Home
Search
Collections
Journals
About
Contact us
My IOPscience
Magnetic topological insulators at finite temperature
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 175601 (http://iopscience.iop.org/0953-8984/26/17/175601) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 129.130.252.222 This content was downloaded on 22/06/2014 at 02:57
Please note that terms and conditions apply.
Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 175601 (6pp)
doi:10.1088/0953-8984/26/17/175601
Magnetic topological insulators at finite temperature Y-X Zhu, J He, C-L Zang, Y Liang and S-P Kou Department of Physics, Beijing Normal University, Beijing, 100875, People's Republic of China E-mail: [email protected] Received 16 January 2014, revised 23 February 2014 Accepted for publication 25 February 2014 Published 11 April 2014 Abstract
In this paper, we study the two-dimensional magnetic topological insulators from the correlated Chern insulator and the correlated Z2 topological insulator at finite temperature. For the 2D correlated Chern insulator, we find that the thermal-fluctuation-induced magnetic topological insulator (MTI) appears in the intermediate interaction region of the correlated Chern insulator. On the contrary, for the correlated Z2 topological insulator, thermal-fluctuation-induced MTI does not exist. Finally, we offer an explanation on the difference between the two cases. Keywords: magnetic topological insulators, correlated topological insulators, finite temperature (Some figures may appear in colour only in the online journal)
1. Introduction
thermal fluctuations will destroy the long-range orders and drive the system into a disordered phase. In Landau's theory, the phase transition between an ordered phase and a disordered phase is always accompanied by a break in symmetry. In this paper we will study two-dimensional (2D) correlated topological insulators, including the correlated Chern insulator and the correlated Z2 topological insulator at finite temperature. For the 2D correlated Chern insulator, the ground states can be understood as a new type of topological quantum state—the topological spin-density-wave (TSDW) state [9]. An interesting issue is the properties of TSDWs at finite temperature. In particular, we found that the thermal-fluctuationinduced magnetic topological insulator (MTI) appears in the intermediate interaction region. By contrast, for the correlated Z2 topological insulator, the TSDW and the MTI do not exist.
In recent years, the physics community has witnessed a series of exciting discoveries. Among them, the topological insulator (TI) is quite impressive and has given rise to a rapidly developing field. For example, the integer quantum Hall (IQH) effect is a remarkable achievement in condensed matter physics [1, 2]. To describe the IQH effect, the Chern number, or so-called TKNN number C, is introduced by integrating over the Brillouin zone (BZ) of the Berry field strength [3]. So, this type of topological insulator displaying the IQH effect is called a Chern insulator. Recently, a new class of topological insulator was discovered displaying time-reversal symmetry and the quantized spin Hall effect [4, 5]. To label this class of TI, Kane and Mele proposed a Z2 topological invariant [4]. For all these TIs, thermal fluctuations will wash out their topological features. Therefore, to observe the topological properties of a TI, low temperature is a necessary condition. In addition, the correlated Chern insulator [8–10] and the correlated Z2 topological insulator [11–17] are studied at zero temperature using different approaches. On the other hand, in condensed-matter physics, different types of (long-range) orders (magnetic order, superconducting order, etc.) have been found. Different local order parameters are defined to describe these ordered phases. In general, 0953-8984/14/175601+6$33.00
2. The correlated Chern insulator Firstly we studied the properties of the 2D-correlated Chern insulator, of which the Hamiltonian is [8–10, 18]
H = HH + H ′ + U ∑ nˆ i ↑nˆ i ↓ − μ ∑ cˆi†σ cˆiσ + h.c. i
i, σ
(1)
where HH is the Hamiltonian of the spinful Haldane model on a honeycomb lattice, which is given by 1
© 2014 IOP Publishing Ltd Printed in the UK
Y-X Zhu et al
J. Phys.: Condens. Matter 26 (2014) 175601
HH = − t ∑ cˆi†σ cˆ jσ + h.c. − t ′ ∑ eiϕijcˆi†σ cˆjσ . (2) i, j , σ
(
)
i, j , σ
Here t and t ′ are the nearest neighbor (NN) hopping and the next-nearest neighbor (NNN) hopping, respectively. There exists a complex phase φij into the NNN hopping which is set ⎛
π⎞
to be the direction of the positive phase clockwise ⎜⎝ ϕij = ⎟⎠. 2 H′ denotes an on-site staggered energy which is defined as H ′ = ε ∑ cˆi†σ cˆiσ − ε ∑ cˆi†σ cˆiσ . U is the on-site i ∈ A, σ
i ∈ B, σ
Coulomb repulsion strength. 〈i, j〉 and 〈〈i, j〉〉 denote the two sites of the NN and NNN links, respectively. nˆ i ↑ and nˆ i ↓ are the number operators of electrons with up-spin and down-spin, respectively. μ denotes chemical potential, and μ = U/2 at halffilling for our purposes in this paper. For free electrons, U = 0, we can see that there are energy 2π gaps Δf1, Δf2 near the two Dirac points k 1 = - 3 (1, 1 / 3 ) and
Figure 1. The phase diagram of the correlated Chern insulator at finite temperature for the case t′/t = 0.15, ε/t = 0.15. There are four phases: C = 2 Chern insulator (QAH), A-MTI, B-MTI and trivial MI. The colour shows the energy gap of the electrons.
2π
k2 = 3 (1 , 1 / 3 ) as Δf1 = 2ε − 6 3 t ′ and Δf2 = 2ε + 6 3 t ′, respectively. There are two phases separated by the phase boundary Δf1 = 0: the Chern insulator, with Chern number C = 2, and the normal band insulator (NI) state. In the Chern insulator, due to the quantum anomalous Hall (QAH) effect with a quantized (charge) Hall conductivity σH = 2e2/h, we denote the Chern insulator by ‘QAH’.
So we get the effective Hamiltonian with staggered magnetization M as H →H = −t +ϵ
As the interaction increases, the correlated Chern insulator becomes unstable against an antiferromagnetic (AF) spindensity-wave (SDW), which is described by
where the local order parameter M denotes staggered magnetization. We set σ = +1 for spin up and σ = −1 for spin down. Using the mean-field approach, we can obtain the self-consistent equation for M by minimizing the free energy at finite temperature in the reduced Brillouin Zone (BZ): ⎡ ξ ′ +ε+Δ 1 M tanh( β E1, k/2) ∑⎢ k Ns M k ⎢⎣ 2E1, k
ϵ
∑ ˆ ˆ −∑ ( − 1)i ΔMˆc i†σ zˆci .
(9) Δf = − 6 3 t ′ + 2ε ± 2ΔM = 0 (see the black lines in figure 1). In figure 1, the colours show the energy gap of the electrons. After determining the phase boundaries, we derive the phase diagram at finite temperature in figure 1 for the parameters t′/t = 0.15, ε = 0.15t. From figure 1, we can see that, at zero temperature for the 2D-correlated Chern insulator, four phases exist: Chern insulator (QAH), A-TSDW (TSDW with Chern number C = 2), B-TSDW (TSDW with Chern number C = 1), and trivial AF-SDW. The Chern insulator exists in the weak-interaction region. With an increase in U/t, the system turns into A-TSDW state. After the electron's energy gap is closed at one Dirac point, the system turns into B-TSDW state. With a further increase in the interaction strength, the electron's energy gap is closed at another Dirac point, and the system turns into a trivial AF-SDW state. At finite temperature, for the 2D-correlated topological insulator with spin rotation symmetry, there is no true long-range SDW order. The nonzero value of M means that
⎤ ξk′ + ε −ΔM tanh( βE2, k / 2) ⎥ ⎥⎦ 2 E2, k
(4) where Ns is the number of unit cells, β = 1 / kBT and ΔM = UM/2. Then the energy spectrums of electrons are (5) E1, k = ( ξk ′ + ε + ΔM )2 + ξk 2 and (6) E2, k = ( ξk ′ + ε −ΔM )2 + ξk 2 where ξk = t 3 + 2cos( 3 k y ) + 4cos(3kx / 2)cos( 3 k y / 2) ξk ′ = 2t ′(sin( 3 k y ) − 4cos(3kx / 2)sin( 3 k y / 2)).
i, j , σ
c i†σ cjσ
To determine the phase diagram at finite temperature, two types of ‘phase transition’ are relevant: one is the ‘magnetic’ phase transition (which is really a crossover according to the following discussions) that separates the magnetic order state with M ≠ 0 and the nonmagnetic state with M = 0 (solving equation (4)); the other is the ‘topological’ phase transition, which is characterized by the condition of zero fermionenergy gaps,
1
∑ ˆˆ
c i†σ cjσ −
4. Phase diagram at finite temperature
cˆi†, σ cˆi, σ = 2 [1 + ( − 1)i σM ] (3)
−
i, j , σ
eiϕij ˆ c i†σ ˆ cjσ
i ∈ B, σ i ∈ B, σ i (8) Finite ΔM will thus contribute to the energy gap of the electrons.
3. Mean field approach
1=
∑ ( ˆc i†σ ˆcjσ + h.c. ) − t ′ ∑
(7) 2
Y-X Zhu et al
J. Phys.: Condens. Matter 26 (2014) 175601
Figure 3. The staggered magnetization M via the temperature at different interaction strength U/t for the case t′/t = 0.15, ε = 0.15t. Figure 2. The edge states of different phases of the correlated Chern insulator with open boundary condition along the y-axis at zero temperature for the case t′/t = 0.15, ε/t = 0.15, T = 0.3t: (a) C = 2 Chern insulator (QAH), (b) A-MTI, (c) B-MTI, (d) trivial MI.
magnetization at first also increases, and the gap of the system becomes smaller; when the temperature further increases, the magnetization decreases, and the gap of the system becomes larger again. This result is confirmed by the results of the energy gap in figure 5(a), at U/t = 3.5. In figures 5(b) and (c), we also give the energy gap for U/t = 3.65 and 4.0 for various temperatures for the case t′/t = 0.15, ε = 0.15t.
effective spin moment only just exists. It does not necessarily imply that the ground state is a long-range AF i † order (that is, denoted by ( − 1) cˆi σcˆi ≠ 0), because the z -axis in direction of spin is chosen to be fixed along the ˆ mean field theory. Hence, at finite temperature, A-TSDW, B-TSDW, and trivial AF-SDW orders evolve into corresponding magnetic insulators with short-range SDW order (that is, denoted by ( − 1)i cˆi†σcˆi = M ≠ 0)—an A-type
5. The ‘topological’ phase transition at finite temperature A related issue is that of ‘topological’ phase transition at finite temperature. To check whether true topological phase transitions exist at finite temperature, we calculate the Hall conductivity and the specific heat. We use the Kubo formula to derive the Hall conductivity [21],
magnetic topological insulator (A-MTI), a B-type magnetic topological insulator (B-MTI), and a trivial magnetic insulator (MI). In figure 2, we show the edge states in different topological states. In particular, in the phase diagram, we find an interesting phenomenon: the it thermal-fluctuation-induced magnetic topological insulator. From figure 1, it can be seen that an MTI exists at finite temperature for about U/t = 3.5 during the temperature T/t = 0.1 ∼ 0.5. In figure 3 we plot the staggered magnetization M against an increase in temperature via the interaction strengths U for the case t′/t = 0.15, ε = 0.15t. From figure 3, we can see that, at U/t = 3.5 (the black line), and the staggered magnetization M is zero at low temperature. But at higher temperature, the staggered magnetization M becomes nonzero, and has its maximum value at T/t ≃ 0.25; with the further increase in temperature, the staggered magnetization M becomes increasingly smaller, eventually reaching zero. Thus, this MTI at finite temperature is assisted by the thermal fluctuations. A brief explanation of the existence of the thermal-fluctuation-induced MTI is in order here. At zero temperature, the DOS inside the energy gap vanishes. At finite temperature, due to the thermal fluctuations, the DOS inside the energy gap increases, which may help to establish the magnetic order. In figure 4, we plot the DOS at various temperatures for the case U/t = 3.5, t′/t = 0.15, ε = 0.15t. From figure 4, we can see that, with an increase in temperature,
σH = lim ωi Qxy ( ω + iδ ) (10) ω→0
where Qxy (iνm ) =
1 ∑ tr[ Jx ( k ) G ( k , i( ωn + νm ) ) Jy ( k ) G ( k , iωn ) ] (11) Ns β k, n
∂ H ( k )σ with the current operator Jx/y ( k ) = ∂ kx/y and G(k, iωn) is the Matsubara Green function, σ is the spin index. In figure 6, we show the Hall conductivity σH via the interaction strength U at different temperatures for the case t′/t = 0.15, ε = 0.15t. At zero temperature, we can use the Hall conductivity to characterize the topological properties of the system. There are three plateaus in the Hall conductivity: σH = 2e2/h in the Chern insulator (QAH) and A-TSDW, σH = e2/h in B-TSDW, σH = 0 in trivial AF-SDW. At finite temperature, the situation changes. From figure 6 we can see that, at finite temperature, the Hall conductivity σH smoothly changes with the interaction and the temperature, and the plateaus of the Hall conductivity are smeared out. That means there is no true ‘topological’ phase transition at finite temperature. The smooth change in Hall conductivity is due to both edge contribution and bulk contribution at finite temperature. 3
Y-X Zhu et al
J. Phys.: Condens. Matter 26 (2014) 175601
Figure 4. The density of state (DOS) at different temperatures for the case U/t = 3.5, t′/t = 0.15, ε = 0.15.
Figure 5. The energy gap of electrons via the temperature for the case t′/t = 0.15, ε = 0.15t.
In addition, we calculated the specific heat at finite temperature, and also did not find a true ‘topological’ phase transition. Instead, the quantum phase transitions at zero temperature turn into ‘crossovers’ at finite temperature, and one can change smoothly from one ‘phase’ in figure 1 to another.
where
E2, k
Next we studied the 2D-correlated Z2 topological insulator. Our starting point was the correlated Kane–Mele (KM) model, which is described by [11–17] H = HKM + H ′ + U ∑ nˆ i ↑nˆ i ↓ − μ ∑ cˆi†cˆi (12) i
where HKM is the Hamiltonian of the KM model, which is given by HKM = − t ∑ cˆi†cˆj + h.c. − t ′ ∑ eiϕijcˆi†σzcˆj , (13)
(
i, j
)
i, j
and H′ denotes an on-site staggered energy, where H′ = ε ∑ cˆ † ciσ̂ − ε ∑ cˆ † cˆiσ. Here we set the on-site i ∈ B , σ iσ i ∈ A, σ iσ staggered energy ε at 0.15t. Without the spin rotation symmetry, the staggered magnetic order is along the XY-plane. In this paper we take a staggered magnetic order along X-direction as an example. Now we derive the self-consistency equation for M by minimizing the free energy at temperature T in the reduced Brillouin zone as 1=
⎪
⎪
2
k
2
2
k
2
k
2
2
For the conventional orders, the competition of different phases at finite temperature is a combination of energy and entropy. For the topological orders, the situation differs. In the latter case, we introduce the concepts ‘mass-gap-competition’ and ‘the mass-gap-coexistence’ to characterize the difference
⎪
⎪
2 M
k
7. Discussion and conclusions
⎧ [1 + ε( Δ2 + ξ′2 )− 12 ] ⎛ βE ⎞ 1 1,k M k ⎟⎟ tanh ⎜⎜ ∑ U⎨ 2Ns k 2 ⎩ − E1, k ⎝ 2 ⎠
1 ⎛ βE ⎞ ⎫ [1 − ε( Δ2M + ξk′ 2 )− 2 ] 2, k ⎟⎟ ⎬ tanh ⎜⎜ + − E2, k ⎝ 2 ⎠⎭
2 M
(15) and ΔM = UM/2. In the phase diagram of the correlated KM model, only one phase transition at zero temperature exists: the magnetic phase transition between a magnetic order state with M ≠ 0 and a nonmagnetic state with M = 0 (solving the equation (14)) [19, 20]. In a nonmagnetic state with M = 0, the ground state is the Z2 topological insulator state with quantum spin Hall (QSH) effect, which is protected by time-reversal symmetry (in the Chern insulator, time-reversal symmetry is broken). In a magnetic state with M ≠ 0, time-reversal symmetry is spontaneously broken, and the quantum spin Hall effect disappears. At finite temperature, the phase transition also turns into a crossover, and there is no true long-range magnetic order [23]1. In this case, time-reversal symmetry is not really broken. As a result, we may call it a Z2 magnetic topological insulator. In figure 7, we plot the phase diagram for the case t′/t = 0.15, ε = 0.15t. From the phase diagram, we can see that there are two phases at finite temperature: the Z2 magnetic topological insulator with M = 0, and the trivial AF-SDW state, with M ≠ 0. From figure 7, we can see that the energy gap will never be closed. The thermal-fluctuation-induced MTI does not exist.
6. The correlated Kane–Mele model at finite temperature
i
( Δ + ξ′ + ε ) + | ξ | , = − ( Δ + ξ′ − ε ) + | ξ |
E1, k = −
1 Since the correlated KM model has a continuous O(2) spin symmetry, Berezinskii–Kosterlitz–Thouless (BKT) transition at finite temperature may exist [22].
(14)
4
Y-X Zhu et al
J. Phys.: Condens. Matter 26 (2014) 175601
Figure 6. The Hall conductivity σH via the interaction for the case
Figure 7. The phase diagram of the correlated Kane–Mele model at finite temperature for the case t′/t = 0.15, ε/t = 0.15. Two phases exist: Z2 topological insulator with quantized spin Hall (QSH) effect with M = 0 and trivial Z2-MTI with M ≠ 0. The colour denotes the energy gap of electrons.
t′/t = 0.15, ε = 0.15t at different temperatures.
between the correlated Chern insulator and the correlated Z2 topological insulator. For the SDW order in the correlated Chern insulator, the effective Hamiltonian becomes
we can determine the energy gap of the electrons to be Δf = ± 2 mT 2 + mM
H0 = vFp · α + ( mT + mM + ε ) β (16)
2
− 2ε
(21) 2 = ± (6 3 t ′) + ( ΔM )2 − 2ε .
where pi = −iℏ∇i is the momentum operator (i ∈ {x, y}), and p 2 = px2 + py2, vF is the Fermi-velocity. Two mass matrices exist: the mass matrix of the parent topological insulator mT = 3 3 t′η3 ⊗ I2 and the mass matrix of the SDW order mM = ΔM(I2 ⊗ σ3), respectively. Here, τ, σ and η are Pauli matrices that denote the indices of the sublattice, spin and node, respectively. I2 is the 2 × 2 unit matrix, and ⊗ represents the Kronecker product. The Dirac matrices can be expressed as a set of 4 × 4 matrices αi = τi ⊗ I2, β = τz ⊗ I2. Here we set the on-site staggered energy ε at 0.15t. Due to
From the Z2 topological insulator state of the correlated KM model ((6 3 t ′ ) − 2ε > 0) [4], when the magnetic order exists along XY plane, the energy gap of the electrons will definitely increase. Thus, the energy gap will never close in the magnetic order. Therefore, we call this system the mass-gap-coexistence. The suppression of the energy gap of the parent topological insulator will not help the formation of the MTI. Thus, there is no thermal-fluctuation-induced MTI in the Z2 topological insulator of the correlated KM model. In addition, it has been found that, for the correlated KM model, it is not the gap but more likely the kinetic energy that competes with magnetic correlations [20]. We may generalize the results in the correlated topological insulators to a long-range order developed in an insulator. For the long-range orders developed in an insulator, there are two different cases: case I, the mass-gap-competition; and case II, the mass-gap-coexistence. For case I, the mass gap induced by a long-range order mO competes with the mass gap in the parent insulator mT. This yields [mT, mM] = 0 and the energy gap of the system is Δf = 2|mT ± mM|. For this case, there may exist a thermal-fluctuation-induced MTI. For case II, the mass gap induced by a long-range order mO coexists with the mass gap in the parent insulator mT. This yields {mT, mM} = 0 and the energy gap of the system is Δf = 2 mT 2 + mM 2 . For this case, there thermal-fluctuation-induced MTI does not exist.
[ mT , mM ] = 0, (17)
we can determine the energy gap of the electrons to be (18) Δf = 2 ± mT ± mM − 2ε = ± 6 3 t ′± 2ΔM − 2ε . Thus, the energy gap from the magnetic order mM competes with that from the parent topological insulator mT. Therefore, we call this system the mass-gap-competition. When we have a small staggered magnetization (or a small ΔM), the energy gap of the electrons shrinks. With an increase in the staggered magnetization, the energy gap will eventually close at the critical point Δf = 0. At finite temperature, the thermal fluctuations will excite the quasiparticle, and may also smear out the energy gap. Thus, due to the suppression of the energy gap of the parent topological insulator, the MTI may be assisted by thermal fluctuations. For the SDW order in the correlated KM model, the effective Hamiltonian becomes H0 = v F p · α +( mT + mM + ε ) β (19)
Acknowledgments
where the mass term of the parent topological insulator is mT = − 3 3 t′η3 ⊗ σ3 and the mass term from the SDW order is mM = ΔM(I2 ⊗ σ1). Due to
This work is supported by the National Basic Research Program of China (973 Program) under grant nos. 2011CB921803, 2012CB921704, 2011cba00102, NFSC grant no.11174035.
{ mT , mM } = 0, (20)
5
Y-X Zhu et al
J. Phys.: Condens. Matter 26 (2014) 175601
[11] Rachel S and Le Hur K 2010 Phys. Rev. B 82 075106 [12] Hohenadler M et al 2011 Phys. Rev. Lett. 106 100403 [13] Yu S-L et al 2011 Phys. Rev. Lett. 107 010401 [14] Zheng D et al 2011 Phys Rev B 84 205121 [15] Wu W et al 2012 Phys. Rev. B 85 205102 [16] Mardani M et al arXiv: 1111.5980 [17] Zong Y H et al 2013 Eur. Phys. J. B 86 28 [18] Haldane F D M 1988 Phys. Rev. Lett. 61 2015 [19] Meng Z Y, Hung H-H and Lang T C 2014 Phys. Lett. B 28 28 143001 [20] Hohenadler M et al 2012 Phys. Rev. B 85 115132 [21] Qi X L et al 2006 Phys. Rev. B 74 085308 [22] Berezinskii V L 1972 Sov. Phys.—JETP 34 610 Kosterlitz J M and Thouless D J 1973 J. Phys. C 6 1181 Kosterlitz J M 1974 J. Phys. C 7 1046 [23] Yoshida T, Fujimoto S and Kawakami N 2012 Phys. Rev. B 85 125113
References [1] Klitzing K V et al 1980 Phys. Rev. Lett. 45 494 [2] Prange R and Girvin S 1987 The Quantum Hall Effect (New York: Springer) Aoki H 1987 Rep. Prog. Phys. 50 655 Morandi G 1988 Quantum Hall Effect (Naples: Bibliopolis) [3] Thouless D J et al 1982 Phys. Rev. Lett. 49 405 [4] Kane C L, Mele E J 2005 Phys. Rev. Lett. 95 146802 Kane C L and Mele E J 2005 Phys. Rev. Lett. 95 226801 [5] Bernevig B A and Zhang S C 2006 Phys. Rev. Lett. 96 106802 [6] Bernevig B A et al 2006 Science 314 1757 [7] Konig M 2007 Science 318 766 [8] He J et al 2011 Phys. Rev. B 83 205116 [9] He J et al 2011 Phys. Rev. B 84 035127 [10] He J, Liang Y and Kou S P 2012 Phys. Rev. B 85 205107
6
Search
Collections
Journals
About
Contact us
My IOPscience
Magnetic topological insulators at finite temperature
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 175601 (http://iopscience.iop.org/0953-8984/26/17/175601) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 129.130.252.222 This content was downloaded on 22/06/2014 at 02:57
Please note that terms and conditions apply.
Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 175601 (6pp)
doi:10.1088/0953-8984/26/17/175601
Magnetic topological insulators at finite temperature Y-X Zhu, J He, C-L Zang, Y Liang and S-P Kou Department of Physics, Beijing Normal University, Beijing, 100875, People's Republic of China E-mail: [email protected] Received 16 January 2014, revised 23 February 2014 Accepted for publication 25 February 2014 Published 11 April 2014 Abstract
In this paper, we study the two-dimensional magnetic topological insulators from the correlated Chern insulator and the correlated Z2 topological insulator at finite temperature. For the 2D correlated Chern insulator, we find that the thermal-fluctuation-induced magnetic topological insulator (MTI) appears in the intermediate interaction region of the correlated Chern insulator. On the contrary, for the correlated Z2 topological insulator, thermal-fluctuation-induced MTI does not exist. Finally, we offer an explanation on the difference between the two cases. Keywords: magnetic topological insulators, correlated topological insulators, finite temperature (Some figures may appear in colour only in the online journal)
1. Introduction
thermal fluctuations will destroy the long-range orders and drive the system into a disordered phase. In Landau's theory, the phase transition between an ordered phase and a disordered phase is always accompanied by a break in symmetry. In this paper we will study two-dimensional (2D) correlated topological insulators, including the correlated Chern insulator and the correlated Z2 topological insulator at finite temperature. For the 2D correlated Chern insulator, the ground states can be understood as a new type of topological quantum state—the topological spin-density-wave (TSDW) state [9]. An interesting issue is the properties of TSDWs at finite temperature. In particular, we found that the thermal-fluctuationinduced magnetic topological insulator (MTI) appears in the intermediate interaction region. By contrast, for the correlated Z2 topological insulator, the TSDW and the MTI do not exist.
In recent years, the physics community has witnessed a series of exciting discoveries. Among them, the topological insulator (TI) is quite impressive and has given rise to a rapidly developing field. For example, the integer quantum Hall (IQH) effect is a remarkable achievement in condensed matter physics [1, 2]. To describe the IQH effect, the Chern number, or so-called TKNN number C, is introduced by integrating over the Brillouin zone (BZ) of the Berry field strength [3]. So, this type of topological insulator displaying the IQH effect is called a Chern insulator. Recently, a new class of topological insulator was discovered displaying time-reversal symmetry and the quantized spin Hall effect [4, 5]. To label this class of TI, Kane and Mele proposed a Z2 topological invariant [4]. For all these TIs, thermal fluctuations will wash out their topological features. Therefore, to observe the topological properties of a TI, low temperature is a necessary condition. In addition, the correlated Chern insulator [8–10] and the correlated Z2 topological insulator [11–17] are studied at zero temperature using different approaches. On the other hand, in condensed-matter physics, different types of (long-range) orders (magnetic order, superconducting order, etc.) have been found. Different local order parameters are defined to describe these ordered phases. In general, 0953-8984/14/175601+6$33.00
2. The correlated Chern insulator Firstly we studied the properties of the 2D-correlated Chern insulator, of which the Hamiltonian is [8–10, 18]
H = HH + H ′ + U ∑ nˆ i ↑nˆ i ↓ − μ ∑ cˆi†σ cˆiσ + h.c. i
i, σ
(1)
where HH is the Hamiltonian of the spinful Haldane model on a honeycomb lattice, which is given by 1
© 2014 IOP Publishing Ltd Printed in the UK
Y-X Zhu et al
J. Phys.: Condens. Matter 26 (2014) 175601
HH = − t ∑ cˆi†σ cˆ jσ + h.c. − t ′ ∑ eiϕijcˆi†σ cˆjσ . (2) i, j , σ
(
)
i, j , σ
Here t and t ′ are the nearest neighbor (NN) hopping and the next-nearest neighbor (NNN) hopping, respectively. There exists a complex phase φij into the NNN hopping which is set ⎛
π⎞
to be the direction of the positive phase clockwise ⎜⎝ ϕij = ⎟⎠. 2 H′ denotes an on-site staggered energy which is defined as H ′ = ε ∑ cˆi†σ cˆiσ − ε ∑ cˆi†σ cˆiσ . U is the on-site i ∈ A, σ
i ∈ B, σ
Coulomb repulsion strength. 〈i, j〉 and 〈〈i, j〉〉 denote the two sites of the NN and NNN links, respectively. nˆ i ↑ and nˆ i ↓ are the number operators of electrons with up-spin and down-spin, respectively. μ denotes chemical potential, and μ = U/2 at halffilling for our purposes in this paper. For free electrons, U = 0, we can see that there are energy 2π gaps Δf1, Δf2 near the two Dirac points k 1 = - 3 (1, 1 / 3 ) and
Figure 1. The phase diagram of the correlated Chern insulator at finite temperature for the case t′/t = 0.15, ε/t = 0.15. There are four phases: C = 2 Chern insulator (QAH), A-MTI, B-MTI and trivial MI. The colour shows the energy gap of the electrons.
2π
k2 = 3 (1 , 1 / 3 ) as Δf1 = 2ε − 6 3 t ′ and Δf2 = 2ε + 6 3 t ′, respectively. There are two phases separated by the phase boundary Δf1 = 0: the Chern insulator, with Chern number C = 2, and the normal band insulator (NI) state. In the Chern insulator, due to the quantum anomalous Hall (QAH) effect with a quantized (charge) Hall conductivity σH = 2e2/h, we denote the Chern insulator by ‘QAH’.
So we get the effective Hamiltonian with staggered magnetization M as H →H = −t +ϵ
As the interaction increases, the correlated Chern insulator becomes unstable against an antiferromagnetic (AF) spindensity-wave (SDW), which is described by
where the local order parameter M denotes staggered magnetization. We set σ = +1 for spin up and σ = −1 for spin down. Using the mean-field approach, we can obtain the self-consistent equation for M by minimizing the free energy at finite temperature in the reduced Brillouin Zone (BZ): ⎡ ξ ′ +ε+Δ 1 M tanh( β E1, k/2) ∑⎢ k Ns M k ⎢⎣ 2E1, k
ϵ
∑ ˆ ˆ −∑ ( − 1)i ΔMˆc i†σ zˆci .
(9) Δf = − 6 3 t ′ + 2ε ± 2ΔM = 0 (see the black lines in figure 1). In figure 1, the colours show the energy gap of the electrons. After determining the phase boundaries, we derive the phase diagram at finite temperature in figure 1 for the parameters t′/t = 0.15, ε = 0.15t. From figure 1, we can see that, at zero temperature for the 2D-correlated Chern insulator, four phases exist: Chern insulator (QAH), A-TSDW (TSDW with Chern number C = 2), B-TSDW (TSDW with Chern number C = 1), and trivial AF-SDW. The Chern insulator exists in the weak-interaction region. With an increase in U/t, the system turns into A-TSDW state. After the electron's energy gap is closed at one Dirac point, the system turns into B-TSDW state. With a further increase in the interaction strength, the electron's energy gap is closed at another Dirac point, and the system turns into a trivial AF-SDW state. At finite temperature, for the 2D-correlated topological insulator with spin rotation symmetry, there is no true long-range SDW order. The nonzero value of M means that
⎤ ξk′ + ε −ΔM tanh( βE2, k / 2) ⎥ ⎥⎦ 2 E2, k
(4) where Ns is the number of unit cells, β = 1 / kBT and ΔM = UM/2. Then the energy spectrums of electrons are (5) E1, k = ( ξk ′ + ε + ΔM )2 + ξk 2 and (6) E2, k = ( ξk ′ + ε −ΔM )2 + ξk 2 where ξk = t 3 + 2cos( 3 k y ) + 4cos(3kx / 2)cos( 3 k y / 2) ξk ′ = 2t ′(sin( 3 k y ) − 4cos(3kx / 2)sin( 3 k y / 2)).
i, j , σ
c i†σ cjσ
To determine the phase diagram at finite temperature, two types of ‘phase transition’ are relevant: one is the ‘magnetic’ phase transition (which is really a crossover according to the following discussions) that separates the magnetic order state with M ≠ 0 and the nonmagnetic state with M = 0 (solving equation (4)); the other is the ‘topological’ phase transition, which is characterized by the condition of zero fermionenergy gaps,
1
∑ ˆˆ
c i†σ cjσ −
4. Phase diagram at finite temperature
cˆi†, σ cˆi, σ = 2 [1 + ( − 1)i σM ] (3)
−
i, j , σ
eiϕij ˆ c i†σ ˆ cjσ
i ∈ B, σ i ∈ B, σ i (8) Finite ΔM will thus contribute to the energy gap of the electrons.
3. Mean field approach
1=
∑ ( ˆc i†σ ˆcjσ + h.c. ) − t ′ ∑
(7) 2
Y-X Zhu et al
J. Phys.: Condens. Matter 26 (2014) 175601
Figure 3. The staggered magnetization M via the temperature at different interaction strength U/t for the case t′/t = 0.15, ε = 0.15t. Figure 2. The edge states of different phases of the correlated Chern insulator with open boundary condition along the y-axis at zero temperature for the case t′/t = 0.15, ε/t = 0.15, T = 0.3t: (a) C = 2 Chern insulator (QAH), (b) A-MTI, (c) B-MTI, (d) trivial MI.
magnetization at first also increases, and the gap of the system becomes smaller; when the temperature further increases, the magnetization decreases, and the gap of the system becomes larger again. This result is confirmed by the results of the energy gap in figure 5(a), at U/t = 3.5. In figures 5(b) and (c), we also give the energy gap for U/t = 3.65 and 4.0 for various temperatures for the case t′/t = 0.15, ε = 0.15t.
effective spin moment only just exists. It does not necessarily imply that the ground state is a long-range AF i † order (that is, denoted by ( − 1) cˆi σcˆi ≠ 0), because the z -axis in direction of spin is chosen to be fixed along the ˆ mean field theory. Hence, at finite temperature, A-TSDW, B-TSDW, and trivial AF-SDW orders evolve into corresponding magnetic insulators with short-range SDW order (that is, denoted by ( − 1)i cˆi†σcˆi = M ≠ 0)—an A-type
5. The ‘topological’ phase transition at finite temperature A related issue is that of ‘topological’ phase transition at finite temperature. To check whether true topological phase transitions exist at finite temperature, we calculate the Hall conductivity and the specific heat. We use the Kubo formula to derive the Hall conductivity [21],
magnetic topological insulator (A-MTI), a B-type magnetic topological insulator (B-MTI), and a trivial magnetic insulator (MI). In figure 2, we show the edge states in different topological states. In particular, in the phase diagram, we find an interesting phenomenon: the it thermal-fluctuation-induced magnetic topological insulator. From figure 1, it can be seen that an MTI exists at finite temperature for about U/t = 3.5 during the temperature T/t = 0.1 ∼ 0.5. In figure 3 we plot the staggered magnetization M against an increase in temperature via the interaction strengths U for the case t′/t = 0.15, ε = 0.15t. From figure 3, we can see that, at U/t = 3.5 (the black line), and the staggered magnetization M is zero at low temperature. But at higher temperature, the staggered magnetization M becomes nonzero, and has its maximum value at T/t ≃ 0.25; with the further increase in temperature, the staggered magnetization M becomes increasingly smaller, eventually reaching zero. Thus, this MTI at finite temperature is assisted by the thermal fluctuations. A brief explanation of the existence of the thermal-fluctuation-induced MTI is in order here. At zero temperature, the DOS inside the energy gap vanishes. At finite temperature, due to the thermal fluctuations, the DOS inside the energy gap increases, which may help to establish the magnetic order. In figure 4, we plot the DOS at various temperatures for the case U/t = 3.5, t′/t = 0.15, ε = 0.15t. From figure 4, we can see that, with an increase in temperature,
σH = lim ωi Qxy ( ω + iδ ) (10) ω→0
where Qxy (iνm ) =
1 ∑ tr[ Jx ( k ) G ( k , i( ωn + νm ) ) Jy ( k ) G ( k , iωn ) ] (11) Ns β k, n
∂ H ( k )σ with the current operator Jx/y ( k ) = ∂ kx/y and G(k, iωn) is the Matsubara Green function, σ is the spin index. In figure 6, we show the Hall conductivity σH via the interaction strength U at different temperatures for the case t′/t = 0.15, ε = 0.15t. At zero temperature, we can use the Hall conductivity to characterize the topological properties of the system. There are three plateaus in the Hall conductivity: σH = 2e2/h in the Chern insulator (QAH) and A-TSDW, σH = e2/h in B-TSDW, σH = 0 in trivial AF-SDW. At finite temperature, the situation changes. From figure 6 we can see that, at finite temperature, the Hall conductivity σH smoothly changes with the interaction and the temperature, and the plateaus of the Hall conductivity are smeared out. That means there is no true ‘topological’ phase transition at finite temperature. The smooth change in Hall conductivity is due to both edge contribution and bulk contribution at finite temperature. 3
Y-X Zhu et al
J. Phys.: Condens. Matter 26 (2014) 175601
Figure 4. The density of state (DOS) at different temperatures for the case U/t = 3.5, t′/t = 0.15, ε = 0.15.
Figure 5. The energy gap of electrons via the temperature for the case t′/t = 0.15, ε = 0.15t.
In addition, we calculated the specific heat at finite temperature, and also did not find a true ‘topological’ phase transition. Instead, the quantum phase transitions at zero temperature turn into ‘crossovers’ at finite temperature, and one can change smoothly from one ‘phase’ in figure 1 to another.
where
E2, k
Next we studied the 2D-correlated Z2 topological insulator. Our starting point was the correlated Kane–Mele (KM) model, which is described by [11–17] H = HKM + H ′ + U ∑ nˆ i ↑nˆ i ↓ − μ ∑ cˆi†cˆi (12) i
where HKM is the Hamiltonian of the KM model, which is given by HKM = − t ∑ cˆi†cˆj + h.c. − t ′ ∑ eiϕijcˆi†σzcˆj , (13)
(
i, j
)
i, j
and H′ denotes an on-site staggered energy, where H′ = ε ∑ cˆ † ciσ̂ − ε ∑ cˆ † cˆiσ. Here we set the on-site i ∈ B , σ iσ i ∈ A, σ iσ staggered energy ε at 0.15t. Without the spin rotation symmetry, the staggered magnetic order is along the XY-plane. In this paper we take a staggered magnetic order along X-direction as an example. Now we derive the self-consistency equation for M by minimizing the free energy at temperature T in the reduced Brillouin zone as 1=
⎪
⎪
2
k
2
2
k
2
k
2
2
For the conventional orders, the competition of different phases at finite temperature is a combination of energy and entropy. For the topological orders, the situation differs. In the latter case, we introduce the concepts ‘mass-gap-competition’ and ‘the mass-gap-coexistence’ to characterize the difference
⎪
⎪
2 M
k
7. Discussion and conclusions
⎧ [1 + ε( Δ2 + ξ′2 )− 12 ] ⎛ βE ⎞ 1 1,k M k ⎟⎟ tanh ⎜⎜ ∑ U⎨ 2Ns k 2 ⎩ − E1, k ⎝ 2 ⎠
1 ⎛ βE ⎞ ⎫ [1 − ε( Δ2M + ξk′ 2 )− 2 ] 2, k ⎟⎟ ⎬ tanh ⎜⎜ + − E2, k ⎝ 2 ⎠⎭
2 M
(15) and ΔM = UM/2. In the phase diagram of the correlated KM model, only one phase transition at zero temperature exists: the magnetic phase transition between a magnetic order state with M ≠ 0 and a nonmagnetic state with M = 0 (solving the equation (14)) [19, 20]. In a nonmagnetic state with M = 0, the ground state is the Z2 topological insulator state with quantum spin Hall (QSH) effect, which is protected by time-reversal symmetry (in the Chern insulator, time-reversal symmetry is broken). In a magnetic state with M ≠ 0, time-reversal symmetry is spontaneously broken, and the quantum spin Hall effect disappears. At finite temperature, the phase transition also turns into a crossover, and there is no true long-range magnetic order [23]1. In this case, time-reversal symmetry is not really broken. As a result, we may call it a Z2 magnetic topological insulator. In figure 7, we plot the phase diagram for the case t′/t = 0.15, ε = 0.15t. From the phase diagram, we can see that there are two phases at finite temperature: the Z2 magnetic topological insulator with M = 0, and the trivial AF-SDW state, with M ≠ 0. From figure 7, we can see that the energy gap will never be closed. The thermal-fluctuation-induced MTI does not exist.
6. The correlated Kane–Mele model at finite temperature
i
( Δ + ξ′ + ε ) + | ξ | , = − ( Δ + ξ′ − ε ) + | ξ |
E1, k = −
1 Since the correlated KM model has a continuous O(2) spin symmetry, Berezinskii–Kosterlitz–Thouless (BKT) transition at finite temperature may exist [22].
(14)
4
Y-X Zhu et al
J. Phys.: Condens. Matter 26 (2014) 175601
Figure 6. The Hall conductivity σH via the interaction for the case
Figure 7. The phase diagram of the correlated Kane–Mele model at finite temperature for the case t′/t = 0.15, ε/t = 0.15. Two phases exist: Z2 topological insulator with quantized spin Hall (QSH) effect with M = 0 and trivial Z2-MTI with M ≠ 0. The colour denotes the energy gap of electrons.
t′/t = 0.15, ε = 0.15t at different temperatures.
between the correlated Chern insulator and the correlated Z2 topological insulator. For the SDW order in the correlated Chern insulator, the effective Hamiltonian becomes
we can determine the energy gap of the electrons to be Δf = ± 2 mT 2 + mM
H0 = vFp · α + ( mT + mM + ε ) β (16)
2
− 2ε
(21) 2 = ± (6 3 t ′) + ( ΔM )2 − 2ε .
where pi = −iℏ∇i is the momentum operator (i ∈ {x, y}), and p 2 = px2 + py2, vF is the Fermi-velocity. Two mass matrices exist: the mass matrix of the parent topological insulator mT = 3 3 t′η3 ⊗ I2 and the mass matrix of the SDW order mM = ΔM(I2 ⊗ σ3), respectively. Here, τ, σ and η are Pauli matrices that denote the indices of the sublattice, spin and node, respectively. I2 is the 2 × 2 unit matrix, and ⊗ represents the Kronecker product. The Dirac matrices can be expressed as a set of 4 × 4 matrices αi = τi ⊗ I2, β = τz ⊗ I2. Here we set the on-site staggered energy ε at 0.15t. Due to
From the Z2 topological insulator state of the correlated KM model ((6 3 t ′ ) − 2ε > 0) [4], when the magnetic order exists along XY plane, the energy gap of the electrons will definitely increase. Thus, the energy gap will never close in the magnetic order. Therefore, we call this system the mass-gap-coexistence. The suppression of the energy gap of the parent topological insulator will not help the formation of the MTI. Thus, there is no thermal-fluctuation-induced MTI in the Z2 topological insulator of the correlated KM model. In addition, it has been found that, for the correlated KM model, it is not the gap but more likely the kinetic energy that competes with magnetic correlations [20]. We may generalize the results in the correlated topological insulators to a long-range order developed in an insulator. For the long-range orders developed in an insulator, there are two different cases: case I, the mass-gap-competition; and case II, the mass-gap-coexistence. For case I, the mass gap induced by a long-range order mO competes with the mass gap in the parent insulator mT. This yields [mT, mM] = 0 and the energy gap of the system is Δf = 2|mT ± mM|. For this case, there may exist a thermal-fluctuation-induced MTI. For case II, the mass gap induced by a long-range order mO coexists with the mass gap in the parent insulator mT. This yields {mT, mM} = 0 and the energy gap of the system is Δf = 2 mT 2 + mM 2 . For this case, there thermal-fluctuation-induced MTI does not exist.
[ mT , mM ] = 0, (17)
we can determine the energy gap of the electrons to be (18) Δf = 2 ± mT ± mM − 2ε = ± 6 3 t ′± 2ΔM − 2ε . Thus, the energy gap from the magnetic order mM competes with that from the parent topological insulator mT. Therefore, we call this system the mass-gap-competition. When we have a small staggered magnetization (or a small ΔM), the energy gap of the electrons shrinks. With an increase in the staggered magnetization, the energy gap will eventually close at the critical point Δf = 0. At finite temperature, the thermal fluctuations will excite the quasiparticle, and may also smear out the energy gap. Thus, due to the suppression of the energy gap of the parent topological insulator, the MTI may be assisted by thermal fluctuations. For the SDW order in the correlated KM model, the effective Hamiltonian becomes H0 = v F p · α +( mT + mM + ε ) β (19)
Acknowledgments
where the mass term of the parent topological insulator is mT = − 3 3 t′η3 ⊗ σ3 and the mass term from the SDW order is mM = ΔM(I2 ⊗ σ1). Due to
This work is supported by the National Basic Research Program of China (973 Program) under grant nos. 2011CB921803, 2012CB921704, 2011cba00102, NFSC grant no.11174035.
{ mT , mM } = 0, (20)
5
Y-X Zhu et al
J. Phys.: Condens. Matter 26 (2014) 175601
[11] Rachel S and Le Hur K 2010 Phys. Rev. B 82 075106 [12] Hohenadler M et al 2011 Phys. Rev. Lett. 106 100403 [13] Yu S-L et al 2011 Phys. Rev. Lett. 107 010401 [14] Zheng D et al 2011 Phys Rev B 84 205121 [15] Wu W et al 2012 Phys. Rev. B 85 205102 [16] Mardani M et al arXiv: 1111.5980 [17] Zong Y H et al 2013 Eur. Phys. J. B 86 28 [18] Haldane F D M 1988 Phys. Rev. Lett. 61 2015 [19] Meng Z Y, Hung H-H and Lang T C 2014 Phys. Lett. B 28 28 143001 [20] Hohenadler M et al 2012 Phys. Rev. B 85 115132 [21] Qi X L et al 2006 Phys. Rev. B 74 085308 [22] Berezinskii V L 1972 Sov. Phys.—JETP 34 610 Kosterlitz J M and Thouless D J 1973 J. Phys. C 6 1181 Kosterlitz J M 1974 J. Phys. C 7 1046 [23] Yoshida T, Fujimoto S and Kawakami N 2012 Phys. Rev. B 85 125113
References [1] Klitzing K V et al 1980 Phys. Rev. Lett. 45 494 [2] Prange R and Girvin S 1987 The Quantum Hall Effect (New York: Springer) Aoki H 1987 Rep. Prog. Phys. 50 655 Morandi G 1988 Quantum Hall Effect (Naples: Bibliopolis) [3] Thouless D J et al 1982 Phys. Rev. Lett. 49 405 [4] Kane C L, Mele E J 2005 Phys. Rev. Lett. 95 146802 Kane C L and Mele E J 2005 Phys. Rev. Lett. 95 226801 [5] Bernevig B A and Zhang S C 2006 Phys. Rev. Lett. 96 106802 [6] Bernevig B A et al 2006 Science 314 1757 [7] Konig M 2007 Science 318 766 [8] He J et al 2011 Phys. Rev. B 83 205116 [9] He J et al 2011 Phys. Rev. B 84 035127 [10] He J, Liang Y and Kou S P 2012 Phys. Rev. B 85 205107
6
