arXiv:1610.08869v1 [cond-mat.str-el] 27 Oct 2016

Magnonic Dirac Semimetal in Quantum Magnets S. A. Owerre1, 2

arXiv:1610.08869v1 [cond-mat.str-el] 27 Oct 2016

2

1 Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada. African Institute for Mathematical Sciences, 6 Melrose Road, Muizenberg, Cape Town 7945, South Africa. (Dated: October 28, 2016)

The concept of topological metals has been intensively investigated. Recently, these concepts have been generalized to magnonic systems in insulating ferromagnets with spin-orbit coupling (SOC) proportional to the Dzyaloshinskii-Moriya interaction (DMI). Analogues of two-dimensional (2D) topological metals, three-dimensional (3D) Weyl semimetals, and nodal-line semimetals have been proposed in many insulating quantum ferromagnets. The 2D Dirac semimetal is a new state of matter characterized by the existence of Dirac points in the presence of SOC. This state appears as a quantum critical point between 2D topological and trivial insulators. Here, we propose a 2D magnonic Dirac semimetal in the distorted honeycomb ferromagnets and coplanar antiferromagnets with symmetry breaking DMI. We show that gapless Dirac magnon points exist in the Brillouin zone independent of the symmetry breaking DMI. By breaking the symmetries of the magnonic Dirac semimetal, we are able to access 2D topological magnon and trivial magnon insulators. The former exhibits gapless magnon edge modes with integer Chern numbers, whereas the latter possesses no Chern number protected magnon edge modes. We also discuss possible experimental realizations. PACS numbers: 73.43.Nq, 75.10.Jm, 66.70.-f, 05.30.Jp

I.

INTRODUCTION

Topological metals have been the most dominant field in different branches of physics over the past decade [1– 4]. The concept of topological band theory is realized in insulating electronic systems with a nontrivial gap in the energy band structures. A common feature of topological systems is the existence of gapless edge modes protected by a topological invariant quantity such as the Chern number and Z2 index [2]. This quantity distinguishes a nontrivial topological system from a trivial (non-topological) one. However, the transition between the two regimes requires a gap closing point which signifies a topological critical point [5, 6]. In many systems a topological critical point is another nontrivial state. For instance, Weyl semimetals [7, 8] can be considered as an intermediate state between 3D topological and trivial insulators. They exhibit Weyl points by breaking either time-reversal or inversion symmetry. In systems that preserve both symmetries 3D Dirac semimetals are possible [9, 10] and they are robust against SOC. Recently, a nonsymmorphic symmetry protected 2D Dirac semimetal has been proposed in the presence of SOC [11]. Also in this system it is possible to access 2D topological and trivial insulators by breaking the symmetries of the Dirac semimetal. Quite recently, the extension of topological concepts to nonelectronic systems such as magnons [12–25] and phonons [26–31] has emerged as a rapidly developing field. In the former, the SOC in electronic systems is replaced with the DMI [32, 33] and the resulting magnon band structures [19, 20, 23–25] have similar features as topological insulators in electronic systems [1–4]. However, magnons are charge-neutral bosonic quasiparticles and the edge modes of the noninteracting systems are insusceptible to dissipation. These magnonic topological

features has been studied in ferromagnetic kagomé lattice with a nearest neighbour DMI [19, 20]. The honeycomb ferromagnets with a staggered second nearest-neighbour DMI was recently proposed [23, 24]. Magnonic Weyl semimetals have recently been proposed in pyrochlore ferromagnets with DMI [34, 35] and without DMI [36], as well as magnon nodal-line semimetals [37]. Insulating ferromagnets with DMI are also known to exhibit magnon thermal Hall effect which has been observed experimentally in different ferromagnets [12–14]. Usually, in 2D systems the DMI lifts the degeneracy of the magnon bands and induces a Berry curvature which gives rise to topological magnon Hall effect [16, 17, 20, 23, 24]. In this report, we show the existence of gapless Dirac points in the presence of SOC or DMI in a distorted honeycomb ferromagnet and coplanar antiferromagnets in frustrated lattices. In the former, the Dirac points appear at certain critical values of the couplings independent of the DMI. Thus, the system is dubbed “2D magnonic Dirac semimetal” and has similar features to the electronic version [11] in that it preserves time-reversal and inversion symmetries at the critical points. By tuning the parameters away from the critical point timereversal symmetry breaking is restored by the DMI and the system exhibits gapped magnon bands. Two insulating regimes are identified comprising topological magnon and trivial magnon phases. While the former possesses Chern number protected gapless magnon edge modes, the latter exhibits vanishing Chern number and no magnon edge modes. The quantum critical point that characterizes the 2D magnonic Dirac semimetal is thus a quantum topological phase transition. Nevertheless, due to the Bose function this system is different from electronic systems in that there is no notion of completely filled bands or Fermi energy. Therefore, the trivial insulator realizes a finite thermal Hall conductivity which is analogous to

2 quantum anomalous Hall conductivity despite vanishing Chern numbers and magnon edge modes. In the latter (coplanar systems), Dirac points are present even at the isotropic point and the degeneracy of the 2D magnonic Dirac semimetal is broken in the presence of an applied magnetic field. We conclude by discussing possible experimental realizations of this phenomenon.

II.

QUANTUM SPIN MODEL

We consider an insulating ferromagnet on the honeycomb lattice governed by the simple Hamiltonian X X H=− Jij Si · Sj + Dij · Si × Sj , (1) ij

hhijii

where Si are the spin moments, Jij > 0 are ferromagnetic distorted interactions with Jij = J1 , J2 , J3 along δ1 , δ2 , δ3 respectively and Jij = J 0 along δ 0 as depicted in Fig. 1. We also consider uniform out-of-plane stagˆ gered DMI between sites i and j, with Dij = νij Dz z and νij = −νji = ±1. This term is allowed because of the inversion symmetry breaking between the bonds of the second-nearest neighbours on the honeycomb lattice. It should be noted that the ground state of Eq. 1 remains a ferromagnetic insulator despite the presence of the DMI. Previously we have studied the isotropic point Jij = J, J 0 = 0 in the context of topological magnon insulator [23] and the associated thermal Hall effect [24]. Other authors also studied spin Nernst effect of this model at the isotropic point [25]. In this limit the 2D magnon Dirac semimetallic behaviour and the topological phase transition we study here are not possible.

A.

We are interested in the low-energy excitations of Eq. 1 at low temperatures which correspond to magnons. In this regime, the standard noninteracting Holsteinz Primakoff spin-boson transformation is valid: Si,α = S− p p † y † x bi,α bi,α , Si,α = i S/2(bi,α − bi,α ), Si,α = S/2(b†i,α + bi,α ), where b†i,α (bi,α ) are the bosonic creation (annihilation) operators and α = A, B label the two sublattices on the honeycomb lattice denoted by different colors in Fig. 1. The resulting magnon tight binding model is given by X X H=− vJij (b†i bj + h.c.) − vD (eiνij φ b†i bj + h.c.) + v0

X

hhi,jii

b†i bi ,

represented by H = Ψ†k = (b†Ak , b†Bk ) and

P

k

Ψ†k · H(k) · Ψk + const., where

H(k) = v0 I2×2 + S

3 X

da (k)σa ,

(2)

i

where v0 = vJij , vD = DS, vJij = SJij , and φ = π/2 is the fictitious magnetic flux encountered by propagating magnons. The Hamiltonian in momentum space can be

(3)

a=1

where √ d1 (k) = −4J cos k˜x cos k˜y − 2 J2 + 2J 0 cos 3kx cos ky − 2J 0 cos 2ky (4) √ d2 (k) = −2 J2 + 2J 0 cos 3kx − 2J 0 cos ky sin ky + 4J cos k˜x sin k˜y d3 (k) = 4Dz cos k˜x − cos 3k˜y sin k˜x ,

Magnonic Tight Binding Model

hi,ji

FIG. 1: Color online. The geometry of honeycomb lattice and the Brillouin zone. The nearest-neighbour √ vectors are √ δ1 = a 23 , − 21 , δ2 = a (0, 1), δ3 = −a 23 , 12 . The red lines denotes the direction of DM-induced fictitious magnetic flux on the second nearest neighbours, and the blue dash √ lines connect the third nearest neighbours with δ10 = a − 3, 1 , √ δ2 = a (0, −2), δ30 = a 3, 1 . Points with the same color are related by symmetry.

(5) (6)

√ where k˜x = 3kx /2 and k˜y = ky /2. We have used J1 = J3 = J and S = 1/2.

B.

Magnonic Dirac Semimetal

We commence with the case of zero DMI, Dz = 0 and J 0 = 0. The isotropic limit (J2 = J, J 0 = 0) exhibits gapless nodes at two inequivalent Dirac points ±K [38]. In the distorted case (J2 6= J, J 0 = 0) the two inequivalent Dirac points remain stable but move away from the ±K points for J < J2 < 2J. They subsequently annihilate each other at J2 = J2c = 2J and emerge as a single Dirac point at M1 as shown in Fig. 2. A gap opens for J2 > J2c and the system becomes a trivial insulator with a gap of ∆M1 = 2|J2 − J2c | and ∆K = 2|J2 − J| at M1 and K respectively. It should be noted that lattice distortion preserves both inversion and time-reversal symmetries. This is similar to graphene

3

4

4 Without DMI

With DMI 3

E/J

E/J

3 J2 = 1.5J

2 1

1

0

0

4

J2 = 1.5J

2

4 Without DMI

With DMI 3

E/J

E/J

3 J2 = 2J

2 1

J2 = 2J

2 1

0

0 Γ

-K

M1

K

M3

Γ

Γ

FIG. 2: Color online. Magnon energy band with protected Dirac points at Dz /J = J 0 /J = 0.

[39, 40]. Nevertheless, the present model is a bosonic system. The shifted J2 < J2c are located at √ for Dirac points 0 0 G0 = ±kx , 0 , where cos( 3kx /2) = −J2 /J2c . The main purpose of this study is the possiblity of Dirac points in the presence of DMI or SOC [11]. Let us consider the effects of nonzero DMI at J 0 = 0. In the magnon tight binding model the DMI breaks the pseudospin time-reversal symmetry of H(k), hence it is expected to gap the magnon semimetallic phase and drives the system into a topological phase [23–25]. Plotted in Fig. 3 are the magnon bands at nonzero DMI for two values of J2 . A nonzero DMI breaks the degeneracy of the bands everywhere except at M1 for J2 = J2c . This critical point realizes Dirac points. Expanding the momentum space Hamiltonian (3) around k = G0 yields H(G0 + q) = v0 I2×2 + Sva σx + S (τ vx σx + v¯z σz )qx + vy σy qy + τ vz σz , (7) where √ va = −2 J2 + 2J cos( 3kx0 /2) , √ √ vx = 2 3J sin( 3kx0 /2), √ vy = −2J2 + 2J cos( 3kx0 /2), √ √ vz = 2Dz − 2 sin( 3kx0 /2) + sin( 3kx0 ) √ √ √ v¯z = 2 3Dz − cos( 3kx0 /2) + cos( 3kx0 ) ,

(8) (9) (10) (11) (12)

and τ = ± at ±G0 . It is evident that the Hamiltonian is always a massive Dirac magnon at G0 for any value of

-K

M1

K

M3

Γ

FIG. 3: Color online. Magnon energy band with protected Dirac points at Dz /J = 0.2, J 0 /J = 0.

J2 . On the other hand, expanding around k = M1 yields √ H(M1 + q) = v0 I2×2 + S˜ va (σx − τ 3σy ) + S v˜z σz qx + (τ v˜x σx + v˜y σy )qy ,

(13)

where τ = ± at ±M1 , √ vã = −2J + J2 , v˜x = 3(J + J2 ), √ v˜y = J + J2 , v˜z = 4 3Dz .

(14) (15)

At M1 , the gap is generated by a DMI independent term vã . Thus, at J2 = J2c the gap closes and the system is time-reversal invariant regardless of the DMI, and forms a massless 2D Dirac magnon. We, therefore, dub the system as 2D magnonic Dirac semimetal analogous to electronic system [11]. Now, we show that the 2D magnonic Dirac semimetallic phase is robust against additional spin interactions. We consider the effects of adding a symmetric isotropic third nearest neighbour interaction J 0 . It should be noted that a symmetric second nearest neighbour interaction or any anisotropic Sz term has no effect on the topological properties of this model as they merely rescales v0 . The inclusion of J 0 introduces complexities in this model. We will first consider Dz /J = 0 as plotted in Fig 4. Keeping the isotropic limit J2 = J and varying the third nearest neighbour interaction J 0 , we observe several Dirac points along Γ → K → Mi and no gap magnon bands are observed for any value of J 0 . The Dirac points are located

4 J2 = J

4

J2 = J

4

With DMI

Without DMI

E/J

E/J

2

0

0 6 Without DMI

J2 = 1.5J ′

J′ = J

J = (2J − J2 )/3

E/J

3

4

E/J

With DMI

4

J2 = J

5

3

2 1

2 1

0

0 Γ

-K

M1

K

M3

Γ

FIG. 4: Color online. Magnon energy band with protected Dirac points at Dz /J = 0.

With DMI J ′ = J2 /3

3 2 1 0 5

With DMI

J2 = 1.5J J ′ = J2 /3

4 3 2 1 0 Γ

-K

M1

K

M3

Γ

FIG. 5: Color online. Magnon energy band with protected Dirac points at Dz /J = 0.2.

√ at G00 = ±kx00 , 0 , where cos( 3kx00 /2) = J± and J± =

−J ±

p J 2 + 4J 0 (J 0 − J2 ) . 4J 0

Γ

-K

M1

K

M3

Γ

FIG. 6: Color online. Magnon energy band with protected Dirac points at Dz /J = 0.2.

and kx00 reduces to kx0 . The Dirac points are gapped once the system becomes distorted J2 6= J. The interesting case is the persistence of Dirac points in the presence of DMI or SOC [11]. Plotted in Figs. 5 and 6 are the magnon bands with DMI. The Dirac points at ±K are gapped by the DMI, but the Dirac points at 0 M1 are protected for J 0 = J1c = (2J − J2 )/3 and those 0 at M2,3 are protected for J 0 = J2c = J2 /3. A clear picture of these points can be understood in the lowenergy Hamiltonian. First, we expand the momentum space Hamiltonian around k = G00 which rescales the velocities at k = G0 as √ va0 = va − 2J 0 1 + 2 cos( 3kx00 ) , (17) √ √ 00 0 0 vx = vx + 4J 3 sin( 3kx ), (18) √ vy0 = vy + 4J 0 1 − cos( 3kx00 ) , (19)

J2 = J

4

E/J

2 1

1

E/J

J ′ = (2J − J2 )/3

3

J ′ = J2 /3

3

(16)

At J2 = J 0 = J there are four inequivalent Dirac points √ √ located along ky = 0 at kx = ±4π/3 3 and kx = ±π/ 3 as shown in Fig. 4. As J 0 → 0, J+ reduces to −J2 /J2c

and vz (¯ vz ) remains the same with kx0 → kx00 . Thus, there is no Dirac points at G00 for finite DMI. At M1 , only vã in Eq. 13 is rescaled by 3J 0 , and we recover a 2D magnonic 0 Dirac semimetallic phase at J 0 = J1c . Expanding around k = M2,3 yields √ H(M2,3 + q) = v0 I2×2 + S˜ va0 (σx + 3σy ) 0 0 0 + S (˜ vxz σz + vxx σx + τ vxy σy )qx 0 0 0 + (τ v˜yz σz + τ vyx σx + vyy σy )qy ,

(20)

where τ = ± at M2,3 , vã0 = 3J 0 − J2 , √ √ 0 0 0 v˜xx = 3J, v˜xy = − 3J2 , v˜xz = −2 3Dz , √ 0 0 0 v˜yx = 3J2 , v˜yy = −J2 , v˜yz = 6Dz ,

(21) (22)

5 J2 < J2c

3.5

J2 = J2c

J2 > J2c

4.5

the magnon bulk bands are gapped, however no gapless magnon edge modes are observed. This exhibits the characteristics of a trivial insulator phase. We characterize these phases using the Chern number given by Z 1 Cλ = dkx dky Ωkλ (k), (23) 2π BZ

4 4

3

3.5

2.5

3

E(kx )/J

3.5 3 2.5

2

2.5

2

1 0.5 0

1.5

1.5

1

1

0.5

0.5

0 0

π

kx

2π

where the Berry curvature is given by

2

1.5

Ωkλ = −2Im[h∂kx ψkλ |∂ky ψkλ i], 1 d3 (k) = λµν ∂kµ Φk ∂kν , 2 k

0 0

π

kx

2π

0

π

2π

kx

FIG. 7: Color online. Magnon bands with a zig-zag strip geometry at J 0 = 0, Dz /J = 0.2 and several values of J2 .

0 The gap vã0 closes at J 0 = J2c yielding a 2D magnonic Dirac semimetallic phase. These Dirac points are robust against magnon-magnon interactions as can be easily shown. They are protected by the lattice distortion critical values and time-reversal symmetry, which suggests that away from the critical values the system is likely to break time-reversal symmetry and transits to different magnon phases. We will explore these possibilities in the following section.

C.

Magnonic Topological Phase Transition

A semimetallic phase in 2D and 3D electronic systems [7–11] appear as intermediate phases between topological and trivial insulator. The possibility of these complete phases has not been studied in magnonic systems because one can only add terms to the original spin model as opposed to the tight binding model as in electronic systems. Because of this discrepancy topological phase transitions are rarely studied in magnonic systems. Here, however, we show that the distorted honeycomb ferromagnet is the simplest magnonic system that exhibits a topological magnon phase transition. In order to exemplify this phase transition we consider the case of J 0 = 0 for simplicity. As we have shown above the system is a 2D magnonic Dirac semimetal at J2 = J2c and the regimes J2 < J2c and J2 > J2c are gapped phases. The question we wish to address is: what does these gapped phases represent? In Fig. 7 we plot the evolution of the magnon bands using a zig-zag strip geometry for several values of J2 across the phase boundary. For J2 < J2c there are two counter-propagating gapless magnon edge modes, while at the phase boundary J2 = J2c the magnon bulk bands are gapless and suppress the edge modes which signifies a topological magnon phase transition. For J2 > J2c

(24)

and µν is a 2D antisymmetric tensor. The eigenvectors are given by s T s d3 (k) d (k) 1  3  , 1+λ , −λe−iΦk 1 − λ |ψkλ i = √ k k 2 (25) p where λ = ±, k = d1 (k)2 + d2 (k)2 + d3 (k)2 and tan Φk = d2 (k)/d1 (k). To substantiate the topological distinctions, we have computed the Chern number across the phase boundary and find that in the topological insulator regime (J2 < J2c ) the gapless magnon edge modes are protected by Chern numbers C± = ±1 for the upper and lower bands respectively, whereas the regime of trivial magnon insulator (J2 > J2c ) has vanishing Chern numbers. This sign change in the Chern number is a solid evidence of topological magnon phase transition. The same situation is observed for J 0 6= 0 across the phase boundary. D.

Coplanar Antiferromagnets

The coplanar antiferromagnetic orderings are present in frustrated magnets such as the kagomé and star lattices. This magnetic ordering is present due to geometric frustration and it is stabilized by an out-of plane DMI [41, 42] or a second nearest neighbour antiferromagnetic interaction [43]. In this system the spins on vertices of each triangular plaquette of the lattice are oriented at 120◦ apart. The model Hamiltonian is given by X X H= Jij Si · Sj + Dij · Si × Sj , (26) i,j

hi,ji

where Jij are all possible antiferromagnetic couplings and Dij is the DMI between sites i and j within triangular plaquette of the lattice. Although an in-plane DMI is sometimes present, the coplanar ordering is stablized ˆ [41], where only the out-of-plane DMI, Dij = νDz z ν = ±1 alternates between down and up pointing triangles respectively. It should be noted that only one ground state is selected for each sign with Dz > 0 (positive chirality) and Dz < 0 (negative chirality). The crucial difference between this model and ferromagnetic systems is

6 0.6 J2 < J2c J2 > J2c

0.5 0.4

-κxy /T

that the out-of-plane DMI does not induce topological effects as in ferromagnets. In other words, one can say that the DMI does not break the time-reversal symmetry of the magnon bands in coplanar antiferromagnets. Hence, this model can equally be attributed as a magnonic Dirac semimetal. A magnetic field applied perpendicular to the lattice plane is capable of driving the system into different topological phases, although the explicit analytical analyses are very tedious to break down [44, 45].

0.3 0.2

III.

EXPERIMENTAL REALIZATIONS A.

Magnonic Hall Response

0.1 0 0

The measurement of thermal Hall response in quantum magnets has been utilized effectively for probing the topological nature of magnetic spin excitations [12– 15]. This can be understood theoretically as a result of the DM-induced Berry curvature [14–17, 19–22]. In the presence of a temperature gradient the transverse Hall response of magnon gives rise to a thermal Hall conductivity given by [17, 21] 2 T XX π2 kB c2 [g (Ekλ )] − Ωλk , (27) κxy = − ~V 3 k λ=±

where V is the volume of the system, kB is the Boltzmann constant, T is the temperature, g(Ekλ ) = [eEkλ /kB T − 1]−1 is the Bose function, and c2 (x) is defined as 2 1+x c2 (x) = (1 + x) ln − (ln x)2 − 2Li2 (−x), (28) x with Li2 (x) being the dilogarithm. Evidently, the DMinduced Berry curvature Ωλk is the driving force in this expression. As mentioned in the Introduction, there is no Fermi energy or completely filled band in bosonic systems due to the Bose distribution function. In this regard, the Chern numbers computed above for the topological and trivial insulator phases are rather fictitious. In other words, the trivial insulator with vanishing Chern number is not adiabatically connected to the system with zero DMI or vacuum. The resulting effect is that κxy is finite in both the trivial and nontrivial regimes, despite the absence of magnon edge modes and Chern number in the former. However, the Berry curvature is ill-defined at the critical points, hence κxy is undefined, which signifies a topological phase transition. Using Eq. 27, we compute the transverse thermal Hall conductivity for the distorted honeycomb model. As shown in Fig. 8 the conductivity is finite in both regimes of the system as mentioned above. We can also see the suppression of κxy in the trivial insulator phase. It has also been confirmed that the inclusion of a third nearest neighbour suppresses κxy in each regime. The conductivity is negative with a quadratic peak at T ∼ J, and never changes sign because the Chern number of the

2

4

6

8

10

T /J FIG. 8: Color online. Transverse thermal Hall conductivity in the topological insulator (J2 < J2c ) and trivial insulator J2 > J2c ) phases at Dz /J = 0.2 and J 0 = 0.

bulk bands is either |C± | = 1 in the topological regime or |C± | = 0 in the trivial insulating regime. At low temperatures the lower band is more populated at k = 0 than the upper band and the Berry curvature can be expanded near k = 0. The resulting conductivity in the low temperature regime is |κxy | ∝ T 2 which is evident from Fig. 8. Possible experimentally accessible spin-1/2 honeycomb ferromagnets include the crystal CrBr3 , with strong ferromagnetic intra-layer coupling [46, 47]. In this material distortion can be induced by applying a strain [48]. The spin-1/2 distorted honeycomb antiferromagnets such as Na3 Cu2 SbO6 [49] and β-Cu2 V2 O7 [50] are possible candidates since a ferromagnetic state can be induced in these systems at strong magnetic field.

B.

Bose Gas in Optical Lattice

Optical lattices are indispensable reliable mediums to experimentally probe nontrivial properties of strongly correlated systems [51–62]. The honeycomb lattice is of particular interest [53–55]. This medium has been utilized effectively in the realization of artificial vector gauge potential for neutral bosonic particles trapped in an optical lattice [56–59]. Therefore, it is possible to generate the fictitious magnon phase in (2) by trapping ferromagnetic magnon or Bose atoms in an optical lattice [53– 55] as implemented in the electronic version of Haldane model [63]. The simplicity of this model makes these approaches very promising. To realize bosonic cold atoms in this model, it is worth pointing out that the connection between spin-1/2 quantum magnets and bosonic systems can be achieved via the Matsubara-Matsuda transformation Si+ → a†i , Si− → ai , Siz = a†i ai − 1/2 [64], where S ± = S x ±iS y , a†i and ai are the bosonic creation and annihilation operators respectively. The bosonic operators

7 obey the algebra [ai , a†j ] = 0 for i 6= j and {ai , a†i } = 1. The resulting bosonic Hamiltonian for honeycomb ferromagnets is given by H=−

tij (a†i aj + h.c.) − t0

X

X

(eiνij φ a†i aj + h.c.)

hhijii

hiji

+ (U + Ud )

X hiji

X 1 1 ni − nj − −µ ni , (29) 2 2 i

where ni = a†i ai , tij → Jij /2, µ → H, t0 → −Dz /2, U → −J, Ud → −J2 , and φ = π/2. This model is the bosonic version of the magnonic system in Eq. 2 and it represents a system of Bose gas interacting via a potential [65]. In this system the topological magnon phase for J2 < J2c can be attributed to a topological superfluid phase, whereas the trivial magnon insulator phase for J2 > J2c could be a non-topological Mott phase and the critical point J2 = J2c is the topological phase transition. A superfluid-Mott insulator transition has been previously realized in a non-topological bosonic optical lattice [60, 62]. Thus, the present model can be designed in a gas of ultracold bosonic atoms at sufficiently low temperatures [51, 52], and distortion can be created by anisotropic laser potential. An alternative model with similar phase transition is the harcdore-Bose-Hubbard model on the honeycomb lattice governed by the Hamiltonian X X H=− tij (a†i aj + h.c.) − t0 (eiνij φ a†i aj + h.c.) hiji

−

X

hhijii

(µ − Ui )ni ,

(30)

i

where tij (t0 ) denotes nearest (second nearest)-neighbour hoppings, µ is the chemical potential, and Ui is a staggered on-site potential , with Ui = ∆ on sublattice A, and Ui = −∆ on sublattice B. The resulting quantum spin Hamiltonian is given by X X ⊥ H=− Jij S⊥ Dij · Si × Sj i · Sj +

y x ˆ, S⊥ where Dij = νij Dz z i = (Si , Si ), Jij → tij /2, Dz → 0 −t /2. At zero DMI and Jij = J, this model has been studied by quantum Monte Carlo simulation [66] and mean field analysis [67]. It exhibits three different phases comprising superfluid phase, Mott insulator, and charge density wave insulator. The presence of the DMI and distortion lead to bosonic (magnonic) semimetallic phase and topological states as shown above.

IV.

CONCLUSION

We have studied tunable two-dimensional magnonic Dirac semimetal in a distorted honeycomb ferromagnet and coplanar antiferromagnets with a symmetry breaking Dzyaloshinskii-Moriya interaction (DMI) or spinorbit coupling (SOC). The magnonic Dirac semimetal appeared as a quantum critical point between a topological magnon insulator and a trivial magnon insulator which are characterized by integer and vanishing Chern numbers respectively. It is interesting that the magnonic Dirac semimetallic phase is not manifested by tuning the DMI since this quantity is fixed in real materials. Although lattice distortion is naturally present in various magnetic materials, it can also be tuned by strain [48], and the magnetic field can equally be tuned in the case of coplanar antiferromagnets which drives the system to different topological phases. This suggests that magnonic Dirac semimetal can be probed by bulk sensitive inelastic neutron scattering in various magnetic systems. The magnon edge modes can be probed by edge sensitive methods such as light [68] or electronic [69] scattering method. These systems can also be realized in optical lattices by trapping magnons or Bose gases in distorted laser beam potentials. The experimental study of topological magnon bands and edge state modes are the subjects of current interest [15].

Acknowledgments

(31)

Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation.

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