Strongly Correlated Quantum Matter January 14 2020

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Quantum Matter: Concepts and Models Brief introduction to the ”third module”: Strongly correlated quantum matter January 14 2020

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 human scales ~ … on length scales from a few Bob Laughlin nanometers to

from Laughlin’s Nobel Prize talk 1998 Bob Laughlin

not so relevant for condensed matter Major problem: The ”Theory of Everything” Hamiltonian

1 can’t be solved accurately for more than (at most) ~ 10 particles.

A catastrophe of dimension! N # of particles Required size of computer memory: m

required memory size to represent theDOS wave function of one particle

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

p =( i~/r)@ n n +1 !

Ey h ⇢yx = 0 = (ﬂux quantum) jx e M =0 M =0 6 ⇢xx

⌫ =0 ⌫ =1 P =0modeP= e/2 mod e

a =1 w =0 v =0 v =0 w =0 6 6

2 2 2 2 2 12 1 T =(UT K) = UT K = UT = n n = n n ⇥ ⇥

2 2 2 2 2 T =(UT K) =(1n n ( iy)) K =(1n n ( iy)) = 12n 2n ⇥ ⌦ ⇥ ⌦ ⇥

HF (k) u (k) = ✏ (k) u (k) | n i n | n i

⇡ ⇡ a a n =1, 2: e↵ective time-independent two-band model

✏n kn=1 n =2 Start easy: • neglect the electron-electron (and ion-ion) interaction • treat the ions as ﬁxed (Born-Oppenheimer approximation)

wave functions built from Slater determinants wave of functions single-particle

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

powerful method for systems with weak electron correlations

LDA

W. Kohn P. Hohenberg L. Sham 1

Next: 2 ¯ 1/3 Ee-e e /d n 1/3 Adding the electron-electron interaction 2 2 2/3 = n Ekin ⇠ ~ kF /m ⇠ n

mN DOS

p =( i~/r)@ n n +1 ! B~ = A~ r⇥ B~ = A~ Ey h r⇥ ⇢yx = 0 = (ﬂux quantum) jx e 1 1 M =0 M =0 ⌧ 6 ⌧ ⇢xx (1) (1) ⇠ O ⇠ O

p =( i~/r)@ n n +1 ⌫ =0 ⌫ =1 P =0modeP = e/2 mod! e p =( i /r)@ n n +1 ~ Ey h ! ⇢yx = 0 = (ﬂux quantum) jx e M =0 M =0a =1 w =0 v =0 v =0 w =0 6 6 6 ⇢xx Ey h ⇢yx = 0 = (ﬂux quantum) jx e 2 2 2 2 2 12 1 T =(UT K) = U⌫T=0K =⌫ =1UT =P =0modn n = ePn n= e/2 mod e M =0 M =0 ⇥ ⇥ 6 ⇢xx a =1 w =0 v =0 v =0 w =0 2 2 2 2 6 2 6 T =(UT K) =(1n n ( iy)) K =(1n n ( iy)) = 12n 2n ⇥ ⌦ ⇥ ⌦ ⇥ 2 2 2 2 2 12 1 T =(UT K) = UT K = UT = n n = n n ⌫ =0 ⌫ =1 P =0modeP= e/2 mod e ⇥ ⇥ HF (k) u (k) = ✏ (k) u (k) 2 n 2 n n 2 2 2 T =(U|T K) =(i 1n n ( |iy)) Ki =(1n n ( iy)) = 12n 2n ⇥ ⌦ ⇥ ⌦ ⇥

a =1 w =0 v =0 v =0 w =0 F 6 6 ⇡ H⇡ (k) un(k) = ✏n(k) un(k) | i | i a a n =1, 2: e↵ective time-independent two-band model ⇡ ⇡ 2 2 2 2 2 12 1 a a T =(UT K) = UT K = UT = n n = n n n =1, 2: e↵ective time-independent⇥ two-band⇥ model

✏n kn=1 n =2 ✏n kn=1 n =2 2 2 2 2 2 T =(UT K) =(1n n ( iy)) K =(1n n ( iy)) = 12n 2n ⇥ ⌦ ⇥ ⌦ ⇥

HF (k) u (k) = ✏ (k) u (k) | n i n | n i

⇡ ⇡ a a n =1, 2: e↵ective time-independent two-band model

✏n kn=1 n =2 1

Next: 2 ¯ 1/3 Ee-e e /d n 1/3 Adding the electron-electron interaction 2 2 2/3 = n Ekin ⇠ ~ kF /m ⇠ n

mN DOS

1 B~ = A~ 1 How to assess the importance of the electron-electron interaction?r⇥ When is its effect weak? When is it ”strong”? 2 ¯ 1/3 Ee-e e /d n 1/3 2 2 2/3 = n Ekin ⇠ ~ kF /2m ⇠ n 2 ¯ 1/3 1 ”weak” Ee-e meN /d n 1/3 ⌧ ”Rule of thumb” (3D): = n 2 2DOS 2/3 Ekin ⇠ ~ kF /2m ⇠ n (1) ”strong” ¯ N d average electron separation ⇠ O m n electron density DOS LDA+U,…? DFT fails p =( i~/r)@ n n +1 ! another, more modern, typeB~ =of mean-fieldA~ theory can sometimes be used r⇥ Dynamical mean-field theory E h B~ = A~ ⇢ = y = (flux quantum)r⇥ W. Metzner and D. Vollhardt, Phys. Rev. Lett.yx 62, 324j (1989) 0 e A. Georges and G.Kotliar , Phys. Rev. B 45, 6479 (1992)x 1 1 MMapping=0 M the=0 full lattice problem to a time-dependent single-site ”mean⌧ field” problem 6 ⌧ ⇢xx usually treated by numerical methods (exact diagonalization, Monte Carlo,…) (1) (1) ⇠ O ⇠ O

✏n kn=1 n =2 ✏n kn=1 n =2 2 2 2 2 2 T =(UT K) =(1n n ( iy)) K =(1n n ( iy)) = 12n 2n ⇥ ⌦ ⇥ ⌦ ⇥

HF (k) u (k) = ✏ (k) u (k) | n i n | n i

⇡ ⇡ a a n =1, 2: e↵ective time-independent two-band model

✏n kn=1 n =2 1

2 ¯ 1/3 Ee-e e /d n 1/3 = n E ⇠ 2k2 /2m ⇠ n2/3 kin ~ F 1 mN r

⇤ = ”high-energy”Next: cuto↵ DOS Adding the electron-electron interaction d¯ average2 ¯ electron1/3 separation Ee-e e /d n 1/3 n2 electron2 density2/3 = n Ekin ⇠ ~ kF /2m ⇠ n

e2 U(q) Alternative⇠ q2 strategy: perturbation theory 1 (”many-body theory”)

2...”Low-energy”¯ dr1/...3 approach, smear out the lattice: Uion(rj) const. Ee-e e /d! n = n 1/3 ! X2 2 Z 2/3 Ekin ⇠ ~ kF /Second2m ⇠ n quantization: 1 kin = B~ = A~ Hkin = r⇥ H N 2 ¯ 1/3 m r Ee-e e /d n 1/3 2 2 2/3 = n Ekin ⇠ ~ kF /2m ⇠ n ee = 1 H ⌧ mN rDOS e2 U(q) 2 ⇠ q (1) d¯ average electron separation ⇠ O n electronDOS density ... dr... ! Z d¯ average electron separation X p =( i~/r)@ n n +1 n electron density ! = Hkin mN r Ey h Uion(rj) const. ⇢yx = 0 = (ﬂux quantum) ! j e DOS x M =0 M =0 d¯ average electron separation 6 n electron densityU (r ) ⇢const.B~ =, A~.... dr.... ion j ! xx r⇥ ! j X Z

1 B~ =⌧ A~ r⇥ ⌫ =0 ⌫ =1 P =0modeP= e/2 mod e

(1) Uion(rj) const. ⇠ O ! 1 ⌧ a =1 w =0 v =0 v =0 w =0 B~ = A~ 6 6 p =( i~/r)@ n n +1 r⇥ (1) ! ⇠ O 1 2 2 2 2 2 12 1 E h ⌧ T =(UT K) = UT K = UT = n n = n n ⇢ = y = (flux quantum) ⇥ ⇥ yx j 0 e p =(x i~/r)@ n n +1 (1) ! ⇠ O M =0 M =0 6 2 2 1 2 2 1 2 1 ⇢xx T =(UT K) =( n n ( iy)) K =( n n ( iy)) = 2n 2n p =( i~/r)@ n n +1 ⇥ ⌦ ⇥ ⌦ ⇥ Ey h ! ⇢yx = 0 = (flux quantum) jx e E h ⇢ = y = (flux quantum) F ⌫ =0 ⌫ =1 P =0modyx eP= 0e/2 mod e H (k) un(k) = ✏n(k) un(k) jx e | i | i M =0 M =0 6 ⇢xx a =1 w =0 v =0 v =0 w =0 6 6

⌫ =0 ⌫ =1 P =0modeP= e/2 mod e 2 2 2 2 2 12 1 T =(UT K) = UT K = UT = n n = n n ⇥ ⇥ a =1 w =0 v =0 v =0 w =0 6 6 2 2 2 2 2 T =(UT K) =(1n n ( iy)) K =(1n n ( iy)) = 12n 2n ⇥ ⌦ ⇥ ⌦ ⇥ First-order perturbative contribution to the groundstate energy: Second-order perturbative contribution to the groundstate energy:

⇤ = ”high-energy” cuto↵

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

e2 U(q) ⇠ q2

... dr... ! X Z

= Hkin mN r

DOS

d¯ average electron separation n electron density

U (r ) const. ion j !

B~ = A~ r⇥

1 ⌧

(1) ⇠ O

p =( i~/r)@ n n +1 !

Ey h ⇢yx = 0 = (ﬂux quantum) jx e M =0 M =0 6 ⇢xx

⌫ =0 ⌫ =1 P =0modeP= e/2 mod e Second-order perturbative contribution to the groundstate energy:

⇤ = ”high-energy” cuto↵

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

e2 U(q) ⇠ q2 Divergent contribution for arbitrary weak electron-electronWhat interaction! to do? ... dr... ! X Z

= Hkin mN r

DOS

d¯ average electron separation n electron density

U (r ) const. ion j !

B~ = A~ r⇥

1 ⌧

(1) ⇠ O

p =( i~/r)@ n n +1 !

Ey h ⇢yx = 0 = (ﬂux quantum) jx e M =0 M =0 6 ⇢xx

⌫ =0 ⌫ =1 P =0modeP= e/2 mod e ”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 M. Gell-Mann K. Brueckner and then resum the infinite perturbation series taking only those diagrams into account! ”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 M. Gell-Mann K. Brueckner and then resum the infinite perturbation series taking only those diagrams into account!

… according to the power of rs ”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 M. Gell-Mann K. Brueckner and then resum the infinite perturbation series taking only those diagrams into account!

Most perturbative expansions of the interacting electron liquid are patterned on RPA. Tricky part: … according to the power of r s 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

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 20th 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!

”strongly correlated systems” (”non-Fermi liquids”) However, in the last few decades experimentalists have found lots of realizations of fermionic quantum matter where Landau Fermi liquid theory breaks down!

• fermionic systems undergoing quantum phase transitions expected

• phases of fermionic quantum matter with broken symmetries expected or nontrivial topology

• interacting fermions subject to strong disorder open problem by now well understood • fermions in one dimension (quantum wires, nanotubes, spin arrays,…)

• strange metals, normal phase high-temperature superconductors, open problems heavy fermion materials,….

• and maybe more… Some of this stuff will be discussed in the third course module on strongly correlated quantum matter 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

• fermionic systems undergoing quantum phase transitions expected

• phases of fermionic quantum matter with broken symmetries expected or nontrivial topology

• interacting fermions subject to strong disorder open problem by now, well understood • fermions in one dimension (quantum wires, nanotubes, spin arrays,…)

• strange metals, normal phase high-temperature superconductors, open problems heavy fermion materials,…. Mariana Malard (5 lectures) • and maybe more… Renormalization group, bosonization, ”Luttinger liquids” Ulf Gran (1 lecture) Holography Hans-Peter Eckle (2 lectures) Integrability and Bethe Ansatz