Quantum impurities in a bosonic bath

Ralf Bulla Institut für Theoretische Physik Universität zu Köln 27.11.2008

contents introduction quantum impurity systems numerical renormalization group bosonic NRG spin-boson model bosonic single-impurity Anderson model summary

quantum impurity systems quantum impurity systems impurity bosonic bath fermionic bath small system large system - might have a complicated structure - small number of degrees of freedom - simple structure - continuous spectrum of degrees of freedom

quantum impurity systems impurities in a fermionic bath Kondo effect A.C. Hewson, The Kondo Problem to Heavy Fermions temperature dependence of resistivity ρ metal 0 T

quantum impurity systems impurities in a fermionic bath Kondo effect A.C. Hewson, The Kondo Problem to Heavy Fermions temperature dependence of resistivity scattering processes of conduction electrons at magnetic impurities ρ c V U V c f f f metal 0 T

quantum impurity systems impurities in a bosonic bath example: spin-boson model D bosonic bath A H = 1 2 σx + 1 2 ɛσz + X i + 1 X 2 σz λ i (a i + a i ) i ω i a i a i [A.J. Leggett et al., Rev. Mod. Phys. 59, 1 (1987)] describes two-level systems in a dissipative environment (qubits, electron transfer systems, etc.) σ x oscillations σ z(a i + a i ) friction (dissipation)

quantum impurity systems impurities in a bosonic bath example: spin-boson model 1 0.8 D bosonic bath A describes two-level systems in a dissipative environment (qubits, electron transfer systems, etc.) σ x oscillations σ z(a i + a i ) friction (dissipation) P(t) 0.6 0.4 0.2 α=0.1 α=0.3 α=0.5 α=0.7 α=1.4 0 0 50 100 150 200 t [F.B. Anders, A. Schiller, Phys. Rev. B 74, 245113 (2006)]

quantum impurity systems electron transfer P(t) occupation at donor site D A P(t) e kt donor bridge acceptor t

quantum impurity systems electron transfer P(t) occupation at donor site D A P(t) e kt donor bridge acceptor quantum-mechanical description: tunneling t ψ D ψ A P 2 σ c Dσ c Aσ + c Aσ c Dσ

quantum impurity systems coupling to the environment [R.A. Marcus, J. Chem. Phys. 24, 966 (1956)] D A = dissipation of the energy modeled by the coupling to a bosonic bath [A. Garg, J.N. Onuchic, V. Ambegaokar, J. Chem. Phys. 83, 4491 (1985)] X ω nb nb n + 1 2 (n D n A ) X n n λ n b n + b n

bosonic NRG numerical renormalization group (NRG) 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 (r) V ε 0 ε 0 1/2 Λ E N (r) 3 E N+1 (r) N ε N ε N ε N+1 after truncation bosonic NRG (ω) 2... ω Λ Λ 1 Λ 3 Λ 2 Λ 1 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 (r) V ε 0 ε 0 1/2 Λ E N (r) 3 E N+1 (r) N ε N ε N ε N+1 after truncation bosonic NRG (ω) 2... ω Λ Λ 1 Λ 3 Λ 2 Λ 1 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 (r) V ε 0 ε 0 1/2 Λ E N (r) 3 E N+1 (r) N ε N ε N ε N+1 after truncation bosonic NRG (ω) 2... ω Λ Λ 1 Λ 3 Λ 2 Λ 1 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 (r) V ε 0 1/2 Λ E N (r) 3 E N+1 (r) N ε N after truncation bosonic NRG (ω) 2... ω Λ Λ 1 Λ 3 Λ 2 Λ 1 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 (r) V ε 0 1/2 Λ E N (r) 3 E N+1 (r) N ε N after truncation bosonic NRG (ω) 2... ω Λ Λ 1 Λ 3 Λ 2 Λ 1 1 1 Λ 5. truncation (ω) a) E N (r) b) c) d) 1/2 Λ E N (r) E N+1 (r) 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)

bosonic NRG bosonic NRG bath spectral function J(ω) ε ε ε 0 1 2 3 4 t t t t 0 1 ε 2 3 ε... 0 Λ 2 Λ 1 1 ω in contrast to fermionic case: keep only a finite number of boson states for each added site: N b 10 for details see: R. Bulla, H.-J. Lee, N.-H. Tong, M. Vojta, Phys. Rev. B 71, 045122 (2005)

bosonic NRG important: choice of the basis for the added site "standard" basis: s(n + 1) = { n N+1 } with b N+1 b N+1 n N+1 = n n N+1 n = 0, 1,... N b 1 works very well...

bosonic NRG important: choice of the basis for the added site "standard" basis: s(n + 1) = { n N+1 } with b N+1 b N+1 n N+1 = n n N+1 n = 0, 1,... N b 1 works very well...... except for displacement = general question: how to construct the optimal basis?

bosonic NRG applications of the bosonic NRG spin-boson model bosonic single-impurity Anderson model electron transfer systems - S. Tornow, N.-H. Tong, R. Bulla, Europhys. Lett. 73, 913 (2006) - S. Tornow, R. Bulla, F.B. Anders, A. Nitzan, Phys. Rev. B 78, 035434 (2008) Bose-Fermi Kondo model - M.T. Glossop, K. Ingersent, Phys. Rev. Lett. 95, 067202 (2005) Kondo lattice model within extended DMFT - M.T. Glossop, K. Ingersent, Phys. Rev. Lett. 99, 227203 (2007) - J.-X. Zhu, S. Kirchner, R. Bulla, Q. Si, Phys. Rev. Lett. 99, 227204 (2007)

T = 0 phase diagram of the spin-boson model J (ω) = 2παω s ω s < 1: sub-ohmic s = 1: ohmic s > 1: super-ohmic

T = 0 phase diagram of the spin-boson model J (ω) = 2παω s ω s < 1: sub-ohmic s = 1: ohmic s > 1: super-ohmic α 1.5 1.0 0.5 A.J. Leggett et al., Rev. Mod. Phys. 59, 1 (1987) =10-1 =10-3 =10-5 localized delocalized 0.0 0.0 0.5 1.0 1.5 s phase transition only in the ohmic case (s=1) calculations valid in the limit 0

T = 0 phase diagram of the spin-boson model J (ω) = 2παω s R. Bulla, N.-H. Tong, and M. Vojta Phys. Rev. Lett. 91, 170601 (2003) 1.5 ω s < 1: sub-ohmic s = 1: ohmic s > 1: super-ohmic α 1.0 0.5 =10-1 =10-3 =10-5 localized delocalized 0.0 0.0 0.5 1.0 1.5 s line of quantum critical points for 0 < s < 1 terminating at s = 1 α c 1 s, this means: α c( 0) = 0 for 0 < s < 1

T = 0 phase diagram of the spin-boson model J (ω) = 2παω s R. Bulla, N.-H. Tong, and M. Vojta Phys. Rev. Lett. 91, 170601 (2003) 1.5 ω s < 1: sub-ohmic s = 1: ohmic s > 1: super-ohmic α 1.0 0.5 =10-1 =10-3 =10-5 localized delocalized 0.0 0.0 0.5 1.0 1.5 s existence of a phase transition in the sub-ohmic case: S. Kehrein and A. Mielke, Phys. Lett. A 219, 313 (1996)

evidence for a line of critical points: structure of the fixed points calculation of physical properties example: entropy 0.80 s=0.8, =0.01 S imp 0.60 0.40 0.20 α=0.1233 α=0.1252 α=0.1254 α=0.1255 α=0.1255160 α=0.1255170 α=0.12553 α=0.1257 α=0.1259 α=0.129 0.00 10 12 10 8 10 4 10 0 critical exponents T M. Vojta, N.-H. Tong, and R. Bulla, Phys. Rev. Lett. 94, 070604 (2005)

Failure of quantum-classical mapping? spin-boson model one-dimensional Ising model H I = P ij J ijs z i S z j J(ω) ω s J ij = J/ i j 1+s (long range) Ising model: 0 < s < 1/2 : mean-field exponents, β = 1/2, ν = 1/s 1/2 < s < 1 : non-trivial exponents critical exponents defined by M loc (α > α c, T = 0, ε = 0) (α α c) β, T α α c ν

A. Winter, H. Rieger, M. Vojta, and R. Bulla The quantum phase transition in the sub-ohmic spin-boson model: Quantum Monte-Carlo study with a continuous imaginary time cluster algorithm arxiv:0807.4716

Bose-Hubbard model µ /V H = X i + 1 2 V X i µb i b i J X <ij> b i b j b i b i b i b i 1 3 MI N=3 2 1 MI N=2 SF MI N=1 M. P. A. Fisher et al., Phys. Rev. B 40, 546 (1989) J/V

Is it possible to develop a dynamical mean-field theory for the Bose-Hubbard model? there are many open questions: limit of infinite dimensionality? proper treatment of the superfluid phase? selfconsistency equations? structure of the effective impurity model? K. Byczuk, D. Vollhardt, Phys. Rev. B 77, 235106 (2008)

bosonic single-impurity Anderson model H = ε 0 b b + 1 2 Ub b b b 1 + X k ε k b k b k + X k V k b k b + b b k (ω) = π X k Vk 2 δ(ω ε k ) = 2π α ωc 1 s ω s, 0 < ω < ω c H.-J. Lee and R. Bulla Quantum Phase Transitions in the Bosonic Single-Impurity Anderson Model Eur. Phys. J. B 56, 199 (2007)

phase diagram T = 0, s = 0.6, U = 0.5 Mott phases separated from BEC phase by lines of quantum critical points 4 3 4 -ε 0 /U 2 1 0 3 2 1 BEC n imp =0 n imp =1 n imp =2 n imp =3 n imp =4-1 0-2 0 0.1 0.2 0.3 0.4 αω c /U

impurity occupation 2 1.5 α=0 α=0.007 α=0.014 α=0.028 n imp (T=0) 1 0.5 0-0.5 0 0.5 1 1.5 -ε 0 /U symbols: Mott phase (non-integer values!) dashed lines: BEC phase

impurity spectral function 20 10 V=0.01, U=0.5, ε=-0.7 (b=0.3) V=0.07, U=0.5, ε=-0.7 (b=0.4) V=0.15, U=0.5, ε=-0.7 (b=0.5) A(ω) 0-10 -0.5 0 0.5 1 ω

What do we expect for the spectral function in the Mott phase? Α(ω) µ ω spectral weight below the chemical potential A(ω) density of states of the non-interacting bosonic bath in DMFT = mapping on an effective impurity model does not work!

summary numerical renormalization group calculations for quantum impurities in a bosonic bath spin-boson model bosonic single-impurity Anderson model the next steps: How to construct the optimal basis? Bose-Hubbard model within DMFT generalized spin-boson models: - coupled spins - nonlinear coupling thanks to: F. Anders, T. Costi, H.-J. Lee, Th. Pruschke, N.-H. Tong, S. Tornow, M. Vojta