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.
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.
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