Orbital Order in Two-orbital Hubbard Model
Kojiro Honkawa Seiichiro Onari
cond-mat.str-el cond-mat.supr-con

Nanoscale electrical conductivity imaging using a nitrogen-vacancy center in diamond
    Amila Ariyaratne, Dolev Bluvstein, Bryan A. Myers & Ania C. Bleszynski Jayich
    
====================================================================================





§ INTRODUCTION

In the phase diagram of iron-based superconductors, the electronic nematic state with C_2
symmetry appears
universally under the structural phase transition temperature T_
S, and the superconducting state emerges next to the nematic state <cit.>.
The origin of the nematic state has
stimulated much attention since the mechanism of superconductivity
would be strongly related to the mechanism of the nematic state. 
To explain the nematic order and fluctuations, both the
spin-nematic scenario<cit.> and the orbital-order scenario<cit.> have been
proposed. In the former scenario, the origin of the nematic state is the
spin-nematic order. The spin-fluctuation-induced spin-nematic order
appears slightly above the magnetic transition temperature when the magnetic
frustration is strong. However, the nematic state in FeSe with small
spin fluctuations cannot be explained by the spin-nematic scenario.
In the latter scenario, the origin is the ferro-orbital
 order (n_xz n_yz), which is caused by the
 vertex correction (VC) of the Coulomb interaction
 <cit.>. 
By applying the self-consistent VC (SC-VC) method to the realistic five
Fe d orbital model based on first-principles calculation, the
nematic state in both
LaFeAsO and FeSe has been
naturally reproduced<cit.>. 


Recently, the remarkable
k dependence of the orbital polarization, which cannot be
explained using the mean-field theory, has been observed in FeSe<cit.>.
The k dependence of the orbital polarization is also
reproduced in the orbital-order scenario<cit.>.
In addition, the s-wave state 
without sign reversal (s_++-wave state) mediated by the orbital
fluctuations has also been
proposed in iron-based superconductors <cit.>. 
The SC-VC theory had also succeeded in explaining other multiorbital
systems such as the nematic CDW in cuprates <cit.>,
the nematic state in Sr_3Ru_2O_7 <cit.>, and the spin-triplet
superconductivity in Sr_2RuO_4 <cit.>.








In our previous work based on the SC-VC theory in realistic five-orbital models of iron-based superconductors, we have found that the nematic
orbital order n_xz n_yz is caused by the VC including
d_xz, d_yz, and d_xy orbitals <cit.>. In
the five-orbital models, Fermi surfaces (FSs) are composed of d_xz, d_yz and
d_xy orbitals. Thus, these orbitals are expected to be important in realizing the
nematic orbital order.
However, the necessary conditions for the orbital order are yet unclear.
In order to clarify the essence of the orbital order, we study the
simple two-orbital (d_xz, d_yz) model systematically by the SC-VC method.


We found that the VC for the orbital susceptibility is also enhanced in the simple two-orbital model,
and that the nematic orbital order appears similarly to the case of the realistic
five-orbital models. Thus, the d_xy orbital is not essential to realize
the nematic orbital order. We also found that another type of orbital order,
which corresponds to the 45^∘-rotated orbital order, appears when the
 holes are heavily doped. The obtained orbital order is determined by the orbital dependence and the
topology of the FSs.



§ FORMULATION

We study a simple two-orbital (d_xz, d_yz) Hubbard model in a square
lattice.
The Hamiltonian is given as

    H=∑_k∑_l,m=1,2∑_σ=↑,↓ξ^lm_kc^†_k,l,σc_k,m,σ+H_
     int,

where l,m=1,2 represents the orbital; 1=d_xz and 2=d_yz.
In the tight-binding model, we employ the dispersion relation<cit.> 
ξ^11_k=-2t_1cos(k_x)-2t_2cos(k_y)-4t_3cos(k_x)cos(k_y),
ξ^22_k=-2t_2cos(k_x)-2t_1cos(k_y)-4t_3cos(k_x)cos(k_y),
and ξ^12_k=ξ^21_k=-4t_4sin(k_x)sin(k_y).
We control the band filling n systematically, where n=1 corresponds to
the half-filling.
H_ int is the multiorbital Coulomb interaction including the intra (inter)
orbital interaction U(U') and the exchange interaction J. Hereafter, we
use the relation U=U'+2J from rotational invariance.
The irreducible susceptibility in the orbital representation without the VC is given as
χ_ll',mm'^0(q)=-T/N∑_kG_lm(k+q)G_m'l'(k),
where G is the Green's function, and q and k are denoted as
q=(q,ω_n=2nπ T) and k=(k,ϵ_n=(2n+1)π T),
respectively. 
We take N=64×64 k meshes and 256 Matsubara
frequencies.
Using the RPA, the matrix of the spin (orbital) susceptibility is given
as


    χ̂^s(c)(q)=χ̂^0(q)[1-Γ̂^s(c)χ̂^0(q)]^-1,


where Γ̂^s(c) denotes the bare Coulomb interaction for spin (charge) channel given by 

    Γ_l_1l_2,l_3l_4^s(c) = 
    U(-U),     l_1=l_2=l_3=l_4 
    
    U'(U'-2J) ,     l_1=l_3 ≠ l_2=l_4 
    
    J(-2U'+J),     l_1=l_2 ≠ l_3=l_4 
    
    J(-J) ,     l_1=l_4 ≠ l_2=l_3
    .


By introducing the VC X̂^c for the orbital channel,
the matrix of the orbital susceptibility is given by


    χ̂^c(q)=[χ̂^0(q)+X̂^c(q)][1-Γ̂^c{χ̂^0(q)+X̂^c(q)}]^-1.


In this study, we neglect the VC X̂^s for the spin channel
for simplicity
since X̂^s is smaller than X̂^c, as reported previously<cit.>.
The spin Stoner factor α_s is given by
 the maximum eigenvalue of Γ̂^sχ̂^0(q,0),
 while the charge Stoner factor α_c is given by the maximum
 eigenvalue of 
 Γ̂^c[χ̂^0(q,0)+X̂^c(q,0)]. 
 The ordered state is realized when α_s=1 or
α_c=1.

The RPA is recovered by putting X̂^c=0.
It is clear that χ̂^c(q) is enhanced over the value
in the RPA when X̂^c(q) is large.
In the SC-VC method, χ̂^c,s and X̂^c are obtained self-consistently.
The VC is given by the Maki-Thompson (MT) and Aslamazov-Larkin (AL)
terms, which are the first- and
second-order terms, respectively, with respect to χ̂^s(c). 
The MT term for the orbital susceptibility X_ll',mm'^
MT,c(q) is given by 


    X_ll',mm'^
     MT,c(q)   =   -(T/N)^2∑_k,k'∑_a,b,c,dG_la(k+q)G_cl'(k)G_bm(k'+q)G_m'd(k')
          [1/2V^c_ab,cd(k-k')+3/2V^s_ab,cd(k-k')-1/2Γ^c_ab,cd-3/2Γ^s_ab,cd],


where V̂^s,c(q)≡Γ̂^s,c
+ Γ̂^s,cχ̂^s,c(q)Γ̂^s,c.

The AL term for the orbital susceptibility X_ll',mm'^ AL,c(q)
is given by


    X_ll',mm'^ AL,c(q)   =   T/N∑_q'∑_a∼ hΛ_ll',ab,ef(q;q')[  3/2V_ab,cd^s(q+q')V_ef,gh^s(-q').
    
          .+1/2V_ab,cd^c(q+q')V_ef,gh^c(-q') ]
    Λ_mm',cd,gh'(q;q') ,


where the three-point vertex Λ̂(q;q') is given as


    Λ_ll',ab,ef(q;q')=-T/N∑_kG_la(k+q)G_fl'(k)G_be(k-q'),


 and 
Λ_mm',cd,gh'(q;q')≡Λ_ch,mg,dm'(q;q')+Λ_gd,mc,hm'(q;-q-q').
 When we calculate the total VC, we subtract the
double-counting U^2-terms in Eqs. (<ref>) and (<ref>)<cit.>.
By calculating the above equations self-consistently, we find that the MT
 and U^2-terms are much smaller than the AL terms.







Here, we introduce the diagonal charge quadrupole 
susceptibilities as

    χ_γ^Q(q)   =   ∑_ll'∑_mm'O_γ^l,l'χ_ll',mm'^c(q)O_γ^m,m',

where O_γ^l,m≡⟨ l|Ô_γ|m⟩ is the
matrix element of the γ-type quadrupole operator. In the present
two-orbital system, O^1,1_x^2-y^2=-O^2,2_x^2-y^2=1, 
O^1,2_xy=O^2,1_xy=1, and
O^1,1_3z^2-r^2=O^2,2_3z^2-r^2=1/2,  while other matrix
elements are zero.
The nematic state with n_xz n_yz is explained by the divergent
of χ^Q_x^2-y^2(q=0,0)<cit.>. In
the two-orbital system without the d_xy orbital, χ^Q_3z^2-r^2
is identical to the charge susceptibility. Thus, the enhancement of
χ^Q_3z^2-r^2 driven by the d_xy orbital in the
five-orbital model<cit.> is not realized in the two-orbital system.

 Hereafter, we fix the temperature T=0.05 and J/U=0.05. 
Although J/U=0.05 is smaller than the value obtained by
the first-principles calculation<cit.>, we note that the obtained
orbital fluctuations without the self-energy for a small J/U∼0.05
are similar to those including the self-energy for a large J/U∼0.14 in the five-orbital system<cit.>. 
The VC for the orbital fluctuations is underestimated in the system without the self-energy
since U is underestimated.


§ RESULTS AND DISCUSSION

We employ the two-orbital tight-binding model of iron-based
superconductors with t_1=-1, t_2=1.39, and
t_3=t_4=-0.85, which is similar to the model given in
Ref. Raghu. 
 First, we calculate the susceptibilities near the ordered state by the SC-VC method.
Figure <ref> shows U as a function of n, where
α_s=0.97 or α_c=0.97 is satisfied for each n. 
In the blue region, α_s=0.97 is satisfied and spin fluctuations
are dominant. 
In the red (green) region, α_c=0.97 is satisfied and x^2-y^2-type (xy-type) charge quadrupole
fluctuations are dominant. In this model, the 3z^2-r^2-type charge
quadrupole susceptibility, which is identical to the charge
susceptibility, is
much smaller than other susceptibilities. In
Fig. <ref>, the small U indicates that the corresponding ordered
state is easily realized.
Although we cannot access α_s(c)>0.98 in order
to maintain the  calculation accuracy, we confirmed that the dominant fluctuations
are basically independent of α_s(c) for 0.98≥α_s(c)≥ 0.95. Only the boundary of the
regions slightly depends on α_s(c).
For n=1.0, the x^2-y^2-type charge quadrupole
fluctuation with the peak at q=(0,0) is dominant over the spin
fluctuations and the xy-type
charge quadrupole fluctuations. This
result is consistent with the phase diagram of iron-based superconductors,
where the nematic x^2-y^2-type charge quadrupole (orbital) order
appears at higher temperatures than at which the spin order appears.




In Fig. <ref>(a), we show FSs composed of orbital 1 (green
line) and orbital 2 (red line) for n=1.0. The FSs around k=(0,0) and
k=(π,π) are the hole FSs, while the FSs around k=(π,0) and k=(0,π) are
the electron FSs.
These FSs are consistent with the FSs of
iron-based superconductors except for the FS around k=(π,π), which is
composed of d_xy orbital in iron-based superconductors. Figures
<ref>(b)–<ref>(d) show the q dependences of
χ^s(q,0), χ^Q_x^2-y^2(q,0), and χ^Q_xy(q,0), respectively.
χ^s(q,0) has a peak at q=(π,0),(0,π) due to the
strong intraorbital nesting between the hole FSs and the electron FSs. 
For n=1.0, χ^Q_x^2-y^2 with peak at q=(0,0) is dominant over χ^s and χ^Q_xy, which is
consistent with the nematic order (fluctuation) in low-doped iron-based superconductors.
χ^Q_x^2-y^2(0,0) is enhanced by the AL terms
X^ AL,c_11,11(0,0),X^ AL,c_22,22(0,0) in
Eq. (<ref>). X^ AL,c_11,11(22,22)(0,0) is mainly enlarged by the
intraorbital spin fluctuation terms 
V_11,11(22,22)^s[Q_1(Q_2),0]V_11,11(22,22)^s[-Q_1(-Q_2),0], where
Q_1=(0,π) and Q_2=(π,0) are the nesting vectors for
orbitals 1 and 2, respectively. The enhancement of
χ^Q_xy(q,0) around Q_3=(π,π) is caused by the
AL terms X^ AL,c_12,12(Q_3,0),X^
AL,c_21,21(Q_3,0).
X^ AL,c_12,12(21,21)(Q_3,0) is mainly enlarged by the
terms
V_11,11(22,22)^s[Q_1(Q_2),0]V_12,12(21,21)^s[Q_2(Q_1),0]
and V_11,11(22,22)^c(0,0)V_12,12(21,21)^c(Q_3,0) in
Eq. (<ref>).




Figure <ref> shows the results for n=1.2.
χ^s has a peak around q=(π,π/5),(π/5,π) due to the
intraorbital nesting between the hole FSs and the electron FSs. This
incommensurate spin fluctuation originates from the large
difference in diameter between the hole FSs and the electron-FSs.
In this filling, χ^s is dominant over χ^Q_x^2-y^2 and
χ^Q_xy. The enhancement of χ^Q_x^2-y^2 and
χ^Q_xy by the VC is small.



In Fig. <ref>, we show the results for n=0.7, where the
electron FSs disappear.
χ^s has a small peak around Q_4=(π/2,π/2) due to the weak
nesting between two hole FSs.
In this filling, χ^Q_xy is dominant over χ^s and 
χ^Q_x^2-y^2. The enhancement of χ^Q_x^2-y^2 around
Q_3=(π,π) in the SC-VC method is explained by the AL terms  X^
AL,c_11,11(22,22)(Q_3,0). X^
AL,c_11,11(22,22)(Q_3,0) is mainly enlarged by the
term
V_11,11(22,22)^s(Q_4,0)V_11,11(22,22)^s(Q_4,0).



Here, we discuss the reason why the ferro-orbital fluctuation
χ^Q_xy with the peak at q=(0,0) is dominant for
n=0.7. We introduce the d_XZ orbital denoted by the
45^∘-rotated d_xz
orbital and the d_YZ orbital denoted by the 45^∘-rotated d_yz orbital.
Figure <ref> shows the FSs composed of the d_XZ and d_YZ orbitals
for n=0.7. We see that the nesting vector
q∼(π/2,π/2) is caused by the intraorbital nesting as
shown by the blue two-way arrow in Fig. <ref>. From
the analogy of the case for n=1.0, the intraorbital nesting induces
the ferro-orbital fluctuation χ^Q_X^2-Y^2 defined in the
45^∘ rotated basis. Thus, the ferro-orbital fluctuation
χ^Q_xy, which is identical to the ferro-orbital fluctuation
χ^Q_X^2-Y^2, is dominant for n=0.7.
The orbital dependence of FSs is important for determining the dominant orbital
fluctuation.



From these results, we confirmed that dominant fluctuations strongly depend on the
topology and the shape of FSs. We also confirmed that dominant
fluctuations are valid for the small change in the band parameters. Only the
boundary, where dominant fluctuation changes, depends on the band parameters.



§ SUMMARY

We studied the orbital fluctuations near the orbital ordered state using the SC-VC method in the two-orbital
(d_xz, d_yz) Hubbard model.
We clarified that the nematic order in the non-doped iron-based
superconductors is explained by the two-orbital model around
half filling n=1.0. Thus, the d_xy orbital is not essential to
realize the nematic order.
Moreover, the 45^∘-rotated nematic order
appears in the heavily hole-doped case. 
We confirmed that the dominant orbital fluctuation is determined by the
orbital dependence and the topology of the Fermi surfaces.




We are grateful to H. Kontani for valuable discussions. 
This work was supported by JSPS KAKENHI Grant Number JP17K05543.
Part of numerical calculations was
performed on the Yukawa Institute Computer Facility.






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