Shou-Cheng Zhang 4 June 2008

“The Quantum Spin Hall Effect and the Topological Magneto-Electric Effect.”

Abstract:Search for topologically non-trivial states of matter has become a important goal for condensed matter physics. Recently, a new class of topological insulators has been proposed. These topological insulators have an insulating gap in the bulk, but have topologically protected edge states due to the time reversal symmetry. In two dimensions the edge states give rise to the quantum spin Hall (QSH) effect, in the absence of any external magnetic field. I shall review the theoretical prediction [1] of the QSH state in HgTe/CdTe semiconductor quantum wells, and its recent experimental observation [2]. The QSH effect can be generalized to three dimensions as the topological magneto-electric effect (TME) of the topological insulators. I shall also present realistic experimental proposals to observe fractional charge, spin-charge separation and the deconfinement of the magnetic monopoles in these novel topological states of matter.

Worked with Bernevig and Hughes on theoretical predictions. Discovered by Molenkamp.

QH states not characterized by broken symmetry, but by topological properties. Integer effect, integer n is the first Chern number (flux of a magnetic field obeying Dirac quantization). Can there be dissipationless edge states without magnetic field.

Spin Hall effect: no breaking of time-reversal. Spins accumulate on opposite sides of sample (even though voltage drop is forbidden by symmetry). Up spins on one side and down spins on the other (from strong spin orbit coupling).

There is an anomalous Hall effect. what is its spin analog?

Think of double layer Hall, where up and down spins see opposite B field (no breaking of time reversal). Edge states propagate in opposite directions.

Consider 1D spinless liquid. Get left and right movers. But can these be spatially separated? If so we have QHE. Opposite chiralities are on opposite edges. This prevents back scattering by impurities.

With spin .. two right movers on upper edge, and two left movers on the lower edge? To change the symmetry type ... a left and a right mover on each edge. Two pairs of spatially separated channels, related by time reversal. Why no back scattering? Stable with respect to time-reversal invariant perturbations.

Bulk gap. But on the edge spin down moves one way and spin up moves the other way.

Graphene (Kane) does not work because spin-orbit is way too weak. Also too weak in GaAs (Zhang-Bernevig)?

Band gap can be negative! Occurs in HgTe (mercury telleride). Splitting of P level is larger than splitting between P and S (inversion due to spin-orbit coupling). Spin up state p_x + ip_y has strongest spin-orbit. Spin down p_x - ip_y has strongest spin-orbit. Get the (2+1)-D Dirac equation in a tight binding model. The mass is tunable, but controlling the thickness. Can be tuned through zero from positive to negative level (inversion)

Can't break Kramers deger, at the T-invariant point, so can't open gap.

In 2+1 the Dirac mass term breaks T. Hall conductance changes when m is tuned through zero. Where m < 0 there is a domain wall at the sample boundary.

Measure conductance G_LR) = 2 e^2/h. There are two (out of four) channels that are active.

What if T is broken. This opens a gap by destroying the Kramers degeracy. E.g., turn on a B field. The smoking gun is that "Hall conductance" is destroyed. Tilt the sample to separate orbital effect.

What other materials? InAs/GaSb quantum wells. Both materials have a normal band gap, but relative gap can change sign.

Fractional charge. e/2 can be measured by a Coulomb blockade. (Pattern of conductance as a function of voltage gets shifted by half a period.) There can be a charge e/2 domain wall at the edge. Could be due to a time-reversal breaking perturbation. Like a polyacetylene soliton. Except with 2 channels on top and 2 channels on bottom, we get not only spin-charge separation, but also charge e/2.

Spin charge separation. Insert a pi flux. Four different ways to achieve -- two spin channels and phi can go to either plus or minus pi. Can get charge e/2 for up state and e/2 for down state -> a charge e with no spin. Or a pure spinon.

Couple to superconductor (flux quantum is a pi flux).

What is the TQFT? Consider 3+1 theory? What is EM response of the insulator? There can be a theta term, an E\dot B term. T-invariance means theta is either 0 or pi. That is why a Z2 classification. (More general than Fu-Kane or Moore-Balents). For theta = pi, there are an odd number of dirac fermions.

Theta term is total derivative of Chern-Simons, so describes QH on the boundary?

Consider the surface of a 3D insulator. Image charge or an electron is a magnetic monopole (Witten effect). AFM will find a 1/r^2 law. AB phase is flux of monopole on the upper hemisphere. Witten electic charge is e/2 for theta = pi. Like a widely separated monopole and antimonopole.

See Physics Today search and discovery.

Summary:

The quantum spin Hall effect (QSHE) is analgous to the integer quantum Hall effect (QHE). In both cases, there is a gap in the bulk, but there are massless excitations that propagate at the edge. The stability of these gapless edge excitations is enforced by a topological principle.

In the ordinary QHE, the gapless edge states are chiral, so they can't scatter back or get a mass. To propagate beckward, they need to cross the bulk to reach the other side of the sample.

In the spin QHE, the gapless excitations are not chiral. However, there is gapless spin transport at the edge. Spin up particles propagate one way and spin down particles propagate the other way, so there is no net transport of charge but there is net transport of spin due to the edge current.

A fundamental difference between the QHE and the spin QHE is that the QHE in the presence of an external magnetic field has instinsic breaking of time-reversal invariance -- it is TRB (time-reversal breaking). But the spin QHE is TRI (time-reversal invariant) because the spin currents circulating in opposite directions are time-reversal conjugates of one another. For the QHE, the effective field theory of the (gapped) bulk is a TRB TQFT. But for the spin QHE, the effective field theory must be a TRI TQFT. It is the TRI that protects the nonchiral gapless modes from acquiring mass. In a non-interacting electron picture, this can be understood in terms of the Kramers degeneracy, which disallows the opening of a gap.

But there is a more robust topological reason for the gapless edge modes that applies even when electrons interact. This is true in the QHE, and also in the QSHE. For the QHE we consider the response of the system to twisted boundary conditions (and let the flux enclosed by the two cycles of the torus both increase by 2*pi).

Cf Nielsen-Ninomiya theorem, but for Dirac fermions instead of chiral. There is a Redlich anomaly, where the integer + 1/2 value of the Chern-Simons term must arise from an odd number of fermions, in 2+1 dimension. Cf Callan-Harvey anomaly, which requires the existence of a bulk. And domain wall fermions. There is a zero-energy mode localized on the wall, for topological reasons.

The Redlich anomaly in odd-dimension massless non-abelian gauge theories (1984)

There is a Z2 classification of 2+1 electrodynamics, according to whether the number of two-component fermions is even or odd. For a pair of fermion, we can construct a Lorentz invariant bilinear mass term which is also P and T invariant. But for a single two-component fermion, the only allowed mass term is P and T violating. If we integrate out a single heavy fermion species, a Chern-Simons term is generated. This gives the gauge field a mass, and the low-energy effective theory is a topological theory.

But Redlich considered the case of an SU(N) gauge theory. He pointed out that the coefficient of the Chern-Simons term does not obey the usual quantization condition --- the integer coefficient is shifted by 1/2. This means that under a large gauge transformation -- one with odd winding number of the map S^3 -> SU(N) -- the CS cofficient changes by an odd multiple of Pi rather than an even multiple, so that e^{-Action} changes sign.

Redlich also observes that for a single two-component fermion in 2+1 dimensions there is a global anomaly analogous to Witten's SU(2) anomaly. The phase of the fermion determinant is ill defined because it is the square root of a well-defined positive quantity in the doubled theory. When we consider a large gauge transformation, the 3D Dirac operator has a zero mode. Therefore, there is a closed path in the space of gauge field configurations with nontrivial spectral flow --- a negative eigenvalue crosses zero to become positive, and vice versa. This means that the sign of the fermion determinant for an odd number of species is not gauge invariant --- under a large gauge transformation it changes sign.

If we have an even number of massless fermions, there is no anomaly. We can introduce a mass term for one species that violates T and P, and then increase the mass to decouple that fermion, which generates a T and P violating mass term for the SU(N) gauge field. The resulting theory is gauge invariant, because under a large gauge transformation there are compensating (-1)'s in the path integral coming from the CS term and the fermion determinant.

So Redlich speaks of the "parity anomaly" --- in order to quantize the theory with an odd number of fermion species, we need to introduce a T and P violating gauge field mass term.

Recall Callan-Harvey NP B250, 427-436 (1985). Axion string has chiral zero modes. Domain walls also have zero modes. On the string, because of the anomaly, charge and energy momentum are not conserved. But the fermions cannot escape the string!

If there is electric field along the string, then charge is created on the string due to the anomaly. So there must be an inflow of charge from outside. In the bulk, current flows transverse to the applied field, as in the Hall effect. The gravitational anomaly works similarly.

In 2+1 dim, the "string" is a domain wall. Again there is a chiral fermion on the domain wall. E.g., if sign of a massive fermion's mass differs on the two sides of the wall, induced CS mass term for gauge field has different sign. Again, an anomaly in the bulk matches the anomaly on the wall.

Kaplan -- Simulating chiral fermions using domain walls (1992)

Consider a 2n+1 dimensional theory, where fermion mass changes sign on a "domain wall". Then there are chiral fermions confined to the wall. There are doublers with opposite chirality that can be removed by a Wilson mass term. But will the gauge field be confined to the brane?

In 2n+1 dimensions, the last Dirac gamma matrix is the "gamma_5" of the 2n dimensional theory. As a function of the "extra dimension" s, the mass m(s) changes sign. It becomes m(s) -> m as s -> infinity and m(s) -> -m as s -> -infinity. There is a zero mode localized on the brane, and it is chiral in the sense that if has gamma_5 = +1, and there is no zero mode with gamma_5 = -1.

If we put a fermion on the lattice in 2n + 1 dimensions, the spectrum will be parity doubled. But since the underlying theory is nonchiral, there is no obstruction to removing the doublers with a Wilson mass term. That way the effective theory on the brane really is chiral!

If there are periodic BC in the s direction with length L, then a domain wall at s=0 means a domain wall of opposite orientation at s= L/2. The overall spectum is vectorlike, but we separate the parity conjugate fermions by putting them on distinct walls that are far apart!

Now, couple gauge fields. Now the chiral gauge theory on the domain wall has a gauge anomaly! The resolution is the Callan-Harvey effect. In the bulk, the heavy fermions generate a Chern-Simons action when integrated out, where the coefficient of the CS term depends on the sign of the fermion mass term. So it has opposite sign on the two sides of the brane. The Nielsen-Ninomiya theorem is evaded because the overall theory is anomaly free even though the effective theory on the brane has an anomaly. The flow from bulk to brane resolves this apparent paradox.

And we can construct theories with cancelling anomalies. For example the 3,4,5 Schwinger model in 1+1 dimensions. There are charge 3 and 4 fermions with one chirality and charge 5 fermions with the opposite chirality. The anomalies cancel because 3^2 + 4^2 - 5^2 =0.

How do we confine the gauge fields to the wall? We can tune the action so there is a penalty for leaving the brane. But it works only if anomalies cancel, since in that case there is no net flow of charge from brane to bulk.

But ... global currents can still have anomalies! If we put the standard model on the lattice by introducing the 5th dimension, the proton can decay because its baryon number moves off the brane into the bulk!

Comment on parity: In 2n +1 dimensions there are two different representations of the Clifford algebra, distinguished by the value \pm 1 of the product of all 2n+1 gamma matrices. Parity does not act as an automorphism of these representations, rather it interchages them (the two form a parity doublet). So we need both for parity invariance. And the two bilinears have opposite sign coefficients, so they give cancelling contributions when we do the one loop calculation of the quadratic term in the gauge field effective action.

Berry phase and the QHE. We consider the QH system on a torus, with fluxes Phi_1 and Phi_2 linking the torus. That means that there is a Berry torus, distinct from the physical torus, since Phi_1 and Phi_2 are defined only modulo a flux quantum (i.e. there determine the effect of transport around a cycle of the torus). The Berry connection on the torus has a magnetic flux, which must obey the Dirac condition. This means that the Hall conductivity is quantized.

For the FQHE, the Berry connection is nonabelian, because of the robust ground state degeneracy. That means that the Dirac flux quantum is smaller by a factor of q is the degeneracy is q-fold. (The transport due to the U(1) connection around the Dirac string need only be in the center of SU(q).)

Why is the magnetic flux the same thing as the Hall conductivity? The Hall conductivity tells us how the voltage drop in the y direction is related to the current flowing in the x direction. Voltage in the x direction is induced by a time dependent flux Phi_x through the closed path C_x, which is the same as ramping up the vector potential A_x.

We ramp this flux from 0 to Phi_0. The flux through the complementary cycle C_y is Phi_y; when Phi_y is held fixed, this is like a twisted BC on the torus. But the Hall conductivity sigma_H is a bulk effect that will not be affected by BC, so it does not depend on Phi_y.

Some questions:

How does this observation relate to the existence of chiral edge states?What happens for a time-reversal invariant case, where the Berry phase is +1 or -1?

Kane-Mele: there is a Z2 classification of edge theories, according to whether the number of Kramers pairs of edge modes is even or odd. Even -> ordinary insulator, Odd -> topological insulator. Cf., Avron and Simon.

Petr Horava used K-theory to classify the topology of fermi surfaces.

Free fermion picture of QHE. TKNN, as interpreted by Barry Simon, consider the wave function of a single electron as a function of the band momentum --- i.e. a torus defined by - Pi < k_x < Pi and -Pi < k_y < Pi (or more generally the Brillouin zone). The first Chern class (i.e. monopole number) of the Berry connection for the wavefunction of the band is the TKNN integer of the band.

Avron et al. discussed the case of time-reversal invariant fermionic systems. T^2 = -1 holds for interger + 1/2 reps of SU(2), where T is antiunitary and inverts the angular moment J. If H is time-reversal invariant it commutes with T. Thus there is "Kramers degeneracy": |psi> and T|psi> are orthogonal and have the same energy. Parameter counting shows that level crossings are generically co-dimension 5, because H is hermitian in the quaternionic sense. Now it is the *second* Chern class that classifies the topology of Berry's connection on a *four* dimensional surface.

Another topological viewpoint (Hatsugai, cited by Moore and Balents and by Kane and Mele): edge state winding number can be related to TKNN integer.

How do we understand the connection between the bulk QHE and the existence of edge states that carry the hall current? For this let us use the single electron picture (but it is important to understand the connection in the interacting electron case, too!).

Cut open the torus -- there are periodic BC in the y direction and open BC in the x direction. For each value of k_y, there is a 1D system parametrized by x, and a spectrum of energy states, Since the bulk system is gapped, there is a gap in this spectrum. But .. the edge states lie in the bulk energy gap. If we apply an E field in the y direction, there is current flow in the x direction for each value of k_y. This E field corresponds to a magnetic flux linking the cyllinder that ramps with time from 0 to 2*Pi. If change is adiabatic, the 1D system stays in its ground state for each k_y.

The total charge flow in the x direction, when we integrate k_y, is the flow due to the Hall current, namely 2*Pi*sigma_H, which is an integer. As k_y increases from 0 to 2*Pi, a charge gets pumped through the bulk from one edge to the other (Thouless argument). This adiabatic pumping is possible only because there are ungapped states at the edge. Conclude: if there is a Hall current then there must be gapless edge states. As k_y changes from 0 to 2*Pi, there is spectral flow across the bulk gap from the lower band to the upper band.

This "adiabatic pumping" argument is rather subtle. Usually when we consider the QHE, we imagine that there is dissipationless longitudinal current flow (carried by the edge states), without a voltage drop, accompanied by a transverse voltage drop (which neutralizes the Lorentz force on the charge carriers). Here, though, we consider a cyllinder, and we apply a (longitudinal) voltage around the cyllinder, which causes charge to be transported (transversely) from one open end of the cyllinder to the other open end --- i.e. from one edge to the other. There is a level crossing at k_y=0, where both edge states have zero energy.

The Thouless adiabatic pumping argument is a different way of looking at the Callan-Harvey effect. The edge theory is chiral, so if a longitudinal electric field is applied, left moving positive charge is created on one edge while right moving negative charge is created on the other edge. This nonconservation of charge on the edge is possible only because charge can be tranported across the bulk from one edge to the other, which is possible despite the gap in the bulk theory --- this is the quantum Hall effect!

The spectral flow at the edge explains how charges appear at one edge and holes at the other edge. But charge is locally conserved --- how is transport of charge across the gapped bulk possible? The Callan-Harvey computation shows it occurs.

Now, back to topological insulators and the quantum SHE. Kane and Mele initially suggested a Z2 classification of the time-reversal invariant edge theory of a topological insulator, based on whether the number of Kramers pairs of edge modes is even or odd. How can this classification be understood from the perspective of the time-reversal invariant gapped bulk theory? Moore and Balents illuminated how the bulk theory has a classification related to the TKNN integers, i.e. the first Chern class used to classify bands in the theory of the IQHE.

In TKNN, one identifies Bloch wave functions with a complex line bundle: there is a Bloch Hamiltonian for each point in the Brillouin zone (e.g. each value of k on a torus), and the TKNN is its first Chern class. It is possible to adiabatically connect two IQHE states if the value of the sum of all Chern numbers of occupied bands is the same for

In the case of a time-reversal invariant system. the topology of the base space changes, because T*H(k)*T^{-1}=H(-k). The Brillouin zone (BZ) becomes the effective Brillouin zone (EBZ), because we don't have the freedom to specify H(k) and H(-k) independently. Note that the Bloch Hamiltonian for a given k need not itself be T invariant, except at the special points where k = -k.

The EBZ spans 0 k_x < Pi and 0 < k_y < 2*Pi, i.e., the torus is cut in half, and we keep one copy of the cyllinder. Furthermore, on the boundary circles at k_x=0 and at k_x=Pi, we need to identify k_y with -k_y. Moore and Balents consider obtaining a sphere by contracting these boundary circles -- for the sphere there is an integer Chern number. But the Chern number is not unique. However, two different ways of contracting to a sphere differe by an even Chern number. Thus, the band for the T-invariant system is classified by Z2. Of course, this classification is for noninteracting electrons.

Back to Callan and Harvey (NP B250, 427-436 (1985).) They consider an "axion model" with a complex scalar coupled to a fermion. An axion string or domain wall has chiral fermion zero modes. An electric along the string or wall creates charge. So there must be a compensating inflow of charge from the bulk. But how is that possible if the fermions are massless in the bulk and massless only on the string.

E.g. for a string embedded in 4D:

There is a Goldstone-Wilczek current, even though the fermions are massive in the bulk, which arises because the vacuum of the heavy fermions is distorted.

Goldstone and Wilczek, PRL 47, 986-989 (1981) discussed quantum numbers on solitions arising because the soliton distorts vacuum polarization relative to the vacuum. Jackiw-Rebbi zero mode occured in a model with C invariance, so the symmetry relating state with occupied and unoccupied zero mode required charge to be plus or minus 1/2. Want to break such symmetries to get other fractional charges.

Their method is to "turn on" the solition gradually, and compute the current flow to determined the solition's charge.

Okay, back to Zhang's paper: Qi, Hughes, Zhang, arXiv:0802.3537

His main tool is dimensional reduction. E.g., in the TRB (time-reversal breaking) case, we can start with e.g. a 2+1 dimensional effective theory, then replace a momentum with an adiabatic parameter (field) to address the 1+1 dimensional case. And in the TRI (time-reversal invariant) case, we start with the 4+1 dimensional case and reduce from there.

The 4+1 dimensional Chern-Simons effective theory describes the nonlinear response to an applied U(1) gauge field, much as the 2+1 dimensional theory describes linear response. The main result of the paper is that the Z2 classification of the topological insulator in 2 or 3 spatial dimensions can be understood by means of this dimensional reduction of the 4+1 dimensional CS theory *and* that this classification goes beyond the case of noninteracting fermions. Furthermore, the analysis yields an topological effective action that encodes the phenomenology of the bulk topological insulator. The resons for the different dimensionality of the fundamental TRB and TRI cases is that degeneracy is co-dimension 3 for TRB, and condimension 5 for fermionic TRI.

I had a misunderstanding at first about the meaning of this "dimensional reduction" --- it is *not* claimed that we can think of the d dimensional case as the edge of the d+1 dimensional case. The reduction is more "formal".

In the TRB case, the dimensional reduction fro 2+1 to 1+1 yields a Z2 classification of charge-conjugation invariant (i.e. particle-hole symmetric) insulators in one spatical dimension. (This is a good warmup for understanding the Z2 classification in the TRI case.) There is a rather lengthy discussion on pages 7-10 of "gapped interpolation" for systems of free fermions that admit a charge conjugation symmetry in one spatial dimension. The conclusion is that there is a Z2 classification.

We consider a two-parameter family of free-fermion Hamiltonians. One parameter is the (one-dimensional) momentum k, varying from -Pi to Pi. The other parameter, theta, labels a one-parameter family of 1D Hamiltonians. As theta varies from 0 to 2*Pi, h(k,theta), the momentum k changes sign and C is applied (particles and holes are interchanged). There is a Chern number that depends on the interpolation, but the ambiguity if an *even* integer. This is enforced by the particle hole symmetry: for the Chern number to change as the interpolation changes, a singularity must be encountered, and the singularities are paired because of the C symmetry. The "Chern parity " (even or odd) then classifies C-invariant insulators in 1D.

In the case of nontrivial Chern parity and open boundary conditions, there are gapless "end states" at the end of the chain that carry charge 1/2 because of the C symmetry. We can think of the end of the chain as the domain wall separating the trivial (theta=0) phase from the nontrivial (theta=Pi) phase. We can compute the charge using the Goldstone-Wilczek method. We "turn on" the solition by winding halfway in theta around the torus defined by k and theta. The integrated charge is half the Chern number. which is half an odd integer.

So .. I guess that is what Zhang means here by "dimensional reduction": by considering theta as well as k, we find a Chern number for a theory on a torus, and in this case the parity of that Chern number provides a classification. But .. this discussion seems to be limited to the free fermion case? Maybe not, *if* we can find a more general way to understand the *even* ambiguity in the Chern number.

The 0+1 dimensional C-invariant case (i.e. a theory with a single site). We can consider a one-dimensional loop starting and ending at a given Hamiltonian, where the result of going around the loop is the action of the charge-conjugation operator. Interpreting the interpolation parameter as a momentum, we have a one-dimensional theory, which has a Chern parity. It can be shown that this Chern parity does not depend on the interpolation, so there is a Z2 classification for single-site theories, too.

Now, the TRI case. We start with the 4+1 dimensional version of the QHE. Now there is a 4-torus and four linked fluxes (twisted BC.) The Berry connection has a *second* Chern class C_2. The effective theory is a 4+1 dimensional CS theory, with coefficient proportional to C_2. This is exactly analogous to the 2+1 dimensional TRB case, where the effective theory is 2+1 dimensional CS with coefficient proportional to C_1.

The 3D edge is a chiral theory with a gauge anomaly. Charge conservation is restored by the Hall current that flows in the 4D bulk when there is a nonvanishing E \dot B in the bulk.

Now 3+1 dimensional TRI insulators. We consider a family of 3D Hamiltonians, parametrized by theta. Now, in the 4D case, we have a 4+1 dimensional topological CS action for the *E&M field*, whose coefficient is found from the second Chern class C_2 of the *Berry connection*. This quantity C_2 is found by integrating the Berry curvature over the 4-torus.

Fixing theta is like fixing one of the "momenta" on the torus -- i.e. the twist in the boundary condition in the 4th direction. The result is that the effective action looks like axion electrodynamics, with the field Zhang calls P_3 (the "magneto-electric polarization") coupled to F \wedge F. To construct P_3, we integrate the 3D CS term constructed from the other three components of the Berry connection over three dimensions. So P_3 becomes the vacuum angle theta, and the TRI static backgrounds have theta = 0 or theta = Pi. Is that the origin of the Z2 classification? Reverting to free fermions, Zhang shows that the "Chern parity" is an invariant. We consider a loop in the space of T-invariant Hamiltonians where the path induces the T transformation, and argue that the ambiguity in the induced C_2 is an even interger.

Now ... can we see that for the theta=Pi phase there are toplogically protected gapless states on the 2-dimensional "edge"? This seems to work for the free-fermion picture, but does it work beyond that? We can think of the edge as a domain wall where theta winds from theta=Pi (inside the insulator) to theta=0 (vacuum) --- see Zhang page 22. Is there then a Goldstone Wilczek fractional charge on the wall, with a Z2 classification? This must correspond to whether the number of Dirac fermions on the edge is even or odd. The gauge part of the effective action becomes a 2+1 CS theory on the edge (since E \dot B is the divergence of the CS Lagrangian), where the coefficient corresponds to the IQHE with Hall conductivity integer + 1/2.

Thus, though the bulk theory in 3+1 dimensions is TRI, the boundary theory is a TRB 2+1 dimensional system whose effective action is CS and with a nontrivial QHE.

Now 2+1 dimensional topological insulators. We have two parameters that can be varied, giving rise to the theta and phi components of an external gauge field. The 4+1 dimensional CS action reduces to the 2+1 dimensional action.

The modes spend most of their time near the sample boundaries, where the energy is low. When energy gets pumped up close to the lattice cutoff, the move quickly to the center of the sample, while another level leaves the center and runs quickly to the opposite edge.

19 June 2008.

Is Zhang saying the following? Consider the edge of the 3+1 dimensional TRI system with theta = Pi. We model the edge as a "domain wall" in which theta winds from Pi to the vacuum value, zero. Then the effective gauge theory on the edge is a CS theory, but with coefficient shifted away from integer by 1/2. Therefore, in order to avoid an anomaly, it is required that there be a massless fermion. This edge theory is not T and P invariant --- it suffers the Redlich partiy anomaly -- even though the underlying bulk theory *is* TRI.

He seems to say something like this on page 22 ff. But is it correct? Since the theory on the edge is not parity invariant anyway, why not give the Dirac fermion a (parity violating) mass term? In effect that changes the CS coefficient back to an integer.

And anyway, what is the corresponding observation when the bulk is 2+1 dimensional and the edge is one dimensional?

For the QHE, we normally argue that the bulk effect is robust, because the Berry phase of the many-body wave function is topologically nontrivial (has nonzero first Chern class). *Then* we say that the existence of the bulk QHE requires the existence of gapless edge states, so that there can be an adiabatic flow of charge from one edge to the other. (The charge flow occurs because of spectral flow: on one edge postive energy modes flow to negative energy, creating holes, while on the other edge negative energy modes flow to positive energy, creating particles.

For the QSHE, we'd like to argue similarly that the existence of a QSHE requires dissipationless flow of spin current on the edge. But for that we first need an argument for the bulk effect that is robust against including electron interactions.

Figure is from Kane-Mele cond-mat/0505581. In the topological insulator phase (a), there is a red pair of states at one edge and a green pair of states on the other edge. TRI enforces that the edge states become degenerate at ka=Pi. Even though there are two counterpropagating modes on each edge, elastic backscattering cannot occur. This is because the momenta k= Pi/a \pm q form a Kramers pair, and it turns out that TRI enforces that the S matrix is diagonal!In the ordinary insulator phase (b), there are no edge states that cross the gap.

In general, if there are an even number of Kramers pairs at one edge, then elastic backscattering can occur and in fact a gap can open so there are no edge states connecting the upper and lower bands.

Here the blue and red on the left are two Kramers pairs on the *same* edge. At each energy the modes are paired at Pi + q and Pi - q; the modes cross at Pi. The black crossing points can meet, and the pairs can recomment as shown removing the "mid gap" mode. For a robust mid-gap state, we need to have an *odd* number of Kramers pairs.

So ... in the QHE, the existence of a QHE implies adiabatic charge pumping from one edge to the other, which means there are chiral edge modes. These edge modes can't scatter even if fermions interact. In QSHE, the existence of a QSHE implies adiabatic spin pumping from one edge to the other, which means there are edge modes that come in pairs of opposite chirality. If the fermions don't interact, this pair of modes is robust because of the Kramer degeneracy enforced by T invariance.

Zhang also notes on page 18 that a domain wall in which theta changes by Pi corresponds to a flip in the sign of the fermion mass. He remarks that this axion domain wall traps fermion zero modes. He also remarks that if theta varies in the z direction, then there will be an induced Hall effect in the xy plane.

Perhaps the key point to note is that the effective theory of the 2-dimensional surface for a 3-dimensional topological insulator *is* anomalous. This is like the 2+1 dimensional QHE effect and its 1+1 dimensional edge. In that case the edge theory is anomalous because there are gauged chiral fermions, but the anomaly on the edge is cancelled by the anomaly (i.e. the Goldstone-Wilczek charge inflow) in the bulk. The edge and the bulk need one another.

For the 3+1 dimensional TRI case, the effective theory of the 2+1 dimensional edge is anomalous. It has an odd number of massless Dirac fermions, but perhaps it does not have the compensating CS term, which would break TRI. (Is that right or is there TRB on the edge even though none in the bulk?) But in this case the anomaly is nonperturbative rather than perturbative, so there is not a corresponding charge inflow? How then are we to think about the spin current induced in the bulk by the QSHE?

This is confusing. Zhang says that there is a QHE on the two-dimensional edge, which means that the edge should have a *gap*, and should be TRB. So what does this have to do with gapless fermions? Zhang's discussion of this on page 22 is not very clear. On page 23 he seems to say that a TRB perturbation on the surface is needed to give mass to the fermions, in order that there be a well defined surface QHE.

An example is discussed on page 24: a heterostructure with slabs of ferromagnet (FM), topological insulator (TI), ferromagnet (FM). There is a QHE at the FM-TI interface, and furthermore, the current is carried by a chiral edge state that runs around the interface.

The 2+1 dimensional effective theory has a "Berry" field strength coupled to the em gauge potential.

Now we apply the adiabatic pumping argument but using the Berry flux instead of the EM flux. That is, we can ramp up the Berry flux adiabatically, and observe that charge is transported. This requires the existence of gapless edge modes. But why does it give a Z2 topological classification?

The basics of Kramers degeneracy and the connection with the 2nd Chern class is nicely discussed by Avron et al. PRL 1988.

More generally, a chargeon has odd charge and is a TRI state, and a spinon has even charge and is a Kramers doublet under T. (Two states such that T: 1 -> 2 -> -1). There is a generalization of the flux insertion, and the topological insulator is the state for which the induced charge is *odd*.

That is, the QSHE occurs in the topological insulator: a spin flux pumps an odd number of charges. We have two dual views of the QSHE --- either spin flux pumps charge or else charge current pumps spin. Here spin current is pumping charge. That is because the emf around the inserted flux tube acts in opposite directions on the two layers(the inserted flux has the opposite value). But since the B fields on the two layers are also opposite, the Hall current is in the same direction, and charge of the same sign accumulates on the flux tube on the two layers.

In a more general setting, we may say that a TRI fermionic system exhibits the QSHE if adiabatic charge pumping occurs. It is the TRI that distinguishes the phenomenon from the QHE (TRI means no Hall conductance). Thus:QHE = TRB and adiabatic charge pumpingQSHE= TRI and adiabatic charge pumpingIn the TRI case, we demand that the perturbation that drives the pumping is itself TRI, but no other symmetries need be imposed.Furthermore, the charge that is pumped has odd fermion number and is time reversal invariant (this captures the generalized idea that the chargeon has no spin).

But what is the generalized version of the concept of turning on a Pi flux that preserves TRI? Is it a nontrivial TRI Berry connection, such that ground state wave function has holonomy -1?

Another paper by Qi and Zhang: arXiv:0801.0252 "Spin charge separation in the quantum spin Hall state"

Proposes a Z2 classification for 2-dim TRI insulators, valid even in presence of interactions and disorder.

Example, two layers of IQHE, with opposite conductance. What is we insert half a flux quantum in each layer, adiabatically, with flux in a layer either increasing from 0 to Pi or decreasing from 0 to -Pi. So consider all four possibilities for the flux in the two layers. If one flux increases while the other decreases by the same amount, then TRI is maintained at all times. Because of IQHE, a charge of e/2 or -e/2 is induced, depending on sign of the flux that turns on. We think of the two layers as two spin states, so the spin created is proportional to charge.

In the cases where the flux that turns on is opposite, the same charge is induced on both layers. A total charge of e is created on the two layers, and no spin, because the spins in the two layers cancel. If the fluxes are the same, spin 1/2 is created but not charge. This is spin-charge separation (spinons with no charge and chargeons with no spin) in two dimensions. But here we have assumed that spin along the z axis is a good quantum number, which won't be true if there is spin orbit coupling.

Fu and Kane considered inserting a flux of Pi, since that changes momentum from one TRI value to another -- e.g. from 0 to Pi. In order to consider an adiabatic cycle, we would need to append a path back to the original Hamiltonian that does not insert any flux (and Moore). I guess in the Zhang et al. discussion, we could append the inverse of the path that creates a spinon to the path that creates a chargeon.

Anyway, just as we can interpret the connection between the bulk and the edge for the QHE as a Callan-Harvey phenomenon, we could like to do the same for the QSHE. We should relate the topology of the TRI Berry connection to spectral flow. The goal should be to understand how charge is pumped by a TRI cycle followed by the system Hamiltonian.

Recall (Nelson and Alvarez-Gaume, CMP 99, 103-114, "Hamiltonian formulation of anomalies") a connection between the Berry phase (-1) in a TRI system, and spectral flow. In the case of the SU(2) anomaly in 4-dimensional spacetime, they note that the space G3 of gauge transformations on a three-dimensional slice has Pi_1(G3)=Z2, for an SU(2) gauge theory. Furthermore, Pi_0(G3)=Z (where space is compactified to a sphere). If the number of SU(2) doublet fermions is odd, and the winding number of the gauge transformation is odd, then the gauge transformation changes the fermion number by an odd number (due to spectral flow dictated by the index theorem). So

Nelson and Alvarez-Gaume consider a family of Hamiltonians for the fermions, one for each background gauge field. To impose the gauge constraint, we want to find a trivial representation of the gauge group, but one won't exist if there is a subbundle on which the fermionic Hamiltonian has a nontrivial twist.

Berry's phase relates the nontrivial twist to the oddness of the number of enclosed degeneracies. The oddness of the number of enclosed degeneracies follows from the mod-2 index theorem as Witten observed (i.e. there are an odd number of modes of the 4D Dirac operator that cross zero).

Likewise, there should be a general theorem for the classification of topological insulators that relates the nontrivial twist in the Berry bundle to the spectral flow. But we need to understand how the adiabatic change in the Hamiltonian is characterized! A key point is that for the definition of the QSHE, we don't need to talk about spin transport directly -- rather we need to argue that there is a loop in the TRI family of Hamiltonians with odd spectral flow. The TRI invariant change in the Hamiltonian corresponds to spin transport, and the spectral flow corresponds to adiabatic pumping of charge from one edge to the other.

And can we relate the charge pumping to the Callan-Harvey bulk-edge connection? (Spectral flow guarantees flow from one edge to the other, because there is a bulk gap, and the only way for charge to be transported is due to the gapless edge states.)

Why don't we have to worry about level crossings being generic for TRI systems only in 5 dimensions? Is it important for the adiabatic change to break TRI, and if so, why is the phase necessarily -1?

When we twist the boundary conditions TRI is broken. Under T, the phase in the twist is complex conjugated. So that means that the Hamiltonian as a function of the twist has the same structure as for the Brillouin zone of free

Therefore, we can consider the Berry connection on the effective Brillouin zone, just as we do for free fermions, but where the Hamiltonian depends on the twist rather than the momentum. The EBZ is defined by restricting phi_x to vary from 0 to Pi, while phi_y is allowed to vary from -Pi to Pi. There is a Z2 classification of such connections. So is that not a nonperturbative criterion?

Recall that the Chern number in the case of IQHE is the pumped charge when the one flux varies over a noncontractible cycle and we average over the other flux. Fu and Kane suggested the analogous idea of Z2 pumping. Now we let phi_x increase from 0 to Pi (from one TRI point to the next). At these points there are Kramers pairs of gapless states, which must be edge states if there is a bulk gap. Z2 charge gets pumped from one edge to the other.

Alternatively Essin and Moore (arXiv:0705.0172) consider a closed cycle. After increasing phi_x from 0 to Pi there is a second stage in which the system returns *without applying a TRB flux. Now the pumped charge in a cycle is a well defined integer. Claim: the number of charges pumped in the second stage must be even, so the parity depends

In the first stage, the TR conjugacy between phi_y and -phi_y is imposed only on the boundary circles, *not* at intermediate values of phi_x. In the second stage, the conjugacy condition is satisfied for all intermediate phi_x as phi_x returns from phi_x=Pi to phi_x=0. The physical interpretation is that no flux is inserted through the cyllinder during the second stage.

Is this a nonperturbative definition? Essin and Moore say: "phases can be distinguished by this invariant as long as many-body effects do not prevent description of the ground state as a Slater determinant". What does that mean? Is the point that the twisted boundary conditions are imposed on the individual electrons? But so what?

Moore and Balents had considered contractions of the EBZ to a sphere, in which the boundary circles are filled in by disks. Here the EBZ is contracted to a torus instead. But either way the idea is that the ambiguity in the Chern number due to the arbitrary contraction is even, so that the Chern parity has a well defined value.

In the Moore-Balents argument, we contract the boundary circle to a point with a T invariant Hamiltonian. Each circle in the disk obeys the same conjugacy condition as the boundary circle. Two different contractions, then differ by a sphere, where on this sphere points on the northern hemisphere are conjugate to points on the southern hemisphere. This can be deformed to two spheres, each with the same Chern number -- hence overall an even Chern number.

To see this, on each disk there is a chord along which Hamiltonian is TRI (where the momentum is 0 or Pi). These two chords comprise the equator of the sphere. But the space Q of TRI Hamiltonians has a trivial Pi_1 and Pi_2. Therefore, H on the equator can be deformed to a constant (since Pi_1(Q)=0) and this deformation is unique (since Pi_2(Q)=0). Thus each hemisphere becomes a sphere.

Qi and Zhang (arXiv:0801.0252) reinterpret this construction --- they refer to the second stage as "charge flux threading".

The review article of Zhang et al. (arXiv:0801.0901) discusses why TRI forbids a mass term term in the nonchiral 1+1 dimensional theory. A mass term couples a left mover to a right mover. This bilinear term is T odd. The lowest dimension operator that is T even and chirality violating is an irrelevant four fermion operator. Without fermion interactions, the forbidden mass term is enough to ensure stability with respect to disorder. With interactions, the irrelevance of the leading perturbation ensures pertubative stability.

But .. with two flavors of nonchiral fermions, a TRI invariant mass term becomes possible, that couples the flavors. I guess this is an alternative way of formulating the Kramers degeneracy argument, indicating that the edge theory has a Z2 invariant. There is a TRI version of the Nielsen-Ninomiya theorem that requires doubling, but this can be evaded by coupling the boundary to the bulk. This is just how the NN theorem is evaded in the case of QHE effect: the anomaly of the edge theory is compensated by the anomaly of the bulk theory.

But in this review Zhang says that: However, the generalization to the many-body case with interactions is not so straightforward,

since the Kramers’ degeneracy is only defined for a single electron state.

So, we should ask, how would David Kaplan construct a single species of massless TRI fermions in 1+1 dimension by taking advantage of the 2+1 dimensional bulk? What is the obstruction? On the one hand we have doublers coming from the k=Pi end of the Brillouin zone, and on the other hand we have the Kramers degeneracy. Altogether, then, there are two left-movers and two right-movers. But in the case of QSHE, the partners switch, with a fermion that was on the lower edge forming a Kramers pair with one on the upper edge.

This figure from Zhang's review illustrates how the edge state topology is different for a topological insulator and a normal insulator.

The statement that Kramers degeneracy is defined only for single electron states means what is says ... the topology we are characterizing is that of a pair of bands for a single electron.

But in the case of the NN theorem, there is a more general argument: anomaly cancellation. Does that also apply to TRI 1D systems with one Kramers pair.

That is our hope, as it would clarify why the bulk is necessary for the edge theory to be sensible!

What if we imagine coupling an external gauge field to the Kramers pairs. Would the edge theory become anomalous and require a coupling to the bulk?

There is no anomaly -- that is, no charge is created -- whether the charges of the modes are matching or opposite. If both charges are positive, then we create RM particles and LM holes, so that the charge cancels. If the RM charge is positive and the LM charge is negative, then we produce RM particles and LM particles, which have cancelling charges. In the latter case this fictitious electric field creates something analogous to a spin current, but there is no anomaly, hence no need for the other bulk to evade Nielsen-Ninomiya?

Of course, in the NN argument, we can consider global symmetry currents rather than gauge currents. The model with one species has an axial anomaly

This is already enough to require fermion doubling. But for this global axial anomaly, can we argue that the bulk theory conserves axial charge and that there is a compensating flow of axial charge from bulk to edge?

How does TRI come into this discussion? Why will the classification be Z2 instead of Z?

What if the 2+1 dimensional bulk theory is gapped and TRI? It has massive fermions, and because the masses do not violate TRI, there must be an even number of fermion species. That is, in 3D Euclidean space, there are two inequivalent representations of the Clifford algebra, and parity interchanges them. So in a parity invariant theory, both types of spinors must be present, and a parity invariant mass couples the two types together. (Is this an alternative way to understand the origin of the Kramers degeneracy?). Likewise, in 2D, we need both chiralities in a parity invariant theory, since parity interchanges the L and R movers.

Alternatively, we could have two flavors, each with degenerate parity violating masses, but *of opposite sign*. Then parity flips both masses, but by combining with the 1 <-> 2 flavor swap, we have a symmetry.

Lee and Ryu, in arXiv:0708.1639, proposed a criterion for a Z2 topological insulator that works for interacting fermions. Their idea is to insert flux through the two cycles of the torus and to consider a doublet of states that can be created by letting an annihilation operator act on the ground state that annihilates an electron in every unit cell at one of two possible sites. There is a TRI condition on the doublet, when we reverse the value of the flux. There could be an obstruction to imposing this condition globally, so we divide the torus into two patches. The transition function relating the doublet on the two patches is a U(2) matrix M(alpha)

Nontrivial Z2 order means that there is an obstruction to globally defining the doublet while mantaining the TRI condition (because of the topologically nontrivial transition function, the bundle is twisted).

The SU(2) Berry connection on the torus has a nontrivial Wilson loop around a cycle of the torus: W(C) = -1, where the loop C is TRI (i.e. mapped to itself under T up to orientation). We can choose C to enclose half of the torus:

The main idea here is interesting, and may be fairly general. It is to consider a degenerate pair of states, a generalization of a Kramers pair, and the Berry connection as the boundary conditions are twisted. Now the Berry transport has a U(2) holonomy, which we can separate locally into an SU(2) part and a U(1) part. We identify an appropriate closed path that maps to itself under TR (perhaps with a change of orientation), and then the SU(2) holonomy is either +1 or -1, which determines the Z2 class.

To understand the Nielsen-Ninomiya obstruction beyond perturbation theory, perhaps we need to think about global anomalies rather than perturbative anomalies. After all, it is the nonperturbative anomaly of the 2+1 dim NA gauge theory that seems to be lurking in the background. But *does* the edge theory have any global anomaly?

Suppose that we want the bulk theory to be gapped and parity invariant. Then we have two fermions, with masses of opposite sign. Now we introduce a domain wall, where both masses change sign. In this case, there will be a zero modes on the edge of each chirality, since the chirality depends on whether the mass change is positive or negative.

If the fermions have charge and there is a longitudinal electric field, then chiral charge accumulates on the edge. This arises from a flow of chiral charge through the bulk from one edge to the other. But (1) isn't conservation of chiral charge already broken in the bulk at the classical level by the explicit mass term? Yes, but that is okay. The reason axial charge is not conserved on the edge is that right handed charge can flow off the brane into the bulk. But since the vector charge is exactly conserved in the bulk, when this happens there must be a compensating flow of left handed charge from the bulk to the brane. (Still, one wonders whether we need a second edge for the charge accounting to work.) (2) The flow of charge is proportional to the number of massive species, an integer, so why a Z2

The production of *chiral charge*, but not electric charge, on the edge is a prototypical spin quantum Hall effect. The theory on the edge makes sense as a continuum theory, but because it has an axial anomaly, we need flow of charge from the bulk to understand how it arises from a regulated theory.

Now we might ask --- if the number of flavors is even (i.e. an even number of pairs of fermion modes of opposite mass) how would we perturb the bulk theory so that there will be a gap on the edge? Is there a relevant *mod 2* version of the index theorem? Are there *discrete* symmetries that can be preserved even when the bulk fermions have masses? When there are two species in the bulk, the Z2 that interchanges the two species is not a symmetry, because it changes the sign of the mass term. Same for multiplying psi_L by i and psi_R by (-i). That's why we need to combine it with time reversal to obtain a symmetry.

But wait. With just one species

Is it anomalous in 2+1 dimensional electrodynamics?

In 2+1 dimensions, and "instanton" is the spontaneous nucleation of a quantum of magnetic flux. In order to have a gap *and T invariance* in the bulk, should we consider the 2+1 electrodynamics in its Higgs phase? If so, the instanton changes the vortex number, which therefore is not exactly conserved. Should I brush off my old work on confinement and Higgs phases in 2+1 dimensions? The Higgs phase is superconducting, which does not seem to be what we want.