Pis’ma v ZhETF, vol. 114, iss. 9, pp. 625 - 626

Topological phase transitions in strongly correlated systems: application

to Co₃Sn₂S₂

V. Yu. Irkhin¹⁾, Yu. N. Skryabin

M. N. Mikheev Institute of Metal Physics, 620108 Ekaterinburg, Russia

Submitted 28 September 2021

Resubmitted 30 September 2021

Accepted 30 September 2021

DOI: 10.31857/S1234567821210096

Recently, the layered kagome lattice compound

Co₃Sn₂S₂ has been a subject of numerous experimental

and theoretical investigations. Its electronic structure

contains Weyl points, Fermi arcs and nodal rings, which

play an important role in the anomalous Hall effect.

Single-crystal experimental data on the Co₃In_xSn_2-xS₂

kagome system [1] show that these systems have an al-

most two-dimensional itinerant magnetism and a chi-

ral spin state; in addition, a strongly correlated state

with a high electronic heat capacity is formed. The im-

Fig. 1. (Color online) Schematic Fermi surfaces (solid green

portant role of correlations is confirmed by a consid-

lines) for Co3Sn2S2 in the kx - kz plane at ky = 0 accord-

erable enhancement of γT-linear specific heat even in

ing to [2]. The thin black solid line shows the nodal lines in

the ferromagnetic phase [1], especially at approaching

the absence of spin-orbit coupling for (a) x = 0.2 and (b)

the magnetic-nonmagnetic critical point somewhat be-

x = 0.4. The upper and lower triangles on the nodal lines

low x = 1.

in (a) stand for the Weyl points with topological charges

The ferromagnetism in Co₃Sn₂S₂ breaks time-

+1 and -1 in the presence of spin-orbit coupling

reversal T-symmetry and is necessary for the existence

of topological Weyl points. Above T_C, intrinsic magnetic

provide a description of these transitions within the

field disappears, the Weyl points annihilate and the

topological classification [3].

Dirac points acquire a gap. This restores T -symmetry

The half-metallic ferromagnetism of Co₃Sn₂S₂ oc-

and eliminates the topological behavior. A similar,

curs in the partially filled Co 3d_x2_-y2 band which crosses

but quantum transition occurs with disappearance of

the Fermi level. The associated moment of 1µ_B is spread

ferromagnetism in the Co₃In_xSn_2-xS₂ system at the

over three Co atoms, in agreement with the 0.33µ_B per

hole doping [2]. The doping shifts the Weyl nodes away

Co magnetic moment from first-principle calculations

from the Fermi level. For small doping, the nodal rings

and the experimental moment which is slightly less than

are located around the Fermi energy, and for x ∼ 0.2,

1µ_B/f.u. This enables one to formulate a local Hubbard

the nodal lines surrounding the L point in the Brillouin

model for the Co atom cluster [4].

zone cross the Fermi surfaces (Fig.1). With further

According to [4], across the magnetic transition,

increasing x, the nodal lines are split into two rings as

Co₃Sn₂S₂ evolves from a Mott ferromagnet to a cor-

with the annihilation of Weyl points in the presence of

related metallic state. In fact, the “Mott ferromagnet”

the spin-orbit coupling. For x > 0.6, the nodal lines are

is a half-metallic ferromagnetic state, so that we have a

located far from the Fermi level, resulting in the small

partial Mott transition in the minority spin subband.

Berry curvature on the whole Fermi surfaces [2]. At

The picture of half-metallic ferromagnetism can be

x = 1 the system becomes insulating; according to [1],

qualitatively described by the simplest narrow-band

this anomalous nonmetallic state may originate from

Hubbard model with large on-site repulsion U. In this

the Fermi energy tuning through a Dirac point.

model, doubly occupied states (doubles) are absent ow-

In the present work we treat the model picture of cor-

ing to the Hubbard splitting, but states with both spin

related half-metallic ferromagnetism in Co₃Sn₂S₂ and

projections are still present. Thus the situation is dif-

ferent from the Stoner model where spin splitting be-

comes infinitely large. The physics does not qualitatively

¹⁾e-mail: valentin.irkhin@imp.uran.ru

change in the case of finite Hubbard U, since the dou-

Письма в ЖЭТФ том 114 вып. 9 - 10

2021

625

626

V. Yu. Irkhin, Yu. N. Skryabin

bles are automatically eliminated in the saturated half-

owing to the Berry curvature vanishes. Thus the con-

metallic state [5]. We can use the slave fermion represen-

servation law for the topological charge [3] is fulfilled.

tation the Hubbard projection operators describing mo-

In the insulator case, we have a transition from topo-

tion of holes in the correlated state on the background of

logical to normal insulator with restoring time-reversal

magnetic moments. X_i(0, σ) = |i0〉〈iσ| = f^†ib_iσ where f_i

symmetry. A still more complicated situation occurs in

are fermions and b_iσ are Schwinger boson operators. In

the case of Chern insulators with a change of the Chern

the saturated ferromagnetic state the b_i↑ boson is con-

number [6, 7].

densed, and b_i↓ becomes magnon annihilation operator.

In the half-metallic ferromagnetic state Hubbard

The spin-up (majority) states propagate freely on the

correlations do not result in narrowing of bare bands for

background of strong ferromagnetic ordering and pos-

majority states, but in the paramagnetic state the situa-

sess an exotic spectrum of chiral Weyl fermions in the

tion changes: we come to the regime of narrow correlated

internal magnetic field.

bands for both spin projections. These may be char-

The spin down (minority) Green’s function in the

acterized either by strongly renormalized quasiparticle

leading approximation is obtained as a convolution of

residue, or even by a non-Fermi-liquid (e.g., marginal

the Green’s functions for free fermions and bosons, so

Fermi-liquid) behavior. Besides absence of T -breaking

that

∑

internal magnetic field in the paramagnetic phase, this

N (ω_q) + f(t_k+q)

G_k↓(E) =

(1)

can be important for vanishing of topological effects.

E-t_k-q +ω_q

Thus the topological properties and strong correlations

where N(ω) and f(E) are the Bose and Fermi functions,

in Co₃Sn₂S₂ are intricately linked, so that one cannot

ω_q is the magnon spectrum, t_k the band energy. Similar

be adequately considered without the other [4].

results for a Hubbard ferromagnet were obtained earlier

According to [8], at finite temperatures the magnetic

in the many-electron representation of X-operators [5],

structure includes the out-of-plane ferromagnetism, in-

the analogy with Anderson’s spinons being discussed.

plane antiferromagnetism, and hidden phases. The cor-

The Green’s function (1) has a purely non-quasiparticle

responding values of transition temperatures are T_C =

nature. The number of minority states is equal to the

= 182 K, T_N = 177 K, and T_com = 150 K. The corre-

number of majority states n₀ owing to the sum rule

∑

sponding first-order phase transition may again indicate

〈X_-k(0, σ)X_k(σ, 0)〉 = 〈X_i(0, 0)〉 = n₀

(2)

strong half-metallic magnetism and be important for a

combined description of the non-topological ferromag-

for both projections σ, so that the current carriers (Hub-

netic and topological transitions.

bard’s holes) are in a sense spinless.

The research was carried out within the state assign-

The description of the transition to the half-metallic

ment of FASO of Russia (theme “Flux” # AAAA-A18-

state can be described as a partial (orbital-selective)

118020190112-8 and theme “Quantum” #AAAA-A18-

Mott transition in the minority spin subband. The Lif-

118020190095-4).

shitz transitions with vanishing quasiparticle poles can

This is an excerpt of the article “Topological phase

be viewed as quantum phase transitions with a change

transitions in strongly correlated systems: application to

of the topology of Fermi surface, but without symmetry

Co₃Sn₂S₂”. Full text of the paper is published in JETP

breaking. Indeed, the Fermi surface itself is the singu-

Letters journal. DOI: 10.1134/S0021364021210013

larity in the Green’s function, which is characterized by

topological invariant N₁ and topologically protected: it

1. M. A. Kassem, PhD Dissertation, Kyoto Univ., Kyoto

is the vortex line in the frequency-momentum space [3].

(2016).

In the gapped phase, usual Fermi surface does not ex-

2. Y. Yanagi, J. Ikeda, K. Fujiwara, K. Nomura,

ist, but its topology is preserved if we take into account

A. Tsukazaki, and M.-T. Suzuki, Phys. Rev. B 103,

the Luttinger contribution. Then the Luttinger theorem

205112 (2021).

(the conservation of the volume enclosed by the Fermi

3. G. E. Volovik, Phys.-Uspekhi 61, 89 (2018).

surface) is still valid. Thus the Fermi surface becomes

4. A. Rossi, V. Ivanov, S. Sreedhar, A. L. Gross, Z. Shen,

ghost (hidden) in the Mott phase for both spin pro-

E. Rotenberg, A. Bostwick, Ch. Jozwiak, V. Taufour,

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S. Y. Savrasov, and I. M. Vishik, Phys. Rev. B 104,

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155115 (2021).

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On the contrary, the transition with disappearance

6. L. Muechler, E. Liu, J. Gayles, Q. Xu, C. Felser, and

of the Weyl points is essentially topological: topological

Y. Sun, Phys. Rev. B 101, 115106 (2020).

invariants are changed. In the Weyl semimetal phase,

7. V. Yu. Irkhin, Yu. N. Skryabin, JETP 133, 116 (2021).

the Weyl points have topological charges N₃ = +1 and

8. D.-H. Shin, J.-H. Jun, S.-E. Lee, and M.-H. Jung,

-1 and annihilate in the critical Dirac semimetal. Fur-

arXiv:2105.03892.

ther on, in the normal paramagnetic state the topology

Письма в ЖЭТФ том 114 вып. 9 - 10

2021