Many-body spin-orbit coupling in two dimensional
correlated itinerant systems in the presence of external in-plane magnetic fields
Nóra Kucska1, Zsolt Gulácsi2
1PhD student, 2professor
1,2Department of Theoretical Physics, University of Debrecen, Hungary
ABSTRACT
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, on- site 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. INTRODUCTION
Systems with spin-orbit interactions (SOI) are representing a broad area of current and exciting research (see the review paper by Galitzky and Spielman (1), which has essential effects on numerous condensed matter phenomena of large interest today, ranging from quantum magnets (2), topological phases (3), via ultracold atom experiments (4), to Majorana fermions (5). The applications mostly appear in low dimensional systems (6-12), and during experiences external fields are often present.
The most interesting condensed matter applications being related to strongly correlated systems.
Contrary to its importance, even if exact treatments in 2D relating the effect of SOI on interacting many-body systems have been already started (12), in the presence of external magnetic fields, such types of studies are completely missing from the literature. Our objective in this paper is to fill up this gap. We aim to set up the details of the calculation procedure for such type of situation considering Hamiltonians describing realistic correlated systems.
The main problem in which we blunder in this attempt is that the here studied 2D systems are non-integrable, so special techniques must be used in order to describe them in exact terms. For this reason, we use the method of positive semidefinite operators whose applicability does not depend on dimensionality and integrability (13-16).
MultiScience - XXXIII. microCAD International Multidisciplinary Scientific Conference University of Miskolc, 23-24 May, 2019, ISBN 978-963-358-177-3
DOI: 10.26649/musci.2019.023
One notes, that the method has been previously applied in conditions unimaginable before in the context of exact solutions, as disordered systems in 2D (17); multiband systems in 2D (18) and 3D (19); stripe, checkerboard and droplet states in 2D (20);
delocalization effect of the Hubbard repulsion in 2D (21); or different non-integrable chain structures (14,15,22-24). The method transforms the Hamiltonian first in exact terms in positive semidefinite form 𝐻̂ = 𝑂̂ + 𝐶, where 𝑂̂ is a positive semidefinite operator, while 𝐶 is a scalar. After this transformation, in a second step, one deduces the exact ground state by constructing the most general wave vector |𝜓𝐺⟩ which satisfies the requirement 𝑂̂|𝜓𝐺⟩ = 0. After obtaining |𝜓𝐺⟩, the physical properties of the system in the ground state (and also low lying excited states) can be analysed.
The remaining part of the paper is structured as follows: Section II. describes the studied system, Section III. presents the transformation in positive semidefinite form of the Hamiltonian, Section IV. deduces the exact ground states, while the Summary and Conclusions in Section V. closes the presentation.
II. THE SYSTEM ANALYSED The Hamiltonian of the system has the form
𝐻̂ = ∑ ∑ ∑(𝑘𝐢,𝐢+𝐫𝑝,𝑝′;𝜎,𝜎′ 𝑐̂𝑝,𝐢,𝜎†
𝑐̂𝑝′,𝐢+𝐫,𝜎′
𝑝,𝑝′
+ 𝐻. 𝑐) + ∑ ∑ 𝑈𝑝,𝐢 𝑝 𝐢
𝑛̂𝑝,𝐢,↑𝑛̂𝑝,𝐢,↓
𝜎,𝜎′
+ ∑ ∑ ℎ⃑ 𝑝,𝐢𝑐̂𝑝,𝐢,𝜎
†𝜎 𝜎,𝜎′𝑐̂𝑝,𝐢,𝜎′
𝜎,𝜎′ 𝑝,𝐢
𝐢,𝐫
(1) where the first term represents the kinetic part of the Hamiltonian 𝐻̂𝑘𝑖𝑛, the second term is the interaction part of the Hamiltonian 𝐻̂𝑖𝑛𝑡, while the last term describes the interaction with the external magnetic field 𝐻̂ℎhence, in fact one has 𝐻̂ = 𝐻̂𝑘𝑖𝑛+ 𝐻̂𝑖𝑛𝑡+ 𝐻̂ℎ. At the level of 𝐻̂𝑘𝑖𝑛, in order to have a realistic 2D surface description one takes into account two bands, denoted hereafter by 𝑎, 𝑏, hence the band indices 𝑝, 𝑝′ = 𝑎, 𝑏. Again in order to approach a real system, besides on-site one particle terms (𝐫 = 0), one takes into consideration nearest-neighbour (𝐫 = 𝐱𝟏, 𝐱𝟐, where 𝐱𝟏, 𝐱𝟐 are the Bravais vectors of the 2D lattice), and next nearest-neighbour ( 𝐫 = 𝐱𝟏 + 𝐱𝟐, 𝐱𝟏 − 𝐱𝟐 ) contributions. Furthermore, note that the 𝑘𝐢,𝐢+𝐫𝑝,𝑝′;𝜎,𝜎′ coefficient represents for (p=p', 𝐫 = 0); [(p=p', 𝐫 ≠ 0)] on-site potential; [hopping matrix element], while for (p≠p', 𝐫 = 0); [ (p≠p', 𝐫 ≠ 0)] on-site hybridization;
[hybridization matrix element]. Regarding 𝐻̂𝑖𝑛𝑡, since in itinerant many-body systems strong screening effects are present, we consider the on-site repulsive Coulomb interaction described by a Hubbard term in the correlated band. The many-body spin- orbit interactions being of one-particle type, are introduced in the kinetic part of the Hamiltonian, explicitly in the coefficients 𝑘𝐢,𝐢+𝐫𝑝,𝑝;𝜎,−𝜎, 𝐫 = 𝐱𝟏, 𝐱𝟐, see the detailed description in the text below. We underline, that even if usually the SOI contributions are small, they provide essential effects, since they break the double spin-projection degeneracy of each band, and their perturbational treatment is strongly misleading (12).
Regarding the Hamiltonian we use, the following aspects should be noted:
i) Real materials are of multiband type. Because of this reason, we use a two band system in (1), i.e. 𝑝 = 𝑎, 𝑏. However, we note that this choice not diminishes the applicability of the deduced results, since usually, the theoretical description of muliband systems is given by projecting the multiband structure in a few-band picture (25) projection which is stopped here only for its relative simplicity at two-bands level. We further underline, that in order to present more realistic systems containing transition elements as well, one of the two-bands, namely 𝑝 = 𝑏, is considered correlated (𝑈𝑏 > 0), while the second band (𝑝 = 𝑎) is taken non-correlated (𝑈𝑎 = 0). ii) In order to properly characterize the band structure of the described systems, the kinetic contributions in the Hamiltonian are considered up to next nearest - neighbour contributions both in hopping (𝑝 = 𝑝′), and in hybridization (𝑝 ≠ 𝑝′) matrix elements of 𝑘𝐢,𝐢+𝐫𝑝,𝑝′;𝜎,𝜎′. The next nearest-neighbour (𝐫 = 𝐱𝟏± 𝐱𝟐) does not contain spin-flip type contributions, and all of them are spin symmetric 𝑘𝐢,𝐢+𝐫𝑝,𝑝′;↑↑= 𝑘𝐢,𝐢+𝐫𝑝,𝑝′;↓↓ = 𝑘𝐢,𝐢+𝐫𝑝,𝑝′. iii) The nearest-neighbour kinetic contributions (𝐫 = 𝐱𝟏, 𝐱𝟐), both at hopping and hybridization level are again spin symmetric. The nearest-neighbour hybridizations do not contain spin-flip terms, but their nearest-neighbour hopping counterpart, given by the presence of SOI, has such contributions. These spin-flip hopping terms are of Rashba (𝜆𝑅𝑝, 𝑝 = 𝑎, 𝑏) and Dresselhaus (𝜆𝐷𝑝, 𝑝 = 𝑎, 𝑏) type (26). Consequently, one has for 𝐫 = 𝐱𝟏 the structure 𝑘𝐢,𝐢+𝐱
𝟏
𝑝,𝑝′;↑↓
= 𝜆𝑅𝑝− 𝑖 ∗ 𝜆𝑝𝐷 ,𝑘𝒊,𝒊+𝒙
𝟏
𝑝,𝑝′;↓↑
=
−𝜆𝑝𝑅− 𝑖 ∗ 𝜆𝐷𝑝, while for 𝐫 = 𝐱𝟐 the expressions 𝑘𝒊,𝒊+𝒙
𝟐
𝑝,𝑝′;↑↓
= 𝜆𝐷𝑝− 𝑖 ∗ 𝜆𝑝𝑅 , 𝑘𝒊,𝒊+𝒙
2
𝑝,𝑝′;↓↑
= −𝜆𝐷𝑝− 𝑖 ∗ 𝜆𝑅𝑝.
iv) In order to not diminish the effect of the spin-flip nearest-neighbour hopping terms produced by SOI, the external fields are only applied in-plane, hence without the z-component (ℎ𝑝,𝐢𝑧 = 0, ℎ𝑝,𝐢𝑥 , ℎ𝑝,𝐢𝑦 ≠ 0). v) The on-site local contributions both at on-site one particle potentials and on-site hybridizations level are considered spin- symmetric 𝑘𝒊,𝒊𝑝,𝑝′;↑↑ = 𝑘𝒊,𝒊𝑝,𝑝′;↓↓ = 𝑘𝒊,𝒊𝑝,𝑝′. Furthermore, on-site spin-flip contributions are missing (𝑘𝒊,𝒊𝑝,𝑝′;𝜎,−𝜎=0). We further underline, that the in-plane ℎ𝑝,𝐢𝑥 , ℎ𝑝,𝐢𝑦 contributions will additively renormalize the 𝑘𝐢,𝐢𝑝,𝑝,;𝜎,−𝜎 contributions as 𝑘̅𝐢,𝐢𝑝,𝑝;↑↓= 𝑘̅𝐢,𝐢𝑝,𝑝;↑↓+ ℎ𝑥− 𝑖ℎ𝑦, ( 𝑘̅𝐢,𝐢𝑝,𝑝;↓↑)∗= 𝑘̅𝐢,𝐢𝑝,𝑝;↑↓.
Fig.1. Unit cell defined at the lattice site 𝐢 with in-cell notations of sites 𝑛 = 1,2,3,4.
III. THE TRANSFORMATION OF THE HAMILTONIAN IN POSITIVE SEMIDEFINITE FORM
A. The positive semidefinite form of the Hamiltonian
After all these considerations let us turn back to [1] by presenting the transformation of the Hamiltonian.
In order to cast the 𝐻̂ from [1] in positive semidefinite form, we introduce two block operators 𝑄 = 𝐴, 𝐵 for each site 𝐢, which for a fixed 𝑄 value are defined as
𝑄̂𝐢 = ∑ ∑ ∑ 𝑞𝑄,𝑝,𝑛,𝛼𝑐̂𝑝,𝐢+𝐫𝐧,𝛼 ,
𝑛=1,2,3,4 𝛼=↑,↓
𝑝=𝑎,𝑏
(2) where in order 𝐫𝟏 = 𝟎, 𝐫𝟐 = 𝐱𝟏, 𝐫𝟑 = 𝐱𝟏 + 𝐱𝟐, and 𝐫𝟒 = 𝐱𝟐 ,see Fig.1. At a given lattice site 𝐢, for a fixed 𝑄 and 𝑝 value, the 𝑄̂𝐢 operator has 8 contributions, 4 for spin 𝛼 =↑, and other 4 for spin 𝛼 =↓. For fixed 𝛼 the mentioned 4 values denoted by 𝑛 = 1,2,3,4 are placed in the four corners of an elementary plaquette connected to the lattice site 𝐢.
Using [2], the starting system Hamiltonian 𝐻̂in [1] becomes of the form 𝐻̂ = 𝑃̂ + 𝑆𝑐,
(3) where 𝑃̂ represents a positive semidefinite operator, while 𝑆𝑐 a scalar. Taking into account that 𝑃̂ = 𝑃̂𝑄+ 𝑃̂𝑈 where 𝑃̂𝑈 = 𝑈𝑏∑ 𝑃̂𝐢 𝐢 , for 𝑈𝑏 > 0 one has
𝑃̂𝑄 = ∑ ∑ 𝑄̂𝐢𝑄̂𝐢†
𝑄=𝐴,𝐵 𝐢
, 𝑃̂𝐢 = 𝑛̂𝑏,𝐢,↑𝑛̂𝑏,𝐢,↓− (𝑛̂𝑏,𝐢,↑+𝑛̂𝑏,𝐢,↓) + 1,
𝑆𝑐 = 𝜂𝑁 − 𝑈𝑏𝑁Λ− ∑ 𝑑𝑖,𝑄 ,
𝑄=𝐴,𝐵
(4) where 𝑁 (𝑁Λ) represents the number of electrons (lattice sites), while 𝑑𝑖,𝑄represents the anti-commutator defined by 𝑑𝑖,𝑄={𝑄̂𝐢, 𝑄̂𝐢†}.
The corresponding matching equations which allow the transformation of the starting Hamiltonian from [1] into the form described by 𝐻̂ with components in [4] are as follows: One has 32 equations for nearest-neighbour contributions 𝑚 = 1,2, namely 16 for a fixed 𝑚 ± 1
−𝑘𝐢,𝐢+𝐱
𝐌
𝑝,𝑝′;𝜎,𝜎′
= ∑ (𝑞𝑄,2𝑚,𝑝,𝜎∗ 𝑞𝑄,1,𝑝′,𝜎′− 𝑞𝑄,3,𝑝,𝜎∗ 𝑞𝑄,6−2𝑚,𝑝′,𝜎′)
𝑄=𝐴,𝐵
,
(5)
and similarly one has 32 equations for the next nearest-neighbour contributions, as previously 16 for a fixed 𝑚 ± 1
−𝑘𝐢,𝐢+𝐱
𝟐+𝑚𝐱𝟏 𝑝,𝑝′;𝜎,𝜎′
= ∑ 𝑞𝑄,3+(1−𝑚)/2,𝑝,𝜎∗ 𝑞𝑄,1+(1−𝑚)/2,𝑝′,𝜎′ 𝑄=𝐴,𝐵
,
(6) Finally local (e.g. 𝐫 = 0) contributions give rise to 16 equations which can be written as
−𝑘𝐢,𝐢𝑝,𝑝′;𝜎,𝜎′ = [(1 − 𝛿𝑝,𝑝′) + (1 − 𝛿𝜎,𝜎′)𝛿𝑝,𝑝′+ 𝛿𝑝,𝑝′𝛿𝜎,𝜎′] + (𝜂 − 𝑈𝑝)𝛿𝑝,𝑝′𝛿𝜎,𝜎′
−[ℎ𝑥 − 𝑖𝑘(𝜎)ℎ𝑦]𝛿𝑝,𝑝′(1 − 𝛿𝜎,𝜎′)= ∑ ∑ 𝑞𝑄,𝑛,𝑝,𝜎∗ 𝑞𝑄,𝑛,𝑝′,𝜎′ 𝑄=𝐴,𝐵
𝑛=1,2,3,4
(7) where 𝑘(𝜎) = 𝛿↑,𝜎− 𝛿↓,𝜎. One has here 80 non-linear equations in total, with 32 numerical prefactors 𝑞𝑄,𝑛,𝑝,𝜎 , which are unknown and called „block operator parameters”, and the parameter 𝜂 entering in the ground state energy. The total number of Hamiltonian parameters (taking into account all possible spin dependences as well) is 76, so a proper description for a real material can be provided.
B. Solution of the matching equations
In order to start the deduction of the exact ground states, first we should deduce the numerical prefactors 𝑞𝑄,𝑝,𝑛,𝛼 of the block operators from [2] from the matching equations [5-7]. Starting this job, first we observe from [5-7] that all 𝑞𝑄=𝐴,𝑝,𝑛,𝛼 components can be given in function of the 𝑞𝑄=𝐴,𝑝,𝑛,𝛼 coefficients via the relation 𝑞𝐴,𝑝,𝑛,𝛼 = 𝑑𝑛,𝛼𝑞𝐵,𝑝,𝑛,𝛼, where the coefficients 𝑑𝑛,𝛼have the expression
𝑑𝑛,𝛼 = − (𝛿𝛼,↑
𝑦 +𝛿𝛼,↓
𝑥 ) 𝛿𝑛,1− (𝛿𝛼,↑
𝑣 +𝛿𝛼,↓
𝑧 ) 𝛿𝑛,2+ (𝑥∗𝛿𝛼,↑+ 𝑦∗𝛿𝛼,↓)𝛿𝑛,3+ (𝑧∗𝛿𝛼,↑+ 𝑣∗𝛿𝛼,↓)𝛿𝑛,4,
(8) where 𝑥, 𝑦, 𝑣, 𝑧 are numerical prefactors. After this step it results that the remaining 𝑞𝐵,𝑝,𝑛,𝛼 unknowns with 𝑝 = 𝑎 can be given in term of the 𝑞𝐵,𝑝,𝑛,𝛼 coefficients containing 𝑝 = 𝑏 via the relation 𝑞𝐵,𝑎,𝑛,𝛼 = 𝛼𝑛𝑞𝐵,𝑏,𝑛,𝛼, where one has for the numerical coefficients 𝛼𝑛 the expression
𝛼𝑛 = 𝛼1𝛿𝑛,1+𝛼2𝛿𝑛,2+ 𝛾0
𝛼1∗𝛿𝑛,3 +𝛾0 𝛼2∗𝛿𝑛,4,
(9) where 𝛾0 is an arbitrary real and positive parameter, while 𝛼1, 𝛼2 are two further numerical prefactors. In this manner, up to [9] only 8 unknown coefficients remain, namely 𝑞𝐵,𝑏,𝑛,𝛼 with 𝑛 = 1,2,3,4 and 𝛼 =↑, ↓. But it turns out that these 8 unknown coefficients are interdependent, and all can be expressed in function of one
block operator parameter, namely 𝑞𝐵,𝑏,𝑛=1,↑, via
𝑞𝐵,𝑏,1,↓ = 1
𝑤∗𝑞𝐵,𝑏,1,↑, 𝑞𝐵,𝑏,3,↓ = −𝑢∗
𝑤∗𝑞𝐵,𝑏,1,↑∗ , 𝑞𝐵,𝑏,3,↑ =|𝛼1|2 𝛾0
1
𝑢𝑦𝑥∗𝑞𝐵,𝑏,1,↑
𝑞𝐵,𝑏,2,↓ = 𝜔𝑞𝐵,𝑏,2,↑, 𝑞𝐵,𝑏,4,↓ = −𝛼1∗ 𝛼2∗
𝑢∗𝑦∗
𝑤∗𝑣∗𝑞𝐵,𝑏,2,↑∗ , 𝑞𝐵,𝑏,4,↑ =𝛼2∗𝛼1 𝛾0
𝜔∗
𝑢𝑦𝑧∗𝑞𝐵,𝑏,2,↑∗ (10) where |𝑞𝐵,𝑏,2,↑| = |𝜇||𝑞𝐵,𝑏,1,↑|, 𝜔 = 𝑧𝑤𝑥∗(1+𝑣𝑦∗)
𝑣𝑦∗(1+𝑧𝑥∗) , |𝑤| = |𝑢||𝑦|√𝛾0
|𝛼1| . The remaining unknown prefactors and 𝑞𝐵,𝑏,1,↑ can be determined numerically from four remaining coupled equations.
IV. THE GROUND STATE WAVE FUNCTIONS
The first deduced ground state wave function corresponding to the transformed Hamiltonian from [3] connected to the matching equations [4-7] is of the form
|𝜓1,𝑔⟩ = ∏ ( ∏ 𝑄̂𝐢†
𝑄=𝐴,𝐵
) 𝑄̂1,𝐢†|0⟩
𝐢
(11) where ∏ 𝐢 extends over all 𝑁Λ lattice sites, one has 𝑄̂1,𝐢†= ∑ 𝛼𝜎 𝜎,𝐢𝑐̂†𝑏,𝐢,𝜎, where 𝛼𝜎,𝐢are numerical prefactors, and |0⟩ is the bare vacuum with no fermions present. This |𝜓1,𝑔⟩ solution corresponds to 3/4 system filling.
The presented wave vector from [11] represents the ground state for the following reason: a) As seen from [2] the block operators 𝑄̂𝐢† are linear combinations of canonical Fermi creation operators acting on the finite number of sites of the given block, consequently the 𝑄̂𝐢†𝑄̂𝐢†= 0 equality is satisfied. Hence the relation 𝑃̂𝑄|𝜓1,𝑔⟩ = 0 automatically holds. Furthermore, b) The 𝑃̂𝐢 positive semidefinite operators from the expression of the 𝑃̂𝑈operators in [4] (note that because of 𝑈𝑏 > 0, also 𝑃̂𝑈 is a positive semidefinite operator) attain their minimum eigenvalue zero when at least one b-electron is present on the site 𝐢 . Hence, for the minimum eigenvalue zero of 𝑃̂𝑈, at least one b-electron is needed to be present on all lattice sites. But ∏ 𝑄̂𝐢 1,𝐢† introduces a b-electron on each site, consequently also 𝑃̂𝑈|𝜓1,𝑔⟩ = 0 holds. As a summary of the above presented arguments, also for 𝑃̂ = 𝑃̂𝑈+ 𝑃̂𝐺one has 𝑃̂| 𝜓1,𝑔⟩= 0, i.e. |𝜓1,𝑔⟩ represents the ground state. The uniqueness of this ground state at 3/4 system filling can also be demonstrated on the line of the Appendix 2 of (16).
We note that the ground state [11] can be extended also above ¾ system filling as follows:
|𝜓2,𝑔⟩ = ∏ ( ∏ 𝑄̂𝐢†
𝑄=𝐴,𝐵
)
𝐢
𝑄̂1,𝐢†(∏ 𝑐̂†𝑏,𝐤𝐣,𝛼
,𝐤𝐣
𝑁1
𝑗=1
) |0⟩
(10)
where 𝑁1 < 𝑁Λ, 𝑐̂†𝑏,𝐤,𝛼 is the Fourier transformed 𝑐̂†𝑏,𝐢,𝛼, 𝛼𝐤𝐣 being an arbitrary spin projection for each 𝐤𝐣, and ∏𝑁𝑗=11 is 𝑁1 taken over 𝐤𝐣 values. The filling corresponding to [12] corresponds to 3/4 + 𝑁1/𝑁Λ system filling. The demonstration of the ground state nature follows the line presented above in the case of [11], and is based on the observation that the supplementary product∏ 𝑐̂†𝑏,𝐤𝐣,𝛼
,𝐤𝐣
𝑁1
𝑗=1 not alters the properties 𝑃̂𝑈|𝜓𝑧,𝑔⟩ = 𝑃̂𝑄|𝜓𝑧,𝑔⟩ = 0 for both 𝑧 = 1,2.
In similar manner we have deduced ground state wave vectors also below system half filling. On this line one has
|𝜓3,𝑔⟩ = ∏𝐶̂†𝑗
𝑁1
𝑗=1
|0⟩
(13) where 𝐶̂†𝑗represent block operators which on their turn are linear combinations of 𝑐̂†𝑏,𝐢,𝛼 creation operators, and must satisfy the anti-commutation relations {𝑄̂𝐢, 𝐶̂†𝑗} = 0 for all lattice sites i, and both 𝑄 = 𝐴, 𝐵 values. The 𝑗 index here denotes different (independent) 𝐶̂†𝑗 terms. The number of carriers described by [13] is given by 𝑁𝑠. In the case of the ground state [10] the starting Hamiltonian [1] is transformed in the expression
𝐻̂ = 𝑃̂𝑄,1+ 𝜂𝑁, 𝑃̂𝑄,1= ∑ ∑ 𝑄̂𝑖𝑄̂†𝑖
𝑄=𝐴,𝐵 𝑖
(14) The corresponding matching equations [5-7] remain unaltered in their right hand side, but their left hand side gains a minus sign, and supplementary, in [7] the renormalisation 𝜂 ⟶ 𝜂 + 𝑈𝑝emerges. The ground state energies 𝐸𝑛,𝑔 corresponding to the ground states |𝜓𝑛,𝑔⟩ for 𝑛 = 1,2,3 become
𝐸𝑛,𝑔 = [𝜂(𝑁 − 𝑁1𝛿𝑛,2)− 𝑈𝑏𝑁Λ−∑ ∑ 𝑑𝑖,𝑄](𝛿𝑛,1+ 𝛿𝑛,2)+ 𝜂𝑁𝑠
𝑄=𝐴,𝐵 𝑖
𝛿𝑛,3,
(15)
V. SUMMARY AND CONCLUSION
We started from the observation that surfaces and interfaces have a broad application spectrum in leading technologies, and such two dimensional facets, in the case of realistic materials have potential gradients at their surfaces by their nature. These gradients are generating many-body spin-orbit coupling, which even if small, produces essential effects since breaks the double spin projection degeneracy of each band. Besides, during the applications, these surfaces containing correlated electrons, are under the action of external (mostly in-plane) external magnetic fields. Such processes, contrary to their importance, have not been analysed till now in exact terms
mainly given by the non-integrable nature of the systems they describe. In this work, we fill up this gap by working out and describing in details a procedure which, using positive semidefinite operator properties, deduces exact ground states in 2D for such systems.
VI. ACKNOWLEDGEMENT
N. K. acknowledges the support of ÚNKP-18-3 New National Excellence Program of the Hungarian Ministry of Human Capacities, while Z.G. of the Alexander von Humboldt Foundation.
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