Numerical renormalization group and multi-orbital Kondo physics

Transcription

1 Numerical renormalization group and multi-orbital Kondo physics Theo Costi Institute for Advanced Simulation (IAS-3) and Institute for Theoretical Nanoelectronics (PGI-2) Research Centre Jülich 25th September 2015 Autumn School on Correlated Electrons: From Kondo to Hubbard

2 Outline 1 Introduction 2 Wilson s NRG 3 Complete Basis Set 4 Recent Developments 5 Summary

3 Collaborations Dr Hoa Nghiem (Postdoc, Juelich), Lukas Merker (PhD, Juelich) RWTH Aachen colaborators Dr Dante Kennes, Prof. Volker Meden (complementary methods, FRG/DMRG), Uni. Tuebingen, Prof. Sabine Andergassen (FRG)

4 References on NRG and codes K.G. Wilson, Rev. Mod. Phys [Kondo model] Krishna-murthy et al, PRB 1980 [Anderson model] R. Bulla, T.A. Costi, T. Pruschke, Rev. Mod. Phys [further developments/models] A. C. Hewson, The Kondo Problem to Heavy Fermions, C.U.P. (1997) Recommended public domain NRG codes: flexible DM-NRG code (G. Zarand, et al., Budapest) NRG-Ljubljana code (R. Zitko, Ljubljana)

5 Quantum impurity systems Quantum impurity systems U Environment Impurity: small system, discrete spectrum Environment: large system, quasi-continuous spectrum Coupling: hybridization, Kondo exchange,... General structure: H = H imp + H int + H bath

6 Quantum impurity systems Anderson model Single-level Anderson impurity model (n d = d d ): H AM = X " d n d + Un d" n d# + X V kd (c k d + d c k ) k {z } {z } H imp H int + X k c k c k k {z } H bath We take constant hybridization function: (1) (!) = X k V 2 kd (! k) ( F = 0) non-constant hybridization can also be dealt with in NRG (e.g., for applications to DMFT, pseudogap systems,...)

7 Quantum impurity systems Multi-orbital and multi-channel Anderson Model Rotationally symmetric case, e.g., for transition metal ions l = 2, and m = 2, 1, 0, +1, +2 H = X " dm n m U X n m n m + 1 X 2 U0 n m n m 0 m m m6=m (U0 J) X J X n m n m 0 d m d m d 2 m d 0 m 0 m6=m 0 m6=m 0 J 0 X d m d m 2 d m 0 d m 0 m6=m 0 + X km V km (c km d m + h.c.) + X km km c km c km Crystal field and spin-orbit interactions ) Further lowering of symmetry, fewer channels

8 Experiments Magnetic impurities in non-magnetic metals Fe in Au: de Haas et al 1934 Fe in Ag: Mallet et al 2006 Au - sample 1 Au -sample 2 T [K] de Haas et al: resistivity minimum and increase as T! 0! Kondo 1964: explanation in terms of magnetic impurities Kondo 1964: R K (T )= c imp ln(k B T /D)!1as T! 0!!

9 I Experiments Quantum dots, Kondo effect and nanoscale size SET s 100nm lateral quantum dots ) Tunable Kondo effect Goldhaber-Gordon et al 1998 source ~ ε d +U Gate Dot V ~T K ε d drain V g T K = p U/2e " d (" d +U)/2 U ) exponential sensitivity Single-electron transistor based on Kondo effect

10 Experiments High/low spin molecules in nanogaps C 60 : Roch et al 2009 Underscreened Kondo effect Asymmetric lead couplings: partial screening of S = 1 ) Underscreened Kondo effect (N. Roch et al PRL 2009)

11 Experiments Magnetic atoms on surfaces: Kondo + spin anisotropy Co (S=3/2): Kondo+anisotropy Ti (S=1/2): usual Kondo effect Otte et al 2008 Otte et al 2008 H imp = gµ B S z + DSz 2 for Co adatom with S=3/2 D > 0 ) m = 1/2 $ 1/2 transitions ) Kondo effect m = ±1/2!±3/2 transitions, side-bands to Kondo

12 Linear chain form NRG: linear chain form of the Anderson model H AM = X z } { " d n d + Un d" n d# + V X (f 0 d + d f 0 )+ X k V f 0 = P k V kd c k k c k c k Tridiagonalize H bath = P k k c k c k using Lanczos with starting vector defined by Vf 0 = P k V kdc k to obtain linear chain form: H AM = X " d n d + Un d" n d# + V X (f 0 d + d f 0 ) + 1X n=0, n f + n f n + 1X n=0, t n (f + n f n+1 + H.c.)...suitable for numerical treatment by adding one orbital at a time, starting with d, then f 0, then f 1,...

13 Linear chain form Atomic and zero-bandwidth limits H m= 1 AM (V = 0) = X " d n d + Un d" n d# H m=0 AM ( k = 0) = X " d n d + Un d" n d# + V X (f 0 d + d f 0 ) A (!) = 1 X hp d qi 2 (e Eq + e Ep ) (! (E q E p ) Z V = 0: atomic limit! " d + U F " d p,q k = F = 0: zero bandwidth limit! " d + U F " d O(V 2 /U) A(!) A(!)

14 Linear chain form Anderson model: the Kondo resonance H AM = X " d n d + Un d" n d# + V X k (f 0 d + d f 0 ) + 1X n=0, n f + n f n + 1X n=0, t n (f + n f n+1 + H.c.) Finite bandwidth Add f 1, f 2,... ) broadening O( ) of " d, and " d + U excitations ) Kondo resonance T K e 1/J, J V 2 /U Kondo resonance! " d + U F " d 2 T K e 1/J A(!) 2

15 Linear chain form Iterative diagonalization/recursion relation Define truncated Hamiltonian (m conduction orbitals): H AM H m = " d n d + Un d" n d# + V X (f + 0 d + d + f 0 ) + mx,n=0 n f + n f n + mx 1,n=0 t n (f + n f n+1 + f + n+1 f n ) satisfying recursion relation: H m+1 = H m + X m+1 f m+1 f m + t m X (f + m f m+1 + f + m+1 f m ). diagonalize H m = P p E m p pi mm hp use product basis r, e m+1 i ri m e m+1 i with e m+1 i = 0i, "i, #i, "#i to set up matrix of H m+1 via recursion relation and diagonalize,...

16 Iterative diagonalization/truncation Iterative diagonalization v t 0 t 1 t 2 t m-1... H imp ε 0 ε 1 ε 2 ε m Exact, if all states retained. m... In practice, no. states of H m is O(4 m+1 ) (= 4096 at m = 5) Symmetries help H m = P H m N e,s,sz ([ ˆN e, H m ]=[ ~ S 2, H m ]=[S z, H m ]=0) For m > 5, truncate to O(1000) lowest eigenstates Truncation works, provided t m decay sufficiently fast with m

17 Iterative diagonalization/truncation Logarithmic discretization N(ω) 1 Λ 1 Λ 2 Λ 3... Λ 3 Λ 2 Λ 1 1 ω Logarithmic discretization of band kn /D = ± n, n = 0, 1,... with > 1, e.g. =2 ) t m m/2, m m For 1, average over discretizations (Oliveira), kn /D = ± n z+1, n = 1,...(±1, n = 0)

18 Iterative diagonalization/truncation NRG for multiple-channels Transition metal ions l = 2, and m = 2, 1, 0, +1, +2 H = H imp ({" dm }, U, U 0, J)+ X km V mf 0m = P k V kmc km z } { V km (c km d m + h.c.) + X km km c km c km Convert to linear chain form using Lanczos: H = H imp ({" dm }, U, U 0, J)+ X m V m (f 0m d m + h.c.) + +1X n=0 m X nm f nm f nm + +1X n=0 m X t nm (f nm f n+1m + h.c.) NRG: Hilbert space increases by 4 2l+1 = 1024 per shell added!

19 Iterative diagonalization/truncation NRG for multiple-channels: exploiting symmetries For 3-channel S = 3/2 Kondo model, without symmetries, fraction of kept states = 1/4 3 = 1/64. implementing SU(3) symmetries (Hanl, Weichselbaum et al, PRB 2013) increased this fraction to 1/4 2 = 1/16, making the 3-channel Kondo calculation equivalent to a 2-channel calculation allowed comparing to experimental resistivity (R(T )) and dephasing ( ' (T )) data for Fe/Au for

20 Iterative diagonalization/truncation Multi-channel fully screened/single-channel underscreened resistivities Resistivity for 3 channel S = 3/2 Kondo model, Hanl et al PRB Resistivity for 1-channel spin S 1/2 Kondo model, Parks et al Science 2010 n=1 ρ(t,b=0)/ ρ(0,0) AuFe3 AgFe2 AgFe3 T (1) (K) 0.6± ± ±0.2 K T (2) (K) K 1.0± ± ±0.3 n=2 n=3 0 T (3) (K) K 1.7± ± ± T (K)

21 Physical properties Output from NRG procedure Kept/discarded states Energy Levels Kept states Discarded states (Toth et al) m 0-1 m 0 m Many-body eigenvalues Ep m and eigenvectors pi m on all relevant energy scales t m m/2, m = 0, 1,...,N. From eigenvalues calculate Z = P p e thermodynamics Ep and From eigenstates calculate matrix elements, hence Green functions and transport Discarded states not used in iterative diagonalization, but useful for calculating physical properties (see later)

22 Physical properties Thermodynamics Impurity entropy/specific heat S imp [k B ]/ln(2) C imp [k B ] z-averaged z=1 (a) (b) Merker et al k B T/D Entropy S(T )=(E F)/T, E = hh AM i, F = 1 ln Z Specific heat C(T )=k 2 B hham 2 hh AM i 2 i Discretization oscillations at 1 eliminated by z-averaging (here =4, n z = 2).

23 Physical properties Dynamics d-spectral function 2 NRG spectra are discrete ( E qq 0 = E q 0 E q ): π 0 A(ω,T=0) Merker et al 2013 A d (!, T )= X q,q 0 w qq 0(T )! E qq ω/ 0 broaden with logarithmic Gaussians: (! E q 0 q)! e b2 /4 b E q 0 q p e (ln(!/ E q 0 q )/b)2 Discretization oscillations at 1 eliminated by z-averaging (here =10, n z = 8).

24 Linear transport Linear transport Thermopower for U < 0 Conductance: Merker et al Andergassen et al, ε d = -U/2 ε d = 0 ε d = U/2 S (µv/k) V g /T K =0.1 V g /T K =1 V g /T K =10 V g /T K =100 G[2 e 2 /h] T/T K T/T K Linear response transport from spectral functions A(!, T ): G(T )= e2 h S(T )= Z 1 e T X / A(!,T ) z } { T (!, T R P /@!) T (!, T ) R P d! /@!) T (!, T )

25 Kept states in NRG Kept/discarded states Energy Levels Kept states Discarded states (Toth et al) m 0-1 m 0 m Z = P p e Ep Z 6= P N m=0 =Z m(t ) z X } { e E p m p Double counting Conventional approach: Z (T ) Z m (T ) for T T m t m Use discarded states to build a complete set of eigenstates for summing up unambiguously contributions from different H m (Anders & Schiller 2005) All states must reside in same Hilbert space: that of H N

26 Construction of the complete basis set Environment states e v H imp ε 0 t 0 t 1 t 2 t m-1 ε 1 ε 2 ε m... m... N H m qmi = E m q qmi, q =(k, l) =(kept,discarded) Product states lemi = lmi ei with e =(e m+1,...,e N i, m = m 0,...,N form a complete set (Anders & Schiller 2005) NX m=m 0 X lemihlem = 1 le Allows unambiguous summation of contributions from different H m

27 Full density matrix Full density matrix (FDM) Full density matrix and partition function (Weichselbaum & von Delft 2007): = 1 Z (T ) Z (T )= NX m=m 0 NX m=m 0 4 N X e le m X e l E m l lemihlem, Tr = 1 ) E m l NX m=m 0 4 N m Z m (T ) Rewrite as weighted sum of shell density matrices m : NX X = w m m, m = lemi e m=m 0 Z m Tr [ m ]=1, w m = 4 N m Z m Z, le N X E m l hlem. m=m 0 w m = 1

28 Full density matrix Thermodynamics Local observables Ô = n d, n d" n d#, d f 0, S z,... h i hôi = Tr Ô = X hl 0 e 0 m 0 Ô l0 e 0 m 0 i l 0 e 0 m 0 = X X l 0 e 0 m 0 lem w m ll 0 ee 0 mm z } 0 { hl 0 e 0 m 0 lemi e E m l Z m E m l = X e w m Oll Z m = X 4 N m w m Oll m lem m lm O m ll z } { hlem Ô l0 e 0 m 0 i E m l e 4 N m Z m NX = w m Oll m m=m 0,l e Z m E m l

29 Full density matrix Double occupancy Double occupancy: hn d" n d# i!hn d" ihn d# i, U! 0 Double occupancy: hn d" n d# i 1, U FDM (lines) and conventional (symbols) in excellent agreement: FDM simpler, obviates need to chose best shell for given T U/ 0 = (a) ε d / 0 = (b) T/T K <n d n d > <n d n d >

30 Application to Dynamics Dynamics Evaluation of equilibrium retarded Green function (Weichselbaum & von Delft 2007; Peters et al 2006): G AB (t) = i (t)h[a(t), B] + i i (t)tr [ (A(t)B + BA(t))] = i (t) C A(t)B + C BA(t) Consider correlation function C A(t)B : h i C A(t)B = Tr [ A(t)B] =Tr [ A(t)B] =Tr A(t)ˆ1B = X X hlem e iht Ae iht l 0 e 0 m 0 ihl 0 e 0 m 0 B lemi lem l 0 e 0 m 0 = X X hlem e iht Ae iht l 0 e 0 m 0 ihl 0 e 0 m 0 B lemi {z } lem l 0 e 0 m 0 e i(em0 l 0 E m l )t hl 0 e 0 m 0 A lemi

31 Application to Dynamics Dynamics... Peters et al 2006 C m0 >m A(t)B = kemihkem = X z X } { hlem e iht Ae iht l 0 e 0 m 0 ihl 0 e 0 m 0 B lemi lem l 0 e 0 m 0 >m Avoid off-diagonal matrix elements A m6=m0 ll 0 by using 1 = 1 m m mx X 1 m = l 0 e 0 m 0 ihl 0 e 0 m 0 m 0 =m 0 l 0 e 0 NX X X 1 + m = l 0 e 0 m 0 ihl 0 e 0 m 0 = kemih kem. m 0 =m+1 l 0 e 0 ke

32 Application to Dynamics Dynamics... only shell diagonal matrix elements appear: A m ll 0... and reduced density matrices Hofstetter 2000: R m red (k 0, k) = P e hk 0 em kemi G AB (! + i )= X m + X m + X m w m X Z m X lk lkk 0 A m kl Bm lk 0 A m lk Bm kl w m X Z m ll 0 A m ll 0Bm l 0 l E m l E l m + e E m l 0 e! + i (El m 0 e! + i (Ek m El m ) +... Rm red (k 0, k)! + i (El m Ek m) +... E m l )

33 Time-dependent NRG (TDNRG) Motivation: pump-probe spectroscopies Loth et al 2010 Fe on CuN/Cu Spin relaxation

34 Time-dependent NRG (TDNRG) Transient response following a sudden quench H(t) = ( O(t) =Tr = NX m=m 0 t)h i + (t)h f h i e ihf t e ihf t Ô X i!f sr (m)e i(e s m Er m)t Ors m rs /2KK 0 (Anders & Schiller 2005 FDM gen.: Nghiem & Costi 2014) " d (t) = (t)u/2: Nghiem et al 2014 n d(t) N z = t Γ O(t! 0 + ) exact in TDNRG (Nghiem et al PRB 2014) O(t!1) not exact in TDNRG (Anders & Schiller 2005, Nghiem et al PRB 2014)

35 Time-dependent NRG (TDNRG) Multiple quenches and general pulses n-quantum quenches (t) =e ihq p+1 (t p) e ihqp p...e ihq 1 1 e ihq e ihqp p e ihq p+1 (t p) O(t) =Tr h i e ihf t e ihf t Ô Nghiem et al 2014 H Q2 2 z} {... H f = H Qn+1 z} { H Qn n H Q1 1 H i z} { = H Q n t = /2KK X 0 mrs i!q p+1 rs (m, p )e i(e m r E m s )(t p) O m sr Formalism rests ONLY on NRG (truncation) approximation! Other approaches use hybrid (DMRG-TDNRG) and additional approximations (Eidelstein et al 2013)

36 Time-dependent NRG (TDNRG) General pulses and periodic switching linear ramp: Nghiem & Costi periodic switching ε d (t)/γ n d (t) ε d (t) /Γ Quench Ramp tγ n d (t) tγ Linear ramp: time-delay in occupation n d (t) Periodic switching (U = 0): coherent control of nanodevices (qubits,...)

37 Summary NRG: powerful method for strongly correlated systems NRG: still in development for: time-dependent & non-equilibrium phenomena complex multi-channel models (see Mitchell et al PRB 2013) Plenty of work still to do for smart PhD students!