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Topological magnons in the antiferromagnetic checkerboard lattice
Journal Pre-proof Topological magnons in the antiferromagnetic checkerboard lattice
A.S.T. Pires PII:
S1386-9477(19)31734-5
DOI:
https://doi.org/10.1016/j.physe.2019.113899
Reference:
PHYSE 113899
To appear in:
Physica E: Low-dimensional Systems and Nanostructures
Received Date:
13 November 2019
Accepted Date:
16 December 2019
Please cite this article as: A.S.T. Pires, Topological magnons in the antiferromagnetic checkerboard lattice, Physica E: Low-dimensional Systems and Nanostructures (2019), https://doi. org/10.1016/j.physe.2019.113899
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Journal Pre-proof Topological magnons in the antiferromagnetic checkerboard lattice A. S. T. Pires Departamento de Física, Universidade Federal de Minas Gerais, Belo Horizonte, MG, CP702, 30123-970, Brasil. Abstract Topological magnon excitations have attracted much attention in the latter years, and have become a very active area of research. Motivated by this, we investigate here spin Hall and thermal spin Hall conductivities in the antiferromagnetic checkerboard lattice. We include the Dzyaloshinskii-Morya interaction, and for spin S > 1/2 a single ion anisotropy term. We use the standard spin wave approach and study the effect of magnon-magnon interactions using a self consistent mean-field spin wave theory. Keywords: A. Heisenberg model; D. topological effects; D. transport.
E-mail address: [email protected] 1
Journal Pre-proof I. Introduction Following the large interest caused by fermionic topological insulators, magnons insulators, which are also characterized by the existence of edge modes, came into play [1-40]. Here, a magnetic field gradient or a thermal gradient leads to a transverse spin current or a transverse heat current respectively. The fact that magnons are uncharged and can propagate for a long distance without dissipation makes them an useful “particle” to be used in spintronics devices. As a consequence, magnon transport is a very active area of research. Ferromagnets being well explored, the attention turned to antiferromagnetic systems. Thermal Hall conductivity was studied in the Kagome [4446], honeycomb [42-44, 49], and even in the square lattice [2,60]. The checkerboard lattice, which is the two-dimensional analog of the pyrochlore lattice, is a unique-twodimensional system of great interest [61] and although it has been studied by a large number of authors [62], up to now there was only a short note leading with its topological aspect in magnetic systems [63]. Nevertheless, topological effects in the fermionic checkerboard lattice have received some attention [64-71], and in the tight binding model used to study this lattice, spin- orbit coupling, represented by spin dependent complex hopping energy, leads to a term similar to the DzyaloshinskiiMoryia term used here. As for magnetic systems, the compound A2F2OQ2 (A= Sr, Ba, Q = S, Se) can be described by a checkerboard lattice with spin S = 2 [72], and certainly new compounds will be synthesized in the future. Here we investigate the antiferromagnetic checkerboard lattice described by the following Hamiltonian: H = J1
S
i , j
i
.S j + J 2
(S
i , j
x i
~ S jx S iy S jy S iz S jz ) J 2
D ij ( S ix S jy S iy S jx ) A ( S iz ) 2 . i , j
S
i , j
i
.S j
(1)
i
As can be seen in Figure 1, the checkerboard lattice has two inequivalent sites A and B. We denote the near neighbor spins with interactions J1 and D between spins at sites A and B by. <> indicates next near neighbor interactions between site A and A, ~ and B and B, with J2 along the x direction and J 2 along the y direction. The fourth term is the Dzyaloshinskii-Moryia (DM) interaction constrained to a direction perpendicular to the lattice plane and ij 1 . The DM interaction plays an analogous role for magnons as the Aharanov-Bohm flux for electrons [6]. We have added the last term, since single ion anisotropy is generally present in magnets with spin S > ½. We consider here only the case A 0 , which favors easy-axis alignment along the z direction. ~ Hamiltonian (1) with A = D = 0, λ =1, and J 2 J 2 was studied by Canals [73] at zero temperature, and for these values of parameters our calculations agree with the ones performed by him. In particular, there is a phase transition from an ordered Néel phase, to a disordered phase at C J 2 / J 1 0.98. We also found that C depends on ~ the values of D, J2, J 2 and λ. Therefore, we restrict the values of the parameters to the 2
Journal Pre-proof regime where the Néel order is preserved. The term A > 0 favors easy-axis alignment along the z- axis and leads to an ordered phase, even for T > 0. We start section II by presenting a harmonic spin wave calculation using the Holstein Primakoff representation. In section III we compute the spin Hall and thermal spin Hall conductivities. In section IV we summarize our results, and finally in appendix A we present the self-consistent mean field spin wave theory used in our calculation of section III. II. Spin-wave calculation In the Appendix A, we introduce the Holstein-Primakoff representation and calculate the Hamiltonian up to four operator terms. Here we consider the harmonic Hamiltonian
H H 1 H 2 H DM DSIA ,
(2)
where H 1 2 J 1 S [a k a k bk bk k (a k b k a k bk )] ,
(3)
k
~ H 2 S [ J 2 ( cos k x )(bk bk bk bk ) J 2 (1 cos k y )(a k a k a k a k )] ,
(4)
k
H DM 2 DSi mk (a k b k a k bk ) , H SIA A(2 S 1) (a k a k bk bk ) , k
(5)
k
and ky kx cos . (6) 2 2 We show explicitly the calculation for mk to take into account the factor ij :
k cos
k k k ky k ky k ky ky i( x ) i( x ) i( x ) k 1 i ( 2x 2y ) 2 2 2 2 m k (e e e e 2 2 ) sin x sin , (7) 4 2 2 where we have set the lattice constant to unity. The spin wave formalism, although widely used in the literature, is justified only for large values of the spin. Note, however, that our calculations hold true for any value of the spin. The Hamiltonian (2) can be written as H kT H k k , (8) k
where
T k
k k
k
(a b a k b ) , and the matrix Hk splits into two blocks:
M H k k 0
0 , where M k*
r M k 1 fk
f k* , r2
(9)
with r1 S [2 J 1 J 2 ( cos k y )] A(2 S 1), ~ r2 S [2 J 1 J 2 (1 cos k x )] A(2 S 1), 3
(10)
Journal Pre-proof hx 2 SJ 1 k ,
f k hx ih y ,
h y 2 SDmk
(11)
To ensure that the transformation
k Tk k ,
(12)
where kT ( k k k k ) , preserves the commutation relations, the normal modes are found using a paraunitary transformation [74]. The matrix Tk satisfy the relation Tk Tk , where diag (1,1,1,1). That is, we diagonalize M k . Doing the
calculations we find that the eigenvectors of the α mode are given by
v k* * , uk
u k , vk
(13)
where u k e i k g 1 , tan k (h y / hx ),
g1
v k e i k g 2 ,
1 r w 2 w
(14)
1/ 2
, g2
1 r w 2 w
1/ 2
,
(15)
with
r
r1 r2 , 2
w r 2 fk
2
.
(16)
The eigenvectors of the β mode are the same with negatives eigenvalues. A quantum of α- magnon carries spin -1 and a quantum of β- magnon a spin +1 [42]. The transformation doubles the Hilbert space, an so the eingenvalues of M k show up in pairs . For bosons, only positive energy states are physical and we keep only the positive branches [60]. As pointed out by Cheng et al. [42], the negative branches are redundant because S i e it and S i e it describe the same spin precession. The energy spectrum of the two magnon branches are found to be
( k ) w ,
(k ) w .
(17)
where
r r (18) 1 2 . 2 We remark that in contrast, in the honeycomb lattice the magnon dispersions are asymmetric with , the energy states are degenerate and the thermal spin Hall response vanishes. Here, the two branches are nondegenerate even with D = 0. The degeneracy ~ ~ happens only if λ =1, and J 2 J 2 . However, even if J 2 J 2 , but λ =1, both branches vanish at k = (0, 0) for A = 0, and the Berry curvature also vanishes. If A > 0 both branches have a gap at k = (0, 0) given by
A( A 4 J 1 ) . This gap stabilizes the grond
state as it becomes energetically isolated from the rest of the spectrum. III. Spin transport 4
Journal Pre-proof The Berry curvature is given by [1,40,43 ] u u k* xy (k ) 2 Im k , , . k x k y
(19)
It is interesting to observe that in the honeycomb lattice the Berry curvature is independent of the DM interaction [43]. The spin Hall conductivity xy and the thermal Hall conductivity xy are given respectively by [6,14,40]
xy
xy
1 (2 )
2 BZ
dk x dk y nk xy (k ) ,
(20)
T dk x dk y c 2 (nk ) xy (k ) , 2 BZ (2 )
(21)
2
1 x 2 1 where c 2 ( x) (1 x) ln (ln x) 2 Li 2 ( x) , nk (e 1) , x Li2 is the dilogarithm function [13] given by
xx , for x 1 , 2 n 1 n
Li 2 ( x)
(22)
where
Li 2 ( x)
2 1 2 (1) n 1 ln x 2 n , 16 2 n 1 n x
for x > 1,
(23)
and nk (e 1) 1 . The function c 2 ( x) increases monotonically with x, and it is zero at x = 0 and tends to 2 / 3 as x . As xy (k ) is weighted by c 2 (nk ) , there is a nonvanishing thermal Hall conductivity at finite temperature even if the two bands have zero Chern numbers. In spin wave formalism the temperature dependence of xy rests only on c 2 (nk ) , and xy tends to a constant value at high temperatures where both bands are equally occupied. As T → 0 both bands are nearly empty and xy → 0. First, we present results for spin S = ½ (noting that the calculations hold for any value of spin with the only change in numerical factors) and A = 0, using only the spin wave formalism. In figure 2 we show the spin Hall conductivity xy as a function of T for J1 ~ =1, J2 =0.3, J 2 0.5 , λ = 0.9, and three values of D. At T = 0, xy vanishes due to the absence of magnon excitations. Magnons are thermally excited as the temperature increases, and xy becomes finite. In figure 3 we present the thermal Hall conductivity
xy versus temperature for the same values of the parameters. As we can see, xy and xy increases with D. In figure 4 we present xy as a function of temperature for J2 =0.1 ~ and J 2 = 0.2, 0.4, and 0.6 (keeping D = 0.1 and A = 0). The gap between and is ~ given by Δ, and so decreasing Δ, as for instance turning J 2 close to J2 the value of xy
decreases. The dominant contribution to xy comes from the lowest magnon branch,
5
Journal Pre-proof ~ and as it can be seem from figure 4, the large J 2 is, the steeper is the increase of xy
with growing T. Now, considering that magnetic anisotropy is important for establishing long-range order in two-dimensional systems, we turn to a discussion of the A term and including the effect of magnon-magnon interaction via a self consistent mean field spin wave theory as described in appendix A. Single ion anisotropy has been studied in the honeycomb [28, 42, 75, 76] and Shastry-Sutherland [77] lattices. The advantage of including this term is that one has an ordered phase at low temperatures, and the Holstein-Primakoff approach works better in an ordered regime. Also, the self consistent mean field spin wave used here works only in an ordered phase and for A = 0 can be applied only at T = 0. The sublattice magnetization decreases with increasing temperature, as shown in ~ figure 4 for S = 1, J1 =1, J2 = 0.3, J 2 0.2 , λ = 0.9, A = 0.2 and D = 0.1, up to TN/J1 = 0.976, where a phase transition from the ordered phase induced by the single ion anisotropy, to a disordered phase takes place. We expect that xy should also decrease and vanish at TN [77] and we have found that indeed xy decreases for higher temperatures before no self consistent solution is found anymore, as depicted in figure 5. To complete our discussion, in figure 6 we show the spin Hall conductivity xy as a function of T, for the same values of parameters as used in figure 6. In the linear spin wave treatment, the temperature dependency of xy and xy stems only form nk and c 2 (nk ) respectively, the Berry curvature being independent of temperature. In the self-consistent spin wave approach we include partly the effects of finite T and quantum fluctuations present even at T = 0. IV. Conclusions We have investigated the spin Hall and thermal spin Hall conductivities in the antiferromagnetic checkerboard lattice in the presence of the Dzyaloshinskii-Morya interaction and including a single ion anisotropy term for spin S > ½. Measurement of the thermal spin Hall response presents a mechanism to detect finite Berry curvatures, and we expect our results will lead to an experimental search for topological effects in antiferromagnetic compounds described by a checkerboard lattice. Despite the large number of theoretical studies in topological systems only a few materials have being identified presenting topological properties. Therefore, it is important to study new lattices in the hope that the system studied could be synthesized in the future. We have mentioned in the introduction compounds that could be described by the checkerboard lattice. It is interesting to remark that optical lattices simulating this lattice can be build where the MD interaction is generated using laser beans [78,79]. Appendix A We start with the Holstein-Primakoff transformation written up to four operator terms:
6
Journal Pre-proof ai ai ai ai ai a1 , S i 2 S ai S 2 S ai 4 S 4S on sublattice A, and i
,
S iz S ai ai , (A1)
b j b j b j bb b , S j 2 S b j j j j , S jz S b j b j , (A2) S j 2 S b j 4 S 4 S on sublattice B. Taking (A1) and (A2) into (1), Fourier transforming and neglecting all constant terms we find H H 1 H 2 H DM H SIA ,
(A3)
where H 1 2 J 1 S [a k a k bk bk k (a k b k a k bk )] k
J1
{
(k1 ) 8
k1 , k 2 , k3
(ak1 bk1 k2 k3 bk2 bk3 ak1 k2 k3 ak2 ak3 bk1 ak1 bk2 bk3 bk1 k2 k3
ak2 ak3 ak1 k2 k3 bk1 )
(k 2 k1 ) 4
(ak1 ak2 bk3 bk1 k2 k3 bk1 bk2 ak3 ak1 k2 k3 )} ,
(A4)
H DM 2 Di mk (a k b k a k bk ) k
iD
{
k1 , k 2 , k3
m(k1 ) (ak1 bk1 k2 k3 bk2 bk3 ak1 k2 k3 ak2 ak3 bk1 ak1 bk2 bk3 bk1 k2 k3 8
ak2 ak3 ak1 k2 k3 bk1 ) },
(A5)
and
~ H 2 S [ J 2 ( cos k x )(bk bk bk bk ) J 2 (1 cos k y )(a k a k a k a k )] k
J2
{
~ (k1 ) 8
k1 , k 2 , k3
(bk1 bk2 bk3 bk1 k 2 k3 bk1 k3 k 2 bk2 bk3 bk1 bk1 bk2 bk3 b k1 k 2 k3
+ bk1 k 2 k3 bk 2 bk3 bk1 )
~ J2
{
( k1 )
k1 , k 2 , k3
8
~ (k 2 k1 ) 4
bk1 bk2 bk3 bk1 k2 k3 } ,
(a k1 a k2 a k3 a k1 k 2 k3 a k1 k3 k 2 a k2 a k3 a k1 a k1 a k2 a k3 a k1 k 2 k3
+ a k1 k2 k3 a k2 a k3 a k1 )
( k 2 k1 ) 4
a k1 a k2 a k3 a k1 k2 k3 } ,
where (k ) 4 k , m(k ) 4mk , ~ (k ) 2 cos k x , (k ) 2 cos k y , and
H SIA A (2 S 1)(a k a k bk bk ) A k
(a
k1 , k 2 , k3
k1 k 2
b b bk3 bk1 k2 k3 ). 7
k1
a k2 a k3 a k1 k2 k3
(A6) (A7)
(A8)
Journal Pre-proof
Now we perform a mean field decoupling on quartic terms to obtain an effective quadratic Hamiltonian. We use the well known relation: ABCD=AB+CD+AC+BD+AD+BC, (A9) Defining 1 ( ak ak bk bk ), 2 k ( ak b k ak bk ) ,
(A10)
3 cos k x ( bk bk bk bk ) ,
(A11)
k
k
k
we obtain the renormalized parameters J1r 2 S J1 (2 S 1 2 ), Dr 2 S D(2 S 1 ) , Ar (2 S 1) A[(2 S 1) 1 ]
~ ~ (A12). J 2r S J 2 S 1 3 , J 2r S J 2 S 1 3 . 2 2 2 2 Using now equations (13-15 ) we obtain the self consistent equations
1 [( u k
2
k
v k2 )(nk nk ) 2v k2 ], 2 k [v k (u k u k* )(1 nk nk )], (A13) k
3 cos k x [( u k
2
v )(1 nk nk )]. 2 k
(A14)
k
The effective Hamiltonian becomes temperature dependent. The sublattice magnetization is given by
m S [ u k nk v k2 (1 nk )] . 2
(A15)
k
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Journal Pre-proof [79] T. Paananen and T. Dahm, Phys. Rev. A 91 (2015) 033604. Figure captions Figure 1. (color online) The checkerboard lattice. Figure 2. (color online) The spin Hall conductivity xy as a function of T for S = ½, J1 ~ =1, J2 = 0.3, J 2 0.5 , λ = 0.9, A = 0, and three values of D. Figure 3. (color online) Thermal Hall conductivity xy versus temperature for S = ½, J1 ~ =1, J2 = 0.3, J 2 0.5 , λ = 0.9, A = 0,and three values of D. Figure 4. (color online) Thermal Hall conductivity xy versus temperature for S = ½, J1 ~ =1, J2 = 0.1, D = 0.1, A = 0, and three values of J 2 . Figure 5. The sublattice magnetization as a function of temperature for S = 1, J1 =1, J2 = ~ 0.3, J 2 0.2 , λ = 0.9, D = 0.1 and A = 0.2. Figure 6. (color online) Thermal Hall conductivity xy versus temperature for S = 1, J1 ~ =1, J2 = 0.3, J 2 0.2 , λ = 0.9, D = 0.1 and A = 0.2. The red line is the result with the self consistent spin wave calculation. Figure 7. (color online) The spin Hall conductivity xy as a function of T for S = 1, J1 ~ =1, J2 = 0.3, J 2 0.2 , λ = 0.9, D = 0.1 and A = 0.2.
Figure 1
11
Journal Pre-proof
D=0.3
20
D=0.2
xyx10
-3
15
D=0.1
10
5
0 0,0
0,2
0,4
0,6
0,8
1,0
1,2
1,4
T/J1 Figure2
12
D=0.3 D=0.2
10
xyx10-3
8
D=0.1
6
4
2
0 0,0
0,2
0,4
0,6
T/J1 Figure 3
12
0,8
1,0
1,6
Journal Pre-proof
0.6
16
xy
x10-3
12
0.4
8
4
0.2 0 0,0
0,2
0,4
0,6
0,8
1,0
0,8
1,0
T/J1 Figure 4
1,0
0,8
m
0,6
0,4
0,2
0,0 0,0
0,2
0,4
0,6
T/J1 Figure 5
13
Journal Pre-proof
4
2
xy
x10-3
3
1
0 0,0
0,2
0,4
0,6
0,8
1,0
T/J1 Figure 6
25
20
xyx10-5
15
10
5
0 0,0
0,2
0,4
0,6
0,8
T/J1 Figure 7
14
1,0
Journal Pre-proof I declare that there is no conflict of interest in this paper. Antonio Pires
Journal Pre-proof Highlights - We study topological effects on an antiferromagnetic checkerboard lattice. - We calculate the spin Hall and the thermal Hall conductivities. - We use a self consistent mean-field spin wave theory to include the effect of magnonmagnon interactions.
A.S.T. Pires PII:
S1386-9477(19)31734-5
DOI:
https://doi.org/10.1016/j.physe.2019.113899
Reference:
PHYSE 113899
To appear in:
Physica E: Low-dimensional Systems and Nanostructures
Received Date:
13 November 2019
Accepted Date:
16 December 2019
Please cite this article as: A.S.T. Pires, Topological magnons in the antiferromagnetic checkerboard lattice, Physica E: Low-dimensional Systems and Nanostructures (2019), https://doi. org/10.1016/j.physe.2019.113899
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Journal Pre-proof Topological magnons in the antiferromagnetic checkerboard lattice A. S. T. Pires Departamento de Física, Universidade Federal de Minas Gerais, Belo Horizonte, MG, CP702, 30123-970, Brasil. Abstract Topological magnon excitations have attracted much attention in the latter years, and have become a very active area of research. Motivated by this, we investigate here spin Hall and thermal spin Hall conductivities in the antiferromagnetic checkerboard lattice. We include the Dzyaloshinskii-Morya interaction, and for spin S > 1/2 a single ion anisotropy term. We use the standard spin wave approach and study the effect of magnon-magnon interactions using a self consistent mean-field spin wave theory. Keywords: A. Heisenberg model; D. topological effects; D. transport.
E-mail address: [email protected] 1
Journal Pre-proof I. Introduction Following the large interest caused by fermionic topological insulators, magnons insulators, which are also characterized by the existence of edge modes, came into play [1-40]. Here, a magnetic field gradient or a thermal gradient leads to a transverse spin current or a transverse heat current respectively. The fact that magnons are uncharged and can propagate for a long distance without dissipation makes them an useful “particle” to be used in spintronics devices. As a consequence, magnon transport is a very active area of research. Ferromagnets being well explored, the attention turned to antiferromagnetic systems. Thermal Hall conductivity was studied in the Kagome [4446], honeycomb [42-44, 49], and even in the square lattice [2,60]. The checkerboard lattice, which is the two-dimensional analog of the pyrochlore lattice, is a unique-twodimensional system of great interest [61] and although it has been studied by a large number of authors [62], up to now there was only a short note leading with its topological aspect in magnetic systems [63]. Nevertheless, topological effects in the fermionic checkerboard lattice have received some attention [64-71], and in the tight binding model used to study this lattice, spin- orbit coupling, represented by spin dependent complex hopping energy, leads to a term similar to the DzyaloshinskiiMoryia term used here. As for magnetic systems, the compound A2F2OQ2 (A= Sr, Ba, Q = S, Se) can be described by a checkerboard lattice with spin S = 2 [72], and certainly new compounds will be synthesized in the future. Here we investigate the antiferromagnetic checkerboard lattice described by the following Hamiltonian: H = J1
S
i , j
i
.S j + J 2
(S
i , j
x i
~ S jx S iy S jy S iz S jz ) J 2
D ij ( S ix S jy S iy S jx ) A ( S iz ) 2 . i , j
S
i , j
i
.S j
(1)
i
As can be seen in Figure 1, the checkerboard lattice has two inequivalent sites A and B. We denote the near neighbor spins with interactions J1 and D between spins at sites A and B by
Journal Pre-proof regime where the Néel order is preserved. The term A > 0 favors easy-axis alignment along the z- axis and leads to an ordered phase, even for T > 0. We start section II by presenting a harmonic spin wave calculation using the Holstein Primakoff representation. In section III we compute the spin Hall and thermal spin Hall conductivities. In section IV we summarize our results, and finally in appendix A we present the self-consistent mean field spin wave theory used in our calculation of section III. II. Spin-wave calculation In the Appendix A, we introduce the Holstein-Primakoff representation and calculate the Hamiltonian up to four operator terms. Here we consider the harmonic Hamiltonian
H H 1 H 2 H DM DSIA ,
(2)
where H 1 2 J 1 S [a k a k bk bk k (a k b k a k bk )] ,
(3)
k
~ H 2 S [ J 2 ( cos k x )(bk bk bk bk ) J 2 (1 cos k y )(a k a k a k a k )] ,
(4)
k
H DM 2 DSi mk (a k b k a k bk ) , H SIA A(2 S 1) (a k a k bk bk ) , k
(5)
k
and ky kx cos . (6) 2 2 We show explicitly the calculation for mk to take into account the factor ij :
k cos
k k k ky k ky k ky ky i( x ) i( x ) i( x ) k 1 i ( 2x 2y ) 2 2 2 2 m k (e e e e 2 2 ) sin x sin , (7) 4 2 2 where we have set the lattice constant to unity. The spin wave formalism, although widely used in the literature, is justified only for large values of the spin. Note, however, that our calculations hold true for any value of the spin. The Hamiltonian (2) can be written as H kT H k k , (8) k
where
T k
k k
k
(a b a k b ) , and the matrix Hk splits into two blocks:
M H k k 0
0 , where M k*
r M k 1 fk
f k* , r2
(9)
with r1 S [2 J 1 J 2 ( cos k y )] A(2 S 1), ~ r2 S [2 J 1 J 2 (1 cos k x )] A(2 S 1), 3
(10)
Journal Pre-proof hx 2 SJ 1 k ,
f k hx ih y ,
h y 2 SDmk
(11)
To ensure that the transformation
k Tk k ,
(12)
where kT ( k k k k ) , preserves the commutation relations, the normal modes are found using a paraunitary transformation [74]. The matrix Tk satisfy the relation Tk Tk , where diag (1,1,1,1). That is, we diagonalize M k . Doing the
calculations we find that the eigenvectors of the α mode are given by
v k* * , uk
u k , vk
(13)
where u k e i k g 1 , tan k (h y / hx ),
g1
v k e i k g 2 ,
1 r w 2 w
(14)
1/ 2
, g2
1 r w 2 w
1/ 2
,
(15)
with
r
r1 r2 , 2
w r 2 fk
2
.
(16)
The eigenvectors of the β mode are the same with negatives eigenvalues. A quantum of α- magnon carries spin -1 and a quantum of β- magnon a spin +1 [42]. The transformation doubles the Hilbert space, an so the eingenvalues of M k show up in pairs . For bosons, only positive energy states are physical and we keep only the positive branches [60]. As pointed out by Cheng et al. [42], the negative branches are redundant because S i e it and S i e it describe the same spin precession. The energy spectrum of the two magnon branches are found to be
( k ) w ,
(k ) w .
(17)
where
r r (18) 1 2 . 2 We remark that in contrast, in the honeycomb lattice the magnon dispersions are asymmetric with , the energy states are degenerate and the thermal spin Hall response vanishes. Here, the two branches are nondegenerate even with D = 0. The degeneracy ~ ~ happens only if λ =1, and J 2 J 2 . However, even if J 2 J 2 , but λ =1, both branches vanish at k = (0, 0) for A = 0, and the Berry curvature also vanishes. If A > 0 both branches have a gap at k = (0, 0) given by
A( A 4 J 1 ) . This gap stabilizes the grond
state as it becomes energetically isolated from the rest of the spectrum. III. Spin transport 4
Journal Pre-proof The Berry curvature is given by [1,40,43 ] u u k* xy (k ) 2 Im k , , . k x k y
(19)
It is interesting to observe that in the honeycomb lattice the Berry curvature is independent of the DM interaction [43]. The spin Hall conductivity xy and the thermal Hall conductivity xy are given respectively by [6,14,40]
xy
xy
1 (2 )
2 BZ
dk x dk y nk xy (k ) ,
(20)
T dk x dk y c 2 (nk ) xy (k ) , 2 BZ (2 )
(21)
2
1 x 2 1 where c 2 ( x) (1 x) ln (ln x) 2 Li 2 ( x) , nk (e 1) , x Li2 is the dilogarithm function [13] given by
xx , for x 1 , 2 n 1 n
Li 2 ( x)
(22)
where
Li 2 ( x)
2 1 2 (1) n 1 ln x 2 n , 16 2 n 1 n x
for x > 1,
(23)
and nk (e 1) 1 . The function c 2 ( x) increases monotonically with x, and it is zero at x = 0 and tends to 2 / 3 as x . As xy (k ) is weighted by c 2 (nk ) , there is a nonvanishing thermal Hall conductivity at finite temperature even if the two bands have zero Chern numbers. In spin wave formalism the temperature dependence of xy rests only on c 2 (nk ) , and xy tends to a constant value at high temperatures where both bands are equally occupied. As T → 0 both bands are nearly empty and xy → 0. First, we present results for spin S = ½ (noting that the calculations hold for any value of spin with the only change in numerical factors) and A = 0, using only the spin wave formalism. In figure 2 we show the spin Hall conductivity xy as a function of T for J1 ~ =1, J2 =0.3, J 2 0.5 , λ = 0.9, and three values of D. At T = 0, xy vanishes due to the absence of magnon excitations. Magnons are thermally excited as the temperature increases, and xy becomes finite. In figure 3 we present the thermal Hall conductivity
xy versus temperature for the same values of the parameters. As we can see, xy and xy increases with D. In figure 4 we present xy as a function of temperature for J2 =0.1 ~ and J 2 = 0.2, 0.4, and 0.6 (keeping D = 0.1 and A = 0). The gap between and is ~ given by Δ, and so decreasing Δ, as for instance turning J 2 close to J2 the value of xy
decreases. The dominant contribution to xy comes from the lowest magnon branch,
5
Journal Pre-proof ~ and as it can be seem from figure 4, the large J 2 is, the steeper is the increase of xy
with growing T. Now, considering that magnetic anisotropy is important for establishing long-range order in two-dimensional systems, we turn to a discussion of the A term and including the effect of magnon-magnon interaction via a self consistent mean field spin wave theory as described in appendix A. Single ion anisotropy has been studied in the honeycomb [28, 42, 75, 76] and Shastry-Sutherland [77] lattices. The advantage of including this term is that one has an ordered phase at low temperatures, and the Holstein-Primakoff approach works better in an ordered regime. Also, the self consistent mean field spin wave used here works only in an ordered phase and for A = 0 can be applied only at T = 0. The sublattice magnetization decreases with increasing temperature, as shown in ~ figure 4 for S = 1, J1 =1, J2 = 0.3, J 2 0.2 , λ = 0.9, A = 0.2 and D = 0.1, up to TN/J1 = 0.976, where a phase transition from the ordered phase induced by the single ion anisotropy, to a disordered phase takes place. We expect that xy should also decrease and vanish at TN [77] and we have found that indeed xy decreases for higher temperatures before no self consistent solution is found anymore, as depicted in figure 5. To complete our discussion, in figure 6 we show the spin Hall conductivity xy as a function of T, for the same values of parameters as used in figure 6. In the linear spin wave treatment, the temperature dependency of xy and xy stems only form nk and c 2 (nk ) respectively, the Berry curvature being independent of temperature. In the self-consistent spin wave approach we include partly the effects of finite T and quantum fluctuations present even at T = 0. IV. Conclusions We have investigated the spin Hall and thermal spin Hall conductivities in the antiferromagnetic checkerboard lattice in the presence of the Dzyaloshinskii-Morya interaction and including a single ion anisotropy term for spin S > ½. Measurement of the thermal spin Hall response presents a mechanism to detect finite Berry curvatures, and we expect our results will lead to an experimental search for topological effects in antiferromagnetic compounds described by a checkerboard lattice. Despite the large number of theoretical studies in topological systems only a few materials have being identified presenting topological properties. Therefore, it is important to study new lattices in the hope that the system studied could be synthesized in the future. We have mentioned in the introduction compounds that could be described by the checkerboard lattice. It is interesting to remark that optical lattices simulating this lattice can be build where the MD interaction is generated using laser beans [78,79]. Appendix A We start with the Holstein-Primakoff transformation written up to four operator terms:
6
Journal Pre-proof ai ai ai ai ai a1 , S i 2 S ai S 2 S ai 4 S 4S on sublattice A, and i
,
S iz S ai ai , (A1)
b j b j b j bb b , S j 2 S b j j j j , S jz S b j b j , (A2) S j 2 S b j 4 S 4 S on sublattice B. Taking (A1) and (A2) into (1), Fourier transforming and neglecting all constant terms we find H H 1 H 2 H DM H SIA ,
(A3)
where H 1 2 J 1 S [a k a k bk bk k (a k b k a k bk )] k
J1
{
(k1 ) 8
k1 , k 2 , k3
(ak1 bk1 k2 k3 bk2 bk3 ak1 k2 k3 ak2 ak3 bk1 ak1 bk2 bk3 bk1 k2 k3
ak2 ak3 ak1 k2 k3 bk1 )
(k 2 k1 ) 4
(ak1 ak2 bk3 bk1 k2 k3 bk1 bk2 ak3 ak1 k2 k3 )} ,
(A4)
H DM 2 Di mk (a k b k a k bk ) k
iD
{
k1 , k 2 , k3
m(k1 ) (ak1 bk1 k2 k3 bk2 bk3 ak1 k2 k3 ak2 ak3 bk1 ak1 bk2 bk3 bk1 k2 k3 8
ak2 ak3 ak1 k2 k3 bk1 ) },
(A5)
and
~ H 2 S [ J 2 ( cos k x )(bk bk bk bk ) J 2 (1 cos k y )(a k a k a k a k )] k
J2
{
~ (k1 ) 8
k1 , k 2 , k3
(bk1 bk2 bk3 bk1 k 2 k3 bk1 k3 k 2 bk2 bk3 bk1 bk1 bk2 bk3 b k1 k 2 k3
+ bk1 k 2 k3 bk 2 bk3 bk1 )
~ J2
{
( k1 )
k1 , k 2 , k3
8
~ (k 2 k1 ) 4
bk1 bk2 bk3 bk1 k2 k3 } ,
(a k1 a k2 a k3 a k1 k 2 k3 a k1 k3 k 2 a k2 a k3 a k1 a k1 a k2 a k3 a k1 k 2 k3
+ a k1 k2 k3 a k2 a k3 a k1 )
( k 2 k1 ) 4
a k1 a k2 a k3 a k1 k2 k3 } ,
where (k ) 4 k , m(k ) 4mk , ~ (k ) 2 cos k x , (k ) 2 cos k y , and
H SIA A (2 S 1)(a k a k bk bk ) A k
(a
k1 , k 2 , k3
k1 k 2
b b bk3 bk1 k2 k3 ). 7
k1
a k2 a k3 a k1 k2 k3
(A6) (A7)
(A8)
Journal Pre-proof
Now we perform a mean field decoupling on quartic terms to obtain an effective quadratic Hamiltonian. We use the well known relation: ABCD=AB
(A10)
3 cos k x ( bk bk bk bk ) ,
(A11)
k
k
k
we obtain the renormalized parameters J1r 2 S J1 (2 S 1 2 ), Dr 2 S D(2 S 1 ) , Ar (2 S 1) A[(2 S 1) 1 ]
~ ~ (A12). J 2r S J 2 S 1 3 , J 2r S J 2 S 1 3 . 2 2 2 2 Using now equations (13-15 ) we obtain the self consistent equations
1 [( u k
2
k
v k2 )(nk nk ) 2v k2 ], 2 k [v k (u k u k* )(1 nk nk )], (A13) k
3 cos k x [( u k
2
v )(1 nk nk )]. 2 k
(A14)
k
The effective Hamiltonian becomes temperature dependent. The sublattice magnetization is given by
m S [ u k nk v k2 (1 nk )] . 2
(A15)
k
The equations presented in section III remain the same, but with the renormalized parameters introduced in equation (A12). Acknowledgments This work was supported by CNPQ (Conselho Nacional de Desenvolvimento Cientifico e Tecnologico). References [1] R. Shindou, R. Matsumoto, S. Murakami, J. Ohe. Phys. Rev. B87 (2013) 174 427. [2] L. Zhang, J. Ren. J. S. Wang, and B. Li, Phys. Rev. B87 (2013) 144101; Phys. Rev. Lett. 105 (2010) 225 901. [3] M. Kawano and C. Hotta, Phys. Rev. B99 (2019) 054422. [4] A. Mook, J. Henk, and I. Mertig, Phys. Rev. Lett. 117 (2016) 157204. [5] R. Shindou, J. i. Ohe, R. Matsumoto, S. Muramaki and E. Saittoh, Phys. Rev. B87 (2013) 174402. [6] J. H. Han and H. Lee, J. Phys. Soc. Japan 86 (2017) 011007; H. Lee, J. H. Han, and P. A. Lee. Phys. Rev. B91 (2015) 125413. [7] H. Lee, J. H. Han, and P. A. Lee, Phys. Rev. B91 (2015) 125413. [8] F. Y. Li, Y. D. Li, Y. B. Kim, L. Balents, Y. Yu and G. Chen, Nat. Commun. 7 (2016) 12691. [9] Y. He, J. Moore, and C. M. Varma, Phys. Rev. B85 (2012) 155106. 8
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Journal Pre-proof [79] T. Paananen and T. Dahm, Phys. Rev. A 91 (2015) 033604. Figure captions Figure 1. (color online) The checkerboard lattice. Figure 2. (color online) The spin Hall conductivity xy as a function of T for S = ½, J1 ~ =1, J2 = 0.3, J 2 0.5 , λ = 0.9, A = 0, and three values of D. Figure 3. (color online) Thermal Hall conductivity xy versus temperature for S = ½, J1 ~ =1, J2 = 0.3, J 2 0.5 , λ = 0.9, A = 0,and three values of D. Figure 4. (color online) Thermal Hall conductivity xy versus temperature for S = ½, J1 ~ =1, J2 = 0.1, D = 0.1, A = 0, and three values of J 2 . Figure 5. The sublattice magnetization as a function of temperature for S = 1, J1 =1, J2 = ~ 0.3, J 2 0.2 , λ = 0.9, D = 0.1 and A = 0.2. Figure 6. (color online) Thermal Hall conductivity xy versus temperature for S = 1, J1 ~ =1, J2 = 0.3, J 2 0.2 , λ = 0.9, D = 0.1 and A = 0.2. The red line is the result with the self consistent spin wave calculation. Figure 7. (color online) The spin Hall conductivity xy as a function of T for S = 1, J1 ~ =1, J2 = 0.3, J 2 0.2 , λ = 0.9, D = 0.1 and A = 0.2.
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Journal Pre-proof I declare that there is no conflict of interest in this paper. Antonio Pires
Journal Pre-proof Highlights - We study topological effects on an antiferromagnetic checkerboard lattice. - We calculate the spin Hall and the thermal Hall conductivities. - We use a self consistent mean-field spin wave theory to include the effect of magnonmagnon interactions.