Many Electrons Strongly Avoiding Each Other:Trying to Make Sense of the Strange Goings On

T V RamakrishnanBanaras 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 TF

0 ~ 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 = σop

zi

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) 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)

ii) There are no good momentum quasiparticles iii) 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

iv) 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

v) 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)

vi) 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- Liτ

2 + 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 yetAre 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