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The Numerical Renormalization Group Ralf Bulla Institut für Theoretische Physik Universität zu Köln 4.0 3.0 Q=0, S=1/2 Q=1, S=0 Q=1, S=1 E N 2.0 1.0 Contents 1. introduction to basic rg concepts 2. introduction to quantum impurity physics 0.0 0 20 40 60 80 N 3. the Numerical Renormalization Group application to the single-impurity Anderson model 4. fixed points in quantum impurity models 5. summary 624th Wilhelm und Else Heraeus-Seminar, Simulating Quantum Processes and Devices Physikzentrum Bad Honnef, September 21, 26

1. introduction to basic rg concepts consider a model on a one-dimensional lattice, with operators a i (i: lattice site), parameters J, h,..., and Hamiltonian H(J, h,...). i=1 2 3 4 5 6... J J J J J a 1 a 2 a 3 a 4 a 5 a... 6 h h h h h h combine two sites to give (effectively) one site with operators a i and parameters J, h,.... J J... a 1 a a... 2 3 h h h we assume that the Hamiltonian H of the effective model is of the same form, with renormalized parameters: H = H(J, h,...). then rescale the model to the original lattice spacing:... J J a a... 1 2 a 3... h h h

the mapping H H is a renormalization group transformation H(J, h,...) = R{H(J, h,...)} with K = (J, h,...): R( K ) = K h now consider a sequence of transformations: K K... R K K R K K this results in a trajectory in parameter space: J flow diagrams and fixed points h stable fixed point h unstable fixed point K * R( K ) = K K* J all perturbations are irrelevant J at least one relevant perturbation

the central issue: How does the behaviour of the system change under a scale transformation? the physics of the problem described as a flow between fixed points. A B S: starting point C identify the fixed points of the model (and their physical meaning) identify the relevant/irrelevant perturbations if possible: describe the full flow from S to C and finally: calculate physical properties

some technical issues: how to perform the mapping H H for a given model? Ising model:(1d) in the partition function Z, sum over every second spin but: the whole strategy depends on the details of the model spins/fermions/bosons dimension, lattice structure, etc. = for a given model, it is a priori not clear whether a successful rg scheme can be developed at all in the following: (numerical) renormalization group for quantum impurity models Wilson s NRG for the single-impurity Anderson model interpretation of fixed points and flow diagrams

2. introduction to quantum impurity physics the Kondo effect: magnetic impurities in metals T -dependence of resistivity ρ metal 0 T

2. introduction to quantum impurity physics the Kondo effect: magnetic impurities in metals T -dependence of resistivity scattering processes of conduction electrons at magnetic impurities ρ c V U V c f f f metal 0 T

modelling of magnetic impurities in metals here: single-impurity Anderson model [A.C. Hewson, The Kondo Problem To Heavy Fermions, CUP 1993] H = ε f f σf σ + Uf f f f σ + ε k c kσ c kσ + V kσ kσ ( f σ c kσ + c kσ fσ ) the model describes: formation of local moments: f, f scattering of conduction electrons screening of local moments below temperature scale T K

3. the numerical renormalization group K.G. Wilson, Rev. Mod. Phys. 47, 773 (1975) Kondo problem review: R. Bulla, T. Costi, and Th. Pruschke, Rev. Mod. Phys. 80, 395 (2008) (ω) conduction band 1 0 1 ω impurity

0 E N V Λ 1 3 Λ 3 Λ 2 Λ 1 ε 0 ε 0 1/2 Λ E N E N+1 N ε N ε N ε N+1 after truncation (ω) 2... ω Λ 1 1 Λ (ω) 1. NRG-discretization parameter Λ > 1 (ω) 1 1 ω ε0 ε1 ε2 ε3 V t 0 t 1 t 2 1 Λ 1 Λ 2 Λ 3... Λ 3 Λ 2 Λ 1 1 ω H N : ε0 ε V t 0 t N 1 t 0 t N 1 r,s N+1 : r N s (N+1) H N+1: V t 0 t N 1 t N a) b) c) d)

0 E N V Λ 1 3 Λ 3 Λ 2 Λ 1 ε 0 ε 0 1/2 Λ E N E N+1 N ε N ε N ε N+1 after truncation (ω) 2... ω Λ 1 1 Λ 2. logarithmic discretization (ω) (ω) 1 1 ω ε0 ε1 ε2 ε3 V t 0 t 1 t 2 1 1 ω H N : ε0 ε V t 0 t N 1 t 0 t N 1 r,s N+1 : r N s (N+1) H N+1: V t 0 t N 1 t N a) b) c) d)

0 E N V Λ 1 3 Λ 3 Λ 2 Λ 1 ε 0 ε 0 1/2 Λ E N E N+1 N ε N ε N ε N+1 after truncation (ω) 2... ω Λ 1 1 Λ 3. mapping on semi-infinite chain (ω) 1 1 ω ε ε ε ε 0 1 2 3 ε0 ε1 ε2 ε3 V t 0 t 1 t 2 V t 0 t 1 t 2 H N : ε0 ε V t 0 t N 1 t 0 t N 1 r,s N+1 : r N s (N+1) H N+1: V t 0 t N 1 t N a) b) c) d)

0 E N V Λ 1 3 Λ 3 Λ 2 Λ 1 ε 0 1/2 Λ E N E N+1 N ε N after truncation (ω) 2... ω Λ 1 1 Λ 4. iterative diagonalization (ω) ε 0 εn 1 1 ω H : N V t 0 t N 1 ε 0 ε N ε0 ε1 ε2 ε3 V t 0 t 1 t 2 V t 0 t N 1 H N : ε0 ε V t 0 t N 1 r,s N+1 : r N s (N+1) t 0 t N 1 r,s N+1 : r N s (N+1) ε 0 ε N ε N+1 ε 0 ε N ε N+1 H N+1: V t 0 t N 1 t N a) b) c) d) H N+1 : V t 0 t N 1 t N

0 E N V Λ 1 3 Λ 3 Λ 2 Λ 1 ε 0 1/2 Λ E N E N+1 N ε N after truncation (ω) 2... ω Λ 1 1 Λ 5. truncation (ω) a) E N b) c) d) 1/2 Λ E N E N+1 after truncation 1 1 ω ε ε ε ε V t 0 t 1 t 2 H N : ε0 ε V t 0 t N 1 t 0 t N 1 r,s : r s (N+1) N+1 N ε 0 ε N ε N+1 H N+1: V t 0 t N 1 t N 0 a) b) c) d)

logarithmic discretization starting point: siam in the integral representation: H imp = σ H bath = σ H imp bath = σ ε f f σf σ + Uf f f f, 1 1 1 1 dε g(ε)a εσa εσ, ) dε h(ε) (f σa εσ + a εσf σ. Λ > 1 defines a set of intervals with discretization points ± x n = Λ n, n = 0, 1, 2,.... 1 Λ 1 Λ 2 Λ 3... Λ 3 Λ 2 Λ 1 1 ε

width of the intervals: d n = Λ n (1 Λ 1 ) within each interval: introduce a complete set of orthonormal functions { ψ np(ε) ± 1 = dn e ±iωnpε for x n+1 < ±ε < x n 0 outside this interval. expand the conduction electron operators a εσ in this basis a εσ = [ ] a npσψ np(ε) + + b npσψnp(ε), np assumption: h(ε) = h ± n, x n+1 < ±ε < x n, the hybridization term then takes the form: 1 1 dε h(ε)f σa εσ = 1 π f σ [ γ + n a n0σ + γn ] b n0σ the impurity couples only to the p = 0 components of the conduction band states! n

the conduction electron term transforms to: 1 (ξ + n a npσa npσ + ξ n b npσb npσ ) dε g(ε)a εσa εσ = 1 np + n,p p ( α + n (p, p )a npσa np σ α n (p, p )b npσb np σ ). For a linear dispersion, g(ε) = ε, one obtains: α ± n (p, p ) = 1 Λ 1 2πi ξ ± n = ± 1 2 Λ n (1 + Λ 1 ), Λ n p p exp [ 2πi(p p) 1 Λ 1 ].

structure of the Hamiltonian p.................. 3 b 02 σ 2 1 α (p,p ) 0 n=0 1 2...... 2 1 0 ε impurity couples to p=0 only discretization of the Hamiltonian: drop the terms with p 0 in the conduction band

now: relabel the operators a n0σ a nσ, etc., the discretized Hamiltonian takes the form: [ξ + n a nσa nσ + ξ n b nσb nσ ] H = H imp + nσ [ 1 ( + γ + n a nσ + γ ) ] n b nσ π + σ 1 π σ f σ [ n n ( γ + n a nσ + γ n b nσ) ] f σ (ω) 1 1 ω

mapping on a semi-infinite chain orthogonal transformation of the operators {a nσ, b nσ} to a new set of operators {c nσ} such that the discretized Hamiltonian takes the form: H = H imp + V σ [f σc 0σ + c 0σ fσ ] + )] [ε nc nσc nσ + t n (c nσc n+1σ + c n+1σ cnσ σn=0 ε ε ε ε 0 1 2 3 V t 0 t 1 t 2 the mapping is equivalent to the tridiagonalization of a matrix

for a constant density of states ( ) ( 1 + Λ 1 1 Λ n 1) t n = 2 1 Λ 2n 1 1 Λ 2n 3 Λ n/2. In the limit of large n this reduces to t n 1 2 ( 1 + Λ 1) Λ n/2. this means: in moving along the chain we start from high energies (U, V, D) and go to arbitrary low energies high energies renormalization group low energies in real space: double the system size by adding two sites to the chain (for Λ = 2)

iterative diagonalization the chain Hamiltonian can be viewed as a series of Hamiltonians H N (N = 0, 1, 2,...) which approaches H in the limit N : H = lim N Λ (N 1)/2 H N, with + H N = Λ (N 1)/2 [H imp + N ε nc nσc nσ + σn=0 N 1 σn=0 ξ0 π ) (f σc 0σ + c 0σ fσ σ t n (c nσc n+1σ + c n+1σ cnσ ) ]. two successive Hamiltonians are related by H N+1 = ΛH N + Λ N/2 σ +Λ N/2 σ ε N+1 c N+1σ c N+1σ t N (c Nσ c N+1σ + c N+1σ c Nσ ),

starting point: H 0 = Λ 1/2 [ H imp + σ ε 0 c 0σ c 0σ + ξ0 π σ ( f σc 0σ + c 0σ fσ ) ]. renormalization group transformation: H N+1 = R (H N ) H N : ε 0 εn V t 0 t N 1 ε 0 ε N V t 0 t N 1 r,s N+1 : r N s (N+1) ε 0 ε N ε N+1 H N+1 : V t 0 t N 1 t N

set up an iterative scheme for the diagonalization of H N construct a basis for H N+1 r; s N+1 = r N s(n + 1). diagonalization: new eigenenergies E N+1 (w) and eigenstates w N+1 truncation: a) E N b) c) d) 1/2 Λ E N E N+1 after truncation 0

renormalization group flow and fixed points plot the rescaled many-particle energies E N as a function of N (odd N only) fixed points of the single-impurity Anderson model 4.0 3.0 NRG flow diagram Q=0, S=1/2 Q=1, S=0 Q=1, S=1 FO: free orbital LM: local moment SC: strong coupling E N 2.0 1.0 0.0 0 20 40 60 80 N parameters: ε f = 0.5 10 3, U = 10 3, V = 0.004, and Λ = 2.5

renormalization group flow and fixed points plot the rescaled many-particle energies E N as a function of N (odd N only) fixed points of the single-impurity Anderson model FO: free orbital LM: local moment SC: strong coupling E N NRG flow diagram 4.0 Q=0, S=1/2 Q=1, S=0 3.0 Q=1, S=1 2.0 FO LM SC 1.0 0.0 0 20 40 60 80 N parameters: ε f = 0.5 10 3, U = 10 3, V = 0.004, and Λ = 2.5

4. fixed points in quantum impurity models quantum impurity models show a variety of different fixed points here: quantum critical point in the soft-gap Anderson model single-impurity Anderson model with hybridization function (ω) = ω r interacting fixed point (ω) 1.0 0.5 r=0.5 r=1.0 r=2.0 0.0-1.0 0.0 1.0 ω non-fermi liquid fixed point in the two-channel Kondo model J J Majorana fermions

soft-gap Anderson model phase diagram flow diagrams 0.06 < c = c > c b 3.0 local moment quantum critical strong coupling a b c 0.04 SC 2.0 0.02 symmetric case asymmetric case LM Λ N/2 E N 1.0 0.00 0.00 0.25 0.50 0.75 r quantum phase transition between SC and LM phases 0.0 0 50 100 N 0 50 100 N 0 50 100 N non-trivial structure of the qcp [H.-J. Lee, R. Bulla, M. Vojta J. Phys.: Condens. Matter 17, 6935 (2005)]

two-channel Kondo model J J H = kσ ε k c kσα c kσα + J α α S s α non-fermi liquid fixed point with residual impurity entropy S imp = 1 2 ln 2 2.5 NRG flow diagram characteristic structure of the non-fermi liquid fixed point E N 2 1.5 1 0.5 0 0 10 20 30 40 50 60 N

structure of the fixed point E two sectors of excitations degeneracies 9/8 1... 32 26... 26... 32 5/8 1/2 12 10 10 12 1/8 0 4 2 2 4 sector I sector II the many-particle spectra of each sector can be constructed from single-particle spectra of Majorana fermions

single-particle spectra E sector I E sector II 5/2 5/2 2 2 3/2 3/2 1 1 1/2 1/2 0 0 [R. Bulla, A.C. Hewson, G.-M. Zhang, Phys. Rev. B 56, 11721 (1997)]

where does this structure come from? vector and scalar Majorana fermion chains with different boundary conditions impurity conduction electrons 0 1 2 3 4 vector scalar from the numerical analysis of the two-channel Anderson model we obtain emergent fractionalized degrees of freedom (Majorana fermions) at the low-energy fixed point!

5. summary in this talk: a short introduction to the renormalization group concept quantum impurity physics the NRG method focus on: flow diagrams and fixed point I did not discuss: how to calculate physical properties all the recent developments which considerably extended the power of the NRG method; see the work of F. Anders, Th. Costi, J. von Delft, A. Mitchell, A. Weichselbaum,... relation to other renormalization group methods DMRG frg

(r) 2.0 E N 1.0