Low-energy local density of states of the 1D Hubbard model Imke - PowerPoint PPT Presentation

Low-energy local density of states of the 1D Hubbard model Imke Schneider, Institut für Theoretische Physik, TU Dresden Florence, 30th May 2012 In collaboration with: Stefan Söffing, Michael Bortz, Alexander Struck, and Sebastian Eggert, TU Kaiserslautern – p. 1

Low energy properties of fermionic systems in 1D Strong correlations, interactions dominant, universal behavior no single-particle picture possible, excitations collective bosonic modes Luttinger liquid Spin- and charge excitations decouple – p. 2

Tunneling in quantum wires Photo emission Scanning tunneling spectroscopy e − proportional to dI/dV Density of states: � ∞ − 1 e iωt G r ( t ) dt ρ ( ω ) = π Im 0 |� ω m | ψ † | 0 �| 2 δ ( ω − ω m ) � = m ρ ( ω ) ∝ ω α Luttinger liquid: – p. 3

Momentum resolved tunneling experiments in 1D Experimental signatures of spin-charge separation Semiconductor hetero-structures Auslaender, Steinberg, Yacoby, Tserkovnyak, Halperin, Baldwin, Pfeiffer, and West, Science 308, 88 (2005) Jompol, Ford, Griffiths, Farrer, Jones, Anderson, Ritchie, Silk, and Schofield, Science 325 (2009) Quasi one-dimensional crystals Kim, Koh, Rotenberg, Oh, Eisaki, Motoyama, Uchida, Tohyama, Maekawa, Shen, and Kim, Nature Phys. 2 (2006) Self-organized atomic chains ( Segovia, Purdie, Hengsberger, and Baer, Nature 402 (1999)) – p. 4

Scanning tunneling spectroscopy in 1D Carbon nanotubes Lee, Eggert, Kim, Kahng, Shinorara, and Kuk, Phys. Rev. Lett. 93 (2004) Venema, Wild¨ oer, Janssen, Tans, Tuinstra, Kouwenhoven, and Dekker, Science 283 (1999) Lemay, Janssen, van den Hout, Mooij, Bronikowski, Willis, Smalley, Kouwenhoven, and Dekker, Nature 412 (2001) Self-organized atomic gold chains Sch¨ Blumenstein, afer, Mietke, Meyer, Dollinger, Lochner, Cui, Patthey, Matzdorf, and Claessen, Na- ture Phys. 7 (2011) Signatures of power law density of states → Luttinger liquid behavior – p. 5

Outline Local density of states of interacting fermions in 1D Luttinger liquid: power laws here: Effects of boundaries and finite system sizes DMRG: lattice model of spinless fermions Spectral weight of individual excitations Bosonization: Recursion formula DMRG: Hubbard model – p. 6

Luttinger liquid with impurity Local density of states |� ω m | ψ † ( x ) | 0 �| 2 δ ( ω − ω m ) � ρ ( ω, x ) = m � � � ∞ � � 1 � � e iωt � ψ ( x, t ) ψ † ( x, 0) � dt � � � � = 2 π −∞ strong depletion for small ρ(ω x ) energies and at the boundary ρ(ω x ) 1/2 (ω x ) 1/8 (ω x ) (here K s = 1 , K c = 1 2 ) 0 2 4 6 8 10 ω x/v c Eggert, Johannesson, Mattsson, Phys. Rev. Lett. 76, (1996) – p. 7

Finite Luttinger liquid with boundaries |� ω m | ψ † ( x ) | 0 �| 2 δ ( ω − ω m ) � ρ ( ω, x ) = � � �� �� m � � �� �� � � �� �� � � �� �� � ∞ 1 e iωt � ψ ( x, t ) ψ † ( x, 0) � dt = 2 π −∞ ρ ( ω 0 , x ) Free fermions: single particle wave function: ρ ( ω m , x ) = | Ψ m ( x ) | 2 L 0 ρ ( ω m , x ) Interacting: m = 1 m = 0 Bosonization Anfuso, Eggert, Phys. Rev. B 68 (2003) 0 L – p. 8

Exact lattice model: local density of states in DMRG L − 1 L − 1 � � � x ψ x +1 + ψ † � ψ † n x = ψ † H = − t x +1 ψ x + U n x n x +1 , x ψ x − 1 / 2 x =1 x =1 Approach 1: Dynamical DMRG and tDMRG + entire spectrum - energy levels not resolvable Jeckelmann, arXiv:1111.6545 Approach 2: transition matrix elements in DMRG + energy levels resolvable - only for low energy excitations Schneider, Struck, Bortz, and Eggert, Phys. Rev. Lett. 101, 206401 (2008) S¨ offing, Schneider, and Eggert, arXiv:1204.0003 � ∞ � |� ω m | ψ † x | 0 �| 2 δ ( ω − ω m ) = − 1 e iωt G r ( x, t ) dt ρ ( ω, x ) = π Im 0 m – p. 9

Understanding of individual excitations Free fermions � � � � � � � � � � � � � � � � � � � � |� a | ψ † x | N 0 �| 2 k ǫ ( k ) c † k = π H = � k c k L n n = 0 , 1 , 2 , . . . x | N 0 �| 2 = 2 |� N 0 | c n ψ † L | sin( k F + k n ) x | 2 L 0 lattice model effective theory a = = † † † c 2 N b c c c N − 0 1 0 1 0 ω 3 ω 2 N 0 +1 n=0 k F k F ω 1 ω 0 – p. 10

Local density of states: DMRG results � � �� �� L − 1 L − 1 � � �� �� � � �� �� � � � � �� �� � � � x ψ x +1 + ψ † � ψ † �� �� � � H = − t x +1 ψ x + U n x n x +1 �� �� � � �� �� � � �� �� � � �� �� � � �� �� � � �� �� � � �� �� x =1 x =1 � � �� �� � � �� �� � � �� �� � � – p. 11

Local density of states: DMRG results ρ ( ω 2 , x ) U = 0 . 7 0.03 0.025 0.02 0.015 0.01 0.005 0 0 20 40 60 x DMRG ρ ( ω 2 , x ) = |� a | ψ † ( x ) | 0 �| 2 + |� b | ψ † ( x ) | 0 �| 2 – p. 12

Local density of states: DMRG results ρ ( ω 2 , x ) U = 0 . 7 0.03 0.025 U = 0 . 7 ρ ( ω 2 , x ) 0.03 0.02 0.025 0.02 0.015 0.015 0.01 0.01 0.005 0 x 0 20 40 60 0.005 0 0 20 40 60 x DMRG ρ ( ω 2 , x ) = |� a | ψ † ( x ) | 0 �| 2 + |� b | ψ † ( x ) | 0 �| 2 – p. 12

Local density of states: DMRG results ρ ( ω 2 , x ) U = 0 . 7 0.03 0.025 U = 0 . 7 ρ ( ω 2 , x ) 0.03 0.02 0.025 0.02 0.015 0.015 0.01 0.01 0.005 0 x 0 20 40 60 0.005 0 0 20 40 60 80 x DMRG Bosonization ρ ( ω 2 , x ) = |� a | ψ † ( x ) | 0 �| 2 + |� b | ψ † ( x ) | 0 �| 2 – p. 12

Density of states: position integrated ρ ( ω m ) U = 0 . 7 1 0.8 ρ ( ω m ) = � x ρ ( ω m , x ) 0.6 0.4 DMRG 0.2 0 0 2 4 6 8 10 m – p. 13

Density of states: position integrated ρ ( ω m ) U = 0 . 7 1 0.8 ρ ( ω m ) = � x ρ ( ω m , x ) 0.6 0.4 DMRG DMRG summiert 0.2 0 0 2 4 6 8 10 m – p. 13

Density of states: position integrated ρ ( ω m ) U = 0 . 7 1 0.8 ρ ( ω m ) = � x ρ ( ω m , x ) 0.6 0.4 DMRG DMRG summed Bosonization 0.2 0 0 2 4 6 8 10 m – p. 13

Recursive method for the density of states � ∞ −∞ e iωt � ψ ( x, t ) ψ † ( x, 0) � dt 1 ρ ( ω, x ) = 2 π Correlation functions in standard bosonization � ∞ � 1 R ( x, 0) � = | c | 2 exp � � ψ R ( x, t ) ψ † ℓ e − iℓ ∆ ωt γ ℓ ( x ) ℓ =1 � ∞ � � ∞ � 1 1 � e iℓ ∆ ωt A † � e − iℓ ∆ ωt A ℓ ( x ) ψ † R ( x, t ) := c ( x ) exp √ ℓ ( x ) exp √ i i ℓ ℓ ℓ =1 ℓ =1 γ ℓ ( x ) = [ A ℓ ( x ) , A † ℓ ( x )] , A ℓ ( x ) = α ( K ) e ik ℓ x b R ℓ − β ( K ) e − ik ℓ x b L ℓ Finite systems � ∞ 1 � dt e iωt � ψ R ( x, t ) ψ † R ( x, 0) � = ρ m δ ( w − m ∆ ω ) 2 π −∞ m 1 ρ 0 = | c | 2 ρ m = m ( ρ m − 1 γ 1 + ρ m − 2 γ 2 + · · · + ρ 1 γ m − 1 + ρ 0 γ m ) mit Schneider and Eggert, Phys. Rev. Lett. 104 (2010) earlier recursive approach: Sch¨ onhammer and Meden, Phys. Rev. B 47 (1993) – p. 14

Spinless fermions with periodic b. c. ρ m = 1 Density of states: m ( ρ m − 1 γ 1 + ρ m − 2 γ 2 + · · · + ρ 1 γ m − 1 + ρ 0 γ m ) Commutator mode independent � 1 γ = 1 � K + K Luttinger-parameter K 2 Recursion formula exacty solvable Γ( m + γ ) 1 ρ m = | c | 2 Γ( γ )Γ( m + 1) ≈ | c | 2 Γ( γ ) m γ − 1 well known power law in general γ ℓ ( x ) mode and x dependent – p. 15

Spinful fermions with open b. c. Luttinger liquid picture: States described by integer spin and charge quantum numbers { m s , m c } π Energies: ω m s ,m c = ( m s v s + m c v c ) L +1 with v s ≤ v c Density of states: ρ uni s,m s ( x ) ρ uni c,m c ( x ) − cos(2 k F x ) ρ osc s,m s ( x ) ρ osc | c x | 2 � � ρ m s ,m c ( x ) = c,m c ( x ) Calculate recursively, e.g. : m c c,m c ( x ) = 1 � ρ uni ρ uni c,m c − ℓ ( x ) γ uni c,ℓ ( x ) m c ℓ =1 γ uni c,ℓ ( x ) = (1 /K c + K c ) / 4 + (1 /K c − K c ) cos(2 k ℓ x ) – p. 16

Comparison to DMRG results (1/3) Hubbard model: H = − t � L − 1 + U � L � ψ † � σ,x ψ σ,x +1 + h.c. x =1 n ↑ ,x n ↓ ,x σ, x =1 Energies ∆ ω Total density of states 0s 2c level 2 ∆ω 2s 1c 1 4s 0c level 1 level 2 ρ 0s 2c, S=1/2 1s 1c 0.1 0s 2c, S=1/2 0s 1c, S=1/2 3s 0c 1s 1c, S=1/2 1s 0c, S=1/2 2s 0c, S=1/2 0.5 level 1 0s 1c 2s 0c, S=3/2 2s 0c 0.05 1s 0c 0 U U 0 1 2 3 4 5 6 1 2 3 4 5 6 level 0 0 0s 0c 0 1 2 3 4 5 U 6 Parameter: N ↑ = N ↓ + 1 = 31 and L = 90 – p. 17

Comparison to DMRG results (2/3) Local density of states: ρ 0s 0c 0.02 0.01 0 N ↑ = N ↓ + 1 = 31 0s 1c 0.006 L = 92 0.005 0s 2c 0s 3c U = 1 0.004 0.005 1s 0c 0.004 2s 0c 0.003 3s 0c 0.002 0.001 x 10 20 30 40 50 60 70 80 90 Lines: predictions for K c = 0 . 9081 and K s = 1 . 16 adjusted by shifts Local density of states increases near boundary – p. 18