Quenching and time evolution
Start in H0, turn on interaction at t = 0, propagate with full Hamiltonian. New experiments allow us to do exactly this in cold atoms, nano-devices etc, since the time scales are much longer than in typical condensed matter systems.
Interesting questions:
- how do the observables evolve? correlations functions are now functions of two times (time difference and waiting time).
Example:
Quantum dot with tunneling turned on suddenly, measure appearance of the Kondo effect, how does the resonance evolve with time?
Bosons in one dimension - what are the effects of interactions on the dynamics?
One dimension because strong quantum fluctuations enhance all interactions, plus powerful mathematical methods - RG, bosonization, CFT and *Bethe Ansatz* (BA), which allows one to diagonalize an integrable Hamiltonian and find all its eigenvalues. Many integral models, like Hubbard, Heisenberg etc are experimentably realizable. So how can one use the BA to solve hte quench dynamics of many body systems?
Setting it up:
A given state can be time evolved knowing the energy eigenstates. These are known via the BA.
Standard approach - impose periodic boundary conditions, generate BA equations with quantized momenta, get spectrum and reconstruct thermo. Nonequalibrium requires overlaps, summing over the whole basis and take infinite volume limits, but typically has required heavy numerics, while we want to be elegant.
BA review:
General N-particle state generally very complicated, but BA-wave-function is much simpler - it is a product of single particle wave-functions and S-matrices (describing scattering phase shifts of two particles). Illustrating on Lieb-Linnigar model - bosons on line with delta-function interaction.
Divide configuration space into N! regions (labeled by Q), within each region there is no interaction, wave-function is product of exponentials. Regions are related by S-matrices - interaction on the boundaries. Assign amplitude to each region, A(Q), and S(ij) relating the regions. However, we have to be careful to do this consistently, if say we are exchanging between three regions there are two ways to do it and they must be the same. This consistency relationship is called the Yang-Baxter equation: S(12)S(13)S(23) = S(23)S(13)S(12), which is satisfied only when the system has enough conservation laws (eg - integrable).
Calculating overlaps is still difficult. Directly going to the infinite system can simplify things since momenta are unconstrained, and so can be integrated on a contour in the momenta space - which may be complicated, but allows a certain simplification using the non-interacting overlap. Dominated by the poles in the full-fledged Bethe eigenstates. Unfortunately, implementing the formula is very system-specific, really a guide for how to go about doing the problem.
Piers Q: New way of writing completeness? (yes) Does it only work for integrable systems?
A: Not really - would be similar formulas if we have a set of lambda where we know the poles.
Applications to bosons in optical traps:
(can make predictions for experiments - noise etc)
Lieb-Linniger model - standard model for bosons in 1 or 2 dimensions. Short range interaction - kinetic energy + local interaction when bosons are on top of one another. Coupling constant can be either attractive or repulsive, can be controlled experimentally via a Feshbach resonance.
Conventional BA results: Eigenstates labeled by momenta. Dynamic factor - normally 1, negative if they cross. Momenta real for repulsive, c > 0, complex pairs for attractive, c < 0, forming bound states.
Initial condition 1 - relax periodic lattice
Intial condition 2 - relax trap and allow bosons to expand and interact - interested in how they interact.
Central theorem: Expand state in term of eigenstates in contour representation. Must use real contours for repulsive case, complex strings for attractive case. Using contour represntation, can easily write down time-evolved state analytically.
Q: number of particles? A: any number of particles.
Keldysh: Time-evolving operator, O is non-perturbative Keldysh (forward and backwards in time on contour C) - not expanding in interaction, just solving exactly.
Okay - let's calculate.
1. Evolution of density -> time of flight experiment.
2. Evolution of noise correlation (more physics than in average density) -> time-dependent Hanbury-Brown Twiss effect.
C2 = <rho(1) rho(2)>/<rho(1)><rho(2)>
HBT - vanishing correlations between two sources (stars) - measure photons find bunching (C2 ~ cos x), fermions find anti-bunching (-cosx)
Many free particles gives more structure, time-dependent - curious about effects of interactions -> repulsive bosons evolve into fermions, attractve bosons evolve to a condensate.
Prepare initial state with two bosons, weakly overlapping. Evolve in time. Can carry out all integrals for two particles. Calculate B\dg b. Results: no interaction, then bosons quantum broaden and nothing interesting happens, density just decays in time. Repulsive interaction - pretty much like free bosons. Attractive interaction competes with the diffusion, as it favors more of an overlap, increasing the effect of interactions, and get much more interesting, oscillatory structure of the density(t).
For simplicity look at the long time limit - bosons turn into fermions - expression develops antisymmetric correlations and can rewrite original bosons in terms of fermions interacted on by free fermion Hamiltonian (Tonks-Girardeau Hamiltonian is c = \infinity limit, where hard-core bosons are effectively free fermions - it flows to this fixed point for any c > 0).
Scaling interaction fails for attracive bosons.
Q: How does time flow on the contour?
A: Time flows only forwards - back is just taking bra.
Looking at corrections to long time - can use stationary phase approximation at large times. Attractive has saddle point + poles and bound states. xi = x/2t only t-dependence.
C2(x1,x2,t) -> C2(xi1,xi2). Develop antipeaks in the C2(x1-x2) at zero, indicating fermionic behavior (bosons develop peak that sharpens in time).
Can do for any number of particles - always fermionic physics, with antidip, and then more structure at longer distances.
Q: Piers - origin of periodicity in 10 particle? A: Starting from periodic initial state (lattice)
Q: Bosons as fermions - wavefunction is antisymmetric? A: Really hardcore bosons, just changing variables into fermions. (okay because 1D)
Time-evolution Renormalization Group:
time flow is RG flow - increasing time plays role of increasing bandwidth, since flows to c = +/- infinity in long-time limits. Can argue that there are basins of attraction, so adding perturbations (short range interactions beyond delta-function), dominated by same fixed points - eg - Bose-Hubbard model (putting LL on the lattice) -> hard core bosons hopping on lattice in long time limit.
Conclusions:
Don't need spectrum of H or normalized eigenstates. Just need theorem on initial state. Takes into account bound states without summing over strings, and is mysterious. Can completely calculate all asymptotics, and can compare to experiments, especially as results are universal.
Working on applying to more models, adding impurities, finite volume, density, temperatures. Approach to steady state, testing dynamical RG hypothesis.
Big Questions:
- what drives thermalization of pure states? is it canonical typicality, entanglement entropy?
- general priciples out of equilibrium?
- what is univeral?
Questions:
- HBT results recovered for bosons and fermions?
A - not yet compared to experiments, but should be recovered.
Chubukov: large time asymptotics means what time scales compared to experiments?
A: time scales are milliseconds. t > 1/coupling (and coupling is known)
Comment: von Neumann did work on thermalization - considered quantum ergodic thm, related to quantum central limit thm - system in long time limit goes to thermal limit.
A: not everyone accepts that this has been answered, but this is even more general and we have full control - not all states thermalize - they remember their initial conditions.
Piers: Vacuum is absence of bosons, and is eigenstate. Could do for magnons against FM vacuum. But what about nontrivial background, like AFM?
A: Can start with Neel state (not eigenstate), act with H and propagate - where does it go? Or can take XXZ model, has phase transition so prepare as eigenstate of Ising model, propagate with XY model.
Physicists meet for two weeks to discuss innovations and developments in our understanding of strongly correlated electron systems. The first week of the meeting is a school, introducing students to the ideas and new developments in the field. The second week of the meeting is a research workshop.
Thursday, August 16, 2012
Girsh Blumberg (Rutgers): CaMn2Sb2 (CMS) with buckled hexagonal Mn plane structure: removal of Hund's spin blockade
Blogged by Eoin O'Farrell
The tittle compound is a new material, on which the erstwhile blogger found just 2 papers in the literature; in the spirit of new materials the cover slide showed several single crystals a few mm in dimension. The structure is the 122 structure (made famous by the FeAs superconductors) and the physics is somewhat related to that presented earlier in the week by Christian Haule.
The Mn ion is in the 2+ state and the 3d orbitals are half filled so Hund's rule gives a large spin 5/2 state. The interesting physics is in the MnSb bilayer which can be viewed as a hexagonal bilayer wherein the Mn ion coordinates with 3 Mn ions in the adjacent layer. Dr. Blumberg explained that the reason to consider this material as strongly correlated is the difference between the bare bandwidth and that measured from Raman is about 2 orders of magnitude.
The behavior of CMS is as follows: at 300K the material is a paramagnetic metal, at 200K CMS becomes weakly ferromagnetic and the resistivity increases following a standard activated behavior and then at ~100K orders AFM. Finally at low temperature the resistivity has increased by about 6 orders of magnitude. The main experimental exhibit is Raman spectroscopy and the explanation of this by hopping in the AFM state, this was used to explain the difference between the optical gap (~1eV) and the transport gap (40meV).
Hybridization between the Mn-Sb produces a finite (~10%) occupancy of 3d6 states on the Mn site, so that adjacent Mn sites are like so:
This gives rise to a flat band with bandwidth 10meV, but Dr. Blumberg stuck to a localized, hopping picture in his explanation.
The key to resolving the discrepancy between the optical gap and the transport gap was the binding of pairs of polarons by the Heisenberg interaction. This gives rise to the possibility of certain photon assisted or activated hopping mechanisms which change the component Sz=5/2 -> 2 and reduce the transport gap below the optical gap.
This also explained why DMFT calculations did not capture the full renormalization of the bandwidth.
Questions:
1.
Andrey Chubukov: Where in the Raman spectrum is the 2 magnon peak.
Answer: We think all the features of the Raman spectrum can be explained, but not due to a 2 magnon peak
2.
Unidentified interrogator: How large is J in the Heisenberg interaction?
Answer: It's about 7meV
3.
Piers Coleman: Has the Fe version of this material been made?
Answer: For the pnictides there is a similar [unspecified - ed] material, but it is insulating, The Fe compounds are expected to be less strongly interacting
4.
Hide Takagi: Please say more about the FM phase?
Answer: In the FM phase nearest neighbor interactions are still AFM but there is no phase rigidity. [Note this should be consistent with the resistivity which showed no significant change between at the AFM/FM phase transition.]
5.
Natalia Perkins (chair)
Why is the renormalization so large?
Answer: DMFT finds some renormalization from the bare LDA, the remaining renormalization comes from strong correlations.
Aditi Mitra (New York University, NY)
Quantum quenches in one-dimension: A renormalization group approach
Blogged by Piers coleman
Aditi discusses the challenges of a "Quantum Quench". One of the most exciting areas for Quantum Quenches (QQ) is in cold atoms systems. These are highly tunable systems - one can realize model Hamiltonians and tune the interactions between the atoms. Another method is the use of ultra-fast optical pump probe methods. This allows one to probe time evolutions on femto-pico time scales, when the physics should be governed by quantum unitary time evolution. One applicaion is work by Cavalieri et al.
AM gives an outline:
Quenches in free theories - result - non equlibrium steady state.
What happens then if one adds a periodic potential. Will study case of "irrelevant operators" in eqn. SHow that they become relevant out of eqn.
Then consider interactions that are relevant in eqn. This leads to a new kind of dynamical phase transition. She considers a Luttinger liquid (spinless), and shows that in a quench that suddenly changes the Luttinger constant K0-> K, the power-laws for the density density and phase-phase correlation function acuquire a new exponent. Correlations always decay more rapidly than in equilibrium but with a new powerlaw. Interestingly, the non-equilibrium state is not a Luttinger liquid because the new powerlaws are not the inverse of one-another.
AM now explains the reasoning behind these results. One looks at the bosonization the initial and final Hamiltonian's are non-interacting bosons. The initial distribution is conserved, and the results can be understood in terms of a "Generalized Gibb's ensemble" rhoGGE.
Now what happens with some non-linearities in the system. Look at the effect of a periodic potential. The final partition function is given by
Z = Lim Tr[U(t)rhogge U\dg(t)]
When the periodic potential is there with interactions, dissipation arise. Interestingly enough, now the density density decays exponentially fast. This is because of the generated noise. More importantly, The conductivity acuires a finite value (from an infinite one).
Now AM considers a different quench protocole in which K is shifted and the lattice potential is simultaneously turned on. In this case, the slow and fast modes separate out. Now the correlation function depends on both the relative time and the "center of mass" time. At short times, power-aw in space with exponent K0, but at long times, plaw in space and time with exponent Kneq. Teh crossover between these two times is governed by ...., which means that the scaling dimensions of the lattice depend on time.
Aditi introduces the scaling equations, which he time-dependent scaling dimensions. Providing one is only interested in times tm < 1/eta. The RG reduces to just three equations. One defines a g-eff = g Sqrt[IK(Tm)] which leads us to a Kostlitz phas transition. Four cases
a. periodic pot irrelevant at all times
b. perioidic potential relevant at all times.
c. PP relevant at short, irrelevant at long times
d. PP irrelevant at short times, relevant at long.
Integrate the RG up to the point where g Cos(phi) = 1- gphi^2. There is an exactly solvable point that can be used to check the results.Case b such that geff = epsilon (Tm*) There is a non-analytic behvior in the RG, corresponding to a dynamical phase transition. Even to leading order in the potential, multi-particle scattering processed.
Now its the conclusions.
Quantum quenches in Free Theories can lead ot interesting non-equilibrium behavior.
In non-linearities, an analytic approach to study dynamics is presented that is valid in teh thermodynamic and long time limit where numerical studies are hard to do.
Even when the periodic potential is irrelevan, its effect is no-trivial as it generates dissipation and noise.
When the PP is relevant a new kind of non-equilbrium dynamical phase transition takes place - a kind of non-equiibrium Kosterlitz Thouless transition.
Q: what happens when you include a cubic term?
A: will give new results, but we have not done this yet.
Q: Is there a quantum version of the Komogorov theorem.
A: Now and you have to be very careful about this.
AM now explains the reasoning behind these results. One looks at the bosonization the initial and final Hamiltonian's are non-interacting bosons. The initial distribution is conserved, and the results can be understood in terms of a "Generalized Gibb's ensemble" rhoGGE.
Now what happens with some non-linearities in the system. Look at the effect of a periodic potential. The final partition function is given by
Z = Lim Tr[U(t)rhogge U\dg(t)]
When the periodic potential is there with interactions, dissipation arise. Interestingly enough, now the density density decays exponentially fast. This is because of the generated noise. More importantly, The conductivity acuires a finite value (from an infinite one).
Now AM considers a different quench protocole in which K is shifted and the lattice potential is simultaneously turned on. In this case, the slow and fast modes separate out. Now the correlation function depends on both the relative time and the "center of mass" time. At short times, power-aw in space with exponent K0, but at long times, plaw in space and time with exponent Kneq. Teh crossover between these two times is governed by ...., which means that the scaling dimensions of the lattice depend on time.
Aditi introduces the scaling equations, which he time-dependent scaling dimensions. Providing one is only interested in times tm < 1/eta. The RG reduces to just three equations. One defines a g-eff = g Sqrt[IK(Tm)] which leads us to a Kostlitz phas transition. Four cases
a. periodic pot irrelevant at all times
b. perioidic potential relevant at all times.
c. PP relevant at short, irrelevant at long times
d. PP irrelevant at short times, relevant at long.
Integrate the RG up to the point where g Cos(phi) = 1- gphi^2. There is an exactly solvable point that can be used to check the results.Case b such that geff = epsilon (Tm*) There is a non-analytic behvior in the RG, corresponding to a dynamical phase transition. Even to leading order in the potential, multi-particle scattering processed.
Now its the conclusions.
Quantum quenches in Free Theories can lead ot interesting non-equilibrium behavior.
In non-linearities, an analytic approach to study dynamics is presented that is valid in teh thermodynamic and long time limit where numerical studies are hard to do.
Even when the periodic potential is irrelevan, its effect is no-trivial as it generates dissipation and noise.
When the PP is relevant a new kind of non-equilbrium dynamical phase transition takes place - a kind of non-equiibrium Kosterlitz Thouless transition.
Q: what happens when you include a cubic term?
A: will give new results, but we have not done this yet.
Q: Is there a quantum version of the Komogorov theorem.
A: Now and you have to be very careful about this.
Andrey Chubukov: Collective instabilities of doped graphene
Blogged by Rafael Fernandes
Contrasting the talk given by Oscar Vafek, who looked at many-body instabilities in double-layer graphene, Andrey will talk about instabilities in (doped) single-layer graphene.Glossary: SDW = spin-density wave; SC = superconductivity; CDW = charge-density wave; FS = Fermi surface; VHS = van Hove singularity; RG = renormalization group.
Andrey explains that when the chemical potential of graphene is changed (via electron doping, for instance), the Fermi surface changes from two Dirac points to a hexagonal FS with three saddle points, corresponding to VHS. He mentions that experiments in doped graphene find a hexagonal FS, which is somewhat similar to the one expected theoretically. However, there is no consensus that this is really the VHS of the band structure.
Andrey proposes to study the "ideal" situation of 3/8 doping. Besides the VHS, there are three nesting vectors connecting different sides of the hexagonal FS. He mentions that such a FS is stable even when one includes next-nearest neighbor hopping. Due to the presence of VHS, the particle-particle bubble acquires an extra log, and the SC instability becomes log^2. Furthermore, due to the presence of perfect nesting, the particle-hole bubble also acquires an extra log, and density wave instabilities also becomes log^2. Other exotic instabilities may also appear, but they do not have log^2. Thus, there are several many-body instabilities, and one needs to investigate in which channels there are small attractive interactions that can take advantage of the these log instabilities. Ultimately, Andrey wants to find which of the several instabilities is the leading one.
To address this problem, he will use parquet RG, which treats particle-particle and particle-hole channels on equal footing. In the RPA level, the pairing interaction is repulsive, but SDW and CDW channels are attractive. However, these are the bare interactions, which become renormalized at low energies. Andrey now writes all the possible interactions between the low-energy fermions near the VHS in each of the three FS patches. There are four of them, including density-density interactions, pair-hopping, exchange, etc. Doing the one-loop parquet RG, these interactions are renormalized and change as higher energy modes are integrated out. The question Andrey wants to answer is which of these four coupling constants diverges, and which order it triggers. Formally, one has a set of coupled first-order differential equations. Andrey considers both the cases of perfect nesting and non-perfect nesting - the latter via an effective parameter.
Q: is umklapp considered? A: sure, pair-hopping is an umklapp process.
Andrey shows the solution of the flow equations - all couplings diverge at a certain energy scale, but one of them diverges first. By writing down the SC, SDW, and CDW vertices, and plugging in the renormalized couplings, he can identify which one diverges first. The two leading instabilities are the SDW (whose bare coupling was positive - attractive) and singlet SC (whose bare coupling was negative - repulsive). Eventually (i.e. at lowest energies), SC wins and becomes the leading instability. Several technical questions pop up, from the validity of the RG procedure for log^2 instability to the fate of the flow equations away from perfect nesting.
Now the symmetry of the SC state is discussed. There are three gaps corresponding to the three VH points, and three possible eigenvectors/eigenvalues for the linearized gap equations. Two of these eigenvalues are actually degenerate - the other one, whose eigenvector corresponds to the s-wave solution, remains always negative, i.e. it is not realized. The two degenerate solutions have d-wave symmetry: d_xy and d_{x^2-y^2}. By performing a Ginzburg-Landau expansion, Andrey finds that the solution with lowest energy is actually a combination of these two symmetries: it is a (d+id) state, which breaks time-reveral symmetry. Interestingly, functional RG gives the same result.
Conclusion: graphene doped to the saddle-point of the band structure displays as the leading instability a time-reversal (d+id) SC state!
Maxim Dzero: Topological Kondo Insulators
Your blogger today: Andy Schofield
OK well due to a blogging malfunction - this blog is starting half-way through the talk - no it seems to be the end of the talk!
Max has defined two types of Topological Insulators: weak or strong. A large N analysis has been done.
Ce-based Kondo Insulators expected to be weak and unstable to disorder.
Mixed valence systems are most likely to be the ones where strong TI insulators would be observed. Hybridization with the conduction electrons can create an infinite spin-orbit coupling.
These are adiabatilcally connected to topological band insulators: small bandwidth and infinite spin-orbit.
Question: Could a Hopf term be useful? Possibly.
Question: Why are all topological insulators cubic and does this impact your discussion? Not sure and the questioner does not know either.
Question: Your analysis looks like 2D - aren't the supposed to be 3D creating surface states? Sure - layers will create it.
Question: Why called Dirac point? Answer: the dispersion looks like a Dirac cone.
Question: Why do we need the notion of entanglement entropy? It provides a way of distinguishing between charge density wavesand topological insulators.
Question: Can all of this physics be deduced from the Green function? No you might need a 4 particle correlation.
Question: Why do you get band narrowing from spin orbit? It is nothing to do with spin orbit. Instead it is simply the usual Kondo lattice phenomenon.
Question: Where is this in the periodic table of the topological insulators? Yes - it is A1.
OK well due to a blogging malfunction - this blog is starting half-way through the talk - no it seems to be the end of the talk!
Max has defined two types of Topological Insulators: weak or strong. A large N analysis has been done.
Ce-based Kondo Insulators expected to be weak and unstable to disorder.
Mixed valence systems are most likely to be the ones where strong TI insulators would be observed. Hybridization with the conduction electrons can create an infinite spin-orbit coupling.
These are adiabatilcally connected to topological band insulators: small bandwidth and infinite spin-orbit.
Question: Could a Hopf term be useful? Possibly.
Question: Why are all topological insulators cubic and does this impact your discussion? Not sure and the questioner does not know either.
Question: Your analysis looks like 2D - aren't the supposed to be 3D creating surface states? Sure - layers will create it.
Question: Why called Dirac point? Answer: the dispersion looks like a Dirac cone.
Question: Why do we need the notion of entanglement entropy? It provides a way of distinguishing between charge density wavesand topological insulators.
Question: Can all of this physics be deduced from the Green function? No you might need a 4 particle correlation.
Question: Why do you get band narrowing from spin orbit? It is nothing to do with spin orbit. Instead it is simply the usual Kondo lattice phenomenon.
Question: Where is this in the periodic table of the topological insulators? Yes - it is A1.
Serguei Borisenko: Fermiology and Order Parameter of Fe Based Superconductors
SVBorisenko(Institut fuer Festkorper Physik, Dresden, Germany)
Fermiology and Order Parameter of Fe based superconductors (FeSC)
–
Blogged by Saurabh Maiti
Blogger’s notation-
SV Borisenko (SVB)
SV Borisenko (SVB)
Superconductor (SC)
Anti Ferro Magnetism (AFM)
Spin Density Wave (SDW)
SVB starts by advertising 1-cubed ARPES that can reach to
temperatures below 1K. He explains how low energy excitations can be angle
resolved by rotating the sample. ARPES can be used to probe-
(*)Fermiology— get information abt:Fermi Surface (FS), band
structure, reconstruction due to (e.g magnetic) ordering
(*)Self Energy- Get information abt: V_F renormalization,
scattering, coupling const. (these last two are related to imaginary and real
parts of the self energy respectively)
(*)Order Parameter (mostly talks abt extracting SC order
parameter)
SVB mentions this was very useful for the cuprates in all
the three above mentioned aspects [although Self energy study was a little
involved because the bosonic mode was likely to be of electronic origin]. For
the cuprate case it was clearly established that there was no 3D and the gap
was d-wave in character with clear nodes – max gaps of 20-30mev.
SVB mentions then talks abt how comparing ARPES band
structure and LDA band structure can be
used to measure correlations, velocity renormalizations…
Having mentioned the success in Cuprates, SVB now moves on
to FeSC-specifically the pnictides. He reminds us that the photoemission data
is a product of
<f|p.A|i> A(k,e)f(e) X R(k,E)---the last term is
resolution
Explains step by step how the matrix elements, polarization,
resolution convolutions are taken care of and finally we get A(k,e)*f(e) and
then final division by the Fermi-function gives the electronic spectral
function which is used to extract info abt self energy and gap structure.
SVB then talks about LiFeAs-mentions presence of one large
hole pocket and two crossed elliptical electron pockets at (\pi,\pi) in the two
iron unit cell Brilliouin zone. But what happens very close to \Gamma point its
more involved [different results for different polarizations---resolved by
scanning across kz---he concludes that in a particular region of kz there is
hole pocket… kind of like cigar shaped. Points out the 3D nature of this
material.]
SVB now talks abt Co-NaFeAs shows ARPES results- combines
results from different polarizations and concludes presence of two crossed elliptical
electron pockets and 3D small hole pocket at \Gamma point.
SVB now moves to K-BaFeAs
Three hole pockets at \Gamma point—for optimally doped
material
But !! Result!! At corner there are 4 hole barrels [blogger’s note- star shaped]
with an electron band crossing in the centre.
KFeAs
Same as above but no central electron
band crossing; So three hole pockets at
\Gamma and 4 hole barrels at the corner —
Co-BaFeAs
Two crossed elliptical electron
pockets
One clear hole pocket, but two
other bands possibly cross the FS. Its complicated due to hybridization of xz /yz.
FeSe-
Very tiny two elliptical electron
pocket
And very tiny hole pocket at \Gamma
point of xz/yz character
Rb-FeSe
Rb expels iron and causes vacancy
ordering but will not discuss this.
Want to discuss the metallic
behavior of RbFe2Se2
Tiny hole pocket and two large
elliptical electron pockets
In this case the disordered
vacancies give metallic regions.
Two electron pockets at corner and
3D ELECRON pocket at gamma point (does not cross for all Kz values)
MAIN MESSAGE is that the standard
picture of 2 circular hole pockets and two circular electron pockets is never
experimentally realized.
Also points out that conventional
mapping of FS topology to phase diagram is not realized experimentally…[insert
picture].
SVB now discusses probing order
parameter.
Start with FeSe (T_c=8K)—remind
yourself that it has tiny electron pockets and tiny hole pocket. Then moves on
to K-BaFeAs
Message- the gaps are strongly
orbital dependent and kz dependent.
Points out that---
-xy band is irrelevant (gaps are
small and some times give large FS, sometimes small, etc)
-xz/yz are the important bands
-Wherever the xy content is present
the gap has a minima---
-Absence of xy character—large gap
SVB returns to his favorite
material ‘LiFe As’ to discuss its order
parameter—The important results is the anisotropy of the SC gap which was
claimed to be isotropic before. [blogger’s note: The oscillations are conts+cos4(\theta)
type as expected from the A1g symmetry]
Electron pockets—also anisotropic.
The tiny hole pockets; cannot probe
for anisotropy. [but gap is large]
[blogger’s note—the pocket is of
xz/yz character; the outer FS is of xy character and the gap structure
anisotropic and weaker]
SVB believes that it is likely to
be s++ gap. He tried the fitting with usual form factors for the s+- wave gap
structure like cosKx + cosKy and cosKx*cosKy. The only consistency he finds is
with the gap structure predicted by Kontani et al. which describes the in-phase
oscillations of the electron pockets. And since this required phonon mechanism
in the theory, it might well be that electron-phonon mechanism is likely in
this material.
Nevertheless, SVB points out that
it can also be explained by S+- picture (S. Maiti et al. PRB 85, 014511
(2012)).
SVB thus invites the theorists to
investigate this material more thoroughly because the self energy is known,
band structure is known orbital characters are known, gaps are known; and this
should be enough to find the right theory for this material.
SVB stresses on the fact that
\Gamma point has cigar like features of the hole/electron pocket and suggests
it be taken into account instead of simply
considering cylindrical FS. Simply put 3D is very likely important for SC. Life
becomes hard but needs to be dealt with.---
Thanks collaborators and restates
conclusions—
Q. Did you observe gaps induced by
SDW gap in your?
A. Yes. It is clearly seen. But
involves a lot of work.
Q. Are there data below Tc which
show change electronic spectrum above and below Tc
A. Yes. No problem. Can be provided.
Q. Can your Spectroscopy give
insight into the local quantum chemistry
of these materials?
A. Ok. Periodic potential gives
good qualitative picture.
Local picture needs special
attention and seems to be theorist dependent that is why I am not using that
picture. Band structure picture seems to have some universal results that most people
are getting. If I get a more or less universal local picture I don’t mind using
it.
Q. 11 sample FS looks like 122 (FeAs based); why? Even
though the latter is heavily electron doped.
A. They are not really same. FeSe
has a HOLE like also the pockets are
very Tiny. Doping with electrons makes electron pockets bigger.
Q. Is the phase diagram applicable
to all pnictides.? What abt LiFeAs? No AFM… where does it fall?
A. The phase diagram is really for
122. 111 is different.
Q. Laser v/s your method; comparison?
A. Prof. Shin has an excellent
technique. We have the same resolution as there’s.
Disadvantage of Laser is cannot
scan kz and matrix elements can play crucial role in those measurements.
Q. In Li-111 what are the
bands/orbitals responsible for SC?
A. xz and yz character of the inner
pocket, I think is most relevant.
Q. Is there any criticality
associated with change in FS topology?
A. It is interesting question—the
main problem is that we don’t have data for continuous doping to probe that
feature.
Time Reversal Symmetry Breaking & Charge Ordering in Pseudogap Phase of HTSC
Time Reversal Symmetry Breaking & Charge Ordering in Pseudogap Phase of HTSC
Aharon Khapitulnik, Stanford University
Good morning everybody, the second talk of the morning is by Aharon on time reversal symmetry breaking and charge ordering in the pseudo gap phase of HTSC.
Aharon starts with an intro on Time reversal symmetry, T = i \sigma^y K (K is complex conjugation)
Magneto-optics and T, Kerr effect.
By axial symmetry, the index of refraction for left and right light is related by:
\epsilon_{r,l} = 1 + 4 pi i \sigma_{r,l} \omega^-1
To measure the Faraday effect, they use a Sagnac loop with 2 quarter waveplates to select the circular polarization.
The new feature is the use of interferometry to detect broken rotational symmetry.
Pseudogap Phase in HTSC:
Theory 1. T* represents a cross-over into a state with pre-formed pairs and a d-wave symmetry.
Theory 2. T* marks a true phase transition into a phase with broken symmetry that ends at a QCP, usually within the SC dome.
Recent neutron scattering, Kerr effect and ARPES all show a broken symmetry phase at T*, and the possibilities include a structural phase, electronic nematic, smectic, stripe phase, DDW, etc.
Kerr effect measurement data on YBCO (detwinned):
1. First they cool in high field 5T, and then measure Kerr while warming up at ZF.
2. Kerr effect is seen above T_c, and disappears completely at T_s.
3. The response below T_c is due to trapped vortices, and cooling in ZF eliminates the vortices.
4. Note that the Kerr effect seen below T_c, within the SC phase as well!
Additional experiments on YBCO also signatures of TRSB, including neutron scattering measurements to detect signatures of the Varma loop current state; and resonant ultrasound spectroscopy that measure compression and shear modes, and they see a transition at 280 K and 200 K, which correspond to neutron scattering and Kerr respectively.
Resonant soft X-ray scattering shows 2D charge fluctuations with incommensurate periodicity, and high energy X-ray shows a co-exiting CDW and SC phase in YBCO. This indicates that there is a structural-type transition at a higher temperature with evidence of a short-range CDW state, and then TRSB at a lower temperature T*.
TRSB Data in LSCO:
There is a 1st order transition from LTO to LTT phase measured using birefringence, and when Aharon measures the Kerr effect, he finds a large signal at the LTT phase transition, which peaks at the spin-orbit temperature, and then levels off below T_c, and remains broken in the SC state!
Moreover, there is no training effect for the Kerr, meaning it does not change sign even when we flip the B field. One possibility is that there is TRSB even at high temperatures, but Aharon will discuss another possibility later.
The zero-field Nernst effect also shows TRSB at the charge-ordering temp, T_co = 54 K, and the Nernst effect is seen also in the SC state, i.e. TRSB in SC state! Ong's group tried to change the sign by heating up to 290 K and then cooling in an opposite B field, but the sign of the Nernst signal remains the same!
ARPES data in Bi2201 also shows evidence of TRSB at T*. The energy position of k_F is the measure at which the pseudogap becomes nonzero below T*, and this agrees with the T-dependent data from Kerr effect.
Aharon has also measured the Kerr effect in HgBa2CuO4, and TRSB is also seen in this system.
Aharon predicts that below T*, a weak charge ordering will be found!
Key Observations:
1. The Kerr effect occurs at the charge-order transition T_co.
2. Possibility of TRSB is being broken at higher temperatures, due to a pre-existing magnetic phase that changes its coupling to the off-diagonal conductivity at T*, and changes due to cystal symmetry changing at T*.
3. Another possibility is that HTSC are magnetoelectric below T*, and below T_co, symmetry is further lowered to acquire AHE effect, and this allows for a finite Nernst effect and Kerr signal.
Free energy, F_{ME} = \alpha_{ij} E_i H_j
4. Comments on Varma Loop-Current state with net moment = 0 in unit cell.
Type 1 Varma state: In-plane loops that breaks T and I(Inversion), but not TI.
Type II Loop current state: Breaks T and C(Chirality), and gives rise to AHE.
Mixed State: Breaks T and Chirality (AHE).
Aharon suggests a Possible Scenario:
At higher T*, there is a Varma loop state, and at T_co a Type II state occurs, giving rise to a mixed state below T_co and thus gives rise to a finite Kerr effect.
Another Scenario: Start with Type I at T*, and at T_co there is canting of the moments that gives rise to a Kerr effect.
Consequence of the magnetoelectric state: This means that there should be no sign of Kerr effect alignment with perpendicular B field, which is what is seen and also in the Nernst effect measurement. Furthermore, there should be the same sign on opposite side of sample, which is also seen in Kerr effect.
Qn: Pierls transition can be measured using derivative of resistance, d \rho/ dT. Was this measured?
Ans: In Bi2201, yes this was measured and found at the same T* as Kerr effect.
Qn: Is there a possibiltity of a QCP within SC dome?
Ans: Yes, there is a Kerr effect below the SC dome and also evidence from resonant ultrasound sepctroscopy, indicating that this should terminate at a QCP.
Qn: What is the wavelength used in the Kerr effect, and do they have data for different wavelengths?
Ans: Wavelength is 1.55 microns, and they have a new system at 830 nm. But they do not have wavelength-dependent measurements.
Qn: Does the community have a consensus that fluctutations due to TRSB and current loops are the cause of linear-T resistivity?
Ans; No consensus in community on this point yet.
We thank the speaker for a very interesting talk!
Aharon Khapitulnik, Stanford University
Good morning everybody, the second talk of the morning is by Aharon on time reversal symmetry breaking and charge ordering in the pseudo gap phase of HTSC.
Aharon starts with an intro on Time reversal symmetry, T = i \sigma^y K (K is complex conjugation)
Magneto-optics and T, Kerr effect.
By axial symmetry, the index of refraction for left and right light is related by:
\epsilon_{r,l} = 1 + 4 pi i \sigma_{r,l} \omega^-1
To measure the Faraday effect, they use a Sagnac loop with 2 quarter waveplates to select the circular polarization.
The new feature is the use of interferometry to detect broken rotational symmetry.
Pseudogap Phase in HTSC:
Theory 1. T* represents a cross-over into a state with pre-formed pairs and a d-wave symmetry.
Theory 2. T* marks a true phase transition into a phase with broken symmetry that ends at a QCP, usually within the SC dome.
Recent neutron scattering, Kerr effect and ARPES all show a broken symmetry phase at T*, and the possibilities include a structural phase, electronic nematic, smectic, stripe phase, DDW, etc.
Kerr effect measurement data on YBCO (detwinned):
1. First they cool in high field 5T, and then measure Kerr while warming up at ZF.
2. Kerr effect is seen above T_c, and disappears completely at T_s.
3. The response below T_c is due to trapped vortices, and cooling in ZF eliminates the vortices.
4. Note that the Kerr effect seen below T_c, within the SC phase as well!
Additional experiments on YBCO also signatures of TRSB, including neutron scattering measurements to detect signatures of the Varma loop current state; and resonant ultrasound spectroscopy that measure compression and shear modes, and they see a transition at 280 K and 200 K, which correspond to neutron scattering and Kerr respectively.
Resonant soft X-ray scattering shows 2D charge fluctuations with incommensurate periodicity, and high energy X-ray shows a co-exiting CDW and SC phase in YBCO. This indicates that there is a structural-type transition at a higher temperature with evidence of a short-range CDW state, and then TRSB at a lower temperature T*.
TRSB Data in LSCO:
There is a 1st order transition from LTO to LTT phase measured using birefringence, and when Aharon measures the Kerr effect, he finds a large signal at the LTT phase transition, which peaks at the spin-orbit temperature, and then levels off below T_c, and remains broken in the SC state!
Moreover, there is no training effect for the Kerr, meaning it does not change sign even when we flip the B field. One possibility is that there is TRSB even at high temperatures, but Aharon will discuss another possibility later.
The zero-field Nernst effect also shows TRSB at the charge-ordering temp, T_co = 54 K, and the Nernst effect is seen also in the SC state, i.e. TRSB in SC state! Ong's group tried to change the sign by heating up to 290 K and then cooling in an opposite B field, but the sign of the Nernst signal remains the same!
ARPES data in Bi2201 also shows evidence of TRSB at T*. The energy position of k_F is the measure at which the pseudogap becomes nonzero below T*, and this agrees with the T-dependent data from Kerr effect.
Aharon has also measured the Kerr effect in HgBa2CuO4, and TRSB is also seen in this system.
Aharon predicts that below T*, a weak charge ordering will be found!
Key Observations:
1. The Kerr effect occurs at the charge-order transition T_co.
2. Possibility of TRSB is being broken at higher temperatures, due to a pre-existing magnetic phase that changes its coupling to the off-diagonal conductivity at T*, and changes due to cystal symmetry changing at T*.
3. Another possibility is that HTSC are magnetoelectric below T*, and below T_co, symmetry is further lowered to acquire AHE effect, and this allows for a finite Nernst effect and Kerr signal.
Free energy, F_{ME} = \alpha_{ij} E_i H_j
4. Comments on Varma Loop-Current state with net moment = 0 in unit cell.
Type 1 Varma state: In-plane loops that breaks T and I(Inversion), but not TI.
Type II Loop current state: Breaks T and C(Chirality), and gives rise to AHE.
Mixed State: Breaks T and Chirality (AHE).
Aharon suggests a Possible Scenario:
At higher T*, there is a Varma loop state, and at T_co a Type II state occurs, giving rise to a mixed state below T_co and thus gives rise to a finite Kerr effect.
Another Scenario: Start with Type I at T*, and at T_co there is canting of the moments that gives rise to a Kerr effect.
Consequence of the magnetoelectric state: This means that there should be no sign of Kerr effect alignment with perpendicular B field, which is what is seen and also in the Nernst effect measurement. Furthermore, there should be the same sign on opposite side of sample, which is also seen in Kerr effect.
Qn: Pierls transition can be measured using derivative of resistance, d \rho/ dT. Was this measured?
Ans: In Bi2201, yes this was measured and found at the same T* as Kerr effect.
Qn: Is there a possibiltity of a QCP within SC dome?
Ans: Yes, there is a Kerr effect below the SC dome and also evidence from resonant ultrasound sepctroscopy, indicating that this should terminate at a QCP.
Qn: What is the wavelength used in the Kerr effect, and do they have data for different wavelengths?
Ans: Wavelength is 1.55 microns, and they have a new system at 830 nm. But they do not have wavelength-dependent measurements.
Qn: Does the community have a consensus that fluctutations due to TRSB and current loops are the cause of linear-T resistivity?
Ans; No consensus in community on this point yet.
We thank the speaker for a very interesting talk!
Satoru Nakatsuji (ISSP Tokyo)
"Unconventional quantum criticality, anomalous metal with strong valence/orbital fluctuations"
Blogged by Maxim Dzero
One of the greatest challenges to Landau's Fermi liquid theory - the standard theory of metals - is presented by complex materials with strong electronic correlations. In these materials, non-Fermi liquid transport and thermodynamic properties are often explained by the presence of a continuous quantum phase transition which happens at a quantum critical point (QCP). QCP can be revealed by applying pressure, magnetic field, or changing the chemical composition. In the first part of his inspiring talk, Prof. Nakatsuji has discussed manifestation of quantum criticality in the f-orbital material beta-YbAlB4. Starting with the review of the data on the magnetization and resistivity measurements, he showed that beta-YbAlB4 displays an anomalous (non-Fermi liquid) behavior at low field and temperatures. Increase in magnetic field fully recovers conventional (metallic) Fermi-liquid properties. Prof. Nakatsuji discussed how the data analysis unambiguously points to the presence of the QCP at very small (of the order of 1G) magnetic field. This results show that beta-YbAlB4 displays quantum criticality without an apparent QCP suggesting the non-Fermi liquid metal is a phase in itself rather than is caused by system’s proximity to QCP.
To get further insight into the physics of that system, the results of the hydrostatic and chemical pressure studies have been presented. As it turns out, pressure induces the phase transition in magnetically ordered state, while the quantum criticality remains intact. Therefore, the results of the pressure studies strongly suggest that the microscopic mechanisms, which govern quantum criticality in this material, are different from those responsible for the onset of magnetic order. In addition, measurements of the Seebeck and Zommerfeld coefficients show that quantum criticality anomalous transport and thermodynamic properties are likely governed by the quasi-particle excitations from the same heavy-Fermi surface. To explain the anomalous features in the Hall constant strongly suggest that one needs to invoke the excitations from another Fermi surface. In addition, there is an indication that there may be a line of nodes in the hybridization gap[. These results triggered the questions from the audience as to what extend the existence of the “quantum critical” Fermi surface can be reconciled with the absence of the hybridization for the specific values of momenta in the Brillouin zone. Another intriguing question is whether it would be possible to identify the Fermi surface, which develops the leading superconducting instability.
In the second part of the talk, Prof. Nakatsuji has discussed the manifestation of an interplay between conduction and orbital degrees of freedom which under certain circumstances may lead to a phenomenon of the orbital (or quadrupolar) Kondo effect. The candidate materials, which may display this physics, are PrTr2Al20 with Tr=V, Ti. Prof. Nakatsuji has presented a set of thermodynamic and transport data consistent with the idea of quadrupolar ordering in PrTi2Al20. On the contrary, PrV2Al20 shows the dominant role of the Kondo screening processes.
I think, the presented results mark novel and important developments in the studies of f-orbital materials.
Wednesday, August 15th
Tom Timusk, "The normal state of URu2Si2: spectroscopic evidence for an anomalous Fermi liquid"
Blogged by Andrey Chubukov
Tom Timusk, "The normal state of URu2Si2: spectroscopic evidence for an anomalous Fermi liquid"
Blogged by Andrey Chubukov
Tom starts with the
summary of 1988 view of URu2Si2 (the time of first ICTP workshop on correlated
electrons)
He discusses normal
state properties of URuSi2: mass enhancement, scattering rate, Drude peak.
Scattering rate has a peak
at 20 meV. There is a coherent
Drude peak at 20K, it becomes incoherent
avove 20K. The conclusion was that at
low T/\omega the system is a Fermi liquid with m^* =25m
In 1988 the community believed that hidden order below 18K
is spin-density-wave (SDW). Superconductivity (SC) emerges below 1K. Tom presented arguments for SDW
which sounded reasonable back in 1988 but later were found to be in
disagreement with the measurements.
20 years have passed (but ICTP workshop is still running!)
He cites STM data by S. Davis group, which show that at 18.6K (right above transition into hidden order phase) the system shows normal metallic dispersion with m^* =3m. At
5.9K the gap develops due to hybridization. This gap was not detected in earlier opt
conductivity data because of too much
noise .
He next describes new method – refined thermal reflectance,
which allows one to change T without
moving the sample. He divides all the spectra by reflectance at 25K and argues
that accuracy increases substantially (noise is reduced by 8). All T-independent features in conductivity are lost,
all T-dependent features stay.
Tom shows data for \sigma (w) down from 75K. At T decreases, Drude peak forms. No change in the total spectral weight at 15 meV -> total Drude weight is independent on T (m^*
remains the same). Conclusion – heavy mass is already present at 75K.
Second result – 1/tau
is smaller that \omega up to 30K .
The implication is that a coherent Fermi liquid forms at 30K. He shows
plot of \pho (w,T) [=Re 1/\sigma(w,T)].
\pho (w, T) follows A
(w^2 + b (pi^2 T^2))
Discussion on the value of b follows. In a Fermi liquid, b should be =4. He finds, from high-frequency part, \rho (w,T) = A T^2,
A = 0.3 \muOm* cm/K^2 . He next looks at
DC resistivity, takes derivative with
respect to T , gets the same A.
Shows Matsuda data in a wider range of T d^2 \pho/dT^2 is almost a const up to 100K (hence T^2 form
works).
Discusses the difference between single fermion lifetime (m \Sigma (w,T) \propto (w^2 + \pi^2 T^2) and
optical lifetime (1/tau \propto w^2 + b
\pi^2 T^2) with b=4. He cites many examples of T^2 resistivity with various A
Examples of b=4? There are none
UPt3 b<1
Nd0.9TiO3 b =1.1 Ce0.95Ca0.05TiO3 b=1.7
Tom discusses theory proposal from two Russian gringo about resonance
impurity scattering. If Im \Sigma has no
T^2 term then optical conductivity has
w^2 + b \pi^2 T^2 form with b=1
He argues that resonant impurities are un-hybridized uranium f-electrons.
Last part – hidden order phase
Tom shows data fitted
by Dynes formula for a dirty SC.
He finds one gap along ab axis and
two gaps along c axis (3.1 meV and 2.7 and 1.8 meV) .
Roughly the same gaps have been found from the resistivity fit and from
neutrons.
Questions: (i) about w/T
range and about direct vs indirect gap, (ii) is the two-component electronic model valid in
the hidden order phase,(iii) is b=1 vs b=4 related to vertex corrections?
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