Prof. Shik Shin (ISSP, Univ. of Tokyo)

Laser ARPES study on Ba_(1-x)K_xFe2As2 and KFe2As2

Introduction

Brief review of BaK122 phase diagram and previous ARPES results.  It's clear that the optimally doped system (x = .4) is fully gapped, while the end member KFe2As2 has nodes.  Whether the optimally doped sample is s+- or s++ is still an open question, as is the d-wave vs. s+- nature of KFe2As2. Understanding these gap symmetries is essential for understanding the mechanism.

Most people believe that the mechanism is spin-fluctuations, which comes from intra-orbital nesting.  In favor of this, STM in Fe(Te,Se) finds sign reversal.  Disfavoring it, the insensitivity to impurities and the ARPES on KFe2Se2 showing that there is no nesting despite the high Tc favor s++.

The multiorbital nature, where DOS(zx) > DOS(yz) below the structure transition indicates orbital ordering, and ARPES on Ba122 sees 2-fold symmetry Fermi surface below Ts.  Kontani's theory of orbital fluctuations relies on interorbital pair scattering, which gives s++ superconductivity.

Information about Laser ARPES.  Very high resolution (70 mueV), low temperatures T < 1.5K, bulk sensitive (100 Angstrom electron escape depth), and can change polarization easily to differentiate the different orbital contributions.  The 7eV laser covers the Gamma surface, and they are developing an 8eV laser that should cover the electon surfaces too.

Laser ARPES results on Ba122 doped with K

Optimally doped: They observe three hole Fermi surfaces around the Z-point (measure kz = Z plane), with the orbital dependences: x/y, xz/yz-z^2,x^2-y^2 (from inside out).  For the optimally doped sample, previous ARPES saw a gap around 10meV with a shoulder around 6-10 meV (kz = Gamma plane).  They see smaller gap (3meV), and two sharp peak structures (A,B).  Identify the lower energy peak, A as the SC peak, as it vanishes above Tc, but the higher energy peak B persists up to 100K, becoming very broad but not dispersing in energy, and thus is not a superconducting peak. The smaller gap magnitude at the Z point is consistent with other measurements finding the smallest gap at the Z point.  Borisenko (at Gamma point) sees peak A, but not B.  The FS dependence of the gap is not particularly anisotropic (4-fold anisotropy) and is the same for all three bands.


So how does the anisotropy change with doping?

KFe2As2:  Transport shows line nodes for the KFe2As2 material, while SANS shows the nodes must be along the c-axis or there must be a full gap.   Measuring at the Z-point, they see the three bands as before.After symmetrizing and plotting for different angles around the FS, they see that the inner band is always gapped (2meV), the middle band is sometimes gapped and sometimes not (1meV max), while the outer band is always ungapped (well, .2meV).  The nodes on the middle band exist around +- 5 degrees, giving a total of 8 nodal points around the four-fold symmetric points (0,90,...).  There is also a gap minima around 45 degrees.  Chubukov et al have shown that the gaps can all be fit by the form \Delta_0 (1 + A cos(4 \phi) + B cos(8\phi)) with different A's and B's for each band.  This observation of nodes is direct evidence for s+- over s++, and thus evidence for the spin fluctuation mechanism.

Underdoped (x = .2, .3):
They again measure the SC gap on the three surfaces, finding the inner two to be stronger than the outer, again see two peaks on both FS.  They also see a kink that may correspond to the non-SC peak.  As they get more underdoped, the difference between the gaps increases, the non-SC peak intensity becomes weaker and so does the kink structure.  Since the gaps are different in magnitude, this implies the inter-orbital pairing is stronger than in the optimally doped sample.

Overdoped (x=.7):
There is no gap on the outer sheet, while the inner and middle sheets have near the the same gap size, which is only weakly anisotropic.

There is a clear doping dependence of the SC gap size, and a clear difference between different Fermi surfaces.  The gap on the inner and middle sheets nearly corresponds to Tc, but outer does not (Inner and middle are active, outer(x2-y2) is passive), and the gap on the outer sheet vanishes abruptly at x=.55.   What happens to cause such a drastic change?  The electron FS at X-point vanishes.  It is also composed of x^2-y^2 orbital, so intra-orbital (spin-fluctuation) pairing is important in the over-doped region.

So the picture is then that spin fluctuations are important at low and high doping, while the orbital fluctuations are important at optimally doping.  This can be potentially explained by examining the dependence of the As-Fe-As angle on doping.  It passes through the magic tetrahedral angle around optimal doping, which should lead to stronger orbital fluctuations - Kontani also proposes a (s+-)(s++)(s+-) type phase diagram with the angle, so it might be s++ at optimal doping.

Next he discussed the temperature dependence of the non-SC peak.  It has the form of the SC gap + a temperature independent constant.  Above Tc, they observe a two-kink structure that is T independent.  The non-SC peak doping dependence is a dome around .4, with much weaker dependence above x = .6.

The non-SC peak has some similarities to the magnetic resonance, which also has a magnitude of around 15meV, but disappears above Tc.  It also gets weaker above x = .7 and is maximal at x = .4.  The doping dependence of the magnitude and intensity are similar.  Likely also related to the disappearance of the electron FS at the X point.  SO the non-SC peak/kink is likely related to the magnetic mode.

Open question: Why does on the the x=1 material have 8-fold symmetry?

Questions:
Fernandes - what about claims by dHvA that KFe2As2 has one band with a mass renormalizaton on the order of 10?
A - question about normal state, while this study was in the SC state.
Shibauchi comment - the heavy mass is seen near the X-point, not the Z-point.

Wen - Maximum gap now appears to be 3-4 meV, making 2\Delta/Tc only 1.3 - less than weak coupling BCS.
A - If electron FS has a larger gap, it could compensate, and most people say that it is sufficiently large.
Borosenko comment - they measure gap in Z-plane only, so limited momentum region.  The gamma point is larger, on order of 10 meV.

Maiti - At x = .2, do you see isotropic gaps, even though it's within the coexistence regime?
A - Yes, slight four-fold symmetry.

Chubukov-  From data alone, is there anything to distinguish between s++ and s++ at optimal doping?
A - No Fermi surface is special and there are no nodes.

Prof. Hong Ding  (IOP) Talk:

Probing Iron based superconductivity by photoelectrons. 

(Arpes studies) 

Arpes introduction, energy resolution of the Beijing group 0.9 meV; angular resolution 0.2 degrees

Question to be answered: is there universality in Fermi Surface topology and superconducting gap despite different structures in 1111, 122, 11, 111, 122', 21113 compounds (in the parameters where we have superconductors).  The Selenium compound 122' doesnt even contain As. 

First lets look at the Ba122 phase diagram:  one sees clear electron hole assymetry. One can dope in different ways, with electron (Co compounds) or holes (K compounds), large changes in the phase diagram. 

Pnictides more homogeneous than cuprates, so ARPES should be better. Arpes resolves all of the five bands, two holes and two electrons around the gamma point. Upon changing kz, a third hole FS appears. One can also resolve the orbital character of the bands, electron has mixed xy. Fitting of LDA with data observes rough similarity, but with some important differences in terms of bandwidth, mass renormalization, etc. Changing layers, Se, influences the mass renormalization effect.  

H. Ding's group looked at the FS evolution with doping. At optimal doping, there is good quasinesting of hole and electron hoping. Upon eavily overdoped hole doping, we lose the electron FS, but additional FS emerge at the M point, which is connected to the hole bands at gamma thru the bandwidth. 

In electron doping, the hole FS dissapears much fasted, explaining the much shorter superconducting dome in the electron doped side. 

Several plots of the superconducting gap are shown, with temperature dependence. Gap of 12 meV, 2 Delta/Tc=7. The observation of a shoulder due to impurity scattering (probably) is shown at low energy. It has sample dependence, maybe hence due to impurity. Same shoulder as in laser arpes. 

Plot that showes nodeless SC gap in BaKFeAs (37K), which close at Tc. Gap is smaller on the larger hole FS. Heuristically, one can look at a coskx cosky gap which can come from strong coupling (but not only! - for example, FRG and RG studies can also obtain such a term), and see how it fits quantitatively (it does fit qualitatively, at least in the BaKFeAs) with the data. Comparison with weak coupling is made, pointed out that Spm from weak coupling scenarion. Resonant mode supports the gap with a wavevector 0,pi. A fermi surface kink similar to the one seen in neutron scattering is also observed in arpes. 

H. Ding then shows data in the operdoped hole side, shows clear observation of the superconductor. Superconducting gaps are isotropic around the FS (this is a feature that seems to be true in arpes for the pnictides).

A psudogap is reported above Tc, shadowing the AFM region. 

Ratio of the gap/Tc remains roughly constant through the hole doping region. 

We then move to electron doped, similar gaps. kz dependence of the gap is presented, and fitted to a s_pm (in x,y) + x  (cos kx + cos ky)coskz form, which when the magnitudude x becomes large, is gapless. the value fo x is determined to be roughly 1/6. 

We now move to the electron doped Co samples, which show sharp coherence peaks, with 2 Delta/Tc= 8. Along FS isotropic gaps. Data on 11 compounds FeTeSe is now shown (Tc=13k) - for this case, there is a surpsise: gap cannot be fitted into a cos kx cos ky form obtained from J1 J2 (but again, this could be theoratically obtains ohersiwe in spin fluctuation too) . One has to include J3 in the spin model to obtain good agreement to data. 

The data on A Fe Se is showed. At 20% electron doping, the hole FS is absent. The gap is isotropic around the FS. What is the symmetry (weak coupling would give Tc=0 as there is no hole pocket). Several theoretical scenarios are presented, coming from quasinodelss d, or nodeless pm. At kz=pi, Ding finds a small electron surface which can help identify the gap. On this pocket the gap is smaller, 6.5 ef. The conclusion is that there is a FS in the z=direction at the Z point, which helps differentiate the symmetries. A d-wave would give nodes on this FS (but not on the slectron), but Ding sees nodeless. Claim: nodeless (on the electron FS) gap ruled out 

LiFeAs data consistent to the coskx cosky term, nice fit. 

The case for 3 classes of high Tc. J1 NN spin spin coupling is the cuprates. J2 NNN is pnictides. J2+J3 is FeTe. The possible magnetic phase is different for all 3. The case is made for local AFM exchange pairing of Iron-based superconductors.  

Q: From Heavily doped to underdoped, for optimal doped, what is the evidence for electron pockets ?  What is the difference between system with different electron pockets bottoms? 

A: LiFeAs, theres a difference in the bottoms

Q: Analysis of gap in FeTeSe using J2 J3. Neutron scattering peaks are 0,pi; pi,0

A. J2 J3 can explain the superconducting data.  J2 J3 can give magnetic at pi/2 pi/2, superconducting neutron resonance at 0,pi

Q: Tunneling spectrum can give you smth like a gap when actually the gap is very small. 

A: impurity broadening would not shift the gap size. 

Q: J1 J2 J3 construction, once the hole pocket becomes small you find Spm not stable. 

Nematic Transition and Antiferromagnetic QCP in BaFe2(As_{1-x]P_x)_2

Nematic Transition & Antiferromagnetic QCP in BaFe_2(As_{1-x}P_x)_2

Taka Shibauchi (Department of Physics, Kyoto University)

(Blogged by Tzen Ong)

Good morning everyone, and the next talk is by Taka Shibauchi on nematic transition and AF QCP in P-doped Ba-122 system.

Key Points:

1. AF QCP below the SC dome (K. Hashimoto et. al., Science 336, 1554 (2012))

2. Nematic Transition above the SC dome (S. Kashara et. al., Nature 486, 382 (2012))

3. Summary: A Renewed Phase Diagram

Taka starts by reviewing the phase diagram of the pnictides, specifically the 122 systems.

He introduces a chemical pressure axis with iso-valent doping, where you get very clean and homogenous samples with clear quantum oscillations in dHvA experiments.

He then shows a resistivity (\rho) vs T plot with several kinks, indicating the different phase transitions. There is a kink associated with the nematic phase transition, i.e. tetragonal to orthorhombic, and T_nematic ~ 120 K. There is then the usual resistivity drop at T_c.

Following on from the resistivity plot, Taka shows a \rho vs T above the SC dome, and there is a clear region of T-linear resistivity indicating NFL behaviour. So this raises the possibility of a QCP in the vicinity and perhaps below the SC dome.

Furthermore, from NMR experiments, we see a strong enhancement of 1/T1T due to Curie-Weiss behaviour at low temperatures, and the Curie temperature, \theta changes sign from low to high doping; so by extrapolating, we find a \theta = 0 at x ~ 0.3, implying a AF QCP exists at x ~ 0.3 below the SC dome.

Another piece of evidence if that the effective mass is also strongly enhanced from dHvA measurements, and this is seen in measurements in the normal state, i.e. above T_c.

This then leads to the question: Is there an AF QCP lying below the SC dome, and how does the QCP affect SC below T_c?

Taka then proposes two scenarios:

1. The QCP is avoided by a transition to SC state.

2. An AF QCP lying below the SC dome.

Taka proposes using the superfluid density, n_s, to differentiate between the two scenarios, and he has carried out London penetration depth, \lambda, measurements to measure the superfluid density.

At T = 0,  the penetration depth \lambda^2 ~ m*/n_s e^2

He used three different experimental methods to measure \lambda, including Tunnel diode oscillator measurements, Microwave surface impedance, and the T-dependence of Nodal Superconducting gap structure.

He then shows a plot of \lambda^2 vs doping, x, from all 3 methods, and they indicate that there is a sharp peak at x  ~ 0.3, approaching from both the low and high doping side. This means that there may strong QC fluctuations of n_s at x= 0.3, i.e, 2nd- order QCP may exist.

Qn: Is this an effective mass effect or is the superfluid density actually decreasing?

Ans: Quantum oscillation measurements show an m* enhancement around the same dopings.

Qn: What is the relation between \rho exponent and NMR results.

Ans: NMR shows monotonic behaviours, i.e. FL behaviour, and does not correspond exactly with the NFL behaviour seen in \rho. Taka said he will return to this point later.

Expt evidence indicates a QCP below the SC dome, meaning there are two different SC phases on both sides of the QCP, i.e. SC1 (pure SC state) and SC2 (co-existing phase of SC + AFM). This then implies a finite-temperature tetra-critical point where SC dome meets SDW state.

Taka therefore puts forth the scenario where the NFL behaviour as seen in \rho measurements and mass enhancement are all associated with the finite-temperature tetra-critical point, which is related to the zero-temperature AF QCP.

Taka also contrasts these results with the cuprates, where divergence or enhancement of \lamba^2 is not seen, e.g. Bi-2212 shows a broad max at p = 0.19, i.e. enhancemnet of n_s/m*, whereas Ba-122 shows a peak in \lambda^2, i.,e suppression of n_s/m* at x = 0.3. This is completely opposite behaviour.

Taka then shows the Uemura plot for the Ba-122 system, i.e T_c vs T_F (Fermi temperature).

The plot of T_c/T_F vs x shows a peak at the QCP - indicating that the strongest pairing occurs at the QCP and the SC may be driven by quantum critical fluctuations.

Taka also compares P-doped and Co-doped Ba-122 systems, and such quantum critical behaviour is not seen in Co-doped system. The reason may be due to disorder in Co-doped system, and STM results shows a large inhomogenity in SC gap for Co-doped systems, whereas P-doped is very clean

In addition, there is also experimental evidence for 4-fold nodes in SC gap for P-doped, but not Co-doped Ba-122 systems.

Nodal SCs in the vicinity of QCPs have been found in many systems, including heavy fermion systems such as  CoCoIn_5 and Ce_2PdIn_8. The physics of these SCs is still an open issue, and the effect of the low-energy q.p excitations on the SC state is not well-understood.

He shows that there is a deviation from T-linear behaviour for \rho at x = 0.3 at low T, which is also seen in Ce115 system. It is not likely to be due to disorder because Ce115 is very clean.

Furthermore, a plot of \rho vs T^1.5 shows a good fit, but away from the QCP there are deviations.

Hence it is believed that it is related to the physics of a QCP, and Taka proposes the concept of Nodal Quantum Criticality in Unconventional SCs.

The effect of low energy fermionic quasi-particles may be modeled phenomenologically by a  divergence in m* at QCP, which is cut-off at low-energies by deviations from QCP.

v_f ~ Z_k ~ 1/m*(k)~ \Delta^\beta/2

m*^2 ~ (p - p_QCP)^-\beta, and  \beta ~ 1 in YbAlB4 and YRS has been observed.

Qn: We have different sheets with different m* in the pnictides.

Ans: There is a large v_F on one of the sheets, and \lambda is governed by this sheet, and all the sheets are coupled, so critical fluctuations must couple to the other sheets.

Qn: Is similar behaviour seen in electron -doped cuprates?

Ans: Taka does not know of any data for \lambda^2 divergence or enhancement in the e-doped cuprates.

Second Key Point: Nematic Transition Above SC Dome

Many experiments have indicated an in-plane anisotropy, in crystals detwinned by uniaxial pressure. However, we have to be careful as uniaxial pressure breaks C_4v and may induce in-plane anistropy.

So we need experiments w/o pressure, so we use magnetic torque measurements which have high sensitivity of 5 * 10 ^-12 emu on small pure crystals, and apply an in plane B field to measure the magnetic susceptibility tensor, \chi.

When there is rotational symmetry, i.e C_4v is preserved, we have \chi_aa  = \\chi_bb, \chi_ab = 0

2-fold symmetry. 

When there is nematic order, i.e. C_4v broken: \chi_aa \neq \chi_bb and \chi_ab \neq 0

At x = 0.33, and at low-T, there is a 4 \phi  component in \chi, i.e. nematic order along Fe-Fe bond. Hence we see evidence for C_4v breaking.

Also synchchroton XRD detected small lattice distortions, with a tiny orthorhombic distortion at T*, i.e. the electronic nematic temperature.

T* changes with doping, and this leads to a new electronic nematic T* line above SC dome, and over a wide doping range.

Nematic and meta-nematic transtions:

Landau free energy = Structural term + electronic term + coupling

\delta = a-ba/+b = structural order

\psi = A_{2\phi} = electronic nematic order

T* : Non-zero \delta and \psi -> Broken C_4v

Suggests a renewed phase diagram:

1. A QCP below the SC dome with SC1 (pure SC) and SC2 (SC + AFM coexist). Nodal quantum criticality

2. Nematicity required for SC and high-T_C associated with AF QCP.

3. A new T* line above the AF state and SC dome.

Qn: Can you differentiate between CDW and nematic order?

Ans; Torque magnetometry cannot differentiate between charge and orbital order.

Qn: Do SC2 and SC1 have the same symmetry?

Ans: Based on the T dependence of \lambda^2 and NMR are similar, so the symmetry seems to the same.

Comment: QC fluctuations + fermions RG machinery shows m* enhances but does not diverge. So there is a peak as seen in the penetration depth measurements.

Comment:  In the GL free energy shown, there is a coupling between structural and electronic. However, you can ignore the structural component, and still show that there is a meta-magnetic transition from purely electronic nematic phase transition.

Qn: Have you measured the NFL down to low-T by applying a high B field?

Ans: Not done yet.

Qn: Assume we have a QCP, with q.p. coupled to critical fluctuations, does this imply that the SC gap should be strongly enhanced?

Ans: The QC fluctuations enhances pairing interaction, but also decreases n_s, so it has competing effect.

Qn: What is the meaning of iso-valent doping?

Ans: P is iso-valent to As, so number of electrons and holes are unchanged.

We thank the speaker for a very nice talk!

Electronic multicriticality on the honeycomb bilayer

Electronic multicriticality on the honeycomb bilayer
Oscar Vafek (National High Magnetic Field Laboratory)
(blogged by Kedar Damle)

Good morning blog-watchers. The next talk is by Oscar
Vafek, and he's going to talk about electronic multicriticality in the honeycomb bilayer system.


Oscar starts by remarking that bilayer graphene at and near
the neutrality point is a model system in which much progress
can be made towards understanding an interesting example
of electronic multicriticality.

And this is what he means by that:

With AB plane hopping \gamma_0 and out of plane (vertical) hopping \gamma_1
4 bands, two bands touch at K and K'. Two split off.
Touching of bands is parabolic with just nn hopping.
This is not a stable situation:
Susceptibilities to many instabilities (say charge imbalance between
layers) diverges logarithmically.
Infinitesimal interactions will drive interesting competing tendencies
to different orders.
Model system to study competing orders in an interaction electronic systems.

Few of the channels with \log(\gamma_1/T) susceptibilities
*Layer antiferromagnet (breaks TR and inversion)
*Ferroelectricity (charge imbalance between layers...breaks Inversion)
*Nematic (strengthens hopping along a fixed direction...breaks lattice rotation
only)
*Spontaneously developed quantum anomalous Hall effect (TR and inversion broken)

Question: Which of these possibilities wins??

Before getting to that, Oscar notes that parabolic band touching can be converted to four Dirac cones by trigonal warping. So strictly speaking,
this log divergence is cutoff in the infrared, but still, the log
will make the susceptibilities grow quite large before they are cut off.
So we still expect lots of competing instabilities, not at infinitesimal
interactions (as in the idealized situation), but at some small non-zero
threshold of interactions.

To analyze this, he constructs a low energy effective theory that
keeps only the touching bands, and leaves out the other two bands.
The fermion field in this theory has four components and each of
these has a spin label, making a total of eight. This is basically
the band Hamiltonian in the k.p expansion around the band-touching.

What about the interactions: By symmetry analysis, there are nine
independent couplings (see OV PRB 82 205106 for details) that
fully characterize the density-density interactions in the general
case.
The largest coupling is expected to be in the A1g channel since
it gets a dominant contribution from the screened coulomb interaction
between electrons.

To study the effects of interactions, Oscar now introduces the RG procedure,
which reduces the cut-off \Lambda around the two points K and K'.

Question: Why does he consider only screened Coulomb interaction?
What about the long-range part?
OV: I'm looking at experimentally relevant situations in which
there are metallic gates nearby which provide screening.


The RG equations have two relevant parameters that enter: the dimensionless
temperature t, and the dimensionless trigonal warping \nu_3.
For parabolic band touching z=2.
At leading order, this determines the growth of t
dt/dl = 2t
d\nu_3/dl = \nu_3
And then there are coupled equations for all the interactions g_i.
[blogger: the RHS of the flow equation is of course quadratic in the gs, but
the details went by too quickly to reproduce fully here]

As a technical device to help with the RG analysis, Oscar now introduces
source terms corresponding to all possible symmetry breaking instabilities
(something like 17(!) partice-hole channel instabilities, and 12(!)
instabilities in the pairing channel) and writes
down a flow equation for each of them.
This allows him to work out the physical consequences of his RG
equations by computing physical susceptibilities [by using
the dependence of the free energy on the source terms]

In the end, he needs to complete the calculation numerically, and
now, he is going to give a flavour of the results in various cases:

*If we only keep the interaction in the A1g channel, the gapless
nematic state wins. This seems a bit odd, since one might
imagine a gapped state will have a bigger energy gain, but somehow
the nematic seems to be the best compromise between energetics
and entropic considerations (blogger: This explanation went
by a bit too quickly for this blogger to be sure of it)

* As we approach the QCP at which the nematic order goes
away (as we weaken the interaction in the A1g channel at finite
trigonal warping), the antiferromagnetic and quantum spin hall
channel susceptibilities **also** diverge although this the
the QCP that signals onset of nematic order.

Question (Andrey) What happens to the divergence of nematic susceptibility
at the QCP?
Answer: This is all obtained numerically, and as far as one
can tell from the numerics, the divergence of the nematic suscepbitility
is still there, but weaker than at the finite-temp transition.

[Aside: Nematic phase seems to have been observed in suspended
graphene bilayer in 2011 by the Manchester group]

*If we include large repulsive Hubbard interaction (large backscattering)
then, we would have a particular ratio of our various weak-coupling
gs, and the RG does predict antiferromagnetism, consistent with
what we expect from a direct strong-coupling analysis.

*It is useful to think of repulsive short-range interactions with variable
range, and run the RG. One finds that in the extreme short-range
limit, one gets AF order, which goes over to coexisting AF and nematic
order, which finally gives way to dominant nematic order as
the range of the interaction gets even larger.

*Accessing other phases needs changing the sign of the back-scattering
term

This leads Oscar to the last part of his talk:
What happens in an attractive Hubbard model: CN Yang PRL 89
has noted that there is a SO(4) symmetry on the lattice for the
attractive Hubbard model.

System likes to put pairs on each site. This can favour
a layer polarized state (breaking inversion). This is
SO(4) symmetry related to an s-wave superconductor.

An interesting check on the weak-coupling RG machinery is to
ask whether this is reproduced within the RG framework:
The answer seems to be yes, so the approach seems quite reliable.

Finally, to recap the basic message:
Interactions need not be large to cause a phase transition
in bilayer graphene.
Dominant instability is to a nematic if forward scattering
dominates. With significant backscattering, the AF phase wins
instead.

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.