Quantum impurity models Algorithms and applications

Transcription

1 1 Quantum impurity models Algorithms and applications Collège de France, 5 Mai 21 O. Parcollet Institut de Physique Théorique CEA-Saclay, France Motivations : why do we need specific algorithms? A few algorithms and examples of applications Out of equilibrium physics. Open problems.

2 Standard case Quantum impurity models Magnetic impurity in a metallic host Thermodynamics : C, χ, transport : ρ? Nanostructures 2 I(V) Dynamical Mean Field Methods (DMFT) Approximations of a lattice model or a solid by an impurity model in a self-consistent bath Requires computing Local Green function G(ω) Quantum dots. Non-equilibrium Current : I(V), conductance, noise? How to solve quantum impurity models?

3 Impurity solvers : a rich toolbox 3 Exact analytic methods (e.g. Bethe Ansatz, BCFT) Controllable algorithms : Exact diagonalization (ED) Numerical Renormalization group (NRG) Density Matrix Renormalization group (DMRG). Continuous Time Quantum Monte Carlo family (CT-QMC) Approximate solvers (e.g. NCA).

4 Anderson model : Hamiltonian vs Action H = ɛ d σd σ + Un d n d + V kσ c kσ d σ + h.c. + ɛ kσ c kσ c kσ σ=, }{{} k,σ=, }{{} k,σ=, }{{} 4 Local site with interaction Hybridization Free electronic band Integrate the fermionic bath V kσ d σ c kσ S = Hybridization function β d σ(τ)g 1 σ (τ τ )d σ (τ )+ σ (iω n ) k β V kσ 2 iω n ɛ kσ dτun d (τ)n d (τ) G 1 σ (iω n) iω n + ɛ σ (iω n )

5 General quantum impurity model 5 A local problem coupled to a free fermionic bath S eff = β d a(τ)g 1 ab (τ τ )d b (τ )+ β G 1 ab (iω n) = (iω n + µ)δ ab ab (iω n ) (Local) Interaction Bath a,b = 1,N : degrees of freedom (e.g. spin, orbital index,...) dτh local ({d a,d a })(τ)

6 6 Challenges for the impurity solvers Why do we need specific algorithms?

7 Universal regime 7 Anderson impurity in a metallic host (structureless bath) Typical energy scale of the bath Δ = D ev very high energy scale (U.V. cut-off). Low energy, universal regime: separation of scales, scaling laws T,ω,T K << D -D Δ(ω) D ω at low energy -D Δ(ω) Flat Bath D ω How to handle the large separation of scales numerically?

8 Baths can have a low-energy structure 8 Gapped bath (insulator, superconductor) : no Kondo effect. The bath can be pseudo-gapped Withoff-Fradkin PRL 64,1835 (199) (ω) ω r (screening transition when r varies) DMFT bath is self-consistently determined and has a structure at low energy Cluster DMFT : bath can have pseudo-gap Ferrero et al. EPL and PRB 29 Analytical methods (Bethe Ansatz & CFT) can not handle this. How to solve an impurity model in such a bath? DMFT bath evolution close to Mott transition metal insulator A. Georges et al., Rev. Mod. Phys. 68, 13, (1996)

9 Multiorbital models 9 Realistic electronic structure calculations for correlated systems e.g. LDA + DMFT. The impurity is d or f shell : 5, 7 bands (3 with crystal-field splitting) Local interaction can be complex : not only density-density (e.g. spin-spin, Hund s coupling) Example : Iron-based superconductors : 5 bands at Fermi level. Need to treat 5 orbitals. Fe-d LDA band structure of LaFeAsO How correlated are those materials? DMFT computation : Aichhorn et al. (29), Haule et al. (28) How to handle N=5,7,... and a general (local) interaction? E#<4 #

10 Multiple impurity models 1 A few impurities, interacting with a free fermionic bath Richer physics than single impurity (Kondo effect vs singlets) Review : M. Ferrero et al., J. Phys.Cond. Mat (27) Cluster DMFT : systematic interpolation between DMFT and lattice. Overcome important limitations of DMFT (short range AF fluctuations, singlet nature of the insulator, d-sc order,...) Real space point of view k-space point of view Single Impurity Model 4 Impurities Anderson model Brillouin zone patching A B D C G G Cluster size = momentum resolution of the self-energy. 8 sites cluster How to solve 8, 16 coupled impurity models in a self-consistent bath?

11 Exact Diagonalisation 11

12 Hamiltonian representation of the Bath Represent the bath with a finite number of auxiliary sites Necessary step for all Hamiltonian methods (ED, NRG, DMRG...) S = β d σ(τ)g 1 σ (τ τ )d σ (τ )+ β dτun d (τ)n d (τ) 12 H = pσ ɛ pσ ξ pσξ pσ + σ ɛ d d σd σ + Un d n d + pσ Ṽ pσ ( ξ pσ d σ + h.c. ) Approximation of Im Δ(ω) by a finite set of Dirac peaks. Different possible shapes for the bath Ṽpσ 2 σ (iω n ) p iω ɛ pσ Δ(ω) = continuous fraction The energy and hoppings of the bath are effective

13 Principle : Exact Diagonalization Caffarel-Krauth, Represent the bath with a finite number of sites (fit V and ε) 2. Compute the ground state of H (Lanczos) and physical quantities : thermodynamics, G(ω),... Examples : Anderson impurity (DMFT bath), ok with only a few sites (5-1). Cluster DMFT of 2d-Hubbard (normal and superconducting phases) e.g. Civelli et al PRL. 1, 4642 (28). Limitations Scaling with size of cluster/number of orbitals is exponential! Small bath ω resolution is poor impossible to resolve low energy scales like Tk Numerical Renormalization Group 13

14 Numerical Renormalization Group 14

15 Numerical Renormalization Group : principle K. Wilson, Rev. Mod. Phys.47, 773, (1975); R. Bulla et al., Rev. Mod. Phys 8, 395 (28) Use a better representation of the bath, adapted to low energy physics. a. Divide the bath spectral function into logarithmic intervals with parameter Λ >1 b. Reduce to a discrete spectrum by associating 1 site to each slice. c. Transform the bath structure into a semi-infinite chain. a) (ω) b) (ω) 1 1 ω 1 Λ 1 Λ 3 Λ 3 Λ 2 Λ 1 1 Λ 2... ω c) ε ε ε ε V t t 1 t 2

16 Numerical Renormalization Group : principle K. Wilson, Rev. Mod. Phys.47, 773, (1975); R. Bulla et al., Rev. Mod. Phys 8, 395 (28) 16 The effective hopping decays exponentially with n H = H imp (f, f )+α σ c) ε ε ε ε V t t 1 t 2 f ξ ξ 1 (f σξ σ +ξ σ f σ)+ σn Approach the chain by successive finite size Hamiltonians H N = Λ (N 1)/2 H imp (f, f )+α σ H N+1 = ΛH N + Λ N/2 σ (f σξ σ + ξ σ f σ)+ ( ) ε n ξ nσξ nσ + t n (ξ nσξ n+1σ + h.c.) N σn ( ε N+1 ξ N+1σ ξ N+1σ + t n (ξ Nσ ξ N+1σ + h.c.) t n Λ n/2 ( ε n ξ nσξ nσ + t n (ξ nσξ n+1σ + h.c.)) )

17 Numerical Renormalization Group : principle 17 H : N ε V t εn t N 1 Iterative diagonalization of HN, with truncation to Ns lowest states ε ε N V t t N 1 r,s N+1 : r N s (N+1) H N+1 : ε V t t N 1 ε N t N ε N+1 a) E N (r) b) c) d) 1/2 Λ E N (r) E N+1 (r) after truncation Evolution of low energy spectrum of HN Allows also computation of thermodynamics, spectral function, at finite T. R. Bulla et al., Rev. Mod. Phys 8, 395 (28) NRG describe the RG flow to the I.R. fixed point

18 NRG : applications 18 Solution of the Kondo model (1 band, S=1/2) H = kσα ɛ k c kσ c kσ + J K S Entropy, susceptibility vs temperature kk σσ Wilson (1975) ckσ σ σσ c k σ S/ln(2) 4T K χ imp Screening of the impurity χ imp T a T K T K γ imp R W T/T K R. Bulla et al., RMP (28)

19 RG evolution of the low energy spectra Anderson model (symmetric case), Krishnamurthy et al, PRB 21, 13 (198) Low-lying energy levels of HN for odd N for U/D = 1.e-3, U/πΓ = 12.6, Λ=2.5 Local Moment 19 Free Local Moment Strong coupling Strong coupling = Spectrum N-1 free fermions = π/2 phase shift Nozières (1974) c) Strong coupling Free Schematic RG flow Singlet { ε ε ε ε V t t 1 t 2 f ξ ξ 1

20 NRG : susceptibility of the Anderson model 2 Anderson model (symetric case), for U/D = 1.e-3, Λ=2.5 Krishnamurthy et al, PRB 21, 13 (198) U/πΓ = 12.6 Local Moment regime Curie law χ imp 1 4T Τχ(T) Strong coupling (SC) χ imp T a T K Free U/πΓ 1

21 NRG & Conformal symmetry (Cardy; Affleck, Ludwig, 1991; I. Affleck, Acta Phys.Polon. B26 (1995) 1869; condmat/951299) NRG give the finite size spectra of the I.R. fixed point Δ(ω) Δ(ω) Flat Bath at low energy ω ω -D D -D ε k = ε kf + α(k k F ) E(k 1,k 2 )=ε k1 + ε k2 = ε k1 q + ε k2 +q = E(k 1 q, k 2 + q) Huge degeneracies powerful symmetries of free fermions & IR fixed point (Conformal, Kac-Moody,...) H is part of the symmetry algebra! With finite spectra, identify the representation at the fixed point Then use CFT to compute various low energy properties. D 21

22 NRG : applications Abrikosov-Suhl-Kondo Resonance in the spectral function (Anderson model) "pi _ T/TK = 28.7 TITK = 8.5 T/TK = T/TK = /< '<-. --J----- F: - T/TK=.15! v cl" Costi et al.. J. Cond. Mat (1994) ti I I 1. i Asymmetric case, varying T Resolution of NRG is much better at low energy than high energy

23 NRG : scaling property 23 Scaling property of the resistivity as a function of T/Tk in the universal regime Costi et al.. J. Cond. Mat (1994) TDK Fi- 12. The scaled resistivity in the Kondo regime showing Ihe universal behaviour Q low temperature up to approximately STK. The inset for 1 versus (T/TK) shows the expected Femd liquid behaviour for khe resistivity at low temperahlre T c.1t~. The mefficienb c, of the Tz term in the resistivity is found to lie within 8% of khe exact result in

24 NRG & DMFT 24 NRG can also solve DMFT (1 band) A(ω) a) R. Bulla et al. (1999) U/W=1. U/W=1.42 U/W= Specially useful.5 to compute transport (e.g. resistivity), e.g. Transport in organics compound κ-(bedt-ttf) 2 Cu[N(CN) 2 ]Cl P. Limelette, P. Wzietek, S. Florens, A. Georges, T.A. Costi, C. Pasquier, D. Jérome, C. Meziere, P. Batail PRL 91, 1641 (23) But : ω/w difficult away from half-filling. CT-QMC is a now a serious competitor (see later)

25 NRG : strengths & limitations 25 Strengths : Energy scale separation built in. Low energy fixed points, crossover towards low temperature. Solve directly in real frequency ( QMC) Limitations : Not precise at high energy (e.g. details in the Hubbard bands) Does not scale well with the size of the impurity problem Flat bath, no spectral function : 3-4 maximum DMFT : 1 band, undoped only.

26 Quantum Monte Carlo 26

27 Monte Carlo sampling Cf e.g. Werner Krauth s book Statistical Mechanics : algorithms & computations Partition function and operator averaging : (assume p(x) >) Z = dx p(x), A = 1 dx A(x)p(x) C Z C 27 Configuration space Probability of configuration x e.g. in classical model : p(x) e βe(x) Principle : use a Markov chain in configuration space. Average replaced by average over the Markov chain. Transition rate Wx y : probability to go from x to y Detailed balance : Ergodicity property : W x y W y x = p(y) p(x) It is possible to reach y from x, x,y in a finite number of steps.

28 Metropolis algorithm To build the Markov chain: Propose moves in the configuration space Accept them with some probability, such that : N. Metropolis et al. J. Chem. Phys Proposition probability (chosen) Acceptance probability (computed) W x y = Wx y prop Wx y acc (1, W acc x y min p(y)w prop y x p(x)w prop x y }{{} R x y )

29 A textbook example : the Ising model 29 Ising model : H = J i,j σ i σ j, σ i = ±1 Configuration : the value of all the Ising spins. p({σ i }) e βh[{σ i}] MC Move (simplest) : flip spin k chosen at random The probability ratio is easy to compute since H is local

30 The sign problem 3 What if p(x) is not always positive? Use p(x) as the probability! A = 1 Z C dx A(x)p(x) = C dx ( A(x) sign(p(x))) p(x) C dx ( sign(p(x)) ) p(x) But generically, the denominator (average of sign (p(x)) decays exponentially as temperature is lowered or in large volume limit. The QMC is correct if <sign> 1, but becomes untractable when <sign> (large error bars). A major limitation of Quantum Monte Carlo (specially for fermions) The sign pb is not intrinsic : it depends on the basis/rewriting of Z! We can hope to find better expression for Z.

31 Monte Carlo 31 A QMC algorithm : Rewrite Z, ideally as a sum of positive terms. Find local ergodic moves Advantages : QMC is a very flexible technique QMC is massively parallel by construction. Drawbacks : Convergence is slow, like 1/ time Sign problem may be severe! Monte Carlo is just a technique to compute sums. How to rewrite Z, which move to use, etc... is your choice!

32 Quantum Monte Carlo for impurity models 32 Hirsch-Fye algorithm : Hirsch-Fye PRL (1986) The historical algorithm: uses a fixed time grid, is limited to densitydensity interaction. The Continuous Time Revolution (CT-QMC) : Expansion in interactions (CT-Int): A.N. Rubtsov et al., PRB (25) Expansion around atomic limit (CT-Hyb): P. Werner et al, PRL (26) Auxiliary field (CT-AUX): similar to Hirsch-Fye, but in continuous time) : E.Gull et al., EPL (28) Work in imaginary time (Matsubara formalism) No sign problem for single impurity (and some N orbital cases). Sign problem reappears for large cluster of impurities CT-QMC are several orders of magnitude faster than Hirsch-Fye and exact (up to Monte Carlo error bars) : no time discretization.

33 Continuous time QMC : principle Write a perturbative expansion of the partition function : 33 H = H a + H b Z = Tr T τ e βh a exp = n = n ( 1) n β τ 1 <τ 2 <...τ n [ dτ 1... β β dτh b (τ) ] [ ] dτ n Tr e βh a H b (τ n )H b (τ n 1 )... H b (τ 1 ) τ n 1 ( τ ) n w(n, γ, τ 1,..., τ n ) }{{} γ Γ n p(x) = x C p(x) Configurations Representation of the configurations a) b) x =(n, γ, τ 1, τ 2,...τ n ) c) β d) τ 1 τ 2 β τ 1 β τ 1 τ 2 τ 3 β

34 Continuous time QMC : principle (II) 34 A CT-QMC move τ 1 τ 2 β remove insert τ 1 τ 3 τ 2 β Move : add/remove one interaction term (= change n by 1 ), e.g. x = (n,...) configuration with n vertices y = (n+1,...) configuration with n+1 vertices Wx y prop = τ Wy x prop = 1 β n +1 The Metropolis rate has a finite limit. Prokofiev (1996) R x y = p(y)w prop y x p(x)w prop x y = w(y)( τ ) n+1 w(x)( τ ) n β τ (n + 1) The algorithm can be formulated directly in continuous time

35 Which perturbative expansion? 35 S eff = β c a(τ)g 1 ab (τ τ )c b (τ )+ β G 1 ab (iω n) = (iω n + µ)δ ab ab (iω n ) a,b = 1,N : degree of freedom (e.g. spin, orbital index,...) dτh local ({c a,c a })(τ) Bath Interaction Expansion in power of the interactions : A.N. Rubtsov et al., Phys. Rev. B 72, (25) Expansion in power of hybridization (around atomic limit) : P. Werner, A. Comanac, L. de Medici, M. Troyer, A. J. Millis, PRL 97, 7645 (26); P. Werner, A.J. Millis, Phys. Rev. B 74, (26)

36 Expansion in interaction 36 Standard perturbative technique at finite temperature. S eff = β σ=, dτdτ c σ(τ)g 1 σ (τ τ )c σ (τ)+ β dτun (τ)n (τ) Z Z =1 U β dτ 1 n (τ 1 )n (τ 1 ) + U 2 2 β dτ 1 dτ 2 T τ n (τ 1 )n (τ 1 )n (τ 2 )n (τ 2 )... Using Wick Theorem : Z Z = n 1 n! β dτ 1... dτ n ( U) [ ] n det G σ(τ i τ j ) 1 i,j n σ=, }{{} w(n, {τ i })

37 S eff = β Expansion in hybridization : Z = n < i=1 n dτ i dτ i Expansion in hybridization c a(τ)g 1 ab (τ τ )c b (τ )+ w is positive (in single impurity problem) Hlocal can be anything (but exponential scaling in N!) Green function computation (or higher order correlations functions): G ab (τ) = 1 δz Z δ ba ( τ) G ab (τ) = n < β G 1 ab (iω n) = (iω n + µ)δ ab ab (iω n ) a i,b i =1,N n dτ i dτ i i=1 dτh local ({c a,c a })(τ) 37 P. Werner et al, PRL (26) [ det ai,b j (τ i τ j) ] ( ) n Tr T e βh local c a 1 i,j n i (τ i )c bi (τ i) i=1 }{{} w(n, {a i,b i }, {τ i }) a i,b i =1,N a,b = 1,N [ ] 1 a i,b j (τ i τ j)δ(τ i τ j = τ)δ ai =aδ bj =bw({τ i })/Z

38 CT-QMC : efficient algorithms Histogram R of expansion order (Werners algo, DMFT, βt=1, δ=, various U) 38 p(k) P. Werner et al, Phys. Rev. Lett 97, 7645 (26) <n> order k βU H-F U/t= U/t=3 U/t=4 U/t=5.5βU Rubtsov U/t Matrix Size Typical matrix size vs β (DMFT, U/t=1) E. Gull et al, Phys. Rev. B 76, (27) Weak Coupling Algorithm Hybridization Expansion Hirsch Fye !t Complexity <n>^3 All diverge like 1/T (singular at T=), but huge prefactor differences CT-QMC is much more efficient

39 CT-QMC : some applications 39

40 Comparison NRG-CTQMC Im Σ(ω) by CTQMC (Werner s algorithm) and NRG for DMFT, 1 band, Bethe Lattice, Beta=4, U = 5.2 et D = 1. Continued by Padé method to real axis from Matsubara Im Σ(ω) Little finite 2 NRG temperature effect -2 Im Σ(ω) 4 M. Ferrero & P. Cornaglia -4-6 CTQMC NRG ω ω

41 Back to multiorbital models Example : Iron-based superconductors LaFeAsO, LDA +DMFT, Aichhorn et al. (29). 41 Possible to solve the 5-band impurity model with Werner s algorithm. Degree of correlations of those materials? Moderate. Extract quasi-particle residue Z /effective mass m* from Matsubara self-energy (Z.62, m* 1.62) Using analytic continuation method, spectral function... ImΣ(iω n ) ω n ( 1 1 Z )

42 Cluster DMFT & cuprates 42 Experiments : Nodal-antinodal dichotomy in cuprates. Nodal region : Quasi-Particle A(k, ω = ) ARPES!!"#$%!# Brillouin zone patchings (,!) ky (!,!) () P - Shen et al. Science 37, 91 (25) Antinodal region: No Quasi-Particle Theory : sector selective Mott transition Some sector of the Brillouin Zone (C) become insulating first. *$!" &' %&% )*+,-.+/,+ %&'( /1,-.+/,+ Gull, OP, Werner, Millis PRB (29) Werner, Gull, OP, Millis PRB (29) Ferrero, Cornaglia, De Leo, OP, Kotliar, Georges, EPL and PRB 29 Only possible with CT-QMC (various flavours) $% $ P + (,) 2 sites cluster D B A C 8 sites cluster (!,) kx

43 CTQMC : strengths & limitations 43 Strengths : A lot faster than before. Solve more general interactions (except CT-AUX) Some have good scaling with number of orbitals/sites Limitations : Works in Matsubara : analytical continuation is an ill-posed pb. Still long for complex interactions & low symmetry... Werner s algorithm scales exponentially with size of the local pb. Open question : Even faster/more precise algorithms, other rewriting of Z...

44 Out of equilibrium physics 44

45 Motivations 45 Nanostructures Quenches in cold atoms I(V) Quantum dots. Current : I(V). Steady state computations Lattice to impurity via DMFT e.g. Change interaction at t=, study relaxation, etc... Methods, e.g. : Time dependent NRG F. Anders et al. Phys. Rev. Lett. 1, 8689 (28) Real time QMC...

46 Real Time Quantum Monte Carlo 46 Diagrammatic Monte Carlo : use the Keldysh contour! Start non-equilibrium at t=, and try to relax to steady state study quench Mühlbacher, Rabani (28) Werner et al (29) Schiro and Fabrizio (29) U= V> U> V> t 1 s 1 t 2 s 2 U> V> t 1 t s 2 s 3 s 1 2 t 3 t max t max t 4 s 4 t3 s3 Switch on interaction Sign problem due to real time (i factor in time evolution) Computation limited in time (does not reach the Kondo time) U> V= iβ t s 6 6 t s 7 7 t 5 s 5 t 4 s 4 Switch on voltage

47 Real Time QMC : applications 47 Quench in Hubbard model M. Eckstein et al, ArXiv: DMFT, change U at t=. Dynamical phase transition at U = 3.2 (?) Solution of a quantum dots (in some regimes) Werner et al. (21) U/Γ=6 V/Γ=8 V/Γ=6 I/Γ.4.2 V/Γ=4 V/Γ=2 V/Γ= tγ V/Γ=.5 Time evolution of the current for different voltage biases (U/Γ=6,T=). In the initial state, the current is given by the steady state current through the non-interacting dot. Interaction turned on at t= FIG. 53 Time evolution of the current for different voltage biases (U/Γ = 6, T = ). In the initial state, the current is given by the steady state current through the non-interacting dot. At time t =, the interaction is turned on. From

48 Conclusion 48 A lot of progress recently on Continuous Time QMC Enable us to solve more complex/realistic models. Open issues : Faster QMC for low symmetry realistic atoms? Real frequency QMC solution? (t-)dmrg for small cluster? Use other diagrammatics? e.g. NCA, cf Gull et al. arxiv: Better methods out of equilibrium?