NUMERICAL METODS FOR QUANTUM IMPURITY MODELS http://www.staff.science.uu.nl/~mitch003/nrg.html March 2015 Andrew Mitchell, Utrecht University
Quantum impurity problems Part 1: Quantum impurity problems and theoretical background Part 2: Kondo effect and RG. 1d chain formulation and iterative diagonalization Part 3: Logarithmic discretization and truncation. The RG in NRG Part 4: Physical quantities. Results and discussion. Andrew Mitchell Quantum impurity problems
NUMERICAL METODS FOR QUANTUM IMPURITY MODELS Part 3: Wilson Chain and the RG in NRG. March 2015 Andrew Mitchell, Utrecht University
Overview: Part 3 Particle in a box revisited The problem of truncation Linear discretization Logarithmic discretization The Wilson Chain Analytic structure: RG, flows, fixed points, scaling, universality
The problem of truncation We wish to join two sub-systems into one large system. If we are only interested in the low-energy physics of the combined system, can we neglect the high-energy states of the two sub-systems? Consider simplest possible system: particle in a box!
Aside: particle in a box In the continuum: pb x 2 2 with boundary conditions ( 0) 0 ( L) On the lattice: (1d chain) L pb t L 1 r0 c r c ( r1).c. amiltonian conserves particle number and spin Consider only the 1-particle, spin-σ subspace
Aside: particle in a box Basis states: ; ; ; ; ; ; ; ; ; ;... ;... ;... ; ; ; ;... ie. single-particle states. Easy!
Aside: particle in a box Diagonalize LxL hopping matrix: Energy of states: Eigenstates are particle-in-a-box wavefunctions with coefficients 0 0 0 t t t t 1 cos 2 L j t E j L r r j j vac c r U 1 ) ( 1 sin 1 2 ) ( L jr L r U j
Aside: particle in a box L 50
Aside: particle in a box L 50
Aside: particle in a box L 100
Aside: particle in a box
Aside: particle in a box Join two boxes together: For, boundary condition mismatch!.c. '.c..c. 1) 2 / ( 2) / ( 2 / 1) ( 1 2 / 0 1) ( L L join L L r r r L r r r box c c t c c t c c t 1 / ' t t box join
Aside: particle in a box t' / t 0 L 100
Aside: particle in a box t' / t 1 L 100
Aside: particle in a box This boundary condition mismatch means that lowenergy states of a composite system cannot just be constructed from low-energy states of two sub-systems in general Need to select states with the correct boundary conditions (ie, nodes in the right places) Motivation for development of DMRG But is there another way? A way that exploits RG?
Aside: particle in a box Consider the opposite limit t' / t 1 Can now treat perturbatively! join eff 1ˆ box 1ˆ 1ˆ join 1ˆ... where 1ˆ r r r is the complete set of diagonal states of box Andrew Mitchell Numerical Quantum Renormalization Impurity Problems: Group: Part 3
Aside: particle in a box When t' is the smallest energy scale of the problem, can project into the ground state manifold of r1,2 1ˆ gs gs; r gs; r box eff 1ˆ gs box 1ˆ gs 1ˆ gs join 1ˆ gs... 1ˆ gs E gs t' 2 U (1) gs; r gs; r... 1 r1,2
Aside: particle in a box Ground state of composite (joined) system: E eff gs eff gs E gs 1 2 t' U 2 1 (1) 1 2 To first order: ground state of combined system can be constructed from ground states of sub-systems provided connecting terms are small.
Aside: particle in a box t' / t 1 L 100
Aside: particle in a box t' / t 0.99 L 100
Aside: particle in a box t' / t 0.95 L 100
Aside: particle in a box t' / t 0.8 L 100
Aside: particle in a box t' / t 0.58 L 100
The Truncation Problem Back to iterative diagonalization and truncation! Philosophy in NRG: Ensure coupling to each added site is always small In NRG, this is achieved by a logarithmic discretization of the conduction band. Mapping to a 1d chain produces hoppings that decrease exponentially down the chain. Energy scale separation allows truncation at every step.
Discretization ow does the discretization work in practice? We saw already that truncating the 1d chain representation is a type of discretization Another possibility: discretize the conduction electron density of states. Replace the continuous spectrum by discretized poles
Linear Discretization host k, c k k, c k, 2 0 1 D : : D D Divide band up into N intervals, each of width 2D/N P P, P, P, P, P,...,, 0, 1 1 2 2 3 1 P n D 1 2n / N P N P N
Linear Discretization Discretize the continuous spectrum by replacing with sum over delta-peaks: n N n n disc a 1 disc ND D P P d a P P n n P P n n n n n n 1 2 2 / 1 1 1
Logarithmic Discretization Logarithmic discretization of conduction band: Define intervals N disc an n n1 P n n with n 0,1, 2, 3, 4,... a n n ~ P n n d ~ Pn 1 n
Logarithmic Discretization
Logarithmic Discretization
Logarithmic Discretization disc
Logarithmic Discretization Is this a good approximation to the true continuum? Broaden the spectral poles into (log-) Gaussians to check: approximately recovers flat band
Wilson s formulation Anderson amiltonian: Conduction band assumed to be isotropic: Impurity then just couples to the s-wave states:
Wilson s formulation Assume and V d Vd are independent of energy (and write k / D ):
Wilson s formulation Divide band into logarithmic intervals: P n n with n 0,1, 2, 3, 4,... Set up a Fourier series in each interval: where,
Wilson s formulation Canonical transformation of operators in each interval: ybridization term of amiltonian: Impurity only couples to p=0 fundamental harmonic!
Wilson s formulation BUT: Conduction electron amiltonian: p 0 modes couple to impurity only indirectly, through the modes with p 0 Coupling between p 0 and p 0 modes controlled in the discretization parameter, and vanish as 1
Wilson s formulation Keep only p=0 modes! where,
Logarithmic Discretization disc
Logarithmic Discretization Low-energy excitations around Fermi level are exponentially-well sampled needed to capture Kondo physics on the scale of T K Treats physics on all energy scales on equal footing Logarithmic divergences in perturbative treatment avoided by logarithmic discretization But does this help? Continuous spectrum: uncountably infinite number of states Discretized spectrum: countably infinite but still infinite!
Mapping to 1d chain The Wilson Chain is a 1d tight-binding chain, with the impurity located at one end: impurity zero orbital of the Wilson Chain: Local Density of States seen by the impurity is the logarithmically discretized host density of states, disc
Wilson Chain disc 2 0, k k disc a k disc host k, b k k b k
Wilson Chain But we don t want the diagonal representation! We need the tridiagonal chain representation k k k k disc host b b,.c. 1 n n n n f f h f W f D b b
Wilson Chain But we don t want the diagonal representation! We need the tridiagonal chain representation 4 3 3 3 2 2 2 1 1 1 4 3 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 e h h e h h e h h e D W
Wilson Chain Tridiagonalize by Lanczos method CONSTRAINT: zero-orbital of Wilson chain must have correct LDOS 2 disc a0, k k k pole weights define the transformation for the zero-orbital, connected to the impurity Lanczos starting vector: a 0
Wilson Chain Tridiagonalize by Lanczos method: Starting ingredients: diagonal host amiltonian and zero-orbital vector 1) Compute: 2) Compute: 3) At any step, only non-zero elements are: k k k k disc host b b, a 0 1 1 0 1 0 a h a e a 2 2 1 1 0 1 1 a h a e a h a i i i i i i h a a e a a and 1
Wilson Chain Tridiagonalize by Lanczos method: e e e e e e 0 1 2 3 4 5 h h h h h 0 1 2 3 4 Wilson showed that the hoppings drop off exponentially down the chain, due to the logarithmic discretization: h n ~ n / 2
NRG Iterative diagonalization of Wilson chain Truncation of ilbert space at each step Throw away high lying states, keeping large but fixed number, N s, states per iteration Justified by the energy-scale separation going from iteration to iteration igh-energy states discarded at one iteration do not affect low-energy states at later iterations
NRG: iterative procedure Start from the impurity-zero orbital sub-system Apply recursion relation to add on extra sites N / 2 join N 1 N N 1 Full (discretized) amiltonian recovered as Lim N ( N 1) / 2 N
NRG: iterative procedure One iteration in NRG:
NRG: iterative procedure Flow of (rescaled) many-particle levels:
NRG: iterative procedure Physics of the model at lower and lower energy scales is revealed as more Wilson chain orbitals are added Fixed number of states kept at each iteration, so no explosion of ilbert space: linear scaling with N After a finite number of iterations (say N=100 for Ʌ=3), access ground state information For the Kondo model: Strong coupling spin-singlet ground state Impurity screened by conduction electrons: Kondo effect!
NRG: iterative procedure Flow of (rescaled) many-particle levels: converged levels Why?!
Analytic structure: the RG in NRG View iterative construction of the Wilson chain N / 2 join N 1 N N 1 as an RG transformation: ˆ N 1 R N
The Renormalization Group The sequence of amiltonians, N, form a group Application of the transformation, new member of the group, N 1 R ˆ N, generates a Successive application of the transformation generates a characteristic RG flow through this amiltonian space Flow starts from the initial amiltonian (corresponding to the bare impurity with its original microscopic parameters) At special points in the flow, physics can be understood in terms of the original model, but with renormalized parameters.
Fixed points Fixed points (FPs) of the RG transformation * correspond to special cases where R ˆ * A fixed point amiltonian is thus one that is invariant to the RG transformation, * N 1 Fixed point amiltonians often correspond to the original model, but with special renormalized values of the parameters (often 0 or infinity, but not always) N
Fixed point stability Analyze behavior near to FPs: * N N It follows from the RG transformation that: N Rˆ 1 N Does N get larger or smaller with N? Construct possible perturbations to each FP consistent with model symmetries N N a i i i Oˆ i Can determine eigenvalue of the transformation i Rˆ for operators analytically! Ô i
Fixed point stability Classify perturbations as relevant or irrelevant : A relevant perturbation grows under RG: An irrelevant perturbation diminishes under RG: Classify fixed points as stable or unstable : A stable FP has no relevant perturbations RG flow attracted to the FP An unstable FP has at least one relevant perturbation RG flow repelled by the FP i 1 i 1
RG Flow RG flow between fixed points Repelled by unstable FPs; attracted by stable FPs Seen in many-particle energies and physical quantities Schematically represented by RG flow diagram Kondo model: Unstable FP: Local Moment J ~ Effective renormalized coupling Stable FP: Strong Coupling
RG Flow Number, character and stability of FPs determined by the specific RG transformation and symmetries RG flow (trajectory) determined by starting parameters of bare model
Universality One stable FP: Single ground state! For any starting parameters, always end up at the same FP (although path to reach the stable FP might be different) Universality Irrespective of the details of a model, or its bare parameters, two systems with the same stable fixed point have the same ground state and low-energy physics RG flow between two fixed points is universal, and characterized only by a single crossover energy scale (for example, T K for the Kondo model) Physical quantities for different systems are described by a single universal curve, when rescaled in terms of T/T K or ω/t K
Critical phenomena One stable FP: Single ground state! Two stable FPs: Two possible ground states Starting parameters determine RG trajectory and ultimately which stable FP is reached Quantum phase transition!
Fixed point properties ow to determine fixed points? What are their properties? Are they stable? Free Wilson Chain is invariant under the RG transformation Rˆ 2 host host
Fixed point properties Free Wilson Chain (no impurity): Represented by tridiagonal matrix Diagonalize matrix to obtain single-particle levels:, k k k k WC N b b N n N n n n n n n n N WC N f f h f f e 0 1 0 1) ( 2 / 1) (.c. 2 1 1 1 0 0 0 2 / 1) ( e h h e h h e N N
Fixed point properties Fill up single-particle levels up to the Fermi level (in accordance with Pauli principle) Construct manyparticle excitations above ground state Rapid convergence with Wilson Chain length: adding more sites does not change levels! Free Wilson chain is a FP of the RG transformation!
Fixed point properties Anderson Impurity Model: reminder Bare amiltonian: nˆ nˆ U nˆ nˆ nˆ V d c.c. imp imp imp imp k k k, k, k, Isolated impurity ˆ n imp d d imp Isolated conduction band host discretize to give Wilson Chain nˆ c k k, c k, ybridization hyb
Fixed point properties Anderson Impurity Model: reminder Bare amiltonian: nˆ nˆ U nˆ nˆ nˆ V d c.c. imp imp imp imp k k k, k, k, FPs amiltonians are of the same form, but with renormalized parameters: Free Orbital (FO) FP (high T) * FO with V * FO 0 ; Free Wilson chain with a single decoupled impurity SITE U * FO 0 ; * FO 0
Fixed point properties Anderson Impurity Model: reminder Bare amiltonian: nˆ nˆ U nˆ nˆ nˆ V d c.c. imp imp imp imp k k k, k, k, FPs amiltonians are of the same form, but with renormalized parameters: Local Moment (LM) FP * LM with V * LM (T ~ U : recall Schrieffer-Wolff) 0 ; ; Free Wilson chain with a single decoupled impurity SPIN U * LM * LM U * LM / 2
Fixed point properties Anderson Impurity Model: reminder Bare amiltonian: nˆ nˆ U nˆ nˆ nˆ V d c.c. imp imp imp imp k k k, k, k, FPs amiltonians are of the same form, but with renormalized parameters: Strong Coupling (SC) FP (low temperature T << T K ) * SC with * 2 * * * V / U ; U / 2 Free Wilson chain with zero-orbital removed SC SC SC SC
FO fixed point analysis Anderson Impurity Model Near the FO FP: d f f d O d d O 0, 0, 2 2 1 ˆ 1 ˆ FO N FO FO N * i N i i i FO N O a ˆ relevant FO FP unstable!
LM fixed point analysis Anderson Impurity Model Near the LM FP: LM N * LM LM N Oˆ 1 marginally relevant: LM FP unstable f f 0, 0, ' eff ~ J ( N 1) / 2 N 1 n0 ( N 1) / 2 f 0, h n f n f f 0, ' ( n1).c. discretized form of Kondo model!
SC fixed point analysis Anderson Impurity Model Near the SC FP: SC N * SC SC N Oˆ Oˆ 1 2 f 1, f f 1 2 1, f 2, 1, irrelevant SC FP is stable As N increases, get closer to SC FP MUST reach SC FP at low enough temperatures T=0 ground state is described by SC FP: Kondo singlet f 2, f 1,
The Renormalization Group Anderson Impurity Model Schematic RG flow diagram: U ~ Kondo model V ~
The Renormalization Group Anderson Impurity Model RG flow of many-particle energies: LM SC FO
The Renormalization Group RG flow also seen in physical quantities more in final part!