Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University Collaboration: Frithjof B. Anders, Dortmund University F.B. Anders and AS, Phys. Rev. Lett. 95, (2005) F.B. Anders and AS, Phys. Rev. B 74, (2006)
Avraham Schiller / Seattle 09 Outline Confined nano-structures and dissipative systems: Time-dependent Numerical Renormalization Benchmarks for fermionic and bosonic baths Spin and charge relaxation in ultra-small dots Non-perturbative physics out of equilibrium Group (TD-NRG)
Avraham Schiller / Seattle 09 Coulomb blockade in ultra-small quantum dots
Avraham Schiller / Seattle 09 Quantum dot Coulomb blockade in ultra-small quantum dots
Avraham Schiller / Seattle 09 Leads Coulomb blockade in ultra-small quantum dots
Avraham Schiller / Seattle 09 Lead Coulomb blockade in ultra-small quantum dots
Avraham Schiller / Seattle 09 Lead Coulomb blockade in ultra-small quantum dots
Avraham Schiller / Seattle 09 U Lead Coulomb blockade in ultra-small quantum dots
Avraham Schiller / Seattle 09 U Lead Coulomb blockade in ultra-small quantum dots
Avraham Schiller / Seattle 09 U Lead Coulomb blockade in ultra-small quantum dots i +U U
Avraham Schiller / Seattle 09 U Lead Coulomb blockade in ultra-small quantum dots
Avraham Schiller / Seattle 09 U Lead Conductance vs gate voltage Coulomb blockade in ultra-small quantum dots
Avraham Schiller / Seattle 09 U Lead Conductance vs gate voltage Coulomb blockade in ultra-small quantum dots
Avraham Schiller / Seattle 09 U Lead Conductance vs gate voltage dI/dV (e 2 /h) Coulomb blockade in ultra-small quantum dots
Avraham Schiller / Seattle 09 The Kondo effect in ultra-small quantum dots
Avraham Schiller / Seattle 09 The Kondo effect in ultra-small quantum dots
Avraham Schiller / Seattle 09 The Kondo effect in ultra-small quantum dots Tunneling to leads
Avraham Schiller / Seattle 09 The Kondo effect in ultra-small quantum dots Inter-configurational energies d and U+ d
Avraham Schiller / Seattle 09 The Kondo effect in ultra-small quantum dots Inter-configurational energies d and U+ d
Avraham Schiller / Seattle 09 The Kondo effect in ultra-small quantum dots Inter-configurational energies d and U+ d
Avraham Schiller / Seattle 09 The Kondo effect in ultra-small quantum dots Inter-configurational energies d and U+ d Hybridization width
Avraham Schiller / Seattle 09 The Kondo effect in ultra-small quantum dots Inter-configurational energies d and U+ d Hybridization width
Avraham Schiller / Seattle 09 The Kondo effect in ultra-small quantum dots Inter-configurational energies d and U+ d Condition for formation of local moment: Hybridization width
Avraham Schiller / Seattle 09 The Kondo effect in ultra-small quantum dots Inter-configurational energies d and U+ d Condition for formation of local moment: Hybridization width
Avraham Schiller / Seattle 09 The Kondo effect in ultra-small quantum dots
Avraham Schiller / Seattle 09 The Kondo effect in ultra-small quantum dots EFEF dd d +U TKTK
Avraham Schiller / Seattle 09 The Kondo effect in ultra-small quantum dots EFEF dd d +U TKTK A sharp resonance of width T K develops at E F when T<T K
Avraham Schiller / Seattle 09 The Kondo effect in ultra-small quantum dots EFEF dd d +U Abrikosov-Suhl resonance TKTK A sharp resonance of width T K develops at E F when T<T K
Avraham Schiller / Seattle 09 The Kondo effect in ultra-small quantum dots EFEF dd d +U TKTK A sharp resonance of width T K develops at E F when T<T K Unitary scattering for T=0 and =1
Avraham Schiller / Seattle 09 The Kondo effect in ultra-small quantum dots EFEF dd d +U TKTK A sharp resonance of width T K develops at E F when T<T K Unitary scattering for T=0 and =1 Nonperturbative scale:
Avraham Schiller / Seattle 09 The Kondo effect in ultra-small quantum dots EFEF dd d +U TKTK A sharp resonance of width T K develops at E F when T<T K Unitary scattering for T=0 and =1 Nonperturbative scale: Perfect transmission for symmetric structure
Avraham Schiller / Seattle 09 Electronic correlations out of equilibrium
Avraham Schiller / Seattle 09 Electronic correlations out of equilibrium dI/dV (e 2 /h) Differential conductance in two-terminal devices Steady state van der Wiel et al.,Science 2000
Avraham Schiller / Seattle 09 Electronic correlations out of equilibrium dI/dV (e 2 /h) Differential conductance in two-terminal devices Steady state ac drive Photon-assisted side peaks Kogan et al.,Science 2004van der Wiel et al.,Science 2000
Avraham Schiller / Seattle 09 Electronic correlations out of equilibrium dI/dV (e 2 /h) Differential conductance in two-terminal devices Steady state ac drive Photon-assisted side peaks Kogan et al.,Science 2004van der Wiel et al.,Science 2000
Avraham Schiller / Seattle 09 Nonequilibrium: A theoretical challenge
Avraham Schiller / Seattle 09 Nonequilibrium: A theoretical challenge The Goal: The description of nano-structures at nonzero bias and/or nonzero driving fields
Avraham Schiller / Seattle 09 Nonequilibrium: A theoretical challenge The Goal: The description of nano-structures at nonzero bias Required: Inherently nonperturbative treatment of nonequilibrium and/or nonzero driving fields
Avraham Schiller / Seattle 09 Nonequilibrium: A theoretical challenge The Goal: The description of nano-structures at nonzero bias Required: Inherently nonperturbative treatment of nonequilibrium and/or nonzero driving fields Problem: Unlike equilibrium conditions, density operator is not known in the presence of interactions
Avraham Schiller / Seattle 09 Nonequilibrium: A theoretical challenge The Goal: The description of nano-structures at nonzero bias Required: Inherently nonperturbative treatment of nonequilibrium and/or nonzero driving fields Problem: Unlike equilibrium conditions, density operator is not Most nonperturbative approaches available in equilibrium known in the presence of interactions are simply inadequate
Avraham Schiller / Seattle 09 Nonequilibrium: A theoretical challenge Two possible strategies Work directly at steady state e.g., construct the many- particle Scattering states Evolve the system in time to reach steady state
Avraham Schiller / Seattle 09 Time-dependent numerical RG
Avraham Schiller / Seattle 09 Time-dependent numerical RG Consider a quantum impurity (e.g., quantum dot) in equilibrium, to which a sudden perturbation is applied at time t = 0
Avraham Schiller / Seattle 09 Time-dependent numerical RG Consider a quantum impurity (e.g., quantum dot) in equilibrium, to which a sudden perturbation is applied at time t = 0 Lead VgVg t < 0
Avraham Schiller / Seattle 09 Time-dependent numerical RG Consider a quantum impurity (e.g., quantum dot) in equilibrium, to which a sudden perturbation is applied at time t = 0 Lead VgVg t > 0 Lead VgVg t < 0
Avraham Schiller / Seattle 09 Time-dependent numerical RG Consider a quantum impurity (e.g., quantum dot) in equilibrium, to which a sudden perturbation is applied at time t = 0
Avraham Schiller / Seattle 09 Time-dependent numerical RG Consider a quantum impurity (e.g., quantum dot) in equilibrium, to which a sudden perturbation is applied at time t = 0 Perturbed Hamiltonian Initial density operator
Avraham Schiller / Seattle 09 Wilson’s numerical RG
Avraham Schiller / Seattle 09 Wilson’s numerical RG -1--1 --2--2 --3--3 -1-1 -2-2 -3-3 /D/D Logarithmic discretization of band:
Avraham Schiller / Seattle 09 Wilson’s numerical RG -1--1 --2--2 --3--3 -1-1 -2-2 -3-3 /D/D Logarithmic discretization of band: imp After a unitary transformation the bath is represented by a semi-infinite chain
Avraham Schiller / Seattle 09 Why logarithmic discretization? Wilson’s numerical RG
Avraham Schiller / Seattle 09 Why logarithmic discretization? Wilson’s numerical RG To properly account for the logarithmic infra-red divergences
Avraham Schiller / Seattle 09 Why logarithmic discretization? Wilson’s numerical RG To properly account for the logarithmic infra-red divergences imp Hopping decays exponentially along the chain:
Avraham Schiller / Seattle 09 Why logarithmic discretization? Wilson’s numerical RG imp Hopping decays exponentially along the chain: Separation of energy scales along the chain To properly account for the logarithmic infra-red divergences
Avraham Schiller / Seattle 09 Why logarithmic discretization? Wilson’s numerical RG imp Hopping decays exponentially along the chain: Exponentially small energy scales can be accessed, limited by T only To properly account for the logarithmic infra-red divergences Separation of energy scales along the chain
Avraham Schiller / Seattle 09 Why logarithmic discretization? Wilson’s numerical RG imp Hopping decays exponentially along the chain: Iterative solution, starting from a core cluster and enlarging the chain by one site at a time. High-energy states are discarded at each step, refining the resolution as energy is decreased. To properly account for the logarithmic infra-red divergences Exponentially small energy scales can be accessed, limited by T only Separation of energy scales along the chain
Avraham Schiller / Seattle 09 Equilibrium NRG: Geared towards fine energy resolution at low energies Discards high-energy states Wilson’s numerical RG
Avraham Schiller / Seattle 09 Equilibrium NRG: Problem: Real-time dynamics involves all energy scales Geared towards fine energy resolution at low energies Discards high-energy states Wilson’s numerical RG
Avraham Schiller / Seattle 09 Equilibrium NRG: Problem: Real-time dynamics involves all energy scales Resolution: Combine information from all NRG iterations Geared towards fine energy resolution at low energies Discards high-energy states Wilson’s numerical RG
Avraham Schiller / Seattle 09 Time-dependent NRG imp Basis set for the “environment” statesNRG eigenstate of relevant iteration
Avraham Schiller / Seattle 09 Time-dependent NRG imp Basis set for the “environment” statesNRG eigenstate of relevant iteration For each NRG iteration, we trace over its “environment”
Avraham Schiller / Seattle 09 Time-dependent NRG Sum over discarded NRG states of chain of length m Matrix element of O on the m-site chain Reduced density matrix for the m-site chain (Hostetter, PRL 2000) Sum over all chain lengths (all energy scales) Trace over the environment, i.e., sites not included in chain of length m
Avraham Schiller / Seattle 09 Fermionic benchmark: Resonant-level model
Avraham Schiller / Seattle 09 Fermionic benchmark: Resonant-level model
Avraham Schiller / Seattle 09 Fermionic benchmark: Resonant-level model We focus on and compare the TD-NRG to exact analytic solution in the wide-band limit (for an infinite system) Basic energy scale:
Avraham Schiller / Seattle 09 Fermionic benchmark: Resonant-level model T = 0 Relaxed values (no runaway!)
Avraham Schiller / Seattle 09 Fermionic benchmark: Resonant-level model T = 0 T > 0 Relaxed values (no runaway!)
Avraham Schiller / Seattle 09 Fermionic benchmark: Resonant-level model T = 0 T > 0 Relaxed values (no runaway!) The deviation of the relaxed T=0 value from the new thermodynamic value is a measure for the accuracy of the TD-NRG on all time scales For T > 0, the TD-NRG works well up to
Avraham Schiller / Seattle 09 T = 0 E d (t 0) = = 2 Source of inaccuracies
Avraham Schiller / Seattle 09 T = 0 E d (t 0) = = 2 Source of inaccuracies
Avraham Schiller / Seattle 09 T = 0 E d (t 0) = = 2 Source of inaccuracies
Avraham Schiller / Seattle 09 T = 0 E d (t 0) = = 2 TD-NRG is essentially exact on the Wilson chain Source of inaccuracies Main source of inaccuracies is due to discretization
Avraham Schiller / Seattle 09 Analysis of discretization effects E d (t 0) =
Avraham Schiller / Seattle 09 Analysis of discretization effects E d (t 0) = E d (t 0) = -10
Avraham Schiller / Seattle 09 Bosonic benchmark: Spin-boson model
Avraham Schiller / Seattle 09 Bosonic benchmark: Spin-boson model Setting =0, we start from the pure spin state and compute
Avraham Schiller / Seattle 09 Bosonic benchmark: Spin-boson model Excellent agreement between TD-NRG (full lines) and the exact analytic solution (dashed lines) up to
Avraham Schiller / Seattle 09 Bosonic benchmark: Spin-boson model For nonzero and s = 1 (Ohmic bath), we prepare the system such that the spin is initially fully polarized (S z = 1/2)
Avraham Schiller / Seattle 09 Bosonic benchmark: Spin-boson model For nonzero and s = 1 (Ohmic bath), we prepare the system such that the spin is initially fully polarized (S z = 1/2) Damped oscillations
Avraham Schiller / Seattle 09 Bosonic benchmark: Spin-boson model For nonzero and s = 1 (Ohmic bath), we prepare the system such that the spin is initially fully polarized (S z = 1/2) Monotonic decay
Avraham Schiller / Seattle 09 Bosonic benchmark: Spin-boson model For nonzero and s = 1 (Ohmic bath), we prepare the system such that the spin is initially fully polarized (S z = 1/2) Localized phase
Avraham Schiller / Seattle 09 Anderson impurity model t < 0t > 0
Avraham Schiller / Seattle 09 Anderson impurity model: Charge relaxation Charge relaxation is governed by t ch =1/ 1 TD-NRG works better for interacting case! Exact new Equilibrium values
Avraham Schiller / Seattle 09 Anderson impurity model: Spin relaxation
Avraham Schiller / Seattle 09 Anderson impurity model: Spin relaxation
Avraham Schiller / Seattle 09 Anderson impurity model: Spin relaxation Spin relaxes on a much longer time scale Spin relaxation is sensitive to initial conditions! Starting from a decoupled impurity, spin relaxation approaches a universal function of t/t sp with t sp =1/T K
Avraham Schiller / Seattle 09 Conclusions A numerical RG approach was devised to track the real-time dynamics of quantum impurities following a sudden perturbation Works well for arbitrarily long times up to 1/T Applicable to fermionic as well as bosonic baths For ultra-small dots, spin and charge typically relax on different time scales