Tuesday, 14th August
Andreas Bernevig (Princeton University)
Fractional topological insulators
Here we go, blogging the fastest speaker in the physics community!
Topological Phases of Matter. Andreas points out that a topological phase of matter has a manifold of ground-states that are sensitive to the topology of the space in which they lie. Thus a FQHE is singly degenerate on a sphere, but acquire ground-state degeneracies on a punctured disk. Part of todays talk is non-interacting topological phases, then we'll turn to interacting ones.
Non-interacting TI's are really band insulators, but their topologies manifest that they can not be adiabatically continued into an atomic limit without closing the gap or breaking a symmetry. The simplest example is the integer quantum Hall effect. It breaks time reversal (magnetic field) the qhe also subtley breaks another symmetry - translational symmetry eg in the Landau gauge
H = (1/2m) (py-eB x)^2
Infact, this system has another translational symmetry - the magnetic translation group, corresponding to a lattice with a 2pi flux per unit cell.
Infact the Dirac medal was just awarded here for three folks (Haldane, Kane and Zhang) who asked key questions. The first question asked by Haldane, was whether we can obtain a QHE without breaking the translational symmetry? The answer, he says, is of course "YES" - but one still needs a magnetic field. The simplest model is the 2x2 Dirac Hamiltonian. If I make a domain wall where the mass goes through zero - and when it does, the Hamiltonian becomes gapless. If I now close the sample, by making another domain wall, one gets a quantum hall effect without an applied magnetic field. It still breaks time reversal, but in a very different way to a magnetic field.
Now what I've shown you was in the continuum - but in 1988, Haldane did show a lattice model which doesn't require a net field, but maintains the full translational symmetry of the lattice. Now, with a lattice, the system has a bloch wave that winds across the Brillouin zone. This is the Topological Chern Insulator, but it still breaks time reversal symmetry.
The next stage was the non-time reversal breaking TI - combining two time-reversed pairs and linking them by spin orbit coupling. The big question I want to address today, is whether we can have a fractional version - a fractional topological insulator.
AB gives a summary of the FQHE. The psi 's are holomorphic. There is an angular momentum index "r" and this state corresponds to the gapped state that forms at filling 1/r.
On a torus, the GS is r-fold degenerate. You can make a quasihole by inserting a flux, which pushes particles away from the insertion point. This is the excitation -
psihole = Prod_i (zqh-z_i)|GS>
This is a zero mode of the hamiltonian.
Q: How is it a zero mode - its an excitation?
A: The QH excitations are different flux - they are the same thing as the ground-state with different flux. This is for a special Hamiltonian for which these are exact eigenstates. (Blogger did not understand the answer in full).
Q: the degeneracy r has a physical meaning?
A: yes, as a translational symmetry of the state in a space of higher genus.
Important point is that the quasiholes are neither bosons nor fermions. They are "anyons". The normalization of the 2qh state
psi_2 = (zqh1 -zqh2)^{1/r} * unnormalized 2qh state
so when you exchange, you pick up a phase pi /r - fractional statistics.
But now, I'm going to argue that the FTI is not guaranteed to exist! For the Chern insulator, you can sort of understand them by extending A(r) to an A(k). But for the Aharanov Bohm phase, the Curl is constant. But in a Chern insulator, if I have a Bzone, then the distribution of the Berry curvature F(k) can not be uniform - it has to vary in the Brillouin zone, there's a topological constraing on it, the curvature has to go to zero in at least 2 * C points, if C is the curvature. Its not clear that with this fluctuation that one will still have a gap, so its not clear that a fractional Chern insulator with a wildly varying Berry curvature, could exist. There are other things too - for example on a lattice one has competing instabilities, such as CDW's. The moral of the story, is that basically they are not guaranteed to exist. But I will show you that I believe, there is conclusive evidence that they do infact exist.
I want to go through the Chern Insulator models. First the Haltane Model - the model of Tang et al and Kai Sun et al. We will then add a Hubbard interaction to them. In these cases, one can have just one filled band with chern no 1, 2, 3... One can then take the TR version of them and obtain a FTI QSH (Fractional Topological Insulator Quantum Spin Hall ). You take this, and go for filling 1/3, and you can see that this has a 3-fold degenerate GS, with a gap. This was taken first to conclusively show it is a FQHS - but its by no means sufficient, for a charge density wave would also have a 3-fold degeneracy. But the way the degeneracy goes to zero would be different - 1/L for a CDW, exp for QHE, but can't infer from the numerics. One has to look at the quasihole excitations, reduce no of particles by 3, look at the spectrum - get a lot of states below the gap - and see that it matches the counting of a 1/3 Laughlin state. But still similar to a CDW. So energy plots don't distinguish between the two states.
Flux insertion - has same effect on CDW and FTI. Similar spectral flows. First check is that it is a featureless liquid - would like a diagnostic that is qualitative. To do this, we'll use the entanglement spectrum. Take a region of your space - then you separate into A and B - usually use a particle partition - before - could integrate over positions of N-m particles. Suppose the particles have some repulsive constraint - then
rho(z1... zm, z1'...zm') = 0 if z1 = z2
Then the wavefunctions that diagonalize rho - the states for which the wavefn does not die when you put them together must have "energy" zero.
So now take out N/2 particles, leaving me with N/2 particles, then the eigenstates of this rho are phi1...phihash, then these states must satisfy same clustering.
See that the counting of the states below the gap satisfy the same as a quasihole - and we can prove its different to a CDW - states below the gap would be much less dense. And one can understand the entanglement spectrum in terms of a kind of translational symmetry.
Moral of the story - is that fractional Chern insulators do exist, at least experimentally - energy spectrum can differentiate, but the entanglement spectrum does.
Q: what is the requirement on lattice for QHS? Eg pnictides?
A: Need no mirror symmetry. Have to sit down and analyze the point group symmetry.
Q: Why can't one just calculate the Hall conductivity to test if the state is a FQH?
A: You can calculate, but need to do it for each state individually. For finite size, there will be deviations from 1/3, 1/3, 1/3.
Andreas Bernevig (Princeton University)
Fractional topological insulators
Here we go, blogging the fastest speaker in the physics community!
Topological Phases of Matter. Andreas points out that a topological phase of matter has a manifold of ground-states that are sensitive to the topology of the space in which they lie. Thus a FQHE is singly degenerate on a sphere, but acquire ground-state degeneracies on a punctured disk. Part of todays talk is non-interacting topological phases, then we'll turn to interacting ones.
Non-interacting TI's are really band insulators, but their topologies manifest that they can not be adiabatically continued into an atomic limit without closing the gap or breaking a symmetry. The simplest example is the integer quantum Hall effect. It breaks time reversal (magnetic field) the qhe also subtley breaks another symmetry - translational symmetry eg in the Landau gauge
H = (1/2m) (py-eB x)^2
Infact, this system has another translational symmetry - the magnetic translation group, corresponding to a lattice with a 2pi flux per unit cell.
Infact the Dirac medal was just awarded here for three folks (Haldane, Kane and Zhang) who asked key questions. The first question asked by Haldane, was whether we can obtain a QHE without breaking the translational symmetry? The answer, he says, is of course "YES" - but one still needs a magnetic field. The simplest model is the 2x2 Dirac Hamiltonian. If I make a domain wall where the mass goes through zero - and when it does, the Hamiltonian becomes gapless. If I now close the sample, by making another domain wall, one gets a quantum hall effect without an applied magnetic field. It still breaks time reversal, but in a very different way to a magnetic field.
Now what I've shown you was in the continuum - but in 1988, Haldane did show a lattice model which doesn't require a net field, but maintains the full translational symmetry of the lattice. Now, with a lattice, the system has a bloch wave that winds across the Brillouin zone. This is the Topological Chern Insulator, but it still breaks time reversal symmetry.
The next stage was the non-time reversal breaking TI - combining two time-reversed pairs and linking them by spin orbit coupling. The big question I want to address today, is whether we can have a fractional version - a fractional topological insulator.
AB gives a summary of the FQHE. The psi 's are holomorphic. There is an angular momentum index "r" and this state corresponds to the gapped state that forms at filling 1/r.
On a torus, the GS is r-fold degenerate. You can make a quasihole by inserting a flux, which pushes particles away from the insertion point. This is the excitation -
psihole = Prod_i (zqh-z_i)|GS>
This is a zero mode of the hamiltonian.
Q: How is it a zero mode - its an excitation?
A: The QH excitations are different flux - they are the same thing as the ground-state with different flux. This is for a special Hamiltonian for which these are exact eigenstates. (Blogger did not understand the answer in full).
Q: the degeneracy r has a physical meaning?
A: yes, as a translational symmetry of the state in a space of higher genus.
Important point is that the quasiholes are neither bosons nor fermions. They are "anyons". The normalization of the 2qh state
psi_2 = (zqh1 -zqh2)^{1/r} * unnormalized 2qh state
so when you exchange, you pick up a phase pi /r - fractional statistics.
But now, I'm going to argue that the FTI is not guaranteed to exist! For the Chern insulator, you can sort of understand them by extending A(r) to an A(k). But for the Aharanov Bohm phase, the Curl is constant. But in a Chern insulator, if I have a Bzone, then the distribution of the Berry curvature F(k) can not be uniform - it has to vary in the Brillouin zone, there's a topological constraing on it, the curvature has to go to zero in at least 2 * C points, if C is the curvature. Its not clear that with this fluctuation that one will still have a gap, so its not clear that a fractional Chern insulator with a wildly varying Berry curvature, could exist. There are other things too - for example on a lattice one has competing instabilities, such as CDW's. The moral of the story, is that basically they are not guaranteed to exist. But I will show you that I believe, there is conclusive evidence that they do infact exist.
I want to go through the Chern Insulator models. First the Haltane Model - the model of Tang et al and Kai Sun et al. We will then add a Hubbard interaction to them. In these cases, one can have just one filled band with chern no 1, 2, 3... One can then take the TR version of them and obtain a FTI QSH (Fractional Topological Insulator Quantum Spin Hall ). You take this, and go for filling 1/3, and you can see that this has a 3-fold degenerate GS, with a gap. This was taken first to conclusively show it is a FQHS - but its by no means sufficient, for a charge density wave would also have a 3-fold degeneracy. But the way the degeneracy goes to zero would be different - 1/L for a CDW, exp for QHE, but can't infer from the numerics. One has to look at the quasihole excitations, reduce no of particles by 3, look at the spectrum - get a lot of states below the gap - and see that it matches the counting of a 1/3 Laughlin state. But still similar to a CDW. So energy plots don't distinguish between the two states.
Flux insertion - has same effect on CDW and FTI. Similar spectral flows. First check is that it is a featureless liquid - would like a diagnostic that is qualitative. To do this, we'll use the entanglement spectrum. Take a region of your space - then you separate into A and B - usually use a particle partition - before - could integrate over positions of N-m particles. Suppose the particles have some repulsive constraint - then
rho(z1... zm, z1'...zm') = 0 if z1 = z2
Then the wavefunctions that diagonalize rho - the states for which the wavefn does not die when you put them together must have "energy" zero.
So now take out N/2 particles, leaving me with N/2 particles, then the eigenstates of this rho are phi1...phihash, then these states must satisfy same clustering.
See that the counting of the states below the gap satisfy the same as a quasihole - and we can prove its different to a CDW - states below the gap would be much less dense. And one can understand the entanglement spectrum in terms of a kind of translational symmetry.
Moral of the story - is that fractional Chern insulators do exist, at least experimentally - energy spectrum can differentiate, but the entanglement spectrum does.
Q: what is the requirement on lattice for QHS? Eg pnictides?
A: Need no mirror symmetry. Have to sit down and analyze the point group symmetry.
Q: Why can't one just calculate the Hall conductivity to test if the state is a FQH?
A: You can calculate, but need to do it for each state individually. For finite size, there will be deviations from 1/3, 1/3, 1/3.
The counting argument, which was used to support that one indeed gets a FQH state on a suitable lattice w/o magnetic field, was based on a generalized exclusion principle of quasiholes in Laughlin states. Can this argument be generalized to states at non-Laughlin filling fractions, e.g. 2/3 or 2/5? If yes, is there an intuitive way to illustrate it?
ReplyDelete