Topological Invariants and Ground-State Wave Functions of Topological Insulators on a Torus
We define topological invariants in terms of the ground states wavefunctions on a torus. This approach leads to precisely defined formulas for the Hall conductance in four dimensions and the topological magneto-electric term in three dimensions, and their generalizations in higher dimensions. They are valid in the presence of arbitrary many-body interaction and disorder. These topological invariants systematically generalize the two-dimensional Niu-Thouless-Wu formula, and will be useful in numerical calculations of disordered topological insulators and strongly correlated topological insulators, especially fractional topological insulators.
Topological insulators are among the major recent developments in condensed matter physicsQi and Zhang (2010); Hasan and Kane (2010); Qi and Zhang (2011). The physics of topological insulators started with noninteracting systemsC. L. Kane and E. J. Mele (2005a); B.A. Bernevig and S.C. Zhang (2006); C. L. Kane and E. J. Mele (2005b); B. A. Bernevig et al. (2006); König et al. (2007); Fu and Kane (2006); Moore and Balents (2007); Qi et al. (2008); Fu et al. (2007); Roy (2009); Schnyder et al. (2008); Kitaev (2009), for which simple and calculable topological invariants have been invaluable tools. More recently, it became clear that the interplay between topology and many-body interaction is a still richer fieldRaghu et al. (2008); Shitade et al. (2009); Zhang et al. (2009); Seradjeh et al. (2009); Pesin and Balents (2010); Fidkowski and Kitaev (2010); Hohenadler et al. (2011); Wang et al. (2010a); Sun et al. (2009); Li et al. (2010); Dzero et al. (2010); Rachel and Le Hur (2010); Regnault and Bernevig (2011); Zhang et al. (2012); Guo and Shen (2011); Guo et al. (2012); Sheng et al. (2011); Zheng et al. (2011); Yu et al. (2011); Levin and Stern (2009); Maciejko et al. (2010); Swingle et al. (2011); Varney et al. (2011); Griset and Xu (2012); Neupert et al. (2012); Go et al. (2012); Chen et al. (2013, 2013); Wang et al. (2011a); Budich et al. (2012); Wang et al. (2012, 2011b); Hohenadler et al. (2012); Kargarian et al. (2011); Assaad et al. (2013); Budich et al. (2013); Hung et al. (2013a, b); Werner and Assaad (2013); Lu et al. (2013); Lang et al. (2013); Hohenadler and Assaad (2013); Zhang et al. (2013); Wang et al. (2012a); Araújo et al. (2013); Zhu et al. (2013); Xu et al. (2013); Xu and Senthil (2013); Oon et al. (2012); Bi et al. (2013), therefore, it is highly desirable to develop topological invariants that are valid in the presence of strong interaction.
The root state of three-(spatial)-dimensional (3D) and two-(spatial)-dimensional (2D) topological insulators with time reversal symmetry is the four-(spatial)-dimensional (4D) quantum Hall (QH) stateZhang and Hu (2001); Qi et al. (2008) from which the topological field theory of 3D and 2D insulators can be obtained by the procedure of “dimensional reduction”Qi et al. (2008). The electromagnetic effective action of the 4D QH effect readsBernevig et al. (2002); Qi and Zhang (2011)
where we have adopted the units that the electric charge , the Planck constant , and the light velocity are all unity. The coefficient is referred to as the “4D Hall conductance”(or the 4D Hall coefficient). Physically, the 4D QH effect has the nonlinear topological electromagnetic responseQi et al. (2008) in the bulk, which is described naturally by the Chern-Simons effective action. If a nontrivial 4D QH insulator is cut open in one direction, there are copies of 3D chiral fermions (Weyl fermions) modes localized at the boundary. These boundary modes are close analogues of the 1D chiral edge statesHalperin (1982); Wen (1990) of 2D QH. In fact, the QH effect can be generalized to all even spatial dimensions, whose boundary modes are chiral fermions in odd spatial dimensions.
In the noninteracting limit, the explicit formula for has been obtained by Qi, Hughes and Zhang asQi et al. (2008)
where is the non-Abelian Berry curvature defined in terms of the noninteracting Bloch states. 111The lower-case letter refers to the noninteracting limit, which should not be confused with the upper-case defined in Sec.VI.
Now a natural question arises: Can we find a formula for that is precisely defined in the presence of arbitrary interaction and disorder? Such a formula, if exists, will be especially desirable for the investigation of fractional quantum Hall states in 4D. More importantly, it may also shed light on strongly interacting topological insulators in lower dimensions.
The same question also arises for the 3D topological insulators, whose effective topological responses theory is given byQi et al. (2008)
This topological effective action describes the quantized topological magnetoelectric effect, in which an electric field induces a magnetization with universal constant of proportionalityQi et al. (2008).
which is a 3D Chern-Simons term. In the presence of time-reversal symmetry, this Chern-Simons term is quantized and has been shown to be equivalentWang et al. (2010b) to the topological invariantFu et al. (2007). The natural question is: Is there a formula for that is valid in the presence of arbitrary interaction and disorder? From the experimentalist’s perspective, this question is more urgent than the 4D QH case, because many 3D topological insulators have been realized in experiments, and the electron-electron interaction has been playing more important roles.
To partially answer these questions, interacting topological invariants expressed in terms of Green’s function at zero-frequency (namely the “topological Hamiltonian”Wang and Yan (2013)) for interacting insulators have been proposedWang and Zhang (2012a, b); Wang et al. (2012b), which provide an efficient approach for topological invariants of various topological insulators and superconductors [See, e.g. Ref.Go et al. (2012); Budich et al. (2013); Werner and Assaad (2013); Hung et al. (2013a, b); Lu et al. (2013); Lang et al. (2013); Hohenadler and Assaad (2013); Nakosai et al. (2013); Manmana et al. (2012); Deng et al. (2013); Yoshida et al. (2013); Budich and Trauzettel (2013) for applications]. However, there are several shortcomings of that approach. First, it cannot be directly applied to disordered systems in which the momentum in the single-particle Green’s function is not a good quantum number.222Enlarging the unit cell can partially overcome this difficulty, which enables an extension of this approach to disordered systems. Second, it is unclear whether or not that approach may fail for some fractional topological states.
In Ref.Niu et al. (1985), Niu, Thouless and Wu found for the 2D QH a topological invariant (the first Chern number) expressed in terms of ground state wavefunction under twisted boundary condition, which is valid in the presence of arbitrary interaction and disorder. 333Twisted boundary conditions have also found applications elsewhere, see e.g. Ref.Kohn (1964); Hetényi (2013). To search for the general formulas for in 4D and in 3D, a hopeful approach is to generalize their formula to higher dimensions. However, as we will see later, the most straightforward 4D generalization of their formula, namely the generalization of the 2D phase twisting to the 4D phase twisting [see Eq.(30)], cannot produce the 4D Hall conductance . Due to this difficulty, it is unclear how this approach can be generalized to higher-dimensional topological states.
In this paper we propose general topological invariants for higher dimensional topological insulators in terms of ground state wavefunctions. The boundary conditions adopted here are not the standard one used in Ref.Niu et al. (1985), which is a pure gauge with vanishing field strength. Using these new boundary conditions [see Sec.II and Sec.VIII], we obtain for and simple formulas expressed in terms of the ground state wavefunction on a torus [see Eq.(12),Eq.(29),Eq.(41), Eq.(44), etc]. We also generalize these formulas to higher dimensions[see Eq.(24), etc]. These topological invariants are valid in the presence of arbitrary interaction and disorder, thus they can be applied to topological states with strong disorders and strongly correlated topological states including fractionalized states. Unexpectedly, the generalized formula for appears not as a second Chern number, but as the difference between two first Chern numbers [Eq.(12),Eq.(29)]. Similarly, the formula for does not appear as a Chern-Simons form, but as the difference between two winding numbers[Eq.(41), Eq.(44)].
The rest part of this paper is organized as follows. In Sec.II we study the 4D QH and define the topological invariant for integer QH in 4D. In Sec.III we test this topological invariant in two noninteracting models. We then generalize the 4D topological invariant to higher dimensional QH effects in Sec.IV. In Sec.V we present the topological invariants for fractional quantum Hall effects. A different boundary condition is investigated in Sec.VI, which leads to 4D topological invariant unrelated to the 4D Hall conductance. The next two Sections, namely Sec.VII and Sec.VIII, is devoted to 1D and 3D term respectively.
Ii 4D Hall coefficients expressed in terms of the ground state wavefunction
In this section we describe the topological invariant defined in terms of the ground state wave function of a 4D insulator on a torus with generalized twisted boundary conditions. For simplicity, in this section we assume that the ground sate is unique, while the cases with ground state degeneracy will be studied in Sec.V. We take the system to be a 4D torus with circumference along the direction respectively. We take the generalized twisted boundary condition parameterized by as follows. 444Note that there are two ways to define the twisted boundary condition. The first is to put the twisted phase factor in the wavefunction, as we did in Eq.(5), Eq.(6), and Eq.(7). The second way is to add phase parameters in the Hamiltonian instead of the wavefunction. These two ways are equivalent and can be translated into each other. In fact, a gauge transformation of the wavefunction changes the twisted boundary condition to the periodic boundary condition, at the price of adding the twisted phase factor to the Hamiltonian. First, for ,
where is the coordinate of the -th particle ( other arguments such as spin are not shown here for simplicity of notation ), is the total particle number, and is the unit vector along the direction. This condition is the same as the one adopted in Ref.Niu et al. (1985). Second,
Since on the torus, the flux has to be quantized as , where the unit flux , and is an integer. Lastly
Physically, these twisted boundary conditions tell us that there is a gauge potential along the () direction, and a gauge potential along the direction, in other words, there is a magnetic flux inside any 2D torus whose coordinates are with fixed .
Before proceeding to our central results, let us briefly outline the motivations of the boundary conditions given in Eq.(5), Eq.(6), and Eq.(7). The first motivation is that the boundary condition [see Sec.VI] does not produce the 4D Hall conductance. The second motivation is the intuitive relation between the 4D Hall effect and the 2D Hall effect. In Eq.(1), if we take to be independent on , and at the same time take to be independent on , then there is a “dimensional reduction” 555This is analogous to the Kaluza-Klein compactification. of the 4D Chern-Simons term to the 2D Chern-Simons term: (up to a numerical factor), where , and the indices in “” take value . According to this argument, in our boundary conditions given in Eq.(5), Eq.(6), and Eq.(7), we have taken , thus we have the dimensional reduction . Intuitively, we have the evident identity
Since the right hand side of this equation is a 2D Chern-Simons term, it seems that we can calculate using well-known results of 2D quantum Hall effects. In practice, however, it is impossible to take the derivative with respect to because is quantized, i.e. takes only discrete values. To resolve this difficulty, we will take a difference instead of a derivative (see below).
Now our task is to formulate these intuitive arguments as a precise mathematical framework. We can define the Berry connection
and the Berry curvature
from which we can define a first Chern number
where we have chosen the notation “” instead of “” to distinguish with the first Chern number appearing in the 2D quantum Hall effectsNiu et al. (1985).
With these preparations, the general formula for appearing in Eq.(1) is proposed as
This is the difference between two first Chern numbers, the first of which is the Chern number with a unit flux in , and the second is the Chern number without this flux, in other words, Eq.(12) measures the jump of the first Chern number after inserting a flux in . The necessity of the second term in Eq.(12) can be easily appreciated in a noninteracting model [see Eq.(19)] to be presented in Sec.III. It is also useful to note that may be zero if the ground state has certain symmetries. For instance, if there is time reversal symmetry, we have and .
Eq.(12) is expressed in terms of the Berry phase of ground states wavefunctions on a torus, which is well-defined in the presence of arbitrary interaction and disorder. 666 We can also write for any integer . For simplicity, this will not be pursued in the present paper. Eq.(12) can also be written equivalently as
Several remarks about Eq.(12) are in order. The noninteracting topological invariant for the 2D quantum Hall effect, namely the TKNN invariantThouless et al. (1982), is expressed as the first Chern number in the Brillouin zone. The Niu-Thouless-Wu formulaNiu et al. (1985), as a generalization of the TKNN invariant, is again a first Chern number. Given the second Chern number in Eq.(2) for the 4D noninteracting quantum Hall effect, we may try to express the 4D Hall coefficient as a second Chern number on certain parameter space, for an interacting system. However, this attempt turns out to be unfruitful. Instead, the topological invariant defined in Eq.(12), which gives , is the difference between two first Chern numbers.
Let us conclude this section with a side remark that the Laughlin’s gauge argumentLaughlin (1981) can also be generalized to 4D QH. The boundary condition in the direction are the same as given by Eq.(6) and Eq.(7), but the system is open along the direction. When we do the adiabatic evolution , the charge transferred from the boundary to is denoted as . The Hall conductance is given as .
Iii The noninteracting limit: Two simple models
In this section we will check in two simple noninteracting models [Eq.(14) and Eq.(19)] that Eq.(12) gives the same result as Eq.(2), as it should do in the noninteracting limit. Incorporating well-known results of topological classification of noninteracting insulators, we will show that Eq.(12) reduces to Eq.(2) for all noninteracting 4D insulators.
First let us consider a noninteracting Hamiltonian for 4D QHQi et al. (2008)
where , and being parameters of the Hamiltonian, and is the -th momentum of the free Bloch state (the lattice constant has been taken as unity). The Gamma matrices here satisfy the identities . For our convenience we choose the representation .
Instead of solving the model numerically in the real space, which is less illuminating for our purpose, let us do calculation in the limit that is significantly smaller than unity. In this limit we can keep only the -linear terms near , and the Dirac Hamiltonian reads
In the presence of twisted boundary conditions, the momenta should be replaced by . Let us calculate the first term of Eq.(12) for the Dirac Hamiltonian in Eq.(15). In this linear- limit, we can first solve the Hamiltonian , whose eigenvalues readCastro Neto et al. (2009)
where . The corresponding eigen-wavefunctions are and , where is the wavefunction of the -th Landau level of Schrodinger particlesCastro Neto et al. (2009), whose precise forms do not concern us for our purpose. It is useful to note that when , the existence of the zero mode is guaranteed by the Atiyah-Singer index theorem. Inputting the eigenvalues given in Eq.(16) into the second parenthesis in Eq.(15), we have a serial of 2D Hamiltonians
The value of can be obtained as the summation of the first Chern number of and , namely , thanks to the fact that the ground state wavefunctions is a Slate determinant of Bloch states in the noninteracting cases. In this calculation we have not been careful about the high energy regularization, thus we can only assert that . Since we require as , we have . Similarly we can obtain that , therefore we have
Let us move to the second noninteracting model, which will explain the reason why we must include the second term in Eq.(12). The simple model has the free Hamiltonian
which is independent on and . If we take , then it is obvious that both and are nonzero, however, they are equal, therefore . From Eq.(2), it is obvious that we have , therefore, Eq.(12) and Eq.(2) produce the same result in this example.
Although we have only explicitly checked that Eq.(12) reduces to Eq.(2) in Dirac models, it is possible to make a more general statement that Eq.(12) is always equivalent to Eq.(2) in the noninteracting limit. In fact, as has been shown in Ref.Kitaev (2009); Ryu et al. (2010), there is a Dirac-Hamiltonian representative in each class of the 4D QH insulators, which means that any noninteracting Hamiltonian for 4D insulator can always be smoothly connected to a Dirac Hamiltonian. Therefore, equivalence between Eq.(12) and Eq.(2) in Dirac model implies their equivalence for all noninteracting Hamiltonians. In the presence of interaction, however, Eq.(2) loses definition, while Eq.(12) remains useful.
Iv Quantum Hall effect in spatial dimensions
for . Physically, these conditions means that there is a flux in the 2D torus . We can define the Berry connection
for , and a first Chern number
Now the -dimensional Hall conductance is given by
V Fractional quantum Hall effects
One of the main motivations for introducing the topological invariant in Eq.(12) is its potential applications in fractional quantum Hall states. Before moving to higher dimensions, let us first present a review of the Niu-Thouless-Wu formula of 2D fractional QH. As has been known from the Ref.Niu et al. (1985), fractional quantization of 2D Hall conductance is possible if the ground states are degenerate on a 2D torus.
In 2D, the standard boundary condition is givenNiu et al. (1985) by Eq.(5) except that the argument is absent. Suppose that a 2D fractional quantum Hall system has -fold degenerate ground states . 777It is worth noting that such a basis can be found only locally, namely that they can be defined only in a topologically trivial patch on the 2D torus parameterized by . The quantities we actually need, such as , are independent on the basis choices. The Hall conductance is given by an average over these degenerate ground states asNiu et al. (1985) (recall that we have taken the units )
where the matrix elements of the non-Abelian Berry curvature read , in which is the non-Abelian Berry connection. The average Chern number , where is the standard definition of the first Chern numberNakahara (1990) of the fiber bundle. Note that the term in vanishes after the tracing. It is a mathematical fact that the first Chern number is quantized as an integer, therefore, the Hall conductance is quantized as a rational number with denominator .
where we have picked up a ground state from the degenerate ground state . The parameter space has been enlarged to .
Now let us move to higher dimensions. For a 4D fractional QH system, suppose that the ground states are -fold degenerate on the 4D torus with boundary conditions described in Sec.II, in other words, the ground states form a bundle over the 2D torus with coordinates ( Note that is fixed ). 888For simplicity, we assume that the degenerate ground states cannot be divided into smaller subspaces, with each subspace forming a fiber bundle over the torus parameterized by . Otherwise we can just pick up one subspace that cannot be reduced further into smaller subspaces, and everything discussed in this section will be unchanged. We can define the Berry connection and the Berry curvature . Eq.(11) can be straightforwardly generalized as
By analogy with Eq.(25), the 4D Hall conductance for fractional quantum Hall effects is obtained as
Eq.(29) is among the central results of this paper. In the presence of time reversal symmetry, the second term vanishes. Eq.(29) reduces to Eq.(12) when , namely the case without ground state degeneracy.
To conclude this section, we mention that the generalization of Eq.(24) for dimensional fractional states read .
Vi More topological invariant for 4D fractional QH
Having studied the 4D fractional Hall conductance using the boundary conditions we have chosen, let us investigate other choices of boundary conditions. The simplest choice is
for . Suppose that the ground states are -fold degenerate, then these ground states form an fiber bundle on the 4D torus parameterized by with . We can define a natural topological invariant
where the matrix elements of non-Abelian Berry curvature are defined as , where . Eq.(31) is a second Chern number defined for fractional QH states in 4D. It should be not be confused with the (lower-case) in Eq.(2), which is defined in terms of the free Bloch states of noninteracting systems.
For -dimensional quantum Hall effects, we can straightforwardly generalize to as
which are topological invariants for higher-dimensional fractional QH states.
In 2D, the first Chern number of the bundle is proportional to the Hall conductance . In fact, Eq.(25) tells us that , thus does not give us new topological invariant other than and . However, the 4D case is quite different. The key difference between 2D and 4D is as follows. For the 2D QH, both and are defined under the same boundary condition parameterized by . For 4D quantum Hall insulators, the topological invariants and is defined using different boundary conditions [ Eq.(5), Eq.(6), and Eq.(7) for , but Eq.(30) for ], therefore, there is no direct relation between and . In principle, can take different values given the same value of ground state degeneracy and Hall coefficient . The topological invariants suggests that there are rich structures in 4D quantum Hall effects. Higher dimensional QHs are similar: Higher Chern numbers () are not directly related to the Hall coefficient because they are defined under different boundary conditions.
Vii Topological insulators in one-dimension
In this section we will briefly discuss 1D topological insulators to prepare us for the investigation of 3D topological insulator in Sec.VIII. One-dimensional topological insulators can be characterized by a termQi et al. (2008)
Let us study the 1D insulator on a torus , which is just a circle. We take the boundary condition as
namely that there is a gauge potential .
where the Berry connection is defined as . Eq.(35) is an interacting generalization of the Zak phaseZak (1989). It has been applied to 1D modelsGuo and Shen (2011); Guo et al. (2012), though its relation to term was not discussed. Eq.(35) is defined modulo because a local gauge transformation of the wavefunction can change it by .
When the ground state is not degenerate, the value is given by . Since we are mainly concerned with higher dimensional topological insulators, we will not study applications of this 1D formula in details. It is useful to mention that the quantity , where is a tuning parameter of the many-body Hamiltonian, is usually more useful than itself, because does not have any ambiguity under local gauge transformation of wavefunctionOrtiz and Martin (1994).
Viii Topological insulators in three-dimensions: Integer and fractional
The approach we applied to 4D QH states can be naturally generalized to 3D. The 3D boundary conditions are chosen as follows. First,
where is the coordinate of the -th particle ( other variables such as spin are not shown for simplicity of notation ), and is the unit vector along the direction. Second,
where , and
being the Berry connection defined in terms of the ground state wavefunction. One can derive Eq.(41) by calculating the Berry phase gained by the adiabatic evolution . Due to the topological terms contained in the term, when a flux exists in , as Eq.(39) and Eq.(40) indicate, the adiabatic evolution of generates a topological phase , which should be identified as the Berry phase accumulated by the adiabatic evolution of ground state wavefunction, namely . It follows that Eq.(41) is the formula for . Note that potentially there is another term that can contribute to the Berry phase in the evolution , which is the reason why the second term in Eq.(41) appears.
If the Hamiltonian and the ground state depend on a tuning parameter, which we denote as , then is a function of . The derivative of with respect to is given by the gauge-invariant formula
where , and . Similar to the 1D case discussed in Sec.VII, the quantity is usually more useful than itself, because is invariant under any local gauge transformation of the wavefunction.
In the above calculations we have assumed that the term is isotropic, which is always satisfied if there is time reversal symmetry (though the Maxwell terms are generally still anisotropic). If the term is anisotropicEssin et al. (2010); Malashevich et al. (2010), namely that we have , we should calculate each coefficient separately, which is also given by Eq.(41) except that the twisted phase in Eq.(38) is added in the direction instead of the direction, and the flux [ see Eq.(39) and Eq.(40)] is added in the plane.
For 3D fractional states with -fold ground state degeneracy, we can generalize Eq.(41) as
where . The logic is similar to Sec.V. An important feature is notable here. We have the transformation rule under a local gauge transformation of the basis of ground state wavefunction, where is a unitary matrix. This may change by multiples of , therefore, the angle of fractional topological insulators is determined modulo .
As a digression, let us briefly mention the generalization for (spatial) dimensional (isotropic) term if the system does not have ground state degeneracy on a dimensional torus. The formula reads
which is analogous to Eq.(24). The meanings of the arguments are similar to that of Eq.(24), which we shall not repeat here. If the state is fractional, we have , where , the integer being the ground state degeneracy.
In this paper we have defined precise topological invariants in terms of the ground state wavefunctions on a torus. This approach provides a conceptual framework in which many topological invariants and topological-field-theoretical coefficients, such as (in 4D) and (in 3D), acquire precise definitions even in the presence of arbitrary interaction and disorder.
Numerically, we do not expect that the wavefunction (on a torus) approach followed in the present paper will be as efficient as the topological Hamiltonian approachWang and Zhang (2012a); Wang and Yan (2013) mentioned in Sec.I. However, the present approach has a wider range of validity because it is applicable in the presence of arbitrary interaction and disorder, therefore, the present approach is highly desirable for certain purposes, especially when both interaction and disorder are present, or when the interaction is so strong that exotic fractional states are generated. It is also useful to note that the topological invariants in the present paper can also be applied to bosonic topological insulators, for which other topological invariants are hard to define.
ZW would like to thank Liang Kong and Yong-Shi Wu for helpful discussions. ZW is supported by NSFC under Grant No. 11304175. SCZ is supported by the Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering, under contract DE-AC02-76SF00515.
Appendix A Application in a three-dimensional noninteracting model
In the noninteracting limit, Eq.(41) should give the same as the noninteracting formulaQi et al. (2008). In this appendix we will check this in a simple noninteracting model. This appendix follows similar calculations of Sec.III.
Let us study a simple 3D noninteracting Dirac model given as
where . In the limit that , the low energy physics is dominated by the region, and we can linearly expand as
The boundary conditions are given in Eq.(38), Eq.(39), and Eq.(40), which mean that there is a flux inside the 2D torus . First let us take . By a calculation similar to Sec.III, we can first solve after replacing