Quantum Phase Transitions of Topological Insulators
Abstract
In this paper, starting from a lattice model of topological insulators, we study the quantum phase transitions among different quantum states, including quantum spin Hall state, quantum anomalous Hall state and normal band insulator state by calculating their topological properties (edge states, quantized spin Hall conductivities and the number of zero mode on a flux). We find that there exist universal features for the topological quantum phase transitions (TQPTs) in different cases : the emergence of nodal fermions at high symmetry points, the nonanalytic third derivative of ground state energy and the jumps of the topological ”order parameters”. In particular, the relation between TQPTs and symmetries of the systems are explored : different TQPTs are protected by different (global) symmetries and then described by different topological ”order parameters”.
pacs:
72.10.Bg, 71.70.Ej, 72.25.bpacs:
73.50.Mx, 78.66.w, 81.05.UwI Introduction
As the first topologically ordered phase, the Quantum Hall Effect (QHE) is a remarkable achievement in condensed matter physics(1); (2). In QHE state, at low temperatures and in strong magnetic fields, quantized Hall conductance can be observed due to the Landau levels formed by states of a twodimensional electron gas. Just one year later, people discovered the socalled anomalous Hall effect where additional Hall current in the ferromagnetic material was observed(3); (4). Then, people find that the transverse transport also exists for different spin species, which is called as spin Hall effect(5). Recently, the topological insulators including the quantized spin Hall (QSH) effect(6); (7); (8); (9) and the quantized anomalous Hall (QAH) effect(10); (11) become hot issues.
On the one hand, to describe the quantized anomalous Hall state, a class of topological insulators with timereversal symmetry breaking, the Chern number, so called TKNN integer, is introduced as integrals over the Brillouin zone (BZ) of the Berry field strength (12)
(1) 
Here is a Berry connection for singleelectron Hamiltonian with the periodic part of a Bloch state .
On the other hand, for quantized spin Hall state the spin Chern number is proposed in Ref.(13) to be the topological invariant to characterize the topological insulators. However, due to spin mixing term (the Rashba term), the spin Chern number is not well defined. Thus for a class of topological insulators with timereversal symmetry (Tsymmetry), due to Kramers degeneracy, Kane and Mele proposed a topological invariant(6); (14); (15); (16); (17)
(2) 
where are the four highsymmetry points satisfying , , , , is the parity eigenvalue at each of these points, and is the number of Kramers pairs below Fermi surface. Such topological invariant can be also defined in terms of the spin Chern number
(3) 
Here the integral of the Berry field strength is defined on half of the Brillouin zone. Physically, is identical with the number of pairs of helical edge modes.
Thus, an interesting issue is the nature of the quantum phase transitions between different types of topological insulators. It is well known that in Landau theory different orders are classified by symmetries. The phase transitions accompanied with (global) symmetry breaking are always described by order parameters. However, Landau’s theory fails to describe the topological quantum phase transition (TQPT). Such type of quantum phase transition cannot be classified by symmetries(18). Instead, they may be characterized by some topological ”order parameters”, such as the Chern number or topological invariant(19); (20); (21).
In this paper, we study the TQPTs between topological insulators and show their physical properties. We find that there are universal features of the TQPTs for different cases : the existence of nodal fermions at high symmetry points, the nonanalytic third derivative of ground state energy and the jumps of the topological ”order parameters”. In particular, we find the symmetry protected nature of the TQPTs : different TQPTs are protected by different (global) symmetries and then described by different topological ”order parameters”.
The remainder of the paper is organized as follows. In Sec.II, the twodimensional lattice models of topological insulators are given. In Sec.III, we focus on the TQPTs between different quantum states with conservation (without time reversal symmetry). In this section, the global phase diagram and the critical behavior are obtained. In addition, the topological properties of different quantum states are calculated, including the edge states, the quantized Hall conductivities and the induced quantum numbers on a flux. In Sec.IV, we will study the TQPTs with time reversal symmetry (without conservation). In Sec.V, We will study the TQPTs without time reversal symmetry and conservation. Finally, the conclusions are given in Sec.VI.
Ii The lattice models of topological insulators
As a starting point, we consider a lattice Hamiltonian for twoflavor spin1/2 fermions(22):
(4) 
where
(5)  
(6) 
Here is a twocomponent particle operator where is the spin index, , denote the flavor indices and labels the site on the square lattice. is the real hopping parameters. is the orbital splitting energy. is an ”effective” magnetic field which was first introduced in Ref.(11) phenomenologically which may be induced by local magnetic moments. The bond variables is set to everywhere at the beginning. For the case of , we set the is positive for spinup while the is negative for spin down. Without terms, the model in Eq.[3] is really a lattice realization of the KaneMele model proposed in Ref.(23); (24), of which the ground state is a topological insulator for .
Using the Fourier transform we can transform the four field operators on each site into momentum space Then we get
(7) 
where, for the spin up part
and for the spin down part
The energy spectrum is then given by diagonalizing the Hamiltonian
(8) 
It is obvious that the term of breaks the Tsymmetry,
(9) 
where ( are Pauli matrices and stands for complex conjugation). However, the model has an additional spin rotation symmetry around direction as
(10) 
In addition, by adding the Rashba term to , we get another lattice model where
(11) 
By using the Fourier transform we get
(12) 
where, is the unit matrix. Now there is Tsymmetry
(13) 
However is not a good quantum number due to
(14) 
Furthermore, one may consider a lattice model with both and as . Now the Hamiltonian has neither Tsymmetry nor conservation due to
(15) 
and
(16) 
In the following parts we will study TQPTs by keeping different symmetries based on different lattice models .
Iii TQPTs with conservation :
iii.1 Global phase diagram
In this section, to learn the TQPT of topological insulators with conservation ( )we focus on
By calculating the phase boundary with zero fermion energy, we obtain the phase boundary,
(17) 
In the global phase diagram which was shown in Fig.1, there exist three phases : QAH state, QSH State and NI state. Taking as an example (the horizontal dash dot line in Fig.1), one can see that the QSH state occurs in the region of . Although the Tsymmetry is broken in this region, the topological properties of the QSH state is preserved (the edge states, the quantized Hall conductivities and the induced quantum numbers on a flux). With the decrease of , the system turns into a QAH state in the region of In this region, one may find that the topological properties are different from that in the QSH state. In the region of , the ground state turns into a normal band insulator with trivial topological properties. In Fig.1, the solid, dashed and dotted lines are given by and which denote the phase boundaries obtained above.
iii.2 Topological ”order parameter”  Spin Chern number
When one can not use the topological number to classify the topological insulators. However is still a good quantum number, the Hamiltonian is decoupled for each spin component, then the spin Chern number is suitable to characterize different quantum states. The Chern number of each spin component is defined as
Now we write the Hamiltonian of upspin and downspin fermions into
(18) 
where and . In above Hamiltonian, we have define
(19)  
where
(20)  
and
(21)  
are Pauli matrices.
Now two Chern numbers and for upspin particle and downspin particle are obtained as(21),
(22) 
and
(23) 
Here and are defined as and , respectively. is the area of Brillouin zone.
In the QSH state, for upspin, the Chern number is ; for downspin, it is . So the total Chern number is zero . However there is nonzero spin Chern number . In the QAH state, for upspin, the Chern number is ; for downspin, it is . So the total Chern number is obtained as and spin Chern number . In NI state, the Chern number is zero for each spin component, . From the results of the Chern number of different quantum states, one may imagine that the QAH state may be considered as half of the QSH state. This statement is also supported by the following calculations of other topological parameters.
In order to give a clear comparison of the total Chern number and the spin Chern number in different quantum states, we list the results in Table.1.

From the Table. 1, one can see obviously that the spin Chern number can be regarded as the topological invariant to characterize the topological insulators and their quantum phase transitions for the Hamiltonian with conservation(13); (25).
Edge states
In this part, we calculate the edge states in different states.
From the relationship between the spin Chern number and the number of the (chiral) edge states for each spin component we have
(24)  
Different Chern numbers also characterize different edge states. In the QSH state, the total number of the edge states is obtained as (21)
(25)  
Thus there are two edge states with opposite chirality. In the QAH state, there is only one edge state The number of edge states in different phases are plotted in Table.2.

The numerical results of the edge states in different states are shown in Fig.2.
From the Table.2, one can see that it is the spin Chern number that determines the number of the edge states in different phases.
Quantized Hall conductivity
Next, we study the quantized Hall conductivity of different states.
We could calculate the charge Hall conductance and the spinHall conductance for this model in different phases using the standard representation(21)
(26) 
and
(27) 
where
(28)  
Here one can see that the physical consequence of the edge states is just the quantum Hall conductance in our case. In the QSH state, the quantum Hall conductance is obtained as
(29) 
Thus there is no charge Hall conductance, . Instead, the spinHall conductance is nonzero, In the QAH state, the quantum Hall conductance is obtained as
(30) 
Thus both charge Hall conductance and spinHall conductance in QAH state are half of the spinHall conductance in QSH state
(31)  
It is pointed out that the spinHall conductance is still well defined due to conservation in the QAH state.
The spin Hall conductance in different states were shown in Fig.3. The xaxis corresponded to the horizontal dash dot line in Fig.1. We can see that the spin Hall conductance is , and in NI, QAH and QSH respectively. The spin Hall conductance changed from 0 to at the phase transition point and changed from to at the phase transition point . In this sense, the TQPTs between QSH, QAH and NI are similar to the quantum Hall plateau transition from Dirac fermion in Ref.(26).
The clearly comparison of chargeHall conductance and spinHall conductance in different states are shown in Table.3

This table indicates that spin Chern number and total Chern number determine charge Hall conductance and the spinHall conductance in different states, respectively.
Induced quantum numbers on a flux
In addition, we calculate the induced quantum number on a flux in different quantum states. A flux denotes half a flux quantum on one plaquette of the square lattice which is shown in Fig.4.
We calculate the induced quantum number on a flux in the QSH state firstly. In the QSH state, the flux defects defined above may be considered as the JackiwRebbi soliton(27). From the numerical results, we find that there are two zero modes of a single flux defect. So such a two dimensional point defect possesses four soliton states, two for each spin (occupied and nonoccupied) which are denoted by
(32) 
and
(33) 
Here and are the empty states of the zero modes and Around a flux, the fermionic operators are expanded as
(34)  
where and are operators of modes that are irrelevant to the soliton states discussed below. are annihilation operators of zero modes and is the spin index. Thus we have the relationships as
(35)  
We define the induced fermion number operators of the soliton states, with
From the relation in Eq.(35), we find that or have eigenvalues of of the induced fermion number operators,
(36)  
From we can define two induced quantum number operators, the total induced fermion number operator and the induced quantum spin operator,
Then we calculate two induced quantum numbers defined above. For an insulator state, the ground states are the two degenerate soliton states denoted by and . One can easily check that the total induced fermion number on the solitons is zero from the cancelation effect between different spin components
(37) 
On the other hand, there exists a spin moment on the soliton states and
(38)  
The induced spin moment may be straightforwardly obtained by combining the definition of and Eq. (36) together.
The four different ways of occupation these two zero modes give rise to four different type of fluxons with the following quantum numbers: ( ) and ( ). The results is consistent to that there exist spincharge separated solitons in the presence of flux with induced quantum numbers(28); (23); (29).
The charge density of a fluxon in the QSH state on a lattice is shown in Fig.5. The zero mode is localized around the defect center within a lengthscale Here is the mass gap of the fermions.
Secondly, we study the zero modes and the induced quantum numbers on a flux in the QAH state. In the QAH state, there is only one zero mode of a single flux defect. The charge densities of a pair of defects in the QAH state on a lattice with periodic boundary condition was shown in Fig.6. For the two defects, the two zero modes slightly split due to tunneling effect between them.
Thus there are two soliton states, two for up spin particles (occupied and nonoccupied) which are denoted by and In contrast, there is no zero mode and soliton states of the flux defect for down spin particles. Similarly we can obtain the eigenvalues of the total induced fermion number operator ,
(39) 
And the induced quantum spin number on the soliton states is obtained as
(40)  
The occupation (or unoccupation) of this zero mode lead to charge and bound to the defect(30). Due to conservation for the QAH state, the induced quantum spin number is also well defined.
Thirdly, in the NI state, there is no zero mode on a single flux defect. As a result, all induced quantum numbers are zero.
Therefore, in this part we find that the zero mode’s on a flux may also be used to distinguish different quantum phases. These induced quantum numbers are shown in Table.4.

Finally we give a summary. When because is a good quantum number, we can use the spin Chern number to characterize different quantum states as shown in the Table.5.

For the TQPTs from one quantum state to another, the spin Chern number will jump.
iii.3 Universal critical behavior of TQPTs
Another feature of the TQPTs is the nonanalyticity of the groundstate energy. The groundstate energy is defined as
(41) 
where, denotes the first Brillouin zone and
(42) 
Here is the area of the system. As an example, we show the TQPT of the model along the line with fixing in Fig.1. As illustrated in Fig.7, the ground state energy and its first and second derivatives are continuous for arbitrary , while its third derivative is nonanalytic at points and , corresponding to phase transitions NI – QAH and QAH – QSH. It means that the TQPTs are third order. Similar third order TQPTs have been pointed out in other systems where the nodal fermion appears(31).
Let us explain why TQPTs are always third order. The energy dispersions near the quantum phase transitions are shown in Fig.8 and Fig.9. On the line with fixing one can see that there is one nodal point with zero energy in the Brillouin zone, at the quantum phase transition between QSH state and QAH state. Thus the TQPT is dominated by nodal Dirac fermionic excitations at . For the TQPT between QAH state and NI state on the line with fixing , the nodal fermion is also pinned at the point .
Furthermore, near the critical points of the TQPTs, we get the low energy effective Hamiltonian near the point
(43) 
and
(44) 
where and are the masses of the electrons with upspin and downspin, respectively. One can see that crossing the critical point , , the mass of the electrons with downspin changes sign, Consequently, the Chern number of the electrons with downspin changes from to that corresponds to the TQPTs from NI state () to QAH state ( ). Similarly, crossing the critical point , , the mass of the electrons with upspin changes sign, , that leads to the Chern number of the electrons with upspin changes from to , corresponding to the TQPTs from QAH state ( ) to QSH state ( ). For a special TQPT at (), the mass of the electrons with upspin is equal to the mass of electrons with downspin The changes of mass sign will lead to the jumps of spin Chern number, that corresponds to a direct topological quantum phase transition from NI state () to QSH state ( )(32). Fig.10 is a scheme to illustrate the relationship between the jump of the spin Chern number and the change of mass sign.
On the other hand, due to the existence of the nodal fermion, the TQPTs with mass sign changes () are always third order. One may use the low energy approximation, and then get the ground state energy as
(45) 
where is a constant(31). It is obvious that the third order derivative of to is discontinuous at the point .
In brief, for the TQPTs, NI – QAH and QAH – QSH, the changes of the mass sign reflect the jumps of the (spin) Chern numbers (). As a result, third order phase transitions is a universal feature of the TQPTs.
iii.4 Finite temperature properties
At finite temperature (), the free energy is defined by
(46) 
Here we set . We calculate the derivatives of the free energy which are shown in Fig.11. One can see that the free energy and its first, second, third derivatives are all analytic without singularity at . The results means that at finite temperature, there is no real phase transition, instead one gets crossovers. This result is consistent with that of the TQPT between QAH state and NI state in Ref.(31).
iii.5 Stability of the TQPTs
Finally we discuss the stability of the TQPTs. At the critical points, there always exist nodal fermion with zero energy at . Thus at the critical points, we linearize the dispersion at and get a low energy effective Dirac Hamiltonian of nodal fermions,
(47)  
where the matrices are defined by and .
Then we consider effect of short range interactions. For example, one may add an onsite fourfermi interaction to Now we get an effective three dimensional GrossNeveu model with the Lagrangian(33)
(48) 
It is known in largeN limit, the CallanSymanzik function of is . For a small interaction, the fourfermi interaction is irrelevant. That means the TQPTs is stable against the small onsite fourfermi interaction. Similarly, considering other types of short range interaction with conservation, one may get the same results: they are irrelevant and the nodal fermions are stable.
Iv TQPTs with Tsymmetry :
iv.1 Global phase diagram
In this section, to learn the TQPT of topological insulators with Tsymmetry, we focus on and show the effect of the Rashba term.
From the global phase diagram in Fig.12, one can see that there exist three quantum phases : QSH state, NI state and gapless semimetal. The solid line divides the gapped phases and the gapless semimetal. As shown in Fig.13, when one fixes at , the energy gap will decrease with the increase of and close at However, the phase boundary (the dashed line in Fig.12) between the QSH state and NI state doesn’t change by the Rashba term.
iv.2 Topological ”order parameter”  topological invariant
To see the topological properties of the Hamiltonian with Tsymmetry but without conservation, we calculate the edge states and the zero mode’s number on a flux in different quantum states. In the QSH state, from the numerical results, we find that there also exist two zero modes of a single flux defect. The charge density of a fluxon in the QSH state on a lattice is shown in Fig.14. In the NI state, there is no such zero mode on a single flux defect. On the other hand, the results of the edge states are shown in Fig.15, from which we find that there are two edge states in QSH state, while in the NI state, there is no edge state.
Above results indicate that the Tsymmetry indeed protects the topological properties of the QSH state. Thus we calculate the topological invariant in different gapped quantum state. For the four highsymmetry points, , , , , the eigenvalues below Fermi surface of this system are
(49) 
respectively. TQPT occurs at . For the four eigenvalues are all positive. Using the formula mentioned above, one can see that the topological invariant is which means the system is in the NI phase. For , one eigenvalue is negative while the others are all positive. It is obvious that the topological invariant is which means the system is in the QSH phase.
Table.6 shows the relationship between the topological invariant and the topological properties of the quantum states,

iv.3 Universal critical behavior of TQPT
To study the universal critical behavior of the TQPT between QSH state and NI state without conservation, we calculate the derivatives of the groundstate energy . Here the groundstate energy becomes where
(50) 
We show the TQPT along the line with fixing . As illustrated in Fig.16, the third derivative is nonanalytic at points , corresponding to quantum phase transitions NI – QSH. So the TQPT is also third order.
The energy dispersion near the quantum phase transition is shown in Fig.17. From the dispersion of the electrons near the critical points, we find that the TQPT is also dominated by the nodal fermion near the point
Let us explain why the TQPT between QSH state and NI state are also third order. When there is no conservation and one cannot define the spin Chern number. So we cannot use the jump of the Chern number to understand the TQPT. However, from the definition of the topological invariant, we find that near the TQPT, the mass signs of the two kinds of low energy fermions () are determined by the topological invariant (See Eq.49). Consequently, for the TQPT between NI state () to QSH state (), the sudden change of the topological invariant will lead to the change of mass sign of low energy fermions.
In brief, for the TQPT, NI – QSH, the change of mass sign come from the sudden change of the topological invariant. As a result, the TQPT between NI state and QSH state is always third order.
V TQPTs without Tsymmetry and conservation :
In this section, we study the TQPTs of topological insulators without Tsymmetry and conservation, based on
We take as an example (which corresponds to the horizontal dash dot line in Fig.1). The results are illustrated in Fig.18. One can see that in the phase diagram there exist three phases : two NI states and one QAH state. The QAH state is robust while arbitrary small will destroy the QSH state. To understand the disappearance of the QSH state, on the one hand the evolution of the energy levels of zero modes on a flux with at the point is plotted in Fig.19. The black bars denote the continuum spectrum while the black triangles denote the energy levels of zero modes on a flux. One can see that the degeneracy of zero modes is lifted by the Rashba term. With increasing the energy gap of the continuum spectrum decreases, while the energy splitting of the zero modes increases. On the other hand, the evolution of edge states in QSH state with the Rashba term is shown in Fig.20. When and due to the hybridization of two edge states, the edge states in the QSH state open an energy gap. In contrast, in the QAH phase, the edge state and the zero modes are all stable.
In this case (a system without Tsymmetry and conservation), the TKNN integer can be regarded as an ”order parameter” to characterize different quantum states, Table.7 shows the relationship between the TKNN integer and the topological properties of the quantum states,
Using similar approach in above sections, the universal critical behavior of the TQPTs are studied. The TQPTs are also dominated by nodal Dirac fermionic excitation at the point . Since the TKNN integer (the Chern number) will jump crossing the TQPTs, the mass sign of one low energy fermionic excitation changes. Consequently, the TQPTs are also third order.
Vi Conclusion
In this paper, based on a twodimensional lattice model, we study the TQPTs between the QSH state, QAH state and normal band insulator and show their physical properties, including the edge state, the quantized (spin) Hall conductivity and the induced quantum number on a flux. There are common features of the TQPTs for different cases : the existence of nodal fermions at high symmetry points, the nonanalytic third derivative of ground state energy and the jumps of the topological ”order parameters”. In particular, we find the symmetry protected nature of the TQPTs which is illustrated in Table.8 :
In Table.8 the symbol means the system is invariant of the given symmetry and means not. From Table.8, one can see that for a system with conservation but without Tsymmetry, the spin Chern number plays the role of ”order parameter” to characterize the topological insulators; for a system with Tsymmetry but without conservation, the topological invariant becomes the ”order parameter”; for the system without conservation and Tsymmetry, it is the TKNN integer that becomes the ”order parameter”.
In the end, we give a comment on the relationship between symmetry and topological invariants for the TQPTs. TQPTs are not be classified by symmetries, instead, they may be characterized by some topological invariants, such as the Chern number or topological invariant. However, the ”symmetry” of the systems still plays important role : different topological quantum phase transitions are protected by different (global) symmetries and then described by different topological ”order parameters”.
The authors thank Ying Ran and XiaoLiang Qi for helpful discussions and comments. The authors acknowledges that this research is supported by NCET, NFSC Grant no. 10874017.
Footnotes
 thanks: Corresponding author
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