Strongly correlated quantum matter

bosonic systems

fermionic systems

electronic correlations major focus in condensed matter physics

…

…

Bob Laughlin

from Laughlin’s Nobel Prize talk 1998

Bob Laughlin

few a rom f les a s c e cal n s s a um gth h n ~ e to n l s o r … ete m o nan

from Laughlin’s Nobel Prize talk 1998

Bob Laughlin

t for n a v rele matter o s not ensed cond

Major problem: The ”Theory of Everything” Hamiltonian

can’t be solved accurately for more than (at most) ~ 10 particles. A catastrophe of dimension! Required size of computer memory:

# of particles

N

m

DOS

required memory size to represent the wave function of one particle

Waiting for a quantum computer, what to do? What we have always done: Make approximations & caricatures guided by experiments!

Start easy: • •

neglect the electron-electron (and ion-ion) interaction treat the ions as fixed (Born-Oppenheimer approximation)

from t l i u b ns o i t c of n s u t f n e a ns n o i i t wav m c r n e u ef det v r a e t w a l e S icl t r a p single

potential of a static lattice of background ions (”hard” quantum condensed matter) major schemes: nearly free electron approximation (Bloch state basis) tight-binding approximation (Wannier state basis)

Band theory of metals, semimetals, semiconductors, and insulators (elementary textbook solid state physics) Still, lots of interesting physics! Even more so when bringing in effects from

relativity, lattice distortions, confined geometries,…! topological band theory

mesoscopic physics

Next: adding the electron-electron interaction (the starting point of this course module!)

popular approach: mean-field theory Hartree-Fock (”textbook”) Density Functional Theory (DFT) P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964) W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965)

single-electron Kohn-Sham Hamiltonian ems t s y s ns for o i d t o a h l t e orr l me c u f n r o e r pow ect l e k a e with w

LDA W. Kohn

P. Hohenberg

L. Sham

Next:

Ee-e e2 /d¯ n1/3 ⇠ 2 2 ⇠ 2/3 = n Ekin ~ kF /m n

Adding the electron-electron interaction

1/3

mN DOS ~ =r⇥A ~ B

How to assess the importance of the electron-electron interaction? When is its effect weak? When is it ”strong”? 2 ”Rule of thumb” (3D):

2 ¯ 1/3 Ee-e e n N /d ⇠ 2m 2 ⇠ 2/3 = n DOS Ekin ~ kF /2m n

d¯ average electron separation n electron density

Ee-e e /d¯ n1/3 ⇠ 2 2 ⇠ 2/3 = n Ekin ~ kF /2m n

⌧1

1/3

⇠ O(1)

p = ( i~/r)@

~ =r⇥A ~ B ⇢yx M 6= 0

⇢xx

M =0

⌧1

Ey = jx

0

”weak”

”strong”

n!n+1

~ =r⇥A ~ B h = (flux quantum) e ⌧1 ⇠ O(1)

1/3

Next: Adding the electron-electron interaction

Ee-e e2 /d¯ n1/3 ⇠ 2 2 ⇠ 2/3 = n Ekin ~ kF /m n

1/3

mN DOS ~ =r⇥A ~ B

How to assess the importance of the electron-electron interaction? When is its effect weak? When is it ”strong”? 2 ”Rule of thumb” (3D):

2 ¯ 1/3 Ee-e e n N /d ⇠ 2m 2 ⇠ 2/3 = n DOS Ekin ~ kF /2m n

Ee-e e /d¯ n1/3 ⇠ 2 2 ⇠ 2/3 = n Ekin ~ kF /2m n

⌧1

1/3

d¯ average electron separation n electron density

⇠ O(1)

1/3

”weak”

”strong” …? , U + A LD

p = ( i~/r)@

n!n+1

DFT fails

~ = ~ B r⇥A another, more modern, type of mean-field theory can sometimes be used Dynamical mean-field theory

Ey ⇢yx 62, =324 (1989) W. Metzner and D. Vollhardt, Phys. Rev. Lett. jx A. Georges and G.Kotliar , Phys. Rev. B 45, 6479 (1992)

0

~ =r⇥A ~ B h = (flux quantum) e

⌧1

1 MMapping 6= 0 Mthe=full 0 lattice problem to a time-dependent single-site ”mean⌧field” problem

⇢xx

usually treated by numerical methods (exact diagonalization,⇠Monte Carlo,…) O(1)

”high-energy” Next: cuto↵

DOS

Adding the electron-electron interaction

d¯eaverage electron separation 2 ¯ /d n1/3 1/3 ⇠ = n n electron density ~2 k 2 /2m n2/3 F

e2 U (q) ⇠ 2 Alternative q

strategy:

perturbation theory

1

(”many-body theory”)

X

Z

approach, smear out the lattice: Uion (r j ) ! const. ...”Low-energy” ! dr... e2 /d¯ n1/3 1/3 ⇠ = n Secondn2/3 quantization: ~2 kF2 /2m

~ =r⇥A ~ B

Hkin = Hkin = Ee-e e2 /d¯ n1/3 ⇠ 2 2 ⇠ 2/3 = n Ekin ~ kF /2m n

Hee =

e2 U (q) ⇠ 2 q X

... !

Z

1

1/3

⌧1 ⇠ O(1)

dr...

p = ( i~/r)@

n!n+1

First-order perturbative contribution to the groundstate energy:

Second-order perturbative contribution to the groundstate energy:

1

⇤ = ”high-energy” cuto↵

Ee-e e2 /d¯ n1/3 ⇠ 2 2 ⇠ 2/3 = n Ekin ~ kF /2m n e2 U (q) ⇠ 2 q X

... !

Z

Hkin = mN r

dr...

1/3

Second-order perturbative contribution to the groundstate energy:

ry a 1 tr i on! b r ti a c r a o r f te n ⇤ = ”high-energy” cuto↵ io in t u on b i r t r t c n ele e /d¯ n o ⇠ ⇠ =n c n~ k /2m n t n tro e rg lec e e iv k e U (q) ⇠ D q a e w What to do?

Ee-e Ekin

2

1/3

2 2 F

2/3

2

2

X

... !

Z

Hkin = mN r

dr...

1/3

”Random phase approximation” (RPA) standard approach in the limit of high electron density M. Gell-Mann and K. A. Brueckner, Phys. Rev. 106, 364 (1957), putting earlier work by D. Pines and D. Bohm (Phys. Rev. 85, 338 (1952)) on firm ground.

Select the most important Feynman diagrams and then resum the infinite perturbation series taking only those diagrams into account!

M. Gell-Mann

K. Brueckner

”Random phase approximation” (RPA) standard approach in the limit of high electron density M. Gell-Mann and K. A. Brueckner, Phys. Rev. 106, 364 (1957), putting earlier work by D. Pines and D. Bohm (Phys. Rev. 85, 338 (1952)) on firm ground.

Select the most important Feynman diagrams and then resum the infinite perturbation series taking only those diagrams into account!

… according to the power of rs

M. Gell-Mann

K. Brueckner

”Random phase approximation” (RPA) standard approach in the limit of high electron density M. Gell-Mann and K. A. Brueckner, Phys. Rev. 106, 364 (1957), putting earlier work by D. Pines and D. Bohm (Phys. Rev. 85, 338 (1952)) on firm ground.

Select the most important Feynman diagrams and then resum the infinite perturbation series taking only those diagrams into account!

… according to the power of rs

M. Gell-Mann

K. Brueckner

Most perturbative expansions of the interacting electron liquid are patterned on RPA. Tricky part: Identify (or introduce) a small parameter (like rs in RPA) that can be used to select diagrams.

from R. Mattuck, A Guide to Feynman Diagrams in the Many-Body Problem (McGraw-Hill, 1976)

Basic features of the interacting electron liquid that emerge from RPA (and other perturbative expansions): •

collective excitations, like density oscillations (plasmons), appear at high energies —> not important for thermal and transport properties of metals and semiconductors

•

the long-range Coulomb potential of the electrons is screened to a short-range interaction

•

low-lying excitations of the electron liquid form nearly stable quasiparticles

Basic features of the interacting electron liquid that emerge from RPA (and other perturbative expansions): •

collective excitations, like density oscillations (plasmons), appear at high energies —> not important for thermal and transport properties of metals and semiconductors

•

the long-range Coulomb potential of the electrons is screened to a short-range interaction

•

low-lying excitations of the electron liquid form nearly stable quasiparticles A quasiparticle carries the same quantum numbers as an electron, but with renormalized mass, life time, magnetic moment, ”wave function renormalization” Z(kF),….

T=0 Fermi-Dirac distribution

from P. Coleman, Introduction to Many-Body Physics (Cambridge University Press, 2015)

Basic features of the interacting electron liquid that emerge from RPA (and other perturbative expansions): •

collective excitations, like density oscillations (plasmons), appear at high energies —> not important for thermal and transport properties of metals and semiconductors

•

the long-range Coulomb potential of the electrons is screened to a short-range interaction

•

low-lying excitations of the electron liquid form nearly stable quasiparticles A quasiparticle carries the same quantum numbers as an electron, but with renormalized mass, life time, magnetic moment, ”wave function renormalization” Z(kF),….

T=0 Fermi-Dirac distribution

The quasiparticle concept was first introduced by Lev Landau 1956 in his phenomenological study of 3He (a neutral Fermi liquid, without the complications of long-range Coulomb interactions). L. D. Landau, J. Exp. Theor. Phys. 3, 920 (1957)

A quasiparticle is the adiabatic evolution of a noninteracting fermion into an interacting environment

We can treat most properties of an interacting electron liquid (but not all!) in most metals and semiconductors as an almost ideal gas of quasiparticles! This is how we ”get away” with elementary solid state physics! N. B. Landau Fermi liquid theory applies also to nuclear matter, the interior of neutron stars, ….

A triumph of

th 20 century

physics!

However, in the last few decades experimentalists have found lots of realizations of fermionic quantum matter where Landau Fermi liquid theory breaks down!

” s m e t s y s d e t a l e r r ) o ” c s y d l i g u q n i o l r i t s m ” r e F n o (”n

However, in the last few decades experimentalists have found lots of realizations of fermionic quantum matter where Landau Fermi liquid theory breaks down! • •

• • •

•

ed t c e p fermionic systems undergoing quantum phase transitions ex

phases of fermionic quantum matter with broken symmetries or nontrivial topology blem o r p n interacting fermions subject to strong disorder ope

ed

t expec

ll e w w y no

ood t s r e und

b fermions in one dimension (quantum wires, nanotubes, spin arrays,…)

s

blem o r p n

strange metals, normal phase high-temperature superconductors, ope heavy fermion materials,…. and maybe more…

Some of this stuff will be discussed in the third course module on strongly

quantum matter

correlated

However, in the last few decades experimentalists have found lots of realizations of fermionic quantum matter where Landau Fermi liquid theory breaks down! Hans Hansson (1 lecture) FQHE and topological order •

ed t c e p fermionic systems undergoing quantum phase transitions ex ed

•

• • •

•

t expec

phases of fermionic quantum matter with broken symmetries or nontrivial topology d m o e o l t b s o r e pr d n n e u p l el interacting fermions subject to strong disorder o w , w o by n fermions in one dimension (quantum wires, nanotubes, spin arrays,…) s m e l b o pr n e p strange metals, normal phase high-temperature superconductors, o heavy fermion materials,…. and maybe more… Ulf Gran (1 lecture) Holography

Mariana Malard (5 lectures) Renormalization group, bosonization, ”Luttinger liquids” Hans-Peter Eckle (2 lectures) Integrability and Bethe Ansatz