Orthogonality Catastrophe

Size: px

Start display at page:

Download "Orthogonality Catastrophe"

Bathsheba Cummings
5 years ago
Views:

1 Filiberto Ares Departamento de Física Teórica Universidad de Zaragoza Orthogonality Catastrophe Martes Cuantico, April 17

2 What is Orthogonality Catastrophe (OC)? 2 / 23

3 2 / 23 What is Orthogonality Catastrophe (OC)? Quantum system of N particles described by a Hamiltonian H = H(λ 1,..., λ R )

4 2 / 23 What is Orthogonality Catastrophe (OC)? Quantum system of N particles described by a Hamiltonian H = H(λ 1,..., λ R ) Perturbation of one parameter: λ j λ j + δλ j

5 2 / 23 What is Orthogonality Catastrophe (OC)? Quantum system of N particles described by a Hamiltonian H = H(λ 1,..., λ R ) Perturbation of one parameter: λ j λ j + δλ j Now the Hamiltonian is H = H(λ 1,..., λ j + δλ j,..., λ R )

6 2 / 23 What is Orthogonality Catastrophe (OC)? Quantum system of N particles described by a Hamiltonian H = H(λ 1,..., λ R ) Perturbation of one parameter: λ j λ j + δλ j Now the Hamiltonian is H = H(λ 1,..., λ j + δλ j,..., λ R ) GS : ground state of H GS : ground state of H

7 2 / 23 What is Orthogonality Catastrophe (OC)? Quantum system of N particles described by a Hamiltonian H = H(λ 1,..., λ R ) Perturbation of one parameter: λ j λ j + δλ j Now the Hamiltonian is H = H(λ 1,..., λ j + δλ j,..., λ R ) Orthogonality catastrophe: GS : ground state of H GS : ground state of H GS GS 0, N

8 Free Harmonic Oscillators 3 / 23

9 3 / 23 Free Harmonic Oscillators N non-interacting Harmonic oscillators in 1D H(ω) = 1 2 N j=1 ( ) x 2 j + ω 2 x 2 j

10 3 / 23 Free Harmonic Oscillators N non-interacting Harmonic oscillators in 1D Ground state: Ψ ω ({x j }) = H(ω) = 1 2 N j=1 ( ) x 2 j + ω 2 x 2 j N ( ω ) N/4 ψ(x j ) = e ω 4 π j=1 N j=1 x2 j

11 3 / 23 Free Harmonic Oscillators N non-interacting Harmonic oscillators in 1D Ground state: Ψ ω ({x j }) = H(ω) = 1 2 N j=1 ( ) x 2 j + ω 2 x 2 j N ( ω ) N/4 ψ(x j ) = e ω 4 π j=1 N j=1 x2 j Take two different frequencies ω, ω N A ω,ω (Ψ ω, Ψ ω ) = Ψ ω ({x j })Ψ ω ({x j }) dx i i=1

12 3 / 23 Free Harmonic Oscillators N non-interacting Harmonic oscillators in 1D Ground state: Ψ ω ({x j }) = H(ω) = 1 2 N j=1 ( ) x 2 j + ω 2 x 2 j N ( ω ) N/4 ψ(x j ) = e ω 4 π j=1 N j=1 x2 j Take two different frequencies ω, ω N A ω,ω (Ψ ω, Ψ ω ) = Ψ ω ({x j })Ψ ω ({x j }) dx i [ ] 4 A ω,ω = ω N/4 /ω (1 + ω /ω ) 2 = e ξn ; ξ > 0 i=1

13 1 Anderson s original OC N non-interacting fermions in a 1D box of length L x 0 L P.W. Anderson PRL 18, 24 (1967) 4 / 23

14 1 Anderson s original OC N non-interacting fermions in a 1D box of length L φn(0) = 0 φn(l) = 0 x For each fermion 0 L d2 φ n (x) dx 2 = k 2 nφ n (x) P.W. Anderson PRL 18, 24 (1967) 4 / 23

15 1 Anderson s original OC N non-interacting fermions in a 1D box of length L φn(0) = 0 φn(l) = 0 x For each fermion 0 L d2 φ n (x) dx 2 φ n (x) = N n sin (k n x) ; = k 2 nφ n (x) k n = πn L, n N P.W. Anderson PRL 18, 24 (1967) 4 / 23

16 1 Anderson s original OC N non-interacting fermions in a 1D box of length L φn(0) = 0 φn(l) = 0 x For each fermion 0 L d2 φ n (x) dx 2 φ n (x) = N n sin (k n x) ; = k 2 nφ n (x) k n = πn L, N-fermion ground state Slater determinant Φ N = 1 N! det (φ n (x j )) n,j N n N P.W. Anderson PRL 18, 24 (1967) 4 / 23

17 Anderson s original OC Perturbation: P.W. Anderson PRL 18, 24 (1967) 5 / 23

18 Anderson s original OC λδ(x x 0 ) Perturbation: x 0 0 L x 1 P.W. Anderson PRL 18, 24 (1967) 5 / 23

19 Anderson s original OC λδ(x x 0 ) Perturbation: ψn(0) = 0 ψn(l) = 0 Now for a single fermion x 0 0 L 1 d2 ψ n (x) dx 2 + λδ(x x 0 ) = k 2 nψ n (x) x P.W. Anderson PRL 18, 24 (1967) 5 / 23

20 Anderson s original OC λδ(x x 0 ) Perturbation: ψn(0) = 0 ψn(l) = 0 Now for a single fermion x 0 0 L 1 d2 ψ n (x) dx 2 + λδ(x x 0 ) = k 2 nψ n (x) x { An sin k nx 0 x x 0 ψ n (x) = B n sin k n(x L) x 0 < x L P.W. Anderson PRL 18, 24 (1967) 5 / 23

21 Anderson s original OC λδ(x x 0 ) Perturbation: ψn(0) = 0 ψn(l) = 0 Now for a single fermion x 0 0 L 1 d2 ψ n (x) dx 2 + λδ(x x 0 ) = k 2 nψ n (x) x { An sin k nx 0 x x 0 ψ n (x) = B n sin k n(x L) x 0 < x L N-fermion ground state Ψ N = 1 N! det (ψ n (x j )) n,j N P.W. Anderson PRL 18, 24 (1967) 5 / 23

22 Anderson s original OC Overlap of states before and after turn on λδ(x x 0 ) ( A = (Φ N, Ψ N ) = det A (1) n,n )n,n N A (1) n,n = (φ n, ψ n ) P.W. Anderson PRL 18, 24 (1967) 6 / 23

23 Anderson s original OC Overlap of states before and after turn on λδ(x x 0 ) ( A = (Φ N, Ψ N ) = det A (1) n,n )n,n N A (1) n,n = (φ n, ψ n ) Applying perturbation theory, if λ 0 k n k n + δ n /L; δ n = λ sin2 (k n x 0 ) k n P.W. Anderson PRL 18, 24 (1967) 6 / 23

24 Anderson s original OC Overlap of states before and after turn on λδ(x x 0 ) ( A = (Φ N, Ψ N ) = det A (1) n,n )n,n N A (1) n,n = (φ n, ψ n ) Applying perturbation theory, if λ 0 If x 0 L k n k n + δ n /L; δ n = λ sin2 (k n x 0 ) k n A (1) n,n πn sin δ n π(n + n ) + δ n π(n n) + δ n P.W. Anderson PRL 18, 24 (1967) 6 / 23

25 Anderson s original OC The N-particle overlap is bounded by the 1-particle overlaps A exp[ n>n n <N A (1) nn 2 ] P.W. Anderson PRL 18, 24 (1967) 7 / 23

26 Anderson s original OC The N-particle overlap is bounded by the 1-particle overlaps Since A exp[ n>n n <N A (1) nn 2 ] A (1) n,n πn sin δ n π(n + n ) + δ n π(n n) + δ n P.W. Anderson PRL 18, 24 (1967) 7 / 23

27 Anderson s original OC The N-particle overlap is bounded by the 1-particle overlaps Since A exp[ n>n n <N A (1) nn 2 ] A (1) n,n πn sin δ n π(n + n ) + δ n π(n n) + δ n for large N, states close to the Fermi level dominate A N (sin δ F /π) 2 P.W. Anderson PRL 18, 24 (1967) 7 / 23

28 Anderson s original OC The N-particle overlap is bounded by the 1-particle overlaps Since A exp[ n>n n <N A (1) nn 2 ] A (1) n,n πn sin δ n π(n + n ) + δ n π(n n) + δ n for large N, states close to the Fermi level dominate A N (δ F /π) 2 n P.W. Anderson PRL 18, 24 (1967); PW Anderson, PR 164 (1967); P Nozieres, CT Dominicis, PR (1969) 7 / 23

29 8 / 23 Application I: X-ray absortion spectrum Photon of frequency ω E F + E C excites a core e of a metal to the conduction band ω E F E C

30 8 / 23 Application I: X-ray absortion spectrum Photon of frequency ω E F + E C excites a core e of a metal to the conduction band ω E F E C Interaction of the core hole with the Fermi sea: λδ(x x 0 )

31 8 / 23 Application I: X-ray absortion spectrum Photon of frequency ω E F + E C excites a core e of a metal to the conduction band ω E F E C Interaction of the core hole with the Fermi sea: λδ(x x 0 ) OC between states of the Fermi sea before and after the absortion!

32 9 / 23 Application I: X-ray absorption spectrum How does OC affect the absorption spectrum?

33 Application I: X-ray absorption spectrum How does OC affect the absorption spectrum? λ = 0 A(ω) E F + E C ω G D Manhan, Many-Particle Physics, 3rd Ed 9 / 23

34 Application I: X-ray absorption spectrum How does OC affect the absorption spectrum? λ = 0 λ 0 A(ω) A(ω) E F + E C ω E F + E C ω G D Manhan, Many-Particle Physics, 3rd Ed 9 / 23

35 Application I: X-ray absorption spectrum How does OC affect the absorption spectrum? λ = 0 λ 0 A(ω) A(ω) E F + E C ω E F + E C ω OC tends to suppress the X-ray absortion G D Manhan, Many-Particle Physics, 3rd Ed 9 / 23

36 Application II: Kondo effect 10 / 23

37 10 / 23 Application II: Kondo effect Scattering of conduction e in a metal due to magnetic impurities

38 Application II: Kondo effect Scattering of conduction e in a metal due to magnetic impurities In general, electrical resistivity ρ of metals grows with temp. T ρ(t ) = ρ 0 + AT 2 + BT 5 ρ(t ) T 1 10 / 23

39 Application II: Kondo effect Scattering of conduction e in a metal due to magnetic impurities In some cases, e.g. in gold, ρ(t ) shows a minimum ρ(t ) T 1 11 / 23

40 12 / 23 Application II: Kondo effect Kondo: scattering e /magnetic impurity flipping their spins, k, k

41 12 / 23 Application II: Kondo effect Kondo: scattering e /magnetic impurity flipping their spins, k, k Perturbative computation logarithmic contribution to ρ(t ) ρ(t ) = ρ 0 + AT 2 + BT 5 C log T

42 12 / 23 Application II: Kondo effect Kondo: scattering e /magnetic impurity flipping their spins, k, k Perturbative computation logarithmic contribution to ρ(t ) ρ(t ) = ρ 0 + AT 2 + BT 5 C log T Diverges when T 0 Kondo problem!

43 12 / 23 Application II: Kondo effect Kondo: scattering e /magnetic impurity flipping their spins, k, k Perturbative computation logarithmic contribution to ρ(t ) ρ(t ) = ρ 0 + AT 2 + BT 5 C log T Diverges when T 0 Kondo problem! Below Kondo temperature: perturbative methods are not valid

44 Application II: Kondo effect Anderson: OC between states before and after a flipping, k, k PW Anderson, PR 164 (1967) 13 / 23

45 Application II: Kondo effect Anderson: OC between states before and after a flipping, k, k Coupling e /impurity leads to two competing effects with T : PW Anderson, PR 164 (1967) 13 / 23

46 Application II: Kondo effect Anderson: OC between states before and after a flipping, k, k Coupling e /impurity leads to two competing effects with T : Spin-Flips increase ρ(t ) PW Anderson, PR 164 (1967) 13 / 23

47 Application II: Kondo effect Anderson: OC between states before and after a flipping, k, k Coupling e /impurity leads to two competing effects with T : Spin-Flips increase ρ(t ) OC tends to suppress the flippings PW Anderson, PR 164 (1967) 13 / 23

48 Application II: Kondo effect Anderson: OC between states before and after a flipping, k, k Coupling e /impurity leads to two competing effects with T : } Spin-Flips increase ρ(t ) OC tends to suppress the flippings Minimum of ρ(t) PW Anderson, PR 164 (1967) 13 / 23

49 Application II: Kondo effect Anderson: OC between states before and after a flipping, k, k Coupling e /impurity leads to two competing effects with T : } Spin-Flips increase ρ(t ) OC tends to suppress the flippings Minimum of ρ(t) Kondo temperature: scale at which spin-flips dominate over OC PW Anderson, PR 164 (1967) 13 / 23

50 Sutherland model B Sutherland, J Math Phys 12, 246 (1971) 14 / 23

51 Sutherland model System of N particles in 1D described by H(ω, µ) = 1 2 N j=1 ( ) x 2 j + ω 2 x 2 j + N j 1 j=2 k=1 µ (x j x k ) 2 µ 3/4; λ = µ B Sutherland, J Math Phys 12, 246 (1971) 14 / 23

52 Sutherland model System of N particles in 1D described by H(ω, µ) = 1 2 N j=1 ( ) x 2 j + ω 2 x 2 j + N j 1 j=2 k=1 µ µ 3/4; λ = 2 The ground state is Ψ ω,λ ({x j }) = N ω,λ 1 j<k N µ (x j x k ) 2 x j x k λ e ω 4 N j=1 x2 j B Sutherland, J Math Phys 12, 246 (1971) 14 / 23

53 Sutherland model System of N particles in 1D described by H(ω, µ) = 1 2 N j=1 ( ) x 2 j + ω 2 x 2 j + N j 1 j=2 k=1 µ µ 3/4; λ = 2 The ground state is Ψ ω,λ ({x j }) = N ω,λ 1 j<k N µ (x j x k ) 2 x j x k λ e ω 4 N j=1 x2 j It resembles the Laughlin wave function of the fractional QHE B Sutherland, J Math Phys 12, 246 (1971) 14 / 23

54 Sutherland model System of N particles in 1D described by H(ω, µ) = 1 2 N j=1 ( ) x 2 j + ω 2 x 2 j + N j 1 j=2 k=1 µ µ 3/4; λ = 2 The ground state is Ψ ω,λ ({x j }) = N ω,λ 1 j<k N µ (x j x k ) 2 x j x k λ e ω 4 N j=1 x2 j It resembles the Laughlin wave function of the fractional QHE This system exhibits Haldane s exclusion statistics B Sutherland, J Math Phys 12, 246 (1971) 14 / 23

55 What is Haldane s exclusion statistics? H(ω, µ) = 1 2 N j=1 ( ) x 2 j + ω 2 x 2 j N j 1 µ + j=2 (x j x k ) 2 k=1 If µ = 0 N free Harmonic oscillators with energy: FDM Haldane PRL (1991), H Ujino & M Wadati JPSJ 64, 11 (1995), AP Polychronakos, JPA 39 (2006) 15 / 23

56 What is Haldane s exclusion statistics? H(ω, µ) = 1 2 N j=1 ( ) x 2 j + ω 2 x 2 j N j 1 µ + j=2 (x j x k ) 2 k=1 If µ = 0 N free Harmonic oscillators with energy: E N = ω N n j + ω N 2 j=1 n j : energy level j-particle n j = 0, 1, 2,... n 1 n 2 n N bosons FDM Haldane PRL (1991), H Ujino & M Wadati JPSJ 64, 11 (1995), AP Polychronakos, JPA 39 (2006) 15 / 23

57 What is Haldane s exclusion statistics? H(ω, µ) = 1 2 N j=1 ( ) x 2 j + ω 2 x 2 j + N j 1 j=2 k=1 µ (x j x k ) 2 If µ 0 N interacting Harmonic oscillators with energy: FDM Haldane PRL (1991), H Ujino & M Wadati JPSJ 64, 11 (1995), AP Polychronakos, JPA 39 (2006) 16 / 23

58 What is Haldane s exclusion statistics? H(ω, µ) = 1 2 N j=1 ( ) x 2 j + ω 2 x 2 j + N j 1 j=2 k=1 µ (x j x k ) 2 If µ 0 N interacting Harmonic oscillators with energy: E N = ω N j=1 N(N 1) n j + ωλ + ω N 2 2 n j : energy level j-particle n j = 0, 1, 2,... n 1 n 2 n N (interacting) bosons FDM Haldane PRL (1991), H Ujino & M Wadati JPSJ 64, 11 (1995), AP Polychronakos, JPA 39 (2006) 16 / 23

59 What is Haldane s exclusion statistics? H(ω, µ) = 1 2 N j=1 ( ) x 2 j + ω 2 x 2 j + N j 1 j=2 k=1 µ (x j x k ) 2 However, defining ñ j = n j + λ(j 1) we have the spectrum of N free quasiparticles N E N = ω ñ j + ω N 2 j=1 FDM Haldane PRL (1991), H Ujino & M Wadati JPSJ 64, 11 (1995), AP Polychronakos, JPA 39 (2006) 17 / 23

60 What is Haldane s exclusion statistics? H(ω, µ) = 1 2 N j=1 ( ) x 2 j + ω 2 x 2 j + N j 1 j=2 k=1 µ (x j x k ) 2 However, defining ñ j = n j + λ(j 1) we have the spectrum of N free quasiparticles N E N = ω ñ j + ω N 2 obeying the occupation rules j=1 ñ j ñ j+1 λ exclusion statistics FDM Haldane PRL (1991), H Ujino & M Wadati JPSJ 64, 11 (1995), AP Polychronakos, JPA 39 (2006) 17 / 23

61 OC in the Sutherland model M Rajabpour & S Sotiriadis, PRA 89 (2014); FA, KS Gupta, AR de Queiroz, arxiv: / 23

62 OC in the Sutherland model Take the ground state Ψ ω,λ ({x j }) = N ω,λ x j x k λ e ω N 4 j=1 x2 j 1 j<k N for two different sets (ω, λ), (ω, λ ) and compute M Rajabpour & S Sotiriadis, PRA 89 (2014); FA, KS Gupta, AR de Queiroz, arxiv: / 23

63 OC in the Sutherland model Take the ground state Ψ ω,λ ({x j }) = N ω,λ x j x k λ e ω N 4 j=1 x2 j 1 j<k N for two different sets (ω, λ), (ω, λ ) and compute A (ω,λ),(ω,λ ) ( Ψ ω,λ, Ψ ω,λ ) = N Ψ ω,λ (x j )Ψ ω,λ (x j) dx i i=1 M Rajabpour & S Sotiriadis, PRA 89 (2014); FA, KS Gupta, AR de Queiroz, arxiv: / 23

64 OC in the Sutherland model Take the ground state Ψ ω,λ ({x j }) = N ω,λ x j x k λ e ω N 4 j=1 x2 j 1 j<k N for two different sets (ω, λ), (ω, λ ) and compute A (ω,λ),(ω,λ ) ( Ψ ω,λ, Ψ ω,λ ) = N Ψ ω,λ (x j )Ψ ω,λ (x j) dx i i=1 We have to solve a Selberg s integral... M Rajabpour & S Sotiriadis, PRA 89 (2014); FA, KS Gupta, AR de Queiroz, arxiv: / 23

65 19 / 23 OC in the Sutherland model Selberg s integral: higher-dimensional generalization of Euler B function N i=1 = x α 1 i (1 x i ) β 1 N 1 j=0 1 i,j N x i x j 2γ dx 1 dx N = Γ(α + jγ)γ(β + jγ)γ(1 + (j + 1)γ) Γ(α + β + (N + j 1)γ)Γ(1 + γ) α, β, γ C Re(α), Re(β) > 0 Re(γ) > min{1/n, Re(α)/(N 1), Re(β)/(N 1)}

66 OC in the Sutherland model... after solving the Selberg s integral: M Rajabpour & S Sotiriadis, PRA 89 (2014); FA, KS Gupta, AR de Queiroz, arxiv: / 23

67 OC in the Sutherland model... after solving the Selberg s integral: ω ω, λ = λ M Rajabpour & S Sotiriadis, PRA 89 (2014); FA, KS Gupta, AR de Queiroz, arxiv: / 23

68 OC in the Sutherland model... after solving the Selberg s integral: ω ω, λ = λ A (ω,λ),(ω,λ) = e ξ 4 [λn(n 1)+N] ξ = log (1 + ω /ω ) 2 4 ω /ω > 0 M Rajabpour & S Sotiriadis, PRA 89 (2014); FA, KS Gupta, AR de Queiroz, arxiv: / 23

69 OC in the Sutherland model... after solving the Selberg s integral: ω ω, λ = λ A (ω,λ),(ω,λ) = e ξ 4 [λn(n 1)+N] ξ = log (1 + ω /ω ) 2 4 ω /ω > 0 ω = ω, λ = λ + δλ with δλ 0 M Rajabpour & S Sotiriadis, PRA 89 (2014); FA, KS Gupta, AR de Queiroz, arxiv: / 23

70 OC in the Sutherland model... after solving the Selberg s integral: ω ω, λ = λ A (ω,λ),(ω,λ) = e ξ 4 [λn(n 1)+N] ξ = log (1 + ω /ω ) 2 4 ω /ω > 0 ω = ω, λ = λ + δλ with δλ 0 A (ω,λ)(ω,λ+δλ) e δλ2 λ N(N+1) 16 M Rajabpour & S Sotiriadis, PRA 89 (2014); FA, KS Gupta, AR de Queiroz, arxiv: / 23

71 OC in the Sutherland model... after solving the Selberg s integral: ω ω, λ = λ A (ω,λ),(ω,λ) = e ξ 4 [λn(n 1)+N] ξ = log (1 + ω /ω ) 2 4 ω /ω > 0 ω = ω, λ = λ + δλ with δλ 0 A (ω,λ)(ω,λ+δλ) e δλ2 λ N(N+1) 16 Exponential OC with N 2 M Rajabpour & S Sotiriadis, PRA 89 (2014); FA, KS Gupta, AR de Queiroz, arxiv: / 23

72 Summarizing 21 / 23

73 21 / 23 Summarizing Free Harmonic oscillators [ ] 4 A ω,ω = ω N/4 /ω (1 + ω /ω ) 2 = e ξn ; ξ > 0

74 21 / 23 Summarizing Free Harmonic oscillators [ ] 4 A ω,ω = ω N/4 /ω (1 + ω /ω ) 2 = e ξn ; ξ > 0 Free fermions A N (δ F /π) 2 h

75 21 / 23 Summarizing Free Harmonic oscillators [ ] 4 A ω,ω = ω N/4 /ω (1 + ω /ω ) 2 = e ξn ; ξ > 0 Free fermions A N (δ F /π) 2 h Sutherland model: free quasiparticles obeying exclusion statistics A (ω,λ)(ω,λ+δλ) e δλ2 λ N(N+1) 16

76 OC in other systems 22 / 23

77 22 / 23 OC in other systems Power-law decay, e.g:

78 22 / 23 OC in other systems Power-law decay, e.g: In graphene by a local perturbation in the lattice M Hentschel & F Guinea, Phys Rev B 76 (2007)

79 22 / 23 OC in other systems Power-law decay, e.g: In graphene by a local perturbation in the lattice M Hentschel & F Guinea, Phys Rev B 76 (2007) In quantum dots by a photon that creates an e -hole pair HE Turecci et al, Phys Rev Lett 106, (2011)

80 22 / 23 OC in other systems Power-law decay, e.g: In graphene by a local perturbation in the lattice M Hentschel & F Guinea, Phys Rev B 76 (2007) In quantum dots by a photon that creates an e -hole pair HE Turecci et al, Phys Rev Lett 106, (2011) In an electromagnetic field in a cavity by inserting a small polarizable particle R Merlin, Phys Rev A 95 (2017)

81 22 / 23 OC in other systems Power-law decay, e.g: In graphene by a local perturbation in the lattice M Hentschel & F Guinea, Phys Rev B 76 (2007) In quantum dots by a photon that creates an e -hole pair HE Turecci et al, Phys Rev Lett 106, (2011) In an electromagnetic field in a cavity by inserting a small polarizable particle R Merlin, Phys Rev A 95 (2017) Exponential decay

82 22 / 23 OC in other systems Power-law decay, e.g: In graphene by a local perturbation in the lattice M Hentschel & F Guinea, Phys Rev B 76 (2007) In quantum dots by a photon that creates an e -hole pair HE Turecci et al, Phys Rev Lett 106, (2011) In an electromagnetic field in a cavity by inserting a small polarizable particle R Merlin, Phys Rev A 95 (2017) Exponential decay In localized disordered systems V Khemani et al, Nature Physics 11, 560 (2015) DL Deng et al, Phys Rev B 92, (2015)

83 Muchas gracias! 23 / 23

Fermi liquids and fractional statistics in one dimension

$Fermi liquids and fractional statistics in one dimension$ UiO, 26. april 2017 Fermi liquids and fractional statistics in one dimension Jon Magne Leinaas Department of Physics University of Oslo JML Phys. Rev. B (April, 2017) Related publications: M Horsdal, M