Exactly Soluble Model of Interacting Electrons

We present an algebraic framework for interacting extended quantum systems to study complex phenomena characterized by the coexistence and competition of different states of matter. We start by showing how to connect different (spin-particle-gauge) languages by means of exact mappings (isomorphisms) that we name dictionaries and prove a fundamental theorem establishing when two arbitrary languages can be connected. These mappings serve to unravel symmetries which are hidden in one representation but become manifest in another. In addition, we establish a formal link between seemingly unrelated physical phenomena by changing the language of our model description. This link leads to the idea of universality or equivalence. Moreover, we introduce the novel concept of emergent symmetry as another symmetry guiding principle. By introducing the notion of hierarchical languages, we determine the quantum phase diagram of lattice models (previously unsolved) and unveil hidden order parameters to explore new states of matter. Hierarchical languages also constitute an essential tool to provide a unified description of phases which compete and coexist. Overall, our framework provides a simple and systematic methodology to predict and discover new kinds of orders. Another aspect exploited by the present formalism is the relation between condensed matter and lattice gauge theories through quantum link models. We conclude discussing applications of these dictionaries to the area of quantum information and computation with emphasis in building new models of computation and quantum programming languages.

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“…This transformation was originally introduced by Mattis and Nam [54] to solve the following Hubbard-like model…”

Section: A Su (2) Heisenberg Magnetsmentioning

confidence: 99%

Algebraic approach to interacting quantum systems

Batista

Ortíz

2004

Advances in Physics

132

159

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“…The model is reduced to chain of spinless fermions (2) with the chemical potential U [13]. According to [13], the ground state degeneracy is dependent on whether κ = U τ (δ) > 4 or otherwise. In the thermodynamic limit there are zero energy Majorana states, they are realized in the interval |κ| ≤ 4.…”

Section: The Hofstadter Model Of Interacting Electronsmentioning

confidence: 99%

“…Such a criterion should connect the value of the gap of a topological insulator with the magnitude of the Coulomb repulsion between fermions. Taking into account the exact solution of fermion chain model [13], we calculate the stability of topological state in the Hofstadler model of interacting electrons. Result does not stripe geometry depend on the symmetry of the lattice, we believe that it is generic and can be applicable to different 2D topological insulators.…”

Section: Introductionmentioning

confidence: 99%

Topological states in the Hofstadter model on a honeycomb lattice

Karnaukhov

2019

Physics Letters A

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We provide a detailed analysis of a topological structure of a fermion spectrum in the Hofstadter model with different hopping integrals along the x, y, z-links (tx = t, ty = tz = 1), defined on a honeycomb lattice. We have shown that the chiral gapless edge modes are described in the framework of the generalized Kitaev chain formalism, which makes it possible to calculate the Hall conductance of subbands for different filling and an arbitrary magnetic flux φ. At half-filling the gap in the center of the fermion spectrum opens for t > tc = 2 φ , a quantum phase transition in the 2D-topological insulator state is realized at tc. The phase state is characterized by zero energy Majorana states localized at the boundaries. Taking into account the on-site Coulomb repulsion U (where U << 1), the criterion for the stability of a topological insulator state is calculated at t << 1, t ∼ U . Thus, in the case of U > 4∆, the topological insulator state, which is determined by chiral gapless edge modes in the gap ∆, is destroyed.

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“…The Hubbard model with correlated hopping on a chain has been proposed and solved exactly in [3][4][5]. Mattic and Nam (MN) proposed modification of the Hubbard model for interacting electrons forming pairs, and solved it exactly in special point [6] (better known as the Kitaev point [7]). In contrast to traditional Hubbard model [1], the MN model describes topological states of interacting electrons [7][8][9], quantum topological phase transition between topological trivial and nontrivial phases.…”

Section: Introductionmentioning

confidence: 99%

“…In contrast to traditional Hubbard model [1], the MN model describes topological states of interacting electrons [7][8][9], quantum topological phase transition between topological trivial and nontrivial phases. In this context, it is interesting to discuss a new model, which is a modification [3,6], the exact solution of which takes place for arbitrary one-site interaction, correlated hopping and pairing.…”

Section: Introductionmentioning

confidence: 99%

Exactly solvable chain of interacting electrons with correlated hopping and pairing

Karnaukhov

2019

Physics Letters A

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A generalization of the Mattic-Nam model (J.Math.Phys., 13 (1972), 1185), which takes into account a correlated hopping and pairing of electrons, is proposed, its exact solution is obtained. In the framework of the model the stability of the zero energy Majorana fermions localized at the boundaries is studied in the chain in which electrons interact through both the one-site Hubbard interaction U and the correlated hopping and pairing t. The ground-state phase diagram of the model is calculated in the coordinates t − U , the region of existence of topological states is determined. It is shown that low-energy excitations destroy bonds between fermions in the chain, leading to a dielectric state.

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Exactly Soluble Model of Interacting Electrons

Abstract: We diagonalize a many-fermion Hamiltonian consisting of terms quadratic as well as quartic in the field operators. A dual spectrum of eigenstates is an interesting result. We also derive a formula for obtaining the free energy at finite temperature.

Cited by 13 publications

References 6 publications

Algebraic approach to interacting quantum systems

Algebraic approach to interacting quantum systems

Topological states in the Hofstadter model on a honeycomb lattice

Exactly solvable chain of interacting electrons with correlated hopping and pairing

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