattice with local hopping and density-density interactions if, close to half filling, the system develops a partial energy gap. The necessary regularization of the QFT model is based on this proposed relation to lattice fermions. We use bosonization methods to diagonalize the Hamiltonian and to compute all correlation functions. We also discuss how, after appropriate multiplicative renormalizations, all short- and long distance cutoffs can be removed. In particular, we prove that the renormalized two-point functions have algebraic decay with non-trivial exponents depending on the interaction strengths, which is a hallmark of Luttinger-liquid behavior.

Keyword
Coupled Luttinger Liquids, Arbitrary Dimension, Field Theory, Surface, Bosonization, Renormalization, Behavior, Particle, System, Gas
National Category
Other Physics Topics
Identifiers
urn:nbn:se:kth:diva-50581 (URN)10.1007/s00220-012-1518-8 (DOI)000307271600001 ()2-s2.0-84863311171 (Scopus ID)
Funder
Swedish Research Council
Note

QC 20120910. Updated from submitted to published.

Available from: 2011-12-06 Created: 2011-12-06 Last updated: 2017-12-08Bibliographically approved
3. Gauge invariance, correlated fermions, and Meissner effect in 2+1 dimensions
Open this publication in new window or tab >>Gauge invariance, correlated fermions, and Meissner effect in 2+1 dimensions
(English)Article in journal (Other academic) Submitted
Abstract [en]

We present a 2+1 dimensional quantum gauge theory model with correlated fermions that is exactly solvable by bosonization. This model gives an effective description of partially gapped fermions on a square lattice that have density-density interactions and are coupled to photons. We show that the photons in this model are massive due to gauge-invariant normal-ordering, similarly as in the Schwinger model. Moreover, the exact excitation spectrum of the model has two gapped and one gapless mode. We also compute the magnetic field induced by an external current and show that there is a Meissner effect. We find that the transverse photons have significant effects on the low-energy properties of the model even if the fermion-photon coupling is small.

National Category
Other Mathematics
Identifiers
urn:nbn:se:kth:diva-50645 (URN)
Note
QS 2011 QS 20120326Available from: 2011-12-07 Created: 2011-12-07 Last updated: 2012-03-26Bibliographically approved
4. Partial continuum limit of the 2D Hubbard model
Open this publication in new window or tab >>Partial continuum limit of the 2D Hubbard model
(English)Article in journal (Other academic) Submitted
National Category
Other Mathematics
Identifiers
urn:nbn:se:kth:diva-50646 (URN)
Note
QS 2011 QS 20120326Available from: 2011-12-07 Created: 2011-12-07 Last updated: 2012-03-26Bibliographically approved

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