Polarons in bulk and near surfaces

Mona Berciu

University of British Columbia

Acknowledgements: Glen Goodvin and Lucian Covaci

NSERC, CIfAR

Polaron: if an entity (electron, hole, exciton, …) interacts with bosons (phonons, magnons, electron-hole pairs, etc) from its environment and becomes “dressed” by a cloud of such excitations, the composite object is a polaron.

Today: I will only discuss cases with a single polaron in the system (avoids complications regarding polaron-polaron interactions, etc, although of course those are very interesting, too) à think of very weakly doped insulators. Also, all results are for T=0.

Plan: -- review of Green’s functions (what we want to calculate)

-- a few simple examples that can be solved exactly

-- discussion of the Holstein polaron in bulk

-- what happens to the Holstein polaron near a surface

Quantity of interest: the Green’s function or propagator

Notes:

àthere is a whole “zoo” of propagators or Green’s functions – causal, retarded, advanced, lesser, greater, one-particle, two-particle, many-particle ….

àfor a non-interacting system, any of these contains full information about the system

àfor systems with interactions, each of these contains additional information. Which to calculate depends on what information we are trying to extract.

àfor single-polaron problems and the type of questions I will address today, the one-particle retarded propagator suffices, and from now on this is what I mean by “propagator”.

General definition of a retarded propagator:

i.e., the amplitude of probability that if the system is in ψ i at τ=0, it will be found in ψ f at τ>0.

This meaning is very important to help understand diagrammatics!

( ) ( )i H

i f f iG i eτ

τ τ ψ ψ−

→ = − Θ h

We can choose any ψ i and ψ f, depending on our needs. Eg., for a clean system it is usually

convenient to take à the propagator then gives the amplitude of probability that if initially we started with just the particle with momentum k, we also end up only with the particle at time τ (no boson excitations present).

With this choice + Fourier transform, we then get:

For a system with surfaces, using states of well-defined momentum is not sensible – we will see later on what makes more sense in those cases. Before that, let’s see what information we can get out of this propagator.

1ˆ ˆ( , ) 0 ( ) 0 , where ( ) , >0 and =1.k kG k c G c GH i

ω ω ω ηω η

+ =− +

@ h

ψi = ψ f = ck+ 0

( , ) ( ) 0 0i H

k kG k i c e cτ

τ τ− += − Θ h

1, ,1, , 1, ,kH k E kαα α= ß eigenenergies and eigenfunctions (1 electron, total momentum k, α is collection of other needed quantum numbers)

G(k,ω) @ 0 ck1

ω − H + iηck

+ 0 =Z1,k ,α

ω − E1,k ,α + iηα∑ Z1,k,α = 1,k,α ck

+ 02

A(k,ω)

ω1, ,kE α

η

Area is equal to Z

Z = quasiparticle weight à measures how similar the true wavefunction is to a non-interacting (free electron, no bosons) wavefunction

A(k,ω) @ − 1π

ImG(k ,ω) = Z1,k ,αδ ω − E1,k ,α( )α∑ ß = spectral weight, is measured

(inverse) angle-resolved photoemission spectroscopy (ARPES)

1. Clean (no disorder) N-site chain, tight-binding hopping, no interactions

Then, à because of symmetry, only the state of energy εk is “visible”.

I could also ask for à is related to

amplitude of probability to go from site n to site m.

Simple examples of Green’s functions

0 1 ,

12 cos( ) [ 2 ,2 ] and i

i i k k ki k

ikRk k i

i

H t c c hc c c

t k t t c e cN

ε

ε

+ ++

+ +

= − + =

= − ∈ − =

∑ ∑

∑

( )

0 01ˆ( , , ) 0 ( ) 0

i m n ka

m nkk

eG m n c G c

N iω ω

ω ε η

−+= =

− +∑

01( , )k

G ki

ωω ε η

=− +

Now the entire spectrum is visible. What these propagators look like depends on n-m and on dimensionality. For example, next page shows imaginary and real parts of propagators in 1D, 2D and 3D, when n-m=0 (particle doesn’t move from initial site) and for an infinite chain.

0 0 01 1 1

( ) ( , , ) ( , )kk k

g G n n G kN N i

ω ω ωω ε η

= = =− +∑ ∑

à 1D: both real and imaginary part are singular at band-edges

à 2D: imaginary part has jump discontinuity, real part has weak (logarithmic) singularity

à 3D: no singularities in either real or imaginary part, at band-edge.

2. Lattice with tight-binding hopping, with one impurity at site 0 (very simple type of disorder)

( )0 0 0,

i ji j

H H V t c c hc Uc c+ += + = − + −∑We can link this propagator to the previous one, to avoid calculating all the eigenfunctions.

The only thing needed is the Dyson’s identity

(ask me for proof in the break, if you’ve not seen this before. It’s very short).0 0

ˆ ˆ ˆ ˆ( ) ( ) ( ) ( )G G G VGω ω ω ω= +

0 0,

0 0

0 00

0

( , , ) ( , , ) ( , , ) ( , , )

( , , ) ( ,0, ) (0, , )( ,0, ) (0, , )

( , , ) ( , , )1 (0, 0, )

l p

G m n G m n G m l l V p G p n

G m n UG m G nG m G n

G m n G m n UUG

ω ω ω ω

ω ω ωω ω

ω ωω

= +

= −

→ = −+

∑

Answer looks the same in any lattice, just use proper G0.

There can be new poles (new eigenstates) at energies where the denominator vanishes.

0

0

1Re[ ( )]

Im[ ( )] 0

gU

g

ω

ω

= −

=

Because of local attraction à an eigenstate outside the band, which will mark an “impurity state” bound in the vicinity of site 0, provided that there is a real solution for the equation:

à In 1D, there is a strongly bound solution for any value of U

à In 2D, there is a weakly bound solution for any U

à In 3D, a bound solution appears only if U is large enough (U>4t, I think, for this model).

Note: if U <0 (repulsive), then we may get an “anti-bound” state, also localized near site 0.

-1/U

This solution was pure mathematics (applying Dyson’s identity blindly). Now for some intuition …Remember the meaning of the propagator – we need to sum over amplitudes of probabilities of all possible ways for the electron to go from site n to site m.

Let’s invent some diagrammatic notation:

= amp. of probability for electron to go n à m in clean system

= −U = scattering on impurity (only possible at site 0, in this simple example)

= amp. of probability to go from n à m when impurity present

Then:

= + + + ….

The new denominator comes from summing over an infinite geometric series à appearance of a bound state is a non-perturbative effect.

Mathematically, this comes from using iterations to rewrite Dyson’s identity as:

n m0( , , )G m n ω=

n m ( , , )G m n ω=

n m n m n 0 m n 0 0 m

0 0 0 0 0 0( , , ) ( , , ) ( ,0, )[ ] (0, , ) ( ,0, )[ ] (0,0, )[ ] (0, , )

= precisely the same answer

G m n G m n G m U G n G m U G U G nω ω ω ω ω ω ω= + − + − − +

0 0 0 0 0 0 0 0ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ...G G G VG G G VG G VG VGω ω ω ω ω ω ω ω ω ω= + = + + +

3. Semi-infinite chain – hopping only allowed between sites 0,1,2, …

Strategy: start with infinite chain (G0 already known) and cut it!

Just as before, we find for any positive n,m:

= + + +

where now represents reflection at the surface (cut)

It’s a good exercise for you to derive the diagrammatics rules; you should find:

Here we have a new denominator – it turns out that this NEVER vanishes, so there is no “surface state” for this type of cut.

( ) ( )0 1 0 1 i

c c i iall

H H V t c c hc t c c hc+ ++ −= + = − + + +∑

n m n m n m n m

0 00

0

( , 1, ) (0, , )( , , ) ( , , )

1 (0, 1, )c

G m G nG m n G m n t

tGω ω

ω ωω

−= +

− −

3b. Semi-infinite chain – hopping only allowed between sites 0,1,2, … but also a surface potential

There are various ways to solve this, and you should convince yourselves that all lead to the same final result. Simplest approach à reduce it to previously solved problems:

Gc is already known, and we know what happens when we add this V to it:

Surface states appear if U > t (?) when the surface attraction is sufficiently strong to bind a surface state.

Note: This is truly just 1 surface!

How about in higher dimensions, not just 1D chains?

( ) ( )0 1 0 1 0 0 i

s c i iall

H H V V t c c hc t c c hc Uc c+ + ++ −= + + = − + + + −∑

ˆ ˆ ˆ ˆ( ) ( ) ( ) ( )s c s c s cH H V G G G VGω ω ω ω= + → = +

( ,0, ) (0, , )( , , ) ( , , )

1 (0,0, )c c

s cc

G m G nG m n G m n U

UGω ω

ω ωω

= −+

3c. Surface in a higher dimensional lattice

à for simplicity: (100) like surface in simple cubic –type lattice.

àvery nice read: T.L. Einstein and J.R. Schrieffer, Phys. Rev. B 7, 3629 (1973)

àInvariance to translation in || direction à partial Fourier transform: : state where electron is in layer n and has in-plane momentum k ||. Then:

surface

n=0 layer

n=1 layer

…..

||, , ,n m p n kc c→

( )|| || || || || ||

||

, 1, || || , , 0, 0,0 0

( )s n k n k n k n k k kk n n

H t c c hc k c c Uc cε+ + ++

≥ ≥

= − + + − ∑ ∑ ∑

( )|| || || || || ||

||

, 1, || || , , 0, 0,0 0

( )s n k n k n k n k k kk n n

H t c c hc k c c Uc cε+ + ++

≥ ≥

= − + + −

∑ ∑ ∑

à for each value of k ||, an effective 1D problem precisely like what we solved, up to an overall shift of the energy by the transverse kinetic energy. Therefore:

à (100) surface also won’t bind a state unless surface potential U > t. If a surface state appears, its energy depends on k ||.

|| ||

3D 1D|| , , || ||

ˆ( , , , ) 0 ( ) 0 ( , , ( ))s m k s n k sG k m n c G c G m n kω ω ω ε+= = −

So far, we’ve only looked at some non-interacting problems. Let’s also see a simple example for a particle interacting with bosons, which can be solved exactly.

Note: for a different surface, one needs to redo calculations and see what happens. For example, a (111) surface will have a very different symmetry/behavior, and for example may bind a surface state even if a (100) surface would not.

add extra e à new equilibrium distance

h0 = P2

2M+

MΩ2( X − X0)2

2→ hΩ b+b + 1

2

b = MΩ2h X − X0 + i P

MΩ

b+ = MΩ2h X − X 0 − i P

MΩ

→ X − X0 ∝ (b+ +b)

0

0 0

ˆˆ

ˆh h gnX

X X nα

= +

→ −

new equilibrium length, determined by how many extra electrons there are

4: 1-site polaron, or the Frack-Condon problem (aka Lang Firsov, in polaron physics):

2† † † † †ˆ( ) ( )

gh b b gn b b b b g b b B B= Ω + + → Ω + + = Ω −

Ω

If we’re adding one electron

Side-note: coherent states are defined by:

b α = α α

→ α = e−

α2

2+αb†

0

The Green’s function is:2

22

20

1 1ˆ( ) 0 ( ) 0!

g n

LFn

gG cG c en gn i

ω ωω η

−+ Ω

≥

= = Ω − Ω + +Ω

∑

+

2 ,

2

2

, [ , ] [ , ] 1

Ground state is c where 0

In general, with !

In gs, , etc.

n

n

g gB b B b B B b b

ph B ph

g gb ph ph ph

g B gE n c

ng

b b

+ + + +

++

+

= + = + → = =Ω Ω

=

→ = − → = −Ω Ω

= Ω − −Ω Ω

=Ω

Of course, we can also do diagramatics – we’ll see the details in a bit, but basically the propagator for starting and ending just with the electron (no bosons) is the sum for only having the electron at all times, plus contribution from the electron emitting and then re-absorbing a phonon, plus contribution from electron emitting and then re-absorbing two phonons (different ordering possible, must sum over all possibilities), plus contribution from electron emitting and then re-absorbing 3 phonons, blah blah blah.

Plan: -- review of Green’s functions (what we want to calculate)

-- a few simple examples that can be solved exactly

-- discussion of the Holstein polaron in bulk

-- what happens to the Holstein polaron near a surface

Polaron (lattice polaron) = electron + lattice distortion (phonon cloud) surrounding it

à very old problem: Landau, 1933;

à most studied lattice model = Holstein model (not very realistic)

, ,

( ) ( )i j j i i i i i ii j i i

H c c c c b b n bt g bσ σ σ σσ

+ + + +

< >

= − + + + +Ω∑ ∑ ∑

3 energy scales: t, Ω, g à 2 dimensionless parameters λ = g2/(2dtΩ), Ω/t (d is lattice dimension)

Eigenstates are linear combinations of states with the electron at different sites, surrounded by a lattice distortion (cloud of phonons). Can have any number of phonons à problem cannot be solved exactly for arbitrary t, g, Ω.

t

polaron (small or large, depending) but always mobile, in absence of disorder

weak coupling Lang-Firsov impurity limit

( )( )

0

00 2 2

1( , ) ;

( , )

k

kk

G ki

A k η

ωω ε η

ηω δ ω ε

π ω ε η

→

=− +

= → − − +

2

ng

E n= − + ΩΩ

2

GSg

E = −Ω

How does the spectral weight evolve between these two very different looking limits?

2

0 ( 0)2

gg

dtλ = = =

Ω

2

( 0)2

gt

dtλ = = ∞ =

Ω

Ω

Asymptotic behavior:

à zero-coupling limit, g=0à eigenstates of given k:

with eigenenergies where, for example, εk , εk−q + Ω, εk−q−q' + 2Ω,....

ck+ 0 , ck−q

+ bq+ 0 , ck−q−q '

+ bq+bq '

+ 0 ,...

εk = −2t coskii=1

d

∑

H = −t (ciσ+ c jσ + c jσ

+ ciσ )<i, j>,σ∑ +Ω bi

+bi + gi

∑ ni (bi+ + bi )

i∑

= εkk∑ ck

+ck + Ω bq+bq +

g

Nq∑ ck−q

+ ckk ,q∑ bq

+ + b−q( )(spin is irrelevant, N = number of unit cells, à infinity at the end, all k,q-sums over Brillouin zone)

k

Ek

single electronΩ

el + 1 ph continuum

el + 2 ph continuum

à weak coupling, g = “small”à low-energy eigenstates of known k:

with eigenenergies (for k< kcross)Ek

= εk − 1N

g2

Ω +εk−q( ) −εkq∑ + ...

ψk = ck+ 0 + φqck−q

+ bq+ 0

q∑

k

Ek

polaron band (infinitely long-lived quasiparticle)

low-k: wavefunction dominated by free electron contribution

high-k: wavefunctiondominated by el+1ph

contributions (need to do perturbation properly)

The phonons can be quite far spatially from the electron à large polaron

polaron + one- phonon continuum (finite lifetimes)

Must always start at Ωabove gs energy!!!

Ω

In fact, a polaron state exists everywhere in the BZ only in d=1,2. In d=3 and weak coupling, the polaron exists only near the center of the BZ.

G.L. Goodvin and M. Berciu, EuroPhys. Lett. 92, 37006 (2010)

G(k ,ω ) =1

ω −εk − Σ(k,ω) + iη→ω −εk = Re[Σ(ω)]; Im[Σ(ω )] = 0

à very strong coupling, λ>>1 (tà 0)àsmall polaron energy is

Ek = − g2

Ω+ e

− g2

Ω2 εk + ... → teff = te− g2

Ω2 → meff = meg2

Ω2

and wavefunction is ψk = eik⋅Ri

Ni∑ ci

† − gΩ i

Again, must have a polaron+one-phonon continuum at EGS + Ωà details too nasty

Because here polaron dispersion is so flat, there is a polaron state everywhere in the BZ.

Ø Diagrammatic Quantum Monte Carlo (Prokof’ev, Svistunov and co-workers)

àcalculate Green’s function in imaginary time

Basically, use Metropolis algorithm to sample which diagrams to sum, and keep summing numerically until convergence is reached

( ) 1, ,2† †, 0 0 1, , 0k k

E EHk kk kG k c e c e k c Z eατ ττ

τα

τ α− −−→∞

= = →∑

Ø Quantum Monte Carlo methods (Kornilovitch in Alexandrov group, Hohenadler in Fehskegroup, …) à write partition function as path integral, use Trotter to discretize it, then evaluate. Mostly low-energy properties are calculated/shown.

Ø Exact diagonalization = EDà finite system (still need to truncate Hilbert space) à can get whole spectrum and then build G(k,w)

Ø Variational methods

Ø Cluster perturbation theory: ED finite system, then use perturbation in hopping to “sew” finite pieces together à infinite system.

Ø + 1D, DMRG+DMFT

à…. (lots of work done in these 50 years, as you may imagine)

Analytic approaches (other than perturbation theory) à diagrammatics

= + + + + + ….

Very useful trick: group proper diagrams into the self-energy:

= Σ(k,ω)= + + +…

= + + + … = +

0 0( , ) ( , ) ( , ) ( , ) ( , )1

( , ) ( , )k

G k G k G k k G k

G kk i

ω ω ω ω ω

ωω ε ω η

= + Σ

→ =− − Σ +

This is exact. The advantage is that much fewer diagrams need to be summed to get the self-energy.

1( , )

( , )k

G kk i

ωω ε ω η

=− − Σ +

For Holstein polaron, we need to sum to orders well above g2/Ω2 to get convergence.

n 1 2 3 4 5 6 7 8

Σ, exact 1 2 10 74 706 8162 110410 1708394

Σ, SCBA 1 1 2 5 14 42 132 429

Traditional approach: find a subclass of diagrams that can be summed, ignore the rest

à self-consistent Born approximation (SCBA) – sums only non-crossed diagrams (much fewer)

New proposal: the MA(n) hierarchy of approximations:

Idea: keep ALL self-energy diagrams, but approximate each such that the summation can be carried out analytically. (Alternative explanation: generate the infinite hierarchy of coupled equations of motion for the propagator using Dyson’s identity, keep all of them instead of factorizing and truncating, but simplify coefficients so that an analytical solution can be found).

MA also has variational meaning:

à only certain kinds of bosonic clouds allowed (O. S. Barišic, PRL 98, 209701 (2007))

à what is reasonable depends on the model. In the simplest case (Holstein model):

(needed to describe polaron + one-boson continuum)

( ) ( )

( ) ( )

(0) † †

(1) † † †

MA 0 , , ,

MA 0 , , , ,

n

i j

n

i j l

c b i j n

c b b i j l n

→ ∀

→ ∀

3D Polaron dispersion

L. -C. Ku, S. A. Trugman and S. Bonca, Phys. Rev. B 65, 174306 (2002).

Polaron bandwidth is narrower than Ω!

A(k,ω) in 1D, Ω =0.4 t

G. De Filippis et al, PRB 72, 014307 (2005)

MA becomes exact for small, large λ

λ = 0.5 λ = 1

λ = 2

1D, k=0, W=0.5t

MA(2)

Our answer to how spectral weight evolves as λ increases from weak to strong coupling

Note: this is considered standard polaron

phenomenology. I believe that polarons with quite

different behavior appear in other models.

What happens to the polaron near a surface?

Older answer (aka “instanteneous approximation”):

Propagator for e + cut = + + +….

Propagator for polaron = = + + + +…

So, polaron+cut must be = + + +…

Problem: this cannot possibly be right, because it only allows the electron to reflect at the surface

if no phonons are present. However, unless e-ph coupling is extremely weak, it is very unlikely

not to have phonons present.

What are missing are diagrams such as: , i.e. the electron is reflected at

the surface in the presence of its phonon cloud.

MA allows us to account for all these processes (again, all diagrams are included, up to exponentially

small terms which are dismissed from each of them).

MA answer = + + +…

where: = + + + … + + + + +…

Conclusion: the e-ph coupling renormalized not only the properties of the quasiparticle (polaron is heavier than free electron, etc), but also renormalized its interactions with surfaces, disorder, etc.

This renormalization can be very strong, is strongly dependent on the energy ω (retardation effects) and can either increase or decrease the original (bare) interaction – it can even change its sign, and for example turn an attractive bare interaction into a renormalized repulsive one, or viceversa.

Note: no numerical simulations available for polarons near surface, so we cannot check our results. However, results are available for polaron + impurity, which is a similar type of problem. Comparison between MA and these results shows excellent agreementà see M. Berciu, A. S. Mischchenko and N. Nagaosa, EuroPhys. Lett. 89, 37007 (2010).

What we find: because the interaction potential is strongly renormalized, we may find bound surface states even for parameters where a bare particle with the same mass as the polaron mass would not form bound states!

see G.L. Goodvin, L. Covaci and M. Berciu, Phys. Rev. Lett. 103, 176402 (2009).

The spectrum near surface can be quite different from that in the bulk, and easy to misinterpret if one doesn’t take into account the renormalization of the interactions on top of the renormalization of the quasiparticle properties!

The end …. any questions?

( )

1 0

02 0 0

0

1( ) ( ) ( ) 1 ( 1) 0, if n is

(

odd

) ( )

2

1; 0

2

nn

nnA M

A M M

ωω δ ω ω δ ω ω

ω δ ω >

= − +

= → =

+ → =

=

= + −

Spectral weight sum rules (see PRB 74, 245104 (2006) for details)1

( ) ( , ) Im ( , )n nnM k d A k d G kωω ω ωω ω

π

∞ ∞

−∞ −∞

= = −∫ ∫ ß can be calculated exactly

( ) 0 0nn k kM k c H c+=

MA(0) satisfies exactly the first 6 sum rules, and with good accuracy all the higher ones.

Note: it is not enough to only satisfy a few sum rules, even if exactly. ALL must be satisfied as well as possible.

Examples: 1. SCBA satisfies exactly the first 4 sum rules, but is very wrong for higher order sum rules à fails miserably to predict strong coupling behavior (proof coming up in a minute).

2. Compare these two spectral weights:

0 w0-w0

( )6 2 4 2 2 3 2 2 2 2 26

3 4 4 2 2 2 6

( ) [5 6 2 4 3 6 ( 2 )

2 ] 12 22 25 1518

k k k k k k

k k k

M k g t d d dt

g dt g

ε ε ε ε ε ε

ε ε ε

= + + − + Ω + Ω + + Ω + Ω

+ Ω + Ω + + + Ω + Ω +

r r r r r r

r r r

r

2 46, 6( ) ( ) 2MAM k M k dt g= −

r r

[ ]4 66, 6( ) ( ) .... 10SCBAM k M k g g= − −

r r

found correctly if n=0 diagram kept correctly à dominates if t >> g, λà 0

found correctly if we sum correct no. of diagrams à dominates if g >>t, λ >>1

Since G(k,w) is a sum of diagrams, keeping the correct no. of diagrams is extremely important!

1( ) ( , ) Im ( , )n n

nM k d A k d G kωω ω ωω ωπ

∞ ∞

−∞ −∞

= = −∫ ∫

2

2gdt

λ =Ω