SLIDE 1
Localized spin systems : from BEC to Luttiger liquids
- T. Giamarchi
http://dpmc.unige.ch/gr_giamarchi/
- A. Tsvelik (BNL)
- R. Chitra (Jussieu)
Localized spin systems : from BEC to Luttiger liquids T. Giamarchi - - PowerPoint PPT Presentation
Localized spin systems : from BEC to Luttiger liquids T. Giamarchi - - PowerPoint PPT Presentation
Localized spin systems : from BEC to Luttiger liquids T. Giamarchi http://dpmc.unige.ch/gr_giamarchi/ A. Tsvelik (BNL) R. Chitra (Jussieu) A. Tsvelik (BNL) R. Chitra (Jussieu) E. Orignac (ENS-Lyon) R. Citro (Salerno U.) A. Tsvelik (BNL)
SLIDE 2
SLIDE 3
- A. Tsvelik (BNL)
- R. Chitra (Jussieu)
- E. Orignac (ENS-Lyon)
- R. Citro (Salerno U.)
SLIDE 4
- A. Tsvelik (BNL)
- R. Chitra (Jussieu)
- E. Orignac (ENS-Lyon)
- R. Citro (Salerno U.)
SLIDE 5
- A. Tsvelik (BNL)
- R. Chitra (Jussieu)
- E. Orignac (ENS-Lyon)
- R. Citro (Salerno U.)
- P. Bouillot (Geneva)
- C. Kollath (Polytechnique)
- M. Zvonarev (LPTMS)
SLIDE 6
- A. Tsvelik (BNL)
- R. Chitra (Jussieu)
- E. Orignac (ENS-Lyon)
- R. Citro (Salerno U.)
- P. Bouillot (Geneva)
- C. Kollath (Polytechnique)
- M. Zvonarev (LPTMS)
SLIDE 7
- A. Tsvelik (BNL)
- R. Chitra (Jussieu)
- E. Orignac (ENS-Lyon)
- R. Citro (Salerno U.)
- P. Bouillot (Geneva)
- C. Kollath (Polytechnique)
- M. Zvonarev (LPTMS)
SLIDE 8
- A. Tsvelik (BNL)
- R. Chitra (Jussieu)
- E. Orignac (ENS-Lyon)
- R. Citro (Salerno U.)
- P. Bouillot (Geneva)
- C. Kollath (Polytechnique)
- M. Zvonarev (LPTMS)
SLIDE 9
- A. Tsvelik (BNL)
- R. Chitra (Jussieu)
- E. Orignac (ENS-Lyon)
- R. Citro (Salerno U.)
- P. Bouillot (Geneva)
- C. Kollath (Polytechnique)
- M. Zvonarev (LPTMS)
Collaborations:
- M. Klanjsek +
group C. Berthier (Grenoble)
SLIDE 10
- A. Tsvelik (BNL)
- R. Chitra (Jussieu)
- E. Orignac (ENS-Lyon)
- R. Citro (Salerno U.)
- P. Bouillot (Geneva)
- C. Kollath (Polytechnique)
- M. Zvonarev (LPTMS)
Collaborations:
- M. Klanjsek +
group C. Berthier (Grenoble)
- B. Thielemann +
group C. Ruegg (LCN/PSI) +LANL Group
SLIDE 11
- A. Tsvelik (BNL)
- R. Chitra (Jussieu)
- E. Orignac (ENS-Lyon)
- R. Citro (Salerno U.)
- P. Bouillot (Geneva)
- C. Kollath (Polytechnique)
- M. Zvonarev (LPTMS)
Collaborations:
- M. Klanjsek +
group C. Berthier (Grenoble)
- B. Thielemann +
group C. Ruegg (LCN/PSI) +LANL Group
- D. Poilblanc ,
S Capponi (Toulouse)
- A. Laüchli (Dresden)
- B. Normand
SLIDE 12
Quantum magnetism
SLIDE 13
Quantum magnetism
Mott insulator: charge gapped
SLIDE 14
Quantum magnetism
Mott insulator: charge gapped
SLIDE 15
Quantum magnetism
Mott insulator: charge gapped Spin ½ degrees of freedom remains
SLIDE 16
Quantum magnetism
Mott insulator: charge gapped Spin ½ degrees of freedom remains Superexchange:
SLIDE 17
Quantum magnetism
Mott insulator: charge gapped Spin ½ degrees of freedom remains Superexchange:
SLIDE 18
Quantum magnetism
Mott insulator: charge gapped Spin ½ degrees of freedom remains Superexchange: Highly non trivial physics and phases
SLIDE 19
Itinerant quantum systems
SLIDE 20
Itinerant quantum systems
Very complicated (screened long range Coulomb)
SLIDE 21
Itinerant quantum systems
Very complicated (screened long range Coulomb) Find simplified models (Hubbard, t-J, etc…..)
SLIDE 22
Itinerant quantum systems
Very complicated (screened long range Coulomb) Find simplified models (Hubbard, t-J, etc…..)
SLIDE 23
Itinerant quantum systems
Very complicated (screened long range Coulomb) Find simplified models (Hubbard, t-J, etc…..)
SLIDE 24
Itinerant quantum systems
Very complicated (screened long range Coulomb) Find simplified models (Hubbard, t-J, etc…..) Physics: artefact of the approximations used ?
SLIDE 25
Itinerant quantum systems
Very complicated (screened long range Coulomb) Find simplified models (Hubbard, t-J, etc…..) Physics: artefact of the approximations used ? Use localised spin systems as controlled realizations
SLIDE 26
Dimer systems
SLIDE 27
Dimer systems
Jr À J
E H S T
SLIDE 28
Dimer systems
Jr À J
E H S T
SLIDE 29
Dimer systems
Jr À J
E H S T
SLIDE 30
Dimer systems
Jr À J
E H S T
SLIDE 31
Dimer systems
Jr À J
E H S T
SLIDE 32
Dimer systems
Jr À J
E H S T
SLIDE 33
Dimer systems
Jr À J
E H S T
SLIDE 34
Dimer systems
Jr À J
E H S T
SLIDE 35
Dimer systems
Jr À J
E H S T
SLIDE 36
Dimer systems
Jr À J
E H S T
SLIDE 37
Dimer systems
Jr À J
E H S T
SLIDE 38
Dimer systems
Jr À J
E H S T
SLIDE 39
Dimer systems
Jr À J
E H S T
SLIDE 40
Dimer systems
Jr À J
E H S T
SLIDE 41
Dimer systems
Jr À J
E H S T
SLIDE 42
Dimer systems
Jr À J
E H S T
SLIDE 43
h m
Gapped
SLIDE 44
h m
Gapped
SLIDE 45
h m
Gapped
SLIDE 46
h m
Gapped
hc1
SLIDE 47
h m
Gapped
hc1 hc2
SLIDE 48
h m
Gapped
hc1 hc2
SLIDE 49
h m
Gapped
hc1 hc2
SLIDE 50
h m
Gapped
hc1 hc2
Quantum phase transition
SLIDE 51
h m
Gapped
hc1 hc2
Quantum phase transition Magnetic field: ``chemical potential’’ for the
triplon band (interacting intinerant ``particles’’)
SLIDE 52
Two examples
SLIDE 53
Two examples
Bose Einstein condensation
(d=3,d=2….)
SLIDE 54
Two examples
Bose Einstein condensation
(d=3,d=2….)
Tomonaga-Luttinger liquids
(d=1)
SLIDE 55
Bose Einstein condensation
h T
Gapped
SLIDE 56
Bose Einstein condensation
(TG and A. M. Tsvelik PRB 59 11398 (1999))
h T
Gapped
SLIDE 57
Bose Einstein condensation
(TG and A. M. Tsvelik PRB 59 11398 (1999))
Phase of the boson : order in the X-Y plane
h T
Gapped
SLIDE 58
Bose Einstein condensation
(TG and A. M. Tsvelik PRB 59 11398 (1999))
Phase of the boson : order in the X-Y plane Why useful: close to hc1,hc2 : nearly free bosons
h T
Gapped
mx = 0
SLIDE 59
Bose Einstein condensation
(TG and A. M. Tsvelik PRB 59 11398 (1999))
Phase of the boson : order in the X-Y plane Why useful: close to hc1,hc2 : nearly free bosons
h T
Gapped
Tc » (h – hc)2/d mx = 0 mx ≠ 0
SLIDE 60
Bose Einstein condensation
(TG and A. M. Tsvelik PRB 59 11398 (1999))
Phase of the boson : order in the X-Y plane Why useful: close to hc1,hc2 : nearly free bosons
h T
Gapped
Tc » (h – hc)2/d mx = 0 mx ≠ 0
SLIDE 61
Bose Einstein condensation
(TG and A. M. Tsvelik PRB 59 11398 (1999))
Phase of the boson : order in the X-Y plane Why useful: close to hc1,hc2 : nearly free bosons
h T
Gapped
Tc » (h – hc)2/d mx = 0 mx ≠ 0
SLIDE 62
Non monotonous mz(T)
SLIDE 63
Non monotonous mz(T)
SLIDE 64
Non monotonous mz(T)
Cusp !
SLIDE 65
Non monotonous mz(T)
Cusp !
NMR relaxation time
SLIDE 66
Non monotonous mz(T)
Cusp !
NMR relaxation time
SLIDE 67
Experimental realization: TlCuCl3
SLIDE 68
Experimental realization: TlCuCl3
SLIDE 69
- T. Nikuni et al., PRL 84
5868 (2000)
Experimental realization: TlCuCl3
SLIDE 70
- T. Nikuni et al., PRL 84
5868 (2000)
Experimental realization: TlCuCl3
SLIDE 71
- C. Ruegg et al., PRB 65 132415 (2002)
Neutron evidence
SLIDE 72
- C. Ruegg et al., PRB 65 132415 (2002)
Neutron evidence
SLIDE 73
- C. Ruegg et al., PRB 65 132415 (2002)
- C. Ruegg et al., Nature 423 62 (2003)
Neutron evidence
SLIDE 74
- C. Ruegg et al., PRB 65 132415 (2002)
- C. Ruegg et al., Nature 423 62 (2003)
Neutron evidence
SLIDE 75
Other noteworthy realizations
SLIDE 76
Other noteworthy realizations
SLIDE 77
Other noteworthy realizations
SLIDE 78
S.E. Sebastian et al. Nature 441 617 (2006)
Other noteworthy realizations
SLIDE 79
S.E. Sebastian et al. Nature 441 617 (2006)
Other noteworthy realizations
+ Many theoretical works: Tsunetsugu, Troyer, Rice, Mila, Affleck, Normand, Batista, Oshikawa, Haas, ........
SLIDE 80
S.E. Sebastian et al. Nature 441 617 (2006)
Other noteworthy realizations
+ Many theoretical works: Tsunetsugu, Troyer, Rice, Mila, Affleck, Normand, Batista, Oshikawa, Haas, ........
TG, Ch. Rüegg, O. Tchernyshyov, Nat. Phys. 4 198 (08)
SLIDE 81
Spins – Cold atoms complementary
SLIDE 82
Spins – Cold atoms complementary
Cold atoms:
- control of lattice and parameters
- short range interactions
- inhomogeneous systems
- probes
SLIDE 83
Spins – Cold atoms complementary
Cold atoms:
- control of lattice and parameters
- short range interactions
- inhomogeneous systems
- probes
BEC dimers/spins:
- homogeneous, density control
- probes
- lattice fixed by chemistry
SLIDE 84
Spins – Cold atoms complementary
Cold atoms:
- control of lattice and parameters
- short range interactions
- inhomogeneous systems
- probes
BEC dimers/spins:
- homogeneous, density control
- probes
- lattice fixed by chemistry
SLIDE 85
Spins – Cold atoms complementary
Cold atoms:
- control of lattice and parameters
- short range interactions
- inhomogeneous systems
- probes
BEC dimers/spins:
- homogeneous, density control
- probes
- lattice fixed by chemistry
SLIDE 86
Bose glass phase
TG + H. J. Schulz PRB 37 325 (1988); M.P.A. Fisher et al. PRB 40 546 (1989)
SLIDE 87
Bose glass phase
- T. Hong, A. Zheludev, H. Manaka L.P. Regault,
arXiv/0909.1496
TG + H. J. Schulz PRB 37 325 (1988); M.P.A. Fisher et al. PRB 40 546 (1989)
SLIDE 88
Bose glass phase
- T. Hong, A. Zheludev, H. Manaka L.P. Regault,
arXiv/0909.1496 IPA-Cu(Cl0.95Br0.05)3
TG + H. J. Schulz PRB 37 325 (1988); M.P.A. Fisher et al. PRB 40 546 (1989)
SLIDE 89
Bose glass phase
- T. Hong, A. Zheludev, H. Manaka L.P. Regault,
arXiv/0909.1496 IPA-Cu(Cl0.95Br0.05)3 d m/d h = compressibility
TG + H. J. Schulz PRB 37 325 (1988); M.P.A. Fisher et al. PRB 40 546 (1989)
SLIDE 90
Bose glass phase
- T. Hong, A. Zheludev, H. Manaka L.P. Regault,
arXiv/0909.1496 IPA-Cu(Cl0.95Br0.05)3 d m/d h = compressibility h Sx i = h à i superfluid order parameter
TG + H. J. Schulz PRB 37 325 (1988); M.P.A. Fisher et al. PRB 40 546 (1989)
SLIDE 91
Bose glass phase
- T. Hong, A. Zheludev, H. Manaka L.P. Regault,
arXiv/0909.1496 IPA-Cu(Cl0.95Br0.05)3 d m/d h = compressibility h Sx i = h à i superfluid order parameter
TG + H. J. Schulz PRB 37 325 (1988); M.P.A. Fisher et al. PRB 40 546 (1989)
SLIDE 92
Tomonaga Luttinger Liquid
SLIDE 93
What about d= 1 ?
SLIDE 94
What about d= 1 ?
SLIDE 95
What about d= 1 ?
J
SLIDE 96
What about d= 1 ?
J J’
SLIDE 97
What about d= 1 ?
J J’
1D : hard core bosons are (spinless) fermions ! 1D : no BEC
SLIDE 98
What about d= 1 ?
J J’
1D : hard core bosons are (spinless) fermions ! 1D : no BEC
SLIDE 99
What about d= 1 ?
J J’
1D : hard core bosons are (spinless) fermions ! 1D : no BEC Allow to study interacting fermions/bosons:
Tomonaga Luttinger liquid
SLIDE 100
Tomonaga Luttinger liquid theory ( )
2
2 1 1
( ) (0) cos( / )
K z z x x
S x S x a π = +
SLIDE 101
Tomonaga Luttinger liquid theory ( )
2
2 1 1
( ) (0) cos( / )
K z z x x
S x S x a π = +
Low energy effective description Power law correlation functions
SLIDE 102
Tomonaga Luttinger liquid theory ( )
2
2 1 1
( ) (0) cos( / )
K z z x x
S x S x a π = +
Low energy effective description Power law correlation functions
SLIDE 103
Tomonaga Luttinger liquid theory ( )
2
2 1 1
( ) (0) cos( / )
K z z x x
S x S x a π = +
Low energy effective description All effects described by two parameters:
u and K (Luttinger liquid parameters)
Power law correlation functions Fractional excitations
SLIDE 104
Tomonaga Luttinger liquid theory ( )
2
2 1 1
( ) (0) cos( / )
K z z x x
S x S x a π = +
Low energy effective description All effects described by two parameters:
u and K (Luttinger liquid parameters)
Power law correlation functions Fractional excitations
SLIDE 105
Tomonaga Luttinger liquid theory ( )
2
2 1 1
( ) (0) cos( / )
K z z x x
S x S x a π = +
Low energy effective description All effects described by two parameters:
u and K (Luttinger liquid parameters)
Power law correlation functions Fractional excitations
SLIDE 106
Tomonaga Luttinger liquid theory ( )
2
2 1 1
( ) (0) cos( / )
K z z x x
S x S x a π = +
Low energy effective description All effects described by two parameters:
u and K (Luttinger liquid parameters)
Power law correlation functions Fractional excitations
SLIDE 107
BPCB-HPIP
SLIDE 108
BPCB-HPIP
- B. C. Watson et al., PRL 86 5168 (2001)
SLIDE 109
BPCB-HPIP
- B. C. Watson et al., PRL 86 5168 (2001)
SLIDE 110
BPCB-HPIP
- B. C. Watson et al., PRL 86 5168 (2001)
- M. Klanjsek et al.,
PRL 101 137207 (2008)
SLIDE 111
BPCB-HPIP
- B. C. Watson et al., PRL 86 5168 (2001)
- M. Klanjsek et al.,
PRL 101 137207 (2008)
- B. Thielemann et al.,
PRB 79, 020408(R) (2009)
SLIDE 112
BPCB-HPIP
- B. C. Watson et al., PRL 86 5168 (2001)
- M. Klanjsek et al.,
PRL 101 137207 (2008)
- B. Thielemann et al.,
PRB 79, 020408(R) (2009)
SLIDE 113
Expected phase diagram
T H
- S. Sachdev , T. Senthil, R. Shankar PRB 50 258 (94); R. Chitra, TG PRB 55 5816 (97); TG,
A.M. Tsvelik PRB 59 11398 (99)
SLIDE 114
Expected phase diagram
T H hc1 gapped
- S. Sachdev , T. Senthil, R. Shankar PRB 50 258 (94); R. Chitra, TG PRB 55 5816 (97); TG,
A.M. Tsvelik PRB 59 11398 (99)
SLIDE 115
Expected phase diagram
T H hc1 hc2 gapped
- S. Sachdev , T. Senthil, R. Shankar PRB 50 258 (94); R. Chitra, TG PRB 55 5816 (97); TG,
A.M. Tsvelik PRB 59 11398 (99)
SLIDE 116
Expected phase diagram
T H hc1 hc2 gapped
- S. Sachdev , T. Senthil, R. Shankar PRB 50 258 (94); R. Chitra, TG PRB 55 5816 (97); TG,
A.M. Tsvelik PRB 59 11398 (99)
SLIDE 117
Expected phase diagram
T H hc1 hc2 gapped
- S. Sachdev , T. Senthil, R. Shankar PRB 50 258 (94); R. Chitra, TG PRB 55 5816 (97); TG,
A.M. Tsvelik PRB 59 11398 (99)
SLIDE 118
Expected phase diagram
T H hc1 hc2 LL Thermal gapped
- S. Sachdev , T. Senthil, R. Shankar PRB 50 258 (94); R. Chitra, TG PRB 55 5816 (97); TG,
A.M. Tsvelik PRB 59 11398 (99)
SLIDE 119
Expected phase diagram
T H hc1 hc2 LL Thermal gapped J
- S. Sachdev , T. Senthil, R. Shankar PRB 50 258 (94); R. Chitra, TG PRB 55 5816 (97); TG,
A.M. Tsvelik PRB 59 11398 (99)
SLIDE 120
Expected phase diagram
T H hc1 hc2 LL 3D Thermal gapped J
- S. Sachdev , T. Senthil, R. Shankar PRB 50 258 (94); R. Chitra, TG PRB 55 5816 (97); TG,
A.M. Tsvelik PRB 59 11398 (99)
SLIDE 121
Expected phase diagram
T H hc1 hc2 LL 3D Thermal gapped J
- S. Sachdev , T. Senthil, R. Shankar PRB 50 258 (94); R. Chitra, TG PRB 55 5816 (97); TG,
A.M. Tsvelik PRB 59 11398 (99)
SLIDE 122
Expected phase diagram
T H hc1 hc2 LL 3D Thermal gapped J
- S. Sachdev , T. Senthil, R. Shankar PRB 50 258 (94); R. Chitra, TG PRB 55 5816 (97); TG,
A.M. Tsvelik PRB 59 11398 (99)
SLIDE 123
Expected phase diagram
T H hc1 hc2 LL 3D Thermal gapped J J’
- S. Sachdev , T. Senthil, R. Shankar PRB 50 258 (94); R. Chitra, TG PRB 55 5816 (97); TG,
A.M. Tsvelik PRB 59 11398 (99)
SLIDE 124
Expected phase diagram
T H hc1 hc2 LL 3D Thermal gapped J J’
- S. Sachdev , T. Senthil, R. Shankar PRB 50 258 (94); R. Chitra, TG PRB 55 5816 (97); TG,
A.M. Tsvelik PRB 59 11398 (99)
SLIDE 125
How to compute
SLIDE 126
How to compute
- Analytical calculations (Luttinger liquid +
BA)
SLIDE 127
How to compute
- Analytical calculations (Luttinger liquid +
BA)
- Numerical calculations (DMRG)
SLIDE 128
How to compute
- Analytical calculations (Luttinger liquid +
BA)
- Numerical calculations (DMRG)
SLIDE 129
How to compute
- Analytical calculations (Luttinger liquid +
BA)
- Numerical calculations (DMRG)
- S. White
SLIDE 130
Finite temperature DMRG
SLIDE 131
Finite temperature DMRG
Specific heat of spin chain
SLIDE 132
Magnetization
SLIDE 133
Magnetization
SLIDE 134
Magnetization
- M. Klanjsek et al., PRL 101 137207 (2008)
SLIDE 135
Magnetization
- M. Klanjsek et al., PRL 101 137207 (2008)
Fixes: Jr = 12.9 K J = 3.6 K
SLIDE 136
Magnetization
- M. Klanjsek et al., PRL 101 137207 (2008)
Fixes: Jr = 12.9 K J = 3.6 K
SLIDE 137
Specific heat
SLIDE 138
Specific heat
SLIDE 139
Specific heat
- Ch. Rüegg et al.,
PRL 101, 247202 (2008)
SLIDE 140
Specific heat
- Ch. Rüegg et al.,
PRL 101, 247202 (2008)
Peak : end of LL Luttinger liquid: Cv / ° T
SLIDE 141
Specific heat
- Ch. Rüegg et al.,
PRL 101, 247202 (2008)
Peak : end of LL Luttinger liquid: Cv / ° T
SLIDE 142
Specific heat
- Ch. Rüegg et al., PRL
101, 247202 (2008)
SLIDE 143
Can one control TLL ?
SLIDE 144
Can one control TLL ?
Compute u(h) and K(h) from (Jr,J,h)
SLIDE 145
Can one control TLL ?
Compute u(h) and K(h) from (Jr,J,h) No adjustable parameters !!
SLIDE 146
Can one control TLL ?
Compute u(h) and K(h) from (Jr,J,h) No adjustable parameters !! Get all (several) correlation functions
SLIDE 147
Can one control TLL ?
Compute u(h) and K(h) from (Jr,J,h) No adjustable parameters !! Get all (several) correlation functions
Allows to quantitatively test for TLL!
SLIDE 148
Luttinger parameters
SLIDE 149
Luttinger parameters
- M. Klanjsek et al., PRL 101 137207 (2008)
SLIDE 150
Luttinger parameters
- M. Klanjsek et al., PRL 101 137207 (2008)
SLIDE 151
Luttinger parameters
- M. Klanjsek et al., PRL 101 137207 (2008)
Red : Ladder (DMRG) Green: Strong coupling (Jr → 1 ) (BA)
SLIDE 152
Correlation functions
SLIDE 153
Correlation functions
NMR relaxation rate:
- M. Klanjsek et al., PRL 101 137207 (2008)
SLIDE 154
Correlation functions
NMR relaxation rate:
- M. Klanjsek et al., PRL 101 137207 (2008)
- R. Chitra, TG PRB 55 5816 (97); TG, AM Tsvelik PRB 59 11398 (99)
SLIDE 155
Correlation functions
NMR relaxation rate: Tc to ordered phase: 1/J’ = Â1D(Tc)
- M. Klanjsek et al., PRL 101 137207 (2008)
- R. Chitra, TG PRB 55 5816 (97); TG, AM Tsvelik PRB 59 11398 (99)
SLIDE 156
Correlation functions
NMR relaxation rate: Tc to ordered phase: 1/J’ = Â1D(Tc)
- M. Klanjsek et al., PRL 101 137207 (2008)
- R. Chitra, TG PRB 55 5816 (97); TG, AM Tsvelik PRB 59 11398 (99)
SLIDE 157
Correlation functions
NMR relaxation rate: Tc to ordered phase: 1/J’ = Â1D(Tc)
- M. Klanjsek et al., PRL 101 137207 (2008)
- R. Chitra, TG PRB 55 5816 (97); TG, AM Tsvelik PRB 59 11398 (99)
SLIDE 158
Correlation functions
NMR relaxation rate: Tc to ordered phase: 1/J’ = Â1D(Tc)
- M. Klanjsek et al., PRL 101 137207 (2008)
- R. Chitra, TG PRB 55 5816 (97); TG, AM Tsvelik PRB 59 11398 (99)
SLIDE 159
Ordered phase
SLIDE 160
Ordered phase
Order parameter at T=0
SLIDE 161
Ordered phase
Order parameter at T=0
SLIDE 162
Ordered phase
Order parameter at T=0
SLIDE 163
NMR
SLIDE 164
NMR
SLIDE 165
- M. Klanjsek et al.,
PRL 101 137207 (2008)
NMR
SLIDE 166
- M. Klanjsek et al.,
PRL 101 137207 (2008)
NMR
SLIDE 167
Neutrons
SLIDE 168
Neutrons
SLIDE 169
Neutrons
J’ = 27 mK
- B. Thielemann et al.,
PRB 79, 020408(R) (2009)
SLIDE 170
Neutrons
J’ = 27 mK
- B. Thielemann et al.,
PRB 79, 020408(R) (2009)
SLIDE 171
Neutrons
J’ = 27 mK
- B. Thielemann et al.,
PRB 79, 020408(R) (2009)
SLIDE 172
Beyond Luttinger liquid
SLIDE 173
Beyond Luttinger liquid
Close to the critical points: hc1, hc2:
dimensional crossover (fermions → bosons)
SLIDE 174
Beyond Luttinger liquid
Close to the critical points: hc1, hc2:
dimensional crossover (fermions → bosons)
High energy correlations
(LL valid for !, T ¿ J)
SLIDE 175
High energy dynamical correlations
SLIDE 176
High energy dynamical correlations
- B. Thielemann et al. PRL 102 107204 (2009)
SLIDE 177
High energy dynamical correlations
- B. Thielemann et al. PRL 102 107204 (2009)
SLIDE 178
High energy dynamical correlations
- B. Thielemann et al. PRL 102 107204 (2009)
SLIDE 179
High energy dynamical correlations
- B. Thielemann et al. PRL 102 107204 (2009)
SLIDE 180
High energy dynamical correlations
- B. Thielemann et al. PRL 102 107204 (2009)
Time dependent DMRG
SLIDE 181
Conclusions
SLIDE 182
Conclusions
Localized spin systems have several behaviors
corresponding to itinerant quantum systems.
SLIDE 183
Conclusions
Localized spin systems have several behaviors
corresponding to itinerant quantum systems.
BPCB (Ladders): remarkable system to show
quantitatively Tomonaga Luttinger liquid universality class
Dimers offer several advantages;
BEC describes well systems in d=3, d=2.
SLIDE 184
Conclusions
Localized spin systems have several behaviors
corresponding to itinerant quantum systems.
BPCB (Ladders): remarkable system to show
quantitatively Tomonaga Luttinger liquid universality class
Dimers offer several advantages;
BEC describes well systems in d=3, d=2.
SLIDE 185
Perspectives
SLIDE 186
Perspectives
Behavior close to quantum critical points:
Luttinger (fermions) → BEC (bosons)
SLIDE 187
Perspectives
Behavior close to quantum critical points:
Luttinger (fermions) → BEC (bosons)
Dynamical quantities in the quantum critical
regime
SLIDE 188
Perspectives
Behavior close to quantum critical points:
Luttinger (fermions) → BEC (bosons)
Dynamical quantities in the quantum critical
regime
Other materials, impurities and doping