Polarons in bulk and near surfaces

1. Polarons in bulk and near surfaces
Mona Berciu
University of British Columbia
Acknowledgements: Glen Goodvin and Lucian Covaci
NSERC, CIfAR

2. 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

3. 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”.
i
− Hτ
General definition of a retarded propagator: Gi → f (τ ) = −i Θ(τ ) ψ f e h ψi
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!

4. We can choose any ψ i and ψ f, depending on our needs. Eg., for a clean system it is usually
+
convenient to take ψi = ψ f = ck 0 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).
i
− Hτ
+
G ( k ,τ ) = −iΘ(τ ) 0 ck e h ck 0
With this choice + Fourier transform, we then get:
1
G (k , ω ) @ 0 ck G(ω )ck 0 , where G (ω ) =
ˆ + ˆ , η >0 and h=1.
ω − H + iη
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.

5. H 1, k ,α = E1, k ,α 1, k , α ß eigenenergies and eigenfunctions (1 electron, total momentum
k, α is collection of other needed quantum numbers)
Z1,k ,α 2
ck 0 = ∑
1 + +
G(k,ω) @ 0 ck Z1,k,α = 1, k,α ck 0
ω − H +iη α
ω − E1,k ,α + iη
A(k,ω) @ − ImG(k,ω) = ∑ Z1,k ,αδ ω − E1,k ,α
1
π
( ) ß = spectral weight, is measured
α (inverse) angle-resolved photoemission
A(k,ω) spectroscopy (ARPES)
Area is equal to Z
η
E1, k ,α ω
Z = quasiparticle weight à measures how similar the true wavefunction is to a non-interacting (free
electron, no bosons) wavefunction

6. Simple examples of Green’s functions
1. Clean (no disorder) N-site chain, tight-binding hopping, no interactions
H 0 = −t ∑ ci+ ci +1 + hc = ∑ ε k c k c k ,
+
i k
1
ε k = −2t cos(k ) ∈ [−2t , 2t ] and c = +
k
N
∑i
eikRi ci+
1
Then, G0 (k , ω ) = à because of symmetry, only the state of energy εk is “visible”.
ω − ε k + iη
1 e i (m −n) ka
I could also ask for G0 (m, n, ω ) = 0 +
cmG0 (ω )cn
ˆ 0 =
N
∑ ω − ε + iη à is related to
k k
amplitude of probability to go from site n to site m.
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.

7. 1 1 1
g0 (ω ) = G0 (n , n, ω ) =
N
∑ G0 ( k , ω ) = N
∑ ω − ε + iη
k k k
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.

8. 2. Lattice with tight-binding hopping, with one impurity at site 0 (very simple type of disorder)
H = H0 + V = −t ∑ ( ci+ c j + hc ) − Uc0 c0
+
i, j
We can link this propagator to the previous one, to avoid calculating all the eigenfunctions.
The only thing needed is the Dyson’s identity G (ω ) = G0 (ω ) + G (ω )VG0 (ω )
ˆ ˆ ˆ ˆ
(ask me for proof in the break, if you’ve not seen this before. It’s very short).
G (m, n ,ω ) = G0 ( m, n ,ω ) + ∑ G0 (m, l , ω ) l V p G ( p , n, ω )
l, p
= G0 (m, n, ω ) − UG0 (m,0, ω )G (0, n, ω )
G0 (m,0, ω )G0 (0, n, ω )
→ G (m, n ,ω ) = G0 (m, n ,ω ) − U
1 + UG0 (0, 0, ω )
Answer looks the same in any lattice, just use proper G 0.
There can be new poles (new eigenstates) at energies where the denominator vanishes.

9. 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:
1
Re[ g0 (ω )] = −
U
Im[ g0 (ω )] = 0
-1/U
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.

10. 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:
n m
= G0 ( m, n ,ω ) = 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)
n m = G (m, n, ω ) = amp. of probability to go from n à m when impurity present
Then:
n m m+ n 0 m n 0 0 m
= n + + ….
G (m, n ,ω ) = G0 (m, n ,ω ) + G0 (m,0, ω )[−U ]G0 (0, n, ω ) + G0 (m,0, ω )[−U ]G0 (0,0, ω)[− U ]G0 (0, n ,ω ) +
= precisely the same answer
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:
G (ω ) = G0 (ω ) + G0 (ω )VG (ω ) = G0 (ω ) + G0 (ω )VG0 (ω ) + G0 (ω )VG0 (ω )VG0 (ω ) + ...
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ

11. 3. Semi-infinite chain – hopping only allowed between sites 0,1,2, …
Strategy: start with infinite chain (G0 already known) and cut it!
H c = H 0 + Vc = −t ∑ ( ci+ ci +1 + hc ) + t ( c0 c−1 + hc )
+
all i
Just as before, we find for any positive n,m:
n m m + n m m
= n + n +
where now represents reflection at the surface (cut)
It’s a good exercise for you to derive the diagrammatics rules; you should find:
G0 (m, −1, ω )G0 (0, n, ω )
Gc (m, n ,ω ) = G0 (m, n, ω ) + t
1 − tG0 (0, −1, ω)
Here we have a new denominator – it turns out that this NEVER vanishes, so there is no “surface
state” for this type of cut.

12. 3b. Semi-infinite chain – hopping only allowed between sites 0,1,2, … but also a surface
potential
H s = H 0 + Vc + V = −t ∑ ( ci+ ci +1 + hc ) + t (c 0 c−1 + hc ) − Uc0 c0
+ +
all i
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:
H s = H c + V → Gs (ω ) = Gc (ω ) + Gs (ω )VGc (ω )
ˆ ˆ ˆ ˆ
Gc is already known, and we know what happens when we add this V to it:
Gc ( m,0, ω )Gc (0, n, ω )
Gs ( m, n, ω ) = Gc ( m, n ,ω ) − U
1 + UGc (0,0, ω )
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?

13. 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)
surface
n=0 n=1 …..
layer layer
àInvariance to translation in || direction à partial Fourier transform: cn, m, p → cn,k|| : state
where electron is in layer n and has in-plane momentum k ||. Then:

k||  n ≥0
( n≥ 0
) + + 
H s = ∑ −t ∑ cn ,k|| cn+1,k|| + hc + ∑ ε || (k|| )cn ,k|| cn ,k|| − Uc0,k|| c0,k|| 
+


14. 
k||  n ≥0
( )
n≥ 0
+ + 
H s = ∑ −t ∑ cn ,k|| c n+1,k|| + hc + ∑ ε || (k|| )cn ,k|| cn ,k|| − Uc0,k|| c0,k|| 
+

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:
+
Gs3D (k|| , m, n, ω ) = 0 cm,k|| Gs (ω ) cn ,k|| 0 = Gs1D (m, n, ω − ε || (k|| ))
ˆ
à (100) surface also won’t bind a state unless surface potential U > t. If a surface state appears,
its energy depends on k ||.
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.
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.

15. 4: 1-site polaron, or the Frack-Condon problem (aka Lang Firsov, in polaron physics):
add extra e à new equilibrium distance
h = h0 + gnX
ˆˆ
P2 M Ω (X − X0)
ˆ 2 ˆ 2  1 X0 → X0 −α n
ˆ
h0 = + → hΩ  b+b + 
2M 2  2
  ˆ 
 b = MΩ X − X +i P 
ˆ
 2h  0
MΩ
 → X − X 0 ∝ (b+ + b)
ˆ
new equilibrium length, determined by
 + MΩ  ˆ P 
ˆ
 b = 2h  X − X 0 − i M Ω  how many extra electrons there are
  

16. g2
h = Ωb†b + gn(b† + b) → Ωb†b + g (b† + b) = ΩB † B −
ˆ
Ω
If we’re adding one electron
g + g
B = b+ , B = b + + → [ B, B + ] = [b, b + ] = 1
Ω Ω
Ground state is c+ ph where B ph = 0 Side-note: coherent states
are defined by:
g g
→ b ph = − ph → ph = − b α =α α
Ω Ω
2
2 +,n α
g + B g − +αb †
In general, En = nΩ − with c − → α =e 2 0
Ω n! Ω
g2
In gs, b b = 2 , etc.
+
Ω
The Green’s function is:
g2 2n
− 1g 1
GLF (ω ) = 0 cG (ω )c + 0 = e
ˆ Ω2
∑ n! Ω 
  g2
n≥ 0
ω − nΩ + + iη
Ω

17. 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

18. 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)
polaron (small or large, depending) but
t always mobile, in absence of disorder
H = −t ∑ (ci+ c jσ + c +σ ciσ ) + Ω∑ bi+bi + g ∑ ni (bi+ + bi )
σ j
<i , j > ,σ i i
3 energy scales: t, Ω, g à 2 dimensionless parameters λ = g 2/(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.

19. g2 g2
weak coupling λ = = 0 ( g = 0) Lang-Firsov impurity limit λ = = ∞ (t = 0)
2dtΩ 2dtΩ
1
G0 ( k , ω ) = ; g2
ω − ε k + iη En = − + nΩ
Ω
η
 δ (ω − ε k )
η →0
A0 (k ,ω ) = →
π (ω − ε k ) + η 2 
2
 
Ω
g2
EGS =−
Ω
How does the spectral weight evolve between these two very different looking limits?

20. H = −t ∑ (ciσ c jσ + c+ ciσ ) + Ω∑bi+bi + g ∑ ni (bi+ + bi )
+
jσ
<i, j>,σ i i
= ∑ε k ck ck + Ω∑bq bq + ∑ck+−qck (bq+ + b−q )
+ + g
k q N k ,q
(spin is irrelevant, N = number of unit cells, à infinity at the end, all k,q-sums over Brillouin zone)
Asymptotic behavior:
à zero-coupling limit, g=0 à eigenstates of given k: c+ 0 , ck−q bq 0 , ck−q−q 'bq bq' 0 ,...
k
+ + + + +
d
with eigenenergies εk , εk −q + Ω, εk −q−q' + 2Ω,.... where, for example, ε = −2t cos k
k ∑ i
Ek i=1
el + 2 ph continuum
el + 1 ph continuum
k
Ω
single electron

21. à weak coupling, g = “small” à low-energy eigenstates of known k: ψk = ck 0 +∑ φqck−q bq 0
+ + +
q
g2
with eigenenergies E = εk − ∑ +... (for k< kcross)
1
k
(
N q Ω +εk−q −εk )
Ek
The phonons can be
quite far spatially from the
electron à large polaron
polaron + one- phonon k
continuum (finite lifetimes)
Must always start at Ω
above gs energy!!! high-k: wavefunction
Ω dominated by el+1ph
contributions (need to do
perturbation properly)
polaron band (infinitely low-k: wavefunction dominated
long-lived quasiparticle) by free electron contribution

22. 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.
1
G(k ,ω ) = → ω −ε k = Re[Σ(ω)]; Im[Σ(ω )] = 0
ω −ε k − Σ(k,ω) + iη
G.L. Goodvin and M. Berciu, EuroPhys. Lett. 92, 37006 (2010)

23. à very strong coupling, λ>>1 (tà 0) àsmall polaron energy is
g2 g2 g2
g2 − 2 − 2
Ek = − +e Ω ε +... → teff = te Ω → meff = me Ω
2
Ω k
ik⋅Ri
and wavefunction is ψk = ∑
e g
ci† −
i N Ω i
Again, must have a polaron+one-phonon continuum at E GS + Ω à details too nasty
Because here polaron dispersion is so flat, there is a polaron state everywhere in the BZ.

24. Ø Diagrammatic Quantum Monte Carlo (Prokof’ev, Svistunov and co-workers)
àcalculate Green’s function in imaginary time
G ( k ,τ ) = 0 ck e−τ H ck 0 = ∑ e
−τ E1,k ,α 2
†
1, k ,α ck 0
†
 Z k e−τ Ek
τ →∞
→
α
Basically, use Metropolis algorithm to sample which diagrams to sum, and keep summing
numerically until convergence is reached
Ø Quantum Monte Carlo methods (Kornilovitch in Alexandrov group, Hohenadler in Fehske
group, …) à write partition function as path integral, use Trotter to discretize it, then evaluate.
Mostly low-energy properties are calculated/shown.
Ø Exact diagonalization = ED à 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)

25. Analytic approaches (other than perturbation theory) à diagrammatics
= + + + + + ….
Very useful trick: group proper diagrams into the self-energy:
= Σ(k,ω)= + + +…
= + + +…= +
G ( k ,ω ) = G0 (k ,ω ) + G0 (k , ω )Σ ( k , ω)G (k , ω)
1
→ G( k , ω ) =
ω − ε k − Σ(k , ω ) + iη
This is exact. The advantage is that much fewer diagrams need to be summed to get the self-energy.

26. 1
G( k , ω ) =
ω − ε k − Σ( k , ω ) + iη
For Holstein polaron, we need to sum to orders well above g 2/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)

27. 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):
→ c (b ) ( ∀) i , j , n
n
(0) † †
MA i j 0 ,
(b ) ( ∀ ) i, j , l , n
n
MA (1)
→c †
i
†
j bl† 0 ,
(needed to describe polaron + one-boson continuum)

29. 3D Polaron dispersion
Polaron bandwidth is
narrower than Ω!
L. -C. Ku, S. A. Trugman and S. Bonca, Phys. Rev. B 65, 174306 (2002).

30. λ = 0.5 λ=1
λ=2
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

31. MA(2)
1D, k=0, W=0.5t

32. 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.

33. 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).

34. 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 agreement
à see M. Berciu, A. S. Mischchenko and N. Nagaosa, EuroPhys. Lett. 89, 37007 (2010).

35. 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!

36. The end …. any questions?

37. Spectral weight sum rules (see PRB 74, 245104 (2006) for details)
∞ ∞
1
M n (k ) = ∫ dωω A(k , ω ) = − Im ∫ dωω nG (k , ω )
n
ß can be calculated exactly
−∞
π −∞ +
M n ( k ) = 0 ck H n ck 0
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:
A1 (ω ) = δ (ω ) → M 0 = 1; M n> 0 = 0 -w0 0 w0
1 ω0n
A2 (ω ) = ( δ (ω − ω0 ) + δ (ω + ω0 ) ) → M n = 1 + (−1)n  = 0, if n is odd
2 2  

38. ∞ ∞
1
M n (k ) = ∫ dωω n A( k , ω ) = − Im ∫ dωω n G(k , ω )
−∞
π −∞
Since G(k,w) is a sum of diagrams, keeping the correct no. of diagrams is extremely important!
found correctly if n=0 diagram kept correctly à dominates if t >> g, λ à 0
r
M 6 ( k ) = ε kr + g 2 [5ε kr + 6t 2 ( 2d 2 − d ) + 4ε kr Ω + 3ε kr Ω 2 + 6dt 2 (ε kr + ε kr Ω + 2Ω 2 )
6 4 3 2 2
+ 2ε kr Ω3 + Ω 4 ] + g 4 18dt 2 + 12ε kr + 22ε kr Ω + 25Ω 2  + 15 g 6

2

r r
M 6,MA ( k ) = M 6 ( k ) − 2 dt 2 g 4 found correctly if we sum correct no. of
diagrams à dominates if g >>t, λ >>1
r r
M 6, SCBA ( k ) = M 6 ( k ) − g 4 [....] − 10 g 6

39. g2
λ=
2dt Ω