Quantum Spin Hall Effect as Z 2 Global Gauge Anomaly
QQuantum Spin Hall Effect as Z Global Gauge Anomaly
Yunqin Zheng and Shaolong Wan ∗ Institute for Theoretical Physics and Department of Modern PhysicsUniversity of Science and Technology of China, Hefei, 230026,
P. R. China (Dated: October 29, 2018)We study the relation between the quantum spin Hall effect(QSHE) and the Z global gaugeanomaly and discover that there exists an one-to-one correspondence between them. By constructinga two dimensional non-abelian gauge theory whose non-abelian gauge field is the Berry connectioninduced by the Bloch wave function of the quantum spin Hall system, we prove that if the quantumspin Hall system is topologically nontrivial, the corresponding 2D gauge theory has a global gaugeanomaly. We further generalize our discussion of the zero modes of the Dirac operator which play acentral role in our analysis. We find that the classical ”spin” Hall effect also contains a Z topologicalinvariant which is not noticed before as far as we know. PACS numbers: 73.43.-f, 03.65.Vf, 03.70.+k, 11.15.-q
1. INTRODUCTION
Topological insulator (TI) discovered by C. L. Kaneand E. J. Mele[1] is a new state of matter whose bulkbehaves like an insulator while its edge behaves like ametal. The mechanism under-lying the phenomenon isthe time-reversal symmetry. The edge, where the en-ergy gap is closed, is robust against perturbations aslong as the time-reversal symmetry is preserved. Thisis the reason why it is named a symmetry-protected-topological(SPT) state more precisely. Each TI systemis labelled by a topological invariant, which indicates itstopological properties. Transition between states withdifferent topological invariants happens only if the en-ergy gap closes. The reader can refer to Refs. [2–4] andreferences therein for a complete review.Now, many progress on the topological insulators havebeen made, among which the discovery of the Z topo-logical property of the quantum spin Hall states is a land-mark. This Z topological property differs greatly fromthose ordinary quantum Hall effects which are Z classi-fied. It states that in the Brillouin zone, there can stablyexist only one or zero Dirac-cone-like point which willbe defined more precisely below. The presence of morethan one such point is non-stable, implying that we canperturb the parameters so that they can be annihilatedand created in pairs, until there are only one pair ornone remains in the Brillouin zone. Then, the authorsof [5] proposed another formalism to describe the quan-tum spin Hall effects, and made a generalization to highdimensional topological insulators using a Chern-Simonsfield theory. The method is mainly dimensional reduc-tion, and they acquired a seemingly different set of topo-logical invariants. Wang[6] proved that in three dimen-sions, the two set of topological invariants are equivalent,which unified the two different approaches. ∗ Corresponding author; Electronic address: [email protected]
All above achievements are based on non-interactingsystems, and the band structure analysis is valid. How-ever, when interaction is taken into consideration, manyof the above discussion fail. Some results have been ac-quired by using Green function[7–9]. But much more arestill unknown. Another way to understand the interact-ing phenomena is to study anomalies related to topo-logical states since anomalies are not restricted to non-interacting systems. It has long been believed that var-ious anomalies and topological phases are in one-to-onecorrespondence. In Ref.[10], the authors made a sum-mary of the correspondence between Z topological statesand anomalies, but the situation for most Z states is un-clear. This is our primary motivation in this work and wepropose a relationship between the quantum spin Hall ef-fect (QSHE) Z invariant and the global gauge anomaly.The article is organized as follows. In Sec.2, we give abrief review according to Kane-Mele’s work[11] on a Z topological invariant. In Sec.3, we summarize some mainfeatures of the global gauge anomaly which are relevantto our discussion. In Sec.4, the relationship between thegauge anomaly and the Z topological invariant is estab-lished. As an application of our formalism, in Sec.5, wediscuss the Morse theory associated to the Dirac opera-tor and find a bonus, which indicates that the classical”spin” Hall effect also have a Z internal topological in-variant as in the quantum case predicted nearly a decadeago. Finally, we give some discussion and conclusion, andpropose some further topics to be investigated in Sec.6.
2. TOPOLOGICAL Z INVARIANTS OF A
DIMENSIONAL QUANTUM SPIN HALLSYSTEM
Before studying the relation between the quantum spinhall effect and the global gauge anomaly, it is beneficialto give a brief review on both theories.According to the work by C. L. Kane and E.J. Mele[11],we first consider a two spatial dimensional system, with- a r X i v : . [ c ond - m a t . s t r- e l ] J u l out interaction, with a Hamiltonian H ( k ). The corre-sponding Bloch wave functions | k, α (cid:105) are given by theeigen-equation H ( k ) | k, α (cid:105) = E α | k, α (cid:105) . (1)We require the Hamiltonian to be the time-reversal in-variant: Θ H ( k )Θ † = H ( − k ), where the time-reversaloperator is Θ = T K and K is a complex conjugate oper-ator. Due to such a symmetry, both | − k, α (cid:105) and Θ | k, α (cid:105) are solutions of an eigen-equation with an eigenvalue E α .They may be related by fixing the gauge condition as | − k, α (cid:105) = Θ | k, α (cid:105) .In order to define topological invariants, it is conve-nient to introduce two concepts: an odd space and aneven space. The even space is a set of points in mo-mentum space satisfying the condition: Θ H ( k )Θ † = H ( − k ) = H ( k ), which means that Θ | k, α (cid:105) and | k, β (cid:105) arenot perpendicular to each other, while the odd space is aset that the two vectors are mutually orthonormal. Thetwo space can be distinguished by considering whetherthe matrix (cid:104) k, α | Θ | k, β (cid:105) equals to zero(an odd space) ornot(an even space).Now, we list some properties of the matrix (cid:104) k, α | Θ | k, β (cid:105) :1. It is anti-symmetric, i.e. (cid:104) k, α | Θ | k, β (cid:105) = −(cid:104) k, β | Θ | k, α (cid:105) . And we can define its Pfaffian as P ( k ) = P f ( (cid:104) k, α | Θ | k, β (cid:105) ).2. The points in an odd space appears in pairs at( k, − k ).3. Since points in an odd space are in pairs, whenthere are two such pairs in the Brillouin zone, itis always possible to the move them together toannihilate them.From the above discussion, we know that the number ofpairs in an odd space can only be zero or one. The twocases cannot be deformed to each other since if k wants tomeet − k , the only choice is to meet at the origin, whichobviously belongs to an even space. Therefore, there is atopological obstruction between the two phases. So, it ispossible to define the number of pairs in an odd space asa topological invariant which distinguishes the two cases.In order to calculate it, it is convenient to express it as acontour integral with the help of the Cauchy theorem: I = 12 πi (cid:73) halfBZ d(cid:126)k · ∇ (cid:126)k log( P ( k ) + iδ ) ( mod . (2)Integrating over half of the Brillouin zone means that weonly have to count points instead of pairs with δ being asmall parameter making the integral convergence.In the above discussion, we only require the Hamilto-nian to be time-reversal symmetric, and do not imposeany other restrictions, such as a rotational symmetry orthe specific form of the Hamiltonian. So the Z topolog-ical properties are quite universal.
3. A BRIEF REVIEW OF THE GLOBALGAUGE ANOMALY
The global gauge anomaly, especially an SU (2) gaugeanomaly, was originally discovered by E. Witten[12] byconsidering a chiral Weyl fermion minimally coupled toan SU (2) gauge field. The action of the model is S = (cid:90) d x (cid:20) ¯ ψi /Dψ − g tr F µν F µν (cid:21) (3)where ψ ( x ) is a Weyl fermion, A µ is an SU (2) gaugefield, D µ = ∂ µ + igA µ is a covariant derivative and F µν = ∂ µ A ν − ∂ ν A µ − ig [ A µ , A ν ] is the field strength. In order toquantize the theory, we consider the partition function: Z = (cid:90) dψd ¯ ψ (cid:90) dA µ × exp (cid:26) − (cid:90) d x (cid:20) ¯ ψi /Dψ − g trF µν F µν (cid:21)(cid:27) . (4)Integrating over the fermion degree of freedom, we mayget an effective low energy action only in terms of thegauge fields. To do the integration, we consider the casefor a Dirac fermion coupled to an SU (2) gauge field. Forthis case, the integration is well known to be det( i /D ).Since the number of Weyl spinors is half of the numberof Dirac spinors, thus, the integration is ± (det( i /D )) .Notice that the sign at the front cannot be fixed globally.In order to settle this ambiguity, one needs to choose aspecific configuration, say, the vacuum A µ , to be positiveand other configurations related to this via a gauge trans-formation A µ = U − A µ U − iU − ∂ µ U , where U ∈ SU (2).In four dimensions, more generally for any even di-mension, aside from all the Gamma matrices, there is anadditional one Γ which is proportional to the productof all Gamma matrices and anti-commute with all oth-ers. Thus, when ψ is an eigenstate with energy E , Γ ψ isalso an eigenstate with the opposite energy − E . There-fore, the energy spectrum is symmetric with respect tothe origin. Considering that det( i /D ) is just the productof all eigenvalues, ± (det( i /D )) is determined by a prod-uct of half of the spectrum, picking one value from eachpair. In order to let the sign in the front to be definitive,one needs to exclude all the zero eigenvalues from thedefinition of the determinant.From the above discussion, one can find that there ex-ists some U such that (det i /D [ A ]) = − (det i /D [ A U ]) and others (det i /D [ A ]) = (det i /D [ A U ]) , which isknown as a global gauge anomaly. More detailed dis-cussions show that the measures of U in both cases turnout to be the same. The consequence of such a gaugeanomaly can lead to the above partition mathematicallyill-defined. But we do not spare our efforts on it. AsWitten pointed out, the mathematical origin of a globalgauge anomaly lies in the fact that π ( SU (2)) = Z , (5)where the subindex 4 is the dimension of space-time.Equation (5) shows the appearance of an anomaly is de-termined by the topological property of the gauge group.Indeed, when we deform the transformation matrix a lit-tle, we can prove that the resulting gauge field belongsto the same class as the original one. The result also tellsus that the global gauge anomaly can occur in other evendimensions since Clifford algebra can have symmetric en-ergy spectrum only in even dimensions and its squareroot can have a definitive meaning[13].In the next section, we will in fact use a more generalcase π ( G ) = Z (6)where the subindex 2 is the dimension of momentumspace and G is the symmetry group of a QSHE Hamilto-nian. It is worth to note that Eq.(6) is true and can beproven almost the same as for the four dimensional case. Z TOPOLOGICAL INVARIANT IN A QSHEFROM A GLOBAL GAUGE ANOMALY
The similarities of the topological property betweenthe QSH and the gauge anomaly makes us to ask whetherthere exists a relation between them and what it is. In-deed, by studying, we find that the system described bya two dimensional Lagrangian which carries the topo-logical information of the QSH system exhibits a globalgauge anomaly. In the following, we show how they areobtained.Firstly, we display some correspondence between aQSHE and a global gauge anomaly: • Field contents : Bloch wavefunction | k, α (cid:105) orPfaffian( (cid:104) k, α | Θ | k, β (cid:105) ) ←→ Nonabelian gauge field A µ . • Base Manifold : 2D momentum space ←→ Evendimensional D space-time. • Symmetry : Time Reversal Symmetry ←→ Chiral-Antichrial Symmetry. • Topological origin : π ( G ) = Z , where H ∈ G ←→ π D ( G ) = Z , where A µ ∈ G .Although there appears to be some difference, the similartopological origin indicates both are determined by thehomotopic group of some group which the field contentbelongs to. This motivates us to find a connection be-tween them. In order to connect the non-abelian gaugefield with the Bloch wavefunction, the most natural wayis to define the Berry connection/gauge field as follows: A αβµ ( k ) = − i (cid:104) k, α | ∂ µ | k, β (cid:105) . (7)Notice that it is indeed a gauge field because when theHamiltonian takes an unitary transformation as H ( k ) → U H ( k ) U − , the Bloch wavefunction is transformed by | k, α (cid:105) → U αβ | k, β (cid:105) and the Berry connection is trans-formed by A αβµ ( k ) → ( U ∗ A µ U ∗† − iU ∗ ∂ µ U ∗† ) αβ , whichindicates that A µ transforms similarily as a gauge field.From the above argument, we may find that the repre-sentation of the Hamiltonian and that of the Berry con-nection belong to the same group. Therefore, the topo-logical origin of both theories exactly matches — both ofthem are determined by the symmetry group which thenon-abelian gauge field belongs to.Although the origin is the same, it is still worthto study whether there is a one-to-one correspondencebetween a topological trivialness/nontrivialness and agauge anomaly appearance/vanishness. So further in-vestigations are needed.As we know, for any antisymmetric matrix,[ P f ( X )] = det( X ), so P f ( X ) = ± [det( X )] , wherethe sign ± is indefinitive. In order to understand therole which the Berry connection is playing in the Z topological invariant, we substitute it into Eq.(2), theinvariant can be given as I = 14 πi (cid:73) halfBZ d(cid:126)k · ∇ (cid:126)k log(det( (cid:104) k, α | Θ | k, β (cid:105) ))( mod πi (cid:73) halfBZ d(cid:126)ktr ([ ∗ (cid:104) k, β | T † ∂ (cid:126)k | k, α (cid:105) ] ∗ + (cid:104) k, α | ∂ (cid:126)k T | k, β (cid:105) ∗ )[ 1 (cid:104) k, α | T | k, β (cid:105) ∗ ]( mod . (8)Here the indefinitive sign disappears due to the ∇ op-erator. The first term in the bracket can be rewrittenas [ ∗ (cid:104) k, β | T † ∂ (cid:126)k | k, α (cid:105) ] ∗ = iS βγ ( k )[ (cid:126)A γα ( k )] ∗ , (9) where we define S αβ = (cid:104) k, α | Θ | k, β (cid:105) .As for the second term, some special care is needed.From the second section, we have chosen the gauge fixingcondition as | − k, α (cid:105) = T | k, α (cid:105) ∗ and to expand it interms of gauge field, we need to use this identity. Oneshould pay attention to the fact that the identity canonly be defined locally on the base manifold, otherwise,as one can check without much effort, the contributionsof the first term and the second term just cancel witheach other, indicating that the QSHE system is alwaystopological trivial. We have to remark that the argumentin the second section does not solve this problem sincethe equation only applies to the proof of the orthonormalcondition, which is a purely local argument. But in thiscase, we have to integrate over the Brillouin zone andthus a global effect emerges. The simplest solution totackle the problem is to divide the Brillouin zone into twoparts, namely A and B . In part A , the above condition isvalid, while in part B , the Bloch wave function is relatedto that of A via a transformation matrix t : | k, α (cid:105) A = t αβ | k, β (cid:105) B . When k is in part A , the second term can begiven as A (cid:104) k, α | ∂ (cid:126)k T | k, β (cid:105) ∗ A = − iS αγA ( k ) (cid:126)A γβA ( − k ) . (10)When k is in part B, the second term can be expressedas B (cid:104) k, α | ∂ (cid:126)k T | k, β (cid:105) ∗ B = t ∗ δγ S αγB ( ∂ (cid:126)k t δβ ) − t δβ ∂ (cid:126)k t ∗ δγ S αγB − it δβ t ∗ δγ S αηB (cid:126)A ∗ ηγB ( k ) . (11)Substituting all Eqs.(9)-(11) into Eq.(8), we obtain thetopological invariant as follow I = 12 πi (cid:73) ∂B d(cid:126)k tr( t ∗ δγ ∂ (cid:126)k t δβ )( mod πi (cid:73) ∂B d(cid:126)k tr( t † ∂ (cid:126)k t )( mod . (12)Here, its physical meaning is obvious: when momentum k travels around the part B once, t gains an additionalphase angle, which is integer times 2 πi . Notice that B inthis case may not be simply connected, but it will not af-fect our analysis because we can add different branches ofintegral provided that we carefully keep the same orienta-tion among them. The topological invariant just countsthis integer. Moreover, when we set A µ to be a trivialBerry connection, and let t be the gauge transformation,we get a new Berry gauge field. The invariant can beexpressed in terms of the new gauge field via I = 12 π (cid:73) BZ tr[ A ]( mod . (13)We have to remark here that this result was previouslyobtained by L. Fu and C. Kane[14] from a more compli-cated method. Here we acquire it by a simple and directcalculation. With the help of this simplified invariant, wecan state an important property which directly leads tothe equivalence between the two theories topologically.If the two Hamiltonians H ( k ) and H ( k ) describeQSHE systems, they determine two Berry connections A ( k ) and A ( k ), as well as two transformation matrices t and t . They correspond to two topological invariants I = 12 πi (cid:73) ∂B d(cid:126)k tr( t † ∂ (cid:126)k t )( mod
2) = 0 I = 12 πi (cid:73) ∂B d(cid:126)k tr( t † ∂ (cid:126)k t )( mod
2) = 1 . (14)We define a new Berry connection connecting the abovetwo adiabatically as A sµ = (1 − s ) A µ + sA µ and s ∈ [0 , L = ¯ ψi /Dψ − e tr F µν F µν , (15) F µν is the strength field induced by the Berry gauge fieldand ψ is a Weyl spinor. The Dirac equation, defined onthe momentum space, can be written as i /D [ A s ] ψ s = λ s ψ s . (16)As the s runs adiabatically from 0 to 1, we can follow thetracks of the eigenvalue λ . In the following, we prove thatif there exist odd number of eigen-spectral lines flowingacross zero, they lead to the fact that(det i /D (cid:48) [ A ]) = − (det i /D (cid:48) [ A ]) , (17)where the prime on Dirac operators means they are in-duced by the Dirac fermions, not the Weyl fermions.Therefore, there is a global gauge anomaly in the sys-tem defined by Lagrangian (15).Here, we need to make a few remarks that the Diracequation (16) and the Lagrangian (15) stated above areslightly different from normal ones, which are defined onthe momentum space.Since the topological invariant is Z valued, any evennumber can be adiabatically deformed to zero; simi-larly, any odd number to one. Thus we may remove the( mod
2) operator and just set the integrals to be one andzero. What is more, since H ( k ) is topological trivial,we may set the corresponding Berry connection and thetransformation matrix identically as zero. Therefore, wemay set the gauge transformation relating A and A as t ≡ t , i.e. A µ = t ∗ A µ t ∗† − it ∗ ∂ µ t ∗† . For the twodimensional case, according to the Atiyah-Singer indextheorem[15, 16] ν + − ν − = 12 π (cid:90) BZ tr F = 12 π (cid:73) BZ tr A, (18)we can obtain ν s + − ν s − = s π (cid:73) ∂BZ d(cid:126)k tr( t † ∂ (cid:126)k t )= sπ (cid:73) ∂B d(cid:126)k tr( t † ∂ (cid:126)k t )= 2 sI = 2 s, (19)where in the second line, we switch the integration do-main only in part B . On the left hand side, ν + counts thenumber of zero modes of the dimensional kernel of the lefthand Dirac operator iP + /D where P + = (1 + γ )[13].Similar definition works for ν − . Here, we will use a moreconvenient and straight forward way to compute the in-dex with the help of Fig.1. ν + represents the number of FIG. 1: Zero modes of Dirac operator at s . This zero pointconsists of three spectral flows: two from the above and onefrom below, i.e. ν + = 2 and ν − = 1 flows from up to down and ν − from down to up. There-fore, the left hand side must be an integer. As for theright hand side, s ∈ [0 , s can only be 0or or 1. When I = − s reverse its sign. We claimthat s = is the only solution in the present case be-cause s = 0 or s = 1 implies that there are zero modes inthe original spectrum, inconsistent with our assumption.Therefore, when s goes from 0 to 1, the only possiblepoint for zero mode is at s = . FIG. 2: A typical flowing spectrum. The horizontal axis rep-resents parameter varying from zero to one. The vertical axisrepresents eigenvalues. Eigenvalues run as the parameter s vary. Fig.2 is a typical flowing spectrum of that Dirac opera-tor. The spectrum is symmetric with respect to the zerovalue, as previously stated. As s varies from 0 to 1, theeigenvalues rearrange to form a new spectrum. Althoughthe overall distribution seems unchanged, the relative po-sition may vary. In order to acquire the spectrum of an operator induced by a Weyl fermion, we can choose oneeigenvalue from each pair. Fig.3 are a pair of Dirac andWeyl spectrum. FIG. 3: An example of Dirac and Weyl spectrum. The left isDirac and the right is Weyl.FIG. 4: When there is only one eigenvalue running across thehorizontal axis, the product of all the eigenvalues reverse sign.Here we only list two possibilities.
Now, we search for all the solutions satisfying ν + − ν − = 1 and prove that all these solutions are with aglobal gauge anomaly.First, when there is one spectral line running acrosshorizontal axis, as shown in Fig.4. In order to guaran-tee the spectrum to be Weyl, the only possibilities arethose which satisfies that the product of all the eigenval-ues being the same absolute value of the original ones.Moreover, since there is only one eigenvalue which re-verses its sign, though its absolute value may change asin Fig.2, the product is exactly the minus of the Originalone, which is equivalent to the identity(det i /D (cid:48) [ A ]) = − (det i /D (cid:48) [ A ]) . (20)So, for this case, we prove that there is a global gaugeanomaly.Second, when there are two eigenvalues flowing acrossthe horizontal axis, the two can both go from up to down,which means ν + − ν − = 2, or in the opposite direction, i.e. ν + − ν − = −
2. Another possibility is that one from up todown and the other from down to up, i.e. ν + − ν − = 0.All the above three cases do not match the condition ν + − ν − = 1. FIG. 5: When there are only three eigenvalues running acrossthe horizontal axis. Here only draw one particular diagram.There are two lines goes from up to down and one from downto up, which preserves ν + − ν − = 1. Effectively, there areonly one eigenvalue reverse the sign, and all others mutuallyrearranged. Then, when there are three eigenvalues flowing acrossthe horizontal axis, as shown in Fig.5, similar analysis asin the first case can tell there is a global gauge anomaly.Finally, for any odd number of lines crossing the hori-zontal axis, it is not difficult to draw the conclusion thatfor all the solutions satisfying ν + − ν − = 1, there is aglobal gauge anomaly. For any even number of linescrossing the horizontal axis, there does not exist a so-lution satisfying ν + − ν − = 1. It ends the prove of ourequivalence property.Up to now, we can conclude that the non-trivial topo-logical quantum spin Hall effect corresponds to the globalgauge anomaly whose gauge field is induced by the Berryconnection of the former. Here, a comment on globalgauge anomaly may be helpful. In a paper[17], the au-thor suggested that an SU (2) global gauge anomaly canbe seen as an U (1) anomaly by observing that the special U (1) element e iπ is also the center of an SU (2) trans-formation. So an SU (2) global gauge anomaly can alsobe viewed as a more familiar U (1) anomaly as in thecase of a 3 D topological insulator. However, here, thegauge group we are discussing is more complicated thanan SU (2). But the existence of the above correspondencemay help us understanding our formulation by studyingthe 3 D case.
5. MORSE THEORY AND THE CLASSICALSPIN HALL EFFECT
In this section, we discuss the Witten-deformed Diracindex[18] and find that Z index for quantum theory isthe same as that for the classical theory, and thereforewe conjecture that the classical ”spin” Hall effect alsocontains the Z topological property, which have not been discussed as far as we know.In the above, we have shown a crucial formula, ν + − ν − = 12 π (cid:73) BZ tr A, (21)which is a specific example of Atiyah-Singer index theo-rem. In fact, the left and right hand side of the equationare two different limits of a same quantity[15], i.e. ind( /D [ A ]) = tr γ e − β /D † /D = ν + − ν − , β → ∞ ; π (cid:72) BZ tr A, β →
0. (22)The index for Dirac operator reminds us the similar su-persymmetric case Tr( − F e − βQ † Q (23)where Q is the supersymmetry operator. These two indexcan be related by the following map:( − F → γ , (24) /D → Q, (25) H = /D † /D → H = Q † Q. (26)E. Witten, in his seminal paper, derived the Morse in-equality with the help of supersymmetric quantum me-chanics of the later case. Here we apply his idea to thediscussion of the Dirac index. By parallel analysis, wecan also derive the Morse inequality.For convenience, we set the representation of the Diracmatrix satisfying γ = diag( I, − I ). The Dirac operatorcan be expressed as the form /D = (cid:18) /D − /D + (cid:19) , (27)where /D †− = − /D + , and /D ± = D i Γ i ± . In the previousanalysis, we focus on the Dirac equation i /D + ψ + = λψ + . (28)And other part is similar, i.e.i /D − ψ − = − λψ − . (29)To counting the number of Dirac zero modes which is ofinterest in the previous section, one only needs to countthe chiral part or anti-chiral part. Here, we will mainlyfocus on the ground state of the Hamiltonian H = tr /D † /D = − ( /D − /D + + /D + /D − )= − D i D i . (30)We can show that the number of ground state of theHamiltonian equals to the number of the zero modes ofthe Dirac equation. If a zero mode of a Dirac equation /D + ψ = 0 with /D − ψ = 0 is simultaneously true, it isobvious that ψ is also the ground state of the Hamiltonian H . Conversely, if Hψ = 0, by integrating by parts, wecan obtain (cid:90) ψ ∗ Hψ = − (cid:90) ψ ∗ D i D i ψ = 2 (cid:90) | D i ψ | = 0 , (31)which immediately leads to D i ψ = 0. Thus, ψ is alsothe zero mode of Dirac operator. Therefore, it is proventhat we may consider the ground states or the vacuum ofthe Hamiltonian instead of counting the number of theDirac zero modes. We should point out here that thisequivalence is only valid in the two dimensional case.According to Witten’s idea, we can deform the Diracoperator slightly as the following /D + → /D t + = e − ht /D + e ht , (32) /D − → /D t − = e ht /D − e − ht . (33)Of course, deformations in this way neither introduce ex-tra zero ground states nor eliminate any. We may expressthe deformed Dirac operator by introducing two morephysical raising and lowering operators /D t + = /D + + tD i + ha i , (34) /D t − = /D − + tD i − ha i ∗ . (35)Here we may regard a i and a j ∗ as rising and loweringone chirality of the following operator respectively. Sub-stituting above into the Hamiltonian H , one can rewritethe Hamiltonian as the following form H = − /D − /D + − /D + /D − + t ( iD i h ) + tiD i h iD j h [ a i , a j ∗ ] , (36)where we let i be explicit because the iD is Hermitian andthe product with its conjugate is positive definited. Onecan immediately see from the equation that the Hamilto-nian is composed of three parts, the kinetic term, the os-cillator potential term, and the quantum correction. Thefirst two terms are purely classical contributions. In orderto extract out the classical parts, it is natural to considerlarge t limit under which case the quantum correction,proportional to t , is suppressed down by the quadratic of t . Under this limit, we may expand the eigenvalue λ ofthe total Hamiltonian by the order of 1 /t as λ ( t ) = t (cid:20) A + Bt + Ct + · · · (cid:21) . (37)The t in the front of Eq.(37) is analogous to the solutionof the free oscillator which has the form of ω ( n + 1 / ω in this situation is t . A represents the classicalcontribution and other terms are the quantum effects.For the ground state of the Hamiltonian, correspondingto λ ( t ) = 0, it implies that A , the energy eigenvalue, corresponding the classical system, reaches its vacuum.However, the reverse case is not necessarily true. Namely,for A = 0, one cannot obtain that the total quantumeigenvalue is zero. With this consideration, one can writedown an inequality { λ p = 0 } ≤ { A p = 0 } , (38)where p means the chirality of the particle and the in-equality is valid for any p . It means that the number ofquantum vacuum is always less than that of the classi-cal vacuum. Actually, as Witten claimed, the number ofvacuums on both sides can be interpreted as the Betti-number and the Morse-index, respectively. The argu-ment will not be discussed here. The above inequalitycan be rewritten as B p ≤ M p , (39)which is the weak form of the Morse inequality.There is one more fact that the Morse inequality hasa strong form, saying (cid:88) p M p t p − (cid:88) p B p t p = ( t + 1) (cid:88) p Q p t p , (40)where Q p ’s are non-negative integers. Since, in the pre-vious section, we only count the total number of zeromodes of the Dirac operator, we should sum over all pos-sible chirality. This motivates us to set t = 1. For t = 1,the equality (40) becomes (cid:88) p M p − (cid:88) p B p = 2 (cid:88) p Q p , (41)which gives us a great surprise. Since B p , which is thenumber of quantum vacuum, is stable only up to Z , andthe right hand side of Eq.(41) is always an even integer,so the number of classical vacuum is stable up to Z also.Paraphrasing it into the condensed matter physics lan-guage, the classical ”spin” Hall effect also contains the Z topological invariant which characterizes its topolog-ical nature. It is worth to note that in classical theory,there is no concepts like spin, so we put a quotation markon the spin. It is still difficult to construct this classicalsystem for the time being. The method we try to use isto construct some kind of index which counts the num-ber of states satisfying (cid:104) k, α | Θ | k, β (cid:105) = 0. Then, usuallysuch index can be written as some path integral. Thenclassical version of this is to use the saddle point approx-imation. Then we can investigate such quantity to seewhether this is a Z quantity. However, we are still notclear about how to construct the index. We are tryingto solve this and find other more accessible ways at thesame time. Moreover, this phenomenon is not coveredin the existing literatures, and more concrete analysis interms of the topological properties of the classical sys-tems indicated by the above is expected to be furtherinvestigated.
6. DISCUSSION AND CONCLUSION
In this article, we prove that the topological nontrivialquantum spin Hall effect is equivalent to the Z globalgauge anomaly of the system described by the Lagrangian(15). The Lagrangian is somewhat similar to that de-scribing massless QCD, which also contains a pair offermions and non-abelian gauge fields. But the differenceis that the former is defined on the momentum space,and the fermions is a Weyl. Moreover, the non-abeliangauge group, in order to generate a Z non-trivial topol-ogy, cannot be chosen arbitrarily. In fact, the symmetrygroup is determined by the CPT symmetry of the Hamil-tonian. Similar to the fact that there exist fractionalcharge in QCD, the fractional excitations in the QSHEalso appear[19], which needs further investigation.As we mentioned before, the appearance of a globalgauge anomaly can occur in any even dimension, there-fore, we expect that we may find similar correspondencebetween a 4 + 1 dimensional topological insulator and a4 d non-abelian gauge theory(Euclidean QCD). Moreover,from the approach of Ref.[5], we find the 2+1 dimensionaltopological invariant, which is obtained by a dimensionalreduction, is P ( θ ) = 12 π (cid:73) dφ Ω φ , (42)where Ω φ is defined in Eq.(112) in Ref.[5]. It is of the same form as (13). Therefore, we can conclude that ourformalism can also be applied to a Chern-Simons fieldtheory approach. However, we should mark that the 3+1dimensional case, which is not included in our formal-ism, is related to the U (1) chiral anomaly. Other Z SPTcases were already found to be corresponding to variousanomalies. The relationship between an anomaly andmost of other Z topological SPT states is still not clearyet. We hope that our article can provide some inspira-tion toward the understanding of these issues.Lastly, through some analysis of the spectrum of thedeformed Hamiltonian (36), we find that the classical”spin” Hall effect also has a Z topological classification,which is not discovered before as far as we know. Byfurther investigation of a classical system, we may findmore interesting topological nature which may enable usa better understanding of the topology of strong corre-lated systems. Acknowledgement
The work was supported by National Natural ScienceFoundation of China under Grant No.11275180 and Na-tional Science Fund for Fostering Talents in Basic ScienceNo.J1103207. [1] C.L.Kane, E.J.Mele, Phys. Rev. Lett. , 225801 (2005).[2] X.L.Qi, S.C.Zhang, Physics Today,
33 (2010).[3] X.L.Qi, S.C.Zhang, Rev. Mod. Phys. , 146820 (2005).[12] E.Witten, Phys. Lett. B , 324 (1982).[13] M. Nakahara. Geometry, Topology, and Physics. A.Hilger (1990).[14] L. Fu, C.L.Kane, Phys. Rev. B
395 (1984).[16] C.Vafa, E.Witten Commun. Math. Phys.
257 (1984).[17] S. P. de Alwis, Phys. Rev. D
661 (1982).[19] Y.P. Lan, and S.L.Wan, J. of Phys. C24