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 Bob Laughlin … on length scales from a few nanometers to ~ human scales 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! pφ =( i~/r)@φ n n +1 − ! Ey h ⇢yx = Φ0 = (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 ( iσy)) K =(1n n ( iσy)) = 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 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 pφ =( i~/r)@φ n n +1 − ! B~ = A~ r⇥ B~ = A~ Ey h r⇥ ⇢yx = Φ0 = (flux quantum) jx e 1 1 M =0 M =0 ⌧ 6 ⌧ ⇢xx (1) (1) ⇠ O ⇠ O pφ =( i~/r)@φ n n +1 ⌫ =0 ⌫ =1 P =0modeP− = e/2 mod! e p =( i /r)@ n n +1 φ ~ φ Ey h − ! ⇢yx = Φ0 = (flux quantum) jx e M =0 M =0a =1 w =0 v =0 v =0 w =0 6 6 6 ⇢xx Ey h ⇢yx = Φ0 = (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 ( iσy)) K =(1n n ( iσy)) = 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 ( |iσy)) Ki =(1n n ( iσy)) = 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 ( iσy)) K =(1n n ( iσy)) = 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 pφ =( i~/r)@φ 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 Φ = (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 pφ =( i~/r)@φ n n +1 ⌫ =0 ⌫ =1 P =0modeP− = e/2 mod! e p =( i /r)@ n n +1 φ ~ φ Ey h − ! ⇢yx = Φ0 = (flux quantum) jx e M =0 M =0a =1 w =0 v =0 v =0 w =0 6 6 6 ⇢xx Ey h ⇢yx = Φ0 = (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 ( iσy)) K =(1n n ( iσy)) = 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 ( |iσy)) Ki =(1n n ( iσy)) = 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 ( iσy)) K =(1n n ( iσy)) = 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 pφ =( i~/r)@φ n n +1 n electron density − ! = Hkin mN r Ey h Uion(rj) const. ⇢yx = Φ0 = (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 pφ =( i~/r)@φ 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 Φ = (flux quantum) ⇥ ⇥ yx j 0 e pφ =(x i~/r)@φ n n +1 (1) − ! ⇠ O M =0 M =0 6 2 2 1 2 2 1 2 1 ⇢xx T =(UT K) =( n n ( iσy)) K =( n n ( iσy)) = 2n 2n pφ =( i~/r)@φ n n +1 ⇥ ⌦ − ⇥ ⌦ − − ⇥ Ey h − ! ⇢yx = Φ0 = (flux quantum) jx e E h ⇢ = y Φ = (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 ( iσy)) K =(1n n ( iσy)) = 12n 2n ⇥ ⌦ − ⇥ ⌦ − − ⇥ First-order perturbative contribution to the groundstate energy: Second-order perturbative contribution to the groundstate energy: 1 ⇤ = ”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 ..