Itinerant surfaces with spin-orbit couplings, correlations and external magnetic fields: Exact results
aa r X i v : . [ c ond - m a t . s t r- e l ] D ec Itinerant surfaces with spin-orbit couplings, correlations andexternal magnetic fields: Exact results
N´ora Kucska and Zsolt Gul´acsi
Department of Theoretical Physics, University of Debrecen,H-4010 Debrecen, Bem ter 18/B, Hungary (Dated: December 18, 2019)
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 broadapplicability in leading technologies are non-integrable, we set up an exact solution procedure forthem, which is described in details. Casting the Hamiltonian in positive semidefinite form, thetechnique leads to the ground state, and also characterizes the low lying excitation spectrum.
PACS numbers: . INTRODUCTION Surfaces with spin-orbit interactions (SOI) are the subject for a broad area of cur-rent research (see the review ), SOI providing essential effects in various phenomena oflarge interest today, ranging from quantum magnets , topological phases , ultracold atomexperiments , to Majorana fermions . The applications appear mostly in low dimensionalsystems , and during processing, often external fields are as well present, the most inter-esting applications being related to strongly correlated systems. Contrary to its importance,although exact treatments of 2D strongly correlated systems with spin-orbit coupling arebeing developed , studies including applied external magnetic fields are absent. Our aimin this Letter is to fill up this gap by setting up the details of a calculation procedure forsuch situations, considering Hamiltonians describing realistic correlated systems.The main difficulty encountered 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 reasonwe use the method based on positive semidefinite operator properties whose applicabilitydoes not depend on dimensionality and integrability . The method has been previouslyapplied in conditions unimaginable before in the context of exact solutions in 1-3D, even inthe presence of the disorder . II. THE SYSTEM ANALYSED
The Hamiltonian of the system has the form ˆ H = ˆ H kin + ˆ H int + ˆ H h ,ˆ H = X p,p ′ X i , r X σ,σ ′ ( k p,p ′ ; σ,σ ′ i , i + r ˆ c † p, i ,σ ˆ c p ′ , i + r ,σ ′ + H.c. ) + X p X i U p, i ˆ n p, i , ↑ ˆ n p, i , ↓ + X p, i X σ,σ ′ ~h p, i ˆ c † p, i ,σ ~σ σ,σ ′ ˆ c p, i ,σ ′ . (1)where the first term represents the kinetic part of the Hamiltonian ( ˆ H kin ), the second termis the interaction part ( ˆ H int ), while the last term describes the interaction with the externalmagnetic field ( ˆ H h ). At the level of ˆ H kin , in order to have a realistic 2D surface descrip-tion, two bands are considered, denoted hereafter by p,p’=a,b. However we note, that thischoice not diminishes the applicability of the deduced results, since usually, the theoreticaldescription of muliband systems is given by projecting the multiband structure in a few-band picture , projection which is stopped here only for its relative simplicity at two-bandslevel. Again in order to approach a real systems, besides on-site one particle terms ( r = 0),2ne takes into consideration nearest-neighbor ( r = x , x , where x , x are the Bravais vec-tors), and next nearest-neighbor ( r = x + x , x − x ) contributions. Furthermore, notethat the k p,p ′ ; σ,σ ′ i , i + r 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,(inter-site hybridization). Concerning ˆ H int , since in itinerant many-body systems strongscreening effects are present, we consider at this stage only the on-site Coulomb repulsion(Hubbard interaction term) in the correlated band (p=b, U b > U a = 0). The many-body spin-orbit interactions being ofone-particle type, are introduced in the kinetic part of the Hamiltonian, explicitly in thenearest neighbor spin-flip hopping terms, i.e. coefficients k p,p ; σ, − σ i , i + r , r = x , x . These termsare of Rashba ( λ pR , p = a, b ) and Dresselhaus ( λ pD , p = a, b ) type . Consequently, one hasfor r = x , the structure k p,p ; ↑ , ↓ i , i + x = λ pR − iλ pD , k p,p ; ↓ , ↑ i , i + x = − λ pR − iλ pD , while for r = x theexpressions k p,p ; ↑ , ↓ i , i + x = λ pD − iλ pR , k p,p ; ↓ , ↑ i , i + x = − λ pD − iλ pR . We underline that even if usuallythe SOI contributions are small, they introduce essential effects since they break the doublespin-projection degeneracy of each band. Hence, in the presence of strong correlations, theessential effects introduced cannot be obtained by standard perturbation approximations .We note that other spin-flip terms are not present in ˆ H kin , and one has for all considered r values k p,p ′ ; ↑ , ↑ i , i + r = k p,p ′ ; ↓ , ↓ i , i + r = k p,p ′ i , i + r . Furthermore, in order to not diminish the effect of thespin-flip nearest-neighbor hopping terms produced by SOI, the external fields are only ap-plied in-plane, hence without the z-component ( h zp, i = 0 , h xp, i , h yp, i = 0). We underline, thatthe in-plane h xp, i , h yp, i contributions will additively renormalize the k p,p ; σ, − σ i , i contributions as¯ k p,p ; ↑ , ↓ i , i = k p,p ; ↑ , ↓ i , i + h x − ih y , (¯ k p,p ; ↓ , ↑ i , i ) ∗ = ¯ k p,p ; ↑ , ↓ i , i . n=2n=3n=4n=1 i+r i+r i+r i+r FIG. 1: Unit cell defined at the lattice site i with in-cell notations of sites n = 1 , , , II. THE HAMILTONIAN CAST IN POZITIVE SEMIDEFINITE FORMA. The transformation of the Hamiltonian
Now we turn back to (1), and present the transformation of ˆ H in exact terms. On thisline we introduce two block operators Q=A,B for each site i , which for a fixed Q value aredefined as ˆ Q i = X p = a,b X n =1 , , , X α = ↑ , ↓ q Q,p,n,α ˆ c p, i + r n ,α . (2)Here, in order r = 0 , r = x , r = x + x , and r = x , see Fig.1. At a given lattice site i , for a fixed Q and p value, the ˆ Q i operator has 8 contributions, 4 for spin α = ↑ , and other4 for spin α = ↓ . For fixed α the mentioned 4 values denoted by n = 1 , , , i . Using (2), thestarting system Hamiltonian ˆ H in (1) becomes of the formˆ H = ˆ P + S c , (3)where ˆ P represents a positive semidefinite operator, while S c a scalar. Taking into accountthat ˆ P = ˆ P Q + ˆ P U where ˆ P U = U b P i ˆ P i , for U b > P Q = X i X Q = A,B ˆ Q i ˆ Q † i , ˆ P i = ˆ n b, i , ↑ ˆ n b, i , ↓ − (ˆ n b, i , ↑ + ˆ n b, i , ↓ ) + 1 ,S c = ηN − U b N Λ − X i X Q = A,B d i ,Q , d i ,Q = { ˆ Q i , ˆ Q † i } , (4)where N ( N Λ ) represents the number of electrons (lattice sites).The corresponding matching equations which allows the transformation of the startingHamiltonian from (1) into the form described by ˆ H in (3,4), are as follows: One has 32equations for nearest-neighbor contributions m = 1 ,
2, namely 16 for a fixed m − k p,p ′ ; σ,σ ′ i , i + x m = X Q = A,B ( q ∗ Q, m,p,σ q Q, ,p ′ ,σ ′ + q ∗ Q, ,p,σ q Q, − m,p ′ ,σ ′ ) , (5)and similarly one has 32 equations for the next nearest-neighbor contributions, as previously16 for a fixed m = ± − k p,p ′ ; σ,σ ′ i , i + x + m x = X Q = A,B q ∗ Q, − m ) / ,p,σ q Q, − m ) / ,p ′ ,σ ′ . (6)4inally local (e.g. r = 0) contributions give rise to 16 equations which can be written as − k p,p ′ ; σ,σ ′ i , i [(1 − δ p,p ′ ) + (1 − δ σ,σ ′ ) δ p,p ′ + δ p,p ′ δ σ,σ ′ ] + ( η − U p ) δ p,p ′ δ σ,σ ′ − [ h x − ik ( σ ) h y ] δ p,p ′ (1 − δ σ,σ ′ ) = X Q = A,B X n =1 , , , q ∗ Q,n,p,σ q Q,n,p ′ ,σ ′ , (7)where k ( σ ) = δ ↑ ,σ − δ ↓ ,σ . One has here totally 80 non-linear equations, whose unknown arethe 32 numerical prefactors q Q,n,p,σ called “block operator parameters”, and the parameter η entering in the ground state energy ( E g = S c ). The total number of Hamiltonian parameters(taking into account all possible spin dependences as well) is 76, so a proper description fora real material can be provided. But taking into account the conditions presented below (1)and used in this description, besides SOI couplings and U, one remains with only 10 ˆ H kin parameters per one band in both x , x directions. B. Solution of the matching equations
In order to start the deduction of the exact ground states, first we should deduce thenumerical prefactors q Q,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 q Q = A,p,n,α components can begiven in function of the q Q = B,p,n,α coefficients via the relation q A,p,n,α = d n,α q B,p,n,α , wherethe coefficiens d n,α have the expression d n,α = − ( δ α, ↑ y + δ α, ↓ x ) δ n, − ( δ α, ↑ v + δ α, ↓ z ) δ n, + ( x ∗ δ α, ↑ + y ∗ δ α, ↓ ) δ n, + ( z ∗ δ α, ↑ + v ∗ δ α, ↓ ) δ n, , (8)where x, y, v, z are numerical prefactors. After this step it results that the remaining q B,p,n,α unknowns with p = a can be given in term of the q B,p,n,α coefficients containing p = b via therelation q B,a,n,α = α n q B,b,n,α , where one has for the numerical coefficients α n the expression α n = α δ n, + α δ n, + γ α ∗ δ n, + γ α ∗ δ n, (9)where γ is an arbitrary real and positive parameter, while α , α are two further numericalprefactors. In this manner, up to (9) only 8 unknown coefficients remain, namely q B,b,n,α with n = 1 , , , α = ↑ , ↓ . But it turns out that these eight unknown coefficientsare interdependent, and all can be expressed in function of one block operator parameter,namely q B,b,n =1 , ↑ , via q B,b, , ↓ = 1 w ∗ q B,b, , ↑ , q B,b, , ↓ = − u ∗ w ∗ q ∗ B,b, , ↑ , q B,b, , ↑ = | α | γ uyx ∗ q B,b, , ↑ , B,b, , ↓ = ωq B,b, , ↑ , q B,b, , ↓ = − α ∗ α ∗ u ∗ y ∗ v ∗ w ∗ q ∗ B,b, , ↑ , q B,b, , ↑ = α ∗ α γ ω ∗ uyz ∗ q ∗ B,b, , ↑ , (10)where | q B,b, , ↑ | = | µ || q B,b, , ↑ | , ω = [ zwx ∗ (1 + vy ∗ )] / [ vy ∗ (1 + zx ∗ )], | w | = ( | u || y |√ γ ) / | α | , σ = yv ∗ . Taking σ, k, φ , φ as arbitrary parameters, one obtains three coupled equations in X = vx ∗ , Z = vz ∗ , V = | v | kσ ∗ (1 + σ ) = Z [ k (1 + σ ) + X ∗ − | X | V ] + ( V − X ) ,X − V X e iφ = V σ ∗ − ZV + σZ ,V + ZX ∗ X − Z e iφ = V (1 + σ ∗ ) σ − V . (11)from where, together with the σ expression, the remaining unknown x, y, z, v parameterscan be deduced, and based on them, starting from the relation k = f q B,b,n =1 , ↑ , where k = | q B,b,n =2 , ↑ | is a free parameter, and f ≡ f ( x, y, z, v ) is a known function, see (12), f = V ( | α | − γ ) | y | ( γ − | α | ) k (1 + | y | ) − (1 + | x | ) k (1 + V ) − (1 + | z | ) | v − x | | y − z | , (12) q B,b,n =1 , ↑ can be determined. Then, q B,b,n =2 , ↑ = | q B,b,n =2 , ↑ | exp ( iθ ), where θ is a free param-eter, is given by k exp iθ . The Hamiltonian parameters expressed in k a,a i , i + x units, enter inthe 12 free parameters k, Re ( σ ) , Im ( σ ) , φ , φ , Re ( α ) , Im ( α ) , Re ( α ) , Im ( α ) , γ , k , θ ofthe solution presented above (e.g. k a,b,σ,σ i , i + x = α ∗ α α ∗ + γ , k b,a,σ,σ i , i + x = α α α ∗ + γ , etc.). We furthernote that when the presented solution appears, the relation k b,b,σ,σ i , i + r = (1 /γ ) k a,a,σ,σ i , i + r fixes forall possible r [see the discussion following (1)] the magnitudes ratio of diagonal ˆ H kin overlapelements from the two bands. IV. THE GROUND STATE WAVE FUNCTIONS
The first deduced ground state wave function corresponding to the transformed Hamil-tonian from (3) connected to the matching equations (4-7) is of the form | Ψ ,g i = Y i ( Y Q = A,B ˆ Q † i ) ˆ Q † , i | i (13)where Q i extends over all N Λ lattice sites, one has ˆ Q † , i = P σ α σ, i ˆ c † b, i ,σ , where α σ, i are numer-ical prefactors, and | i is the bare vacuum with no fermions present. This | Ψ ,g i solutioncorresponds to 3 / Q † i are linear combinations of canonical Fermicreation operators acting on the finite number of sites of the given block, consequently theˆ Q † i ˆ Q † i = 0 equality is satisfied. Hence the relation ˆ P Q | Ψ ,g i = 0 automatically holds. Fur-thermore, b) The ˆ P i positive semidefinite operators from the expression of the ˆ P U operatorsin (4) (note that because of U b >
0, also ˆ P U is a positive semidefinite operator) attain theirminimum eigenvalue zero when at least one b-electron is present on the site i . Hence, for theminimum eigenvalue zero of ˆ P U , at least one b-electron is needed to be present on all latticesites. But Q i ˆ Q † , i introduces a b-electron on each site, consequently also ˆ P U | Ψ ,g i = 0 holds.As a summary of the above presented arguments, also for ˆ P = ˆ P Q + ˆ P U one has ˆ P | Ψ ,g i = 0,i.e. | Ψ ,g i represents the ground state. The uniqueness of this ground state at 3 / / | Ψ ,g i = Y i ( Y Q = A,B ˆ Q † i ) ˆ Q † , i ( N Y j =1 ˆ c † b, k j ,α k j ) | i (14)where N < N Λ , ˆ c † b, k ,α is the Fourier transformed ˆ c † b, i ,α , α k j being an arbitrary spin projectionfor each k j , and Q N j =1 is taken over N arbitrary k j values. The filling corresponding to (14)corresponds to 3 / N /N Λ system filling. The demonstration of the ground state naturefollows the line presented above in the case of (13), and is based on the observation that thesupplementary product ( Q N j =1 ˆ c † b, k j ,α k j ) not alters the properties ˆ P Q | Ψ z,g i = ˆ P U | Ψ z,g i = 0,for both z = 1 , | Ψ ,g i = N s Y j ˆ C † j | i (15)where ˆ C † j represent block operators which on their turn are linear combinations of ˆ c † p, i ,σ creation operators, and must satisfy the anti-commutation relations { ˆ Q i , ˆ C † j } = 0 for alllattice sites i , and both Q = A, B values. The j index here denotes different (independent)ˆ C † j terms. The number of carriers described by (15) is given by N s . We underline that in the7ase of the ground state (15) the starting Hamiltonian (1) is transformed in the expressionˆ H = ˆ P Q, + ηN, ˆ P Q, = X i X Q = A,B ˆ Q † i ˆ Q i . (16)The corresponding matching equations (5-7) remain unaltered in their right hand side, buttheir left hand side gains a minus sign, and supplementary, in (7) the renormation η → η + U p emerges. The energies E n,g corresponding to the ground states | Ψ n,g i for n = 1 , , E n,g = [ η ( N + N δ n, ) − U b N Λ − X i X Q = A,B d i ,Q ]( δ n, + δ n, ) + ηN s δ n, . (17) V. SUMMARY AND CONCLUSIONS