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arXiv:1912.07888 · Philosophical Magazine Letters

Itinerant surfaces with spin-orbit couplings, correlations and external magnetic fields: exact results

Nóra Kucska, Zsolt Gulácsi

Published Mar 4, 2019• 1 citation
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Abstract

We analyze, in exact terms, multiband 2D itinerant correlated fermionic systems with many-body spin-orbit interactions, and in-plane external magnetic fields. Even if such systems with broad applicability in leading technologies are non-integrable, we set up an exact solution procedure for them, which is described in details. Casting the Hamiltonian in positive semidefinite form, the technique leads to the ground state, and also characterises the low lying excitation spectrum.

DOI
10.1080/09500839.2019.1634291

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We analyze, in exact terms, multiband 2D itinerant correlated fermionic systems with many-body spin-orbit interactions, and in-plane external magnetic fields. Even if such systems with broad applicability in leading technologies are non-integrable, we set up an exact solution procedure for them, which is described in details. Casting the Hamiltonian in positive semidefinite form, the technique leads to the ground state, and also characterises the low lying excitation spectrum.
We set up and describe in details an exact solution procedure for a 2D itinerant correlated fermionic many-body system in the presence of spin-orbit interactions, onsite Coulomb repulsion in the correlated band, and in-plane external magnetic fields.The model is non-integrable, but using the method based on positive semidefinite operator properties, exact ground states are provided below half filling, at 3/4, and above 3/4 filling.The presented technique provides the possibility to study in exact terms a 2D phenomenon with broad applicability in leading technologies. I.
A 2D square, two-bands, strongly correlated and non-integrable system is analysed exactly in the presence of many-body spin–orbit interactions via the method of Positive Semidefinite Operators. The deduced exact ground states in the high concentration limit are strongly entangled, and given by the spin–orbit coupling are ferromagnetic and present an enhanced carrier mobility, which substantially differs for different spin projections. The described state emerges in a restricted parameter space region, which however is clearly accessible experimentally. The exact solutions are provided via the solution of a matching system of equations containing 74 coupled, non-linear and complex algebraic equations. In our knowledge, other exact results for 2D interacting systems with spin–orbit interactions are not present in the literature.
I discuss many-body models for correlated fermions in two space dimensions which can be solved exactly using group theory. The simplest example is a model of a quantum Hall system: two-dimensional (2D) fermions in a constant magnetic field and a particular non-local four-point interaction. It is exactly solvable due to a dynamical symmetry corresponding to the Lie algebra gl ∞ ⊕ gl ∞ . There is an algorithm to construct all energy eigenvalues and eigenfunctions of this model. The latter are, in general, many-body states with spatial correlations. The model also has a non-trivial zero temperature phase diagram. I point out that this QH model can be obtained from a more realistic one using a truncation procedure generalizing a similar one leading to mean field theory. Applying this truncation procedure to other 2D fermion models I obtain various simplified models of increasing complexity which generalize mean field theory by taking into account non-trivial correlations but nevertheless are treatable by exact methods.
Recently the interest in non-Neel ground states for antiferromagnetically interacting localized spin systems has been revived. We discuss to what extent the generalization of such states is possible for itinerant electrons. Multiple spin correlation functions can be used to enumerate nontrivial order parameters for localized spins. This approach can be applied to itinerant systems without any modification. However, for the transition which breaks translational symmetry, electronic states far away from the Fermi surface are important and, hence, there is no distinct energetic mechanism for the nontrivial magnetic ordering. The only exception is the Fermi surface with the nesting property. As it is shown, among all phases known for localized spins only p-type spin nematic and chiral spin states can appear as a result of a Fermi liquid breakdown with respect to the electron-hole condensation near the Fermi surface. All other phases with zero local average spin known for localized systems, either are accompanied by the charge density wave or persistent orbital currents creating orbital magnetic moments, or can not appear as a result of a phase transition driven by such mechanism. Although two aforementioned phases are equivalent to their counterparts for localized spins from the symmetry viewpoint, the driving order parameter in an itinerant case is not a multiple spin average, but a wave function of the electron-hole condensate.
Orbital magnetism is a purely quantum phenomenon that reflects intrinsic electronic properties of solids, yet its microscopic description in interacting multiband systems remains incomplete. We develop a general quantum many-body framework for orbital magnetic responses based on the Luttinger-Ward functional. Starting from the Dyson equation, we reformulate the thermodynamic potential in a weak magnetic field and construct a controlled expansion in powers of BBB applicable to correlated electron systems. A key technical advance is a modified ``Fourier'' representation using noncommutative coordinates, which allows the thermodynamic potential to be expressed in an effective momentum space where the magnetic field acts perturbatively. This formulation makes analytic progress possible within the Moyal algebra. As an application, we derive the spontaneous orbital magnetization and express it entirely in terms of the zero-field Hamiltonian renormalized by the self-energy. For frequency-dependent but Hermitian self-energies, we generalize the orbital magnetic moment and Berry curvature to momentum-frequency space and identify two gauge-invariant contributions built from these quantities. For frequency-independent self-energies the result reduces to the familiar geometric formula for noninteracting systems. This framework provides a unified foundation for computing orbital magnetic responses in correlated multiband materials.
The microscopic mechanism of itinerant ferromagnetism is a long-standing problem due to the lack of non-perturbative methods to handle strong magnetic fluctuations of itinerant electrons. We have non-pertubatively studied thermodynamic properties and magnetic phase transitions of a two-dimensional multi-orbital Hubbard model exhibiting ferromagnetic ground states. Quantum Monte-Carlo simulations are employed, which are proved in a wide density region free of the sign problem usually suffered by simulations for fermions. Both Hund's coupling and electron itinerancy are essential for establishing the ferromagnetic coherence. No local magnetic moments exist in the system as a priori, nevertheless, the spin channel remains incoherent showing the Curie-Weiss type spin magnetic susceptibility down to very low temperatures at which the charge channel is already coherent exhibiting a weakly temperature-dependent compressibility. For the SU(2) invariant systems, the spin susceptibility further grows exponentially as approaching zero temperature in two dimensions. In the paramagnetic phase close to the Curie temperature, the momentum space Fermi distributions exhibit strong resemblance to those in the fully polarized state. The long-range ferromagnetic ordering appears when the symmetry is reduced to the Ising class, and the Curie temperature is accurately determined. These simulations provide helpful guidance to searching for novel ferromagnetic materials in both strongly correlated ddd-orbital transition metal oxide layers and the ppp-orbital ultra-cold atom optical lattice systems.