Download t_v_ramakrishnan

Many Electrons Strongly Avoiding Each Other:
Trying to Make Sense of the Strange Goings On
T V Ramakrishnan
Banaras Hindu University, Varanasi
Indian Institute of Science, Bengaluru
( Also, National Centre for Biological
Sciences,Bengaluru)
Thanks to colleagues and friends, especially
Debanand Sa, BHU, Varanasi, India
• A part of the immediate past of
the physics of condensed matter, and
the subject of perhaps the most intense and
concerted effort both experimental and theoretical
of a generation ( ~ 1985 – 2005 ) of physicists in the
field. Has brought out many strange phenomena,
perhaps even more theories and strong views. One
does not have a working, comprehensive picture
yet. ( Some believe that the systems are so complex
that nothing like that is possible; possibly the
problem will be shelved because it cannot be
solved)
Maybe we are prisoners of a wildly successful past
Examples of systems in which similar strange effects have been observed:
Cu2+ has unfilled shells with configuration 3d9 4(s,p)1
La2 CuO4 ; La2-x Sr x Cu O4 (lanthanum cuprate, ‘hole’ doped lanthanum cuprate)
(High temperature superconductor : Bednorz and Muller, 1986 .
Qualitatively unlike any metal known earlier, ~ 105 papers and two decades later.
The problem routinely figures as one of the major unsolved physics questions)
Ce3+ has unfilled shells with configuration 4f1 5(s,p)2
CeRhIn5 ( ‘Heavy’ fermion metal, meff ~ 103 me ;
also a Neel antiferromagnet and a superconductor at very low T ~ 2K)
The unfilled d and f orbital states are supposed to be the home of their strangeness
But also, many organics eg (BEDT-TTF)TCNQ (has metal-insulator transition at 330 K)
{ bis(ethylenedithio) tetrathiofulvalene tetracyanoquinodimethane }
Kx C60 (potassium doped fullerene; not quite organic, but a superconductor at 18K)
(Both of these also have only s,p electrons in unfilled shells)
And perhaps many many other systems waiting to be recognized
What are the strange goings on?
(Strangeness depends on one’s expectations)
So, what do we expect?
Soon after electrons were discovered to be the common constituents of
all matter
( eg JJ Thomson,1897)
P Drude (1900) made the bold assumption that in a solid
all outer electrons can be regarded as moving independently of each other
( Atoms are close to each other so that an electron
experiences strong forces from nearby atoms, and consequently
each electron (in an unfilled shell) breaks out
of its parental moorings, roaming freely throughout the solid. Further,
the force due to the other electrons and ions cancels each other )
This is the free electron gas model of the solid ( in the image of… )
It works! ( surprisingly often)
eg all semiconductor devices use this idea (think cell phones!). Many
very sophisticated versions of this have been developed and used
( eg Wilson theory of semiconductors, insulators and metals
Landau theory of a Fermi liquid ; low energy electronic excitations of an interacting
fluid are like a ‘gas’ )
• Free electron ( fermion ) gas
• All the lowest energy states occupied
(one by one; Pauli exclusion principle)
till the Fermi energy εF
• Quantum scale εF = kBTF
• (this is peculiar to electrons, and is not associable with a
classical gas)
• TF ~ 104 K (for typical solids) >> typical
temperatures
• Electronically, for T << TF , one
is in the extreme quantum regime
• Physical properties Universal !
• Simple power laws of (T/TF), eg specific
heat Cv ~ kB (T/TF)1
Qualitatively new effects if U >> t
eg lattice of hydrogen atoms (lattice constant a)
metallic for a<<ao ( U << t ) ( half-filled band)
insulating for a>>a0 (U >> t )
Mott insulator
(each electron stays at home ; one per lattice site)
What if this is not the case? ( eg there is less or
more than one electron per lattice site)
New kind of metal
Many electrons move from lattice site to lattice
site, strongly avoiding certain sites. What is the
low energy quantum dynamics of such
strongly correlated electrons?
I. Some features common (?) to strongly correlated
metallic electron systems
i) Electron ‘quasiparticles’ either faint or nonexistent ( eg ARPES)
ii) ‘Bad’ metals , ie invariably have large electrical resistivities
ρ(T) ~ Tα for T<T* and Tβ for T>T*
where α ≤ 2 and 0≤β ≤1;
(T* is very small, ie ~ 300K, much smaller than TF0 ~ 5x104 K)
(‘non Fermi liquid’ if α < 2; resistivity saturation if β~0)
Often ρ ~ or > the Mott (~Ioffe-Regel) maximum
[electron mean free path l ~ electron wavelength (2πkF-1)]
iii) Optical conductivity σ(ω) anomalous: small low frequency or Drude
peak ( expected to be large in a metal) which rapidly disappears as T
rises beyond T*, but a large nearly flat part extending over a broad
range in frequency ω
iv) Large thermopower, ‘saturating’ to a classical value above T~T*
II. So far, our thinking about SCES has been based on
the Drude model with less or more radical ‘add-ons’
( new fields, eg auxiliary bosons/fermions
nonperturbative single site solutions eg DMFT… )
Do we need a new paradigm? What is it?
In this unfashionable but presumably fundamental direction,
we have started on a new route; perhaps we will end up with a
new paradigm
Two stage theory:
Stage I : High temperature or symmetric state of all strongly
correlated matter, above a quantum coherence temperature T*
Is this the new paradigm different from Drude ? Yes, I think so
Properties can be obtained exactly ( TVR and D Sa, in preparation)
Stage II : Quantum fluctuations /coherence leading to Fermi
liquid/non Fermi liquid behaviour with or without broken symmetry
( including superconductivity, spin density wave etc…)
( TVR and D Sa; in the works for a simple well studied magnetic
impurity model, the Anderson/Kondo model;inspiration D Logan .. )
Background:
Electrons on lattice sites i moving from i to j with amplitude tij and an
onsite effective repulsion U
H = ∑ij,σ tij a+iσajσ + U ∑i ni↑ni↓
with average ni or ‹ni› = (1-x)
(Hubbard model, 1960’s)
(simplest model which describes the competition between
kinetic and potential energies t and U; the latter couples the motion of
an electron to that of other electrons )
(Misses chemical realities eg many orbitals and their anisotropy ;
physical realities eg long range coulomb interaction and coupling to
the vibrating lattice )
Standard approximations : Treat U
i) As a perturbation:
(the unperturbed eigenstates are φkσ = ∑i exp(ik.ri ) φiσ )
ii) As a ‘mean’ field ( average potential); the most successful is
Dynamical Mean Field Theory or DMFT
(Muller-Hartmann,Vollhardt, Georges and Kotliar, late 1980’s)
Since the strong interaction is local, embed a single
site or a small cluster in a ‘medium’. Solve the local
problem of electron(s) with any U in a mean field
due to others exactly . Find the (time or frequency
dependent) mean field self consistently.
( Treats all energy scales from U to the lowest, ~ 10-4 U
in a single approximation)
‘Best’ method in use presently for all strongly
correlated electron systems
We obtain the strong correlation features in stage I;
stage II is for scales T*<<U
III. Outline of our present approach:
Use exp -( U niτ↑ niτ↓) = exp-{(U/4) (niτ 2 – sizτ 2)}
= exp- {(U/4) (niτ2 – (σi.Ωiτ )2)}
(the last is an identity for spin (1/2; Ωiτ is an arbitrary direction )
Do a Hubbard Stratonovich transformation
( a form of exp(x2) = (π)-(1/2)∫ exp(-a2 + 2ax) da for operators )
Converts the problem of a large number of electrons locally
interacting with each other to that of electrons moving in time
dependent fields
namely potential viτ and magnetic field miτΩiτ of size miτ and
direction Ωiτ with Gaussian weight for viτ and miτ
The approach is useful if the fields vary slowly with ‘time’ τ
Formally:
The direction of Ωiτ in general is given by a SU(2) matrix
The charge potential viτ is given by a U(1) term viτ = vi exp( iθiτ)
SU(2)xU(1) lattice gauge field theory
(of coupled chargeless spinon Fermi fields (fiτ) and SU(2)xU(1)
gauge Bose fields , with a Gaussian weight for their size )
Rotate axis of spin quantization such that it points along the z direction
at each site
Riτ+σopiRiτ = σopzi
Rotation is generally site and time dependent
Rotated fermion operators c+I = a+i Riτ ( a+ iτ= f+iτ exp( iθiτ) )
Stage I : Ignore ‘time’ dependence of Ωiτ miτ and viτ
Stage II: Quantum Lattice Gauge Fields Aντ
where ∑νσν Aντ = Riτ∂ Riτ+
and ∂τ θiτ
(Treat effect in the Gaussian, harmonic or large N approx.)
Some results in the ‘static’ or Stage I approximation:
i)
ii)
iii)
iv)
v)
vi)
Can obtain physical properties ( eg single particle Green’s
function) in an approximation which is exact in d= ∞, but
not bad for d=2 or 3 ( eg DMFT experience)
There are no good momentum quasiparticles
The system is a disordered quantum paramagnetic metal
with local magnetic moments at each site, pointing in
random directions at different sites. There are lower and
upper Hubbard bands
The electron mean free path is very short ; the resistivity
is near the Mott limit. It has a x-1 prefactor for small x as
observed
The exactly calculable single particle Green’s function
does not have simple poles (like in quasiparticle models)
but a cut ( actually a square root discontinuity across the
real axis)
All physical properties (eg thermopower, σ(ω) ) calculable
• Stage II : ( see eg D Sa, next talk) Illustration:
Anderson model of magnetic impurity in a metal
Suppose at just one site in a metal, there is an ‘f ’ electron
with energy εf with respect to chemical potential μ,
hybridization Vfk with conduction electron state k ,
local U ( Unf↑nf↓)
Because of the hybridization, the f electron decays ( its energy
is no longer sharp, but broadens) with an energy width
Δ= π| Vfk |2ρ(μ) . Anderson showed ( 1960!) that
for large U, ie U > π Δ, there is a local magnetic moment (!)
The magnetic moment disappears smoothly below a
characteristic Kondo temperature TK ~ U exp(- πaΔ/U)
( > a decade of experiment and theory, 1965 onwards)
Our stage I is like the local moment solution; the idea is that
in stage II, quantum fluctuations which connect the two
strictly degenerate mean field states with +m↑ and –m↓,
lead to the Kondo like disappearance of the local moment as
temperature crosses over to below TK
In this example, TK= T* the quantum coherence temperature
We have analyzed the Anderson impurity model in our
SU(2)xU(1) theory, and find, in a harmonic approximation for
the fluctuations, that the above is true(?). At least for this
case then, it seems that the two stage picture can be
quantitatively and simply implemented.
What is T* ?
Single impurity Kondo temperature TK~ U exp(-a’U/t)
Doping induced moment decay ~ x t (effective hopping)
Polaronic narrowing (Huang-Rhys) scale Tp ~ t exp(-EJT/ħω0)
Is there a small parameter? Can it be (T*/t) ?
A generic two stage, two fluid theory ?
( a coherent quantum fluid emerging at low temperatures from an incoherent
quantum fluid )
The approach starts from high temperatures, in contrast to
QCP based thinking, which starts from low temperatures. Earlier experience ( eg with
interacting quantum spin models like the Heisenberg model ) seems to show that
one can understand and do much more from the former direction.
I have not talked about many things, eg
• Would a NonLinear σ Model (NLσM) description be more natural?
(ie Ωiτ = niτ √1- Liτ2 + Liτ , niτ and Liτ being perpendicular)
• Multi orbital, electron lattice coupling effects
• For small U (certainly for U=0) coherent superposition of onsite
states is either an eigenstate or a good starting point; bandwidth ~zt.
For large U, bandwidth ~ t . Crossover?
• In the static approximation, electrons move in a static random site
potential with random hopping amplitudes. Yet we use extended
states (CPA, exact at d=∞) as if there is no Anderson localization. We
concentrate on the mean; for localization, fluctuations from the
mean are crucial. Do the latter go as (1/d) for large d? Is localization a
(1/d) effect for large d? Are the localized states delocalized by
quantum fluctuations and is this why we do what we do (CPA)?
• What is our ‘Luttinger Ward’ functional for the free energy of the
coupled spinon/ gauge boson fields? ( My guess: it is like the
Eliashberg approximation for the coupled electron phonon system)
Conclusion and Prospect:
Many independent electrons together have been experimentally and
theoretically explored for more than a hundred years
( Some of the unexpected things they do, eg effectively behaving as
massless Dirac fermions in graphene, or in topological insulators, are
very active current areas of research )
Many electrons strongly avoiding each other (strong correlations)
lead to strange behaviour explored increasingly in the last three decades
or so.
Great ferment : no new paradigms yet
Are we on the verge of a new paradigm?
Will it naturally lead to a two stage theory which may replace the one
stage DMFT?
* Logjam in condensed matter physics - Anderson
* Where the clear stream of reason has not
lost its way in the dreary desert sands of dead habit - Tagore