Unparticles in High T_c Superconductors
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1 Unparticles in High T_c Superconductors Thanks to: NSF, EFRC (DOE) Kiaran dave Charlie Kane Brandon Langley J. A. Hutasoit
2 Correlated Electron Matter
3 Correlated Electron Matter What is carrying the current?
4 fractional quantum Hall effect current-carrying excitations e = e q (q odd)
5 fractional quantum Hall effect current-carrying excitations e = e q (q odd)
6 fractional quantum Hall effect current-carrying excitations
7 fractional quantum Hall effect current-carrying excitations e =?
8 How Does Fermi Liquid Theory Breakdown? TRSB / at? LaSrCuO
9 can the charge density be given a particle interpretation?
10 can the charge density be given a particle interpretation? does charge density= total number of electrons?
11 if not?
12 if not? charge stuff= particles + other stuff
13 if not? charge stuff= particles + other stuff unparticles!
14 counting particles
15 counting particles 1
16 counting particles 2 1
17 counting particles 2 3 1
18 counting particles
19 counting particles is there a more efficient way?
20 µ IR 1 UV
21 n = Z µ 1 N(!)d! µ IR 1 UV
22 n = Z µ 1 N(!)d! µ IR 1 UV can n be deduced entirely from the IR(lowenergy) scale?
23 Luttinger s `theorem
24 Each particle has an energy
25 Each particle has an energy particles energy E = " p
26 Each particle has an energy particles energy E = " p G(E) = E 1 " p
27 Each particle has an energy particles energy E = " p G(E) = E 1 " p particle=infinity
28 Each particle has an energy particles energy E = " p Green function (propagator) 1 G(E) = E " p particle=infinity
29 Each particle has an energy particles particle count is deducible from Green function energy E = " p Green function (propagator) 1 G(E) = E " p particle=infinity
30 Luttinger s Theorem Green function (propagator) G(E) = E 1 " p G(E) " p E
31 Luttinger s Theorem Green function (propagator) G(E) = E 1 " p G(E) E>" p " p E
32 Luttinger s Theorem Green function (propagator) G(E) = E 1 " p G(E) E>" p " p E E<" p
33 Luttinger s Theorem Green function (propagator) G(E) = E 1 " p G(E) E>" p " p E E<" p particle density= number of sign changes of G
34 Counting sign changes?
35 Counting sign changes? (x)
36 Counting sign changes? (x) = { 0 x<0
37 Counting sign changes? (x) = { 0 x<0 1 x>0
38 Counting sign changes? (x) = { 0 x<0 1 x>0 1/2 x =0
39 Counting sign changes? (x) = { 0 x<0 1 x>0 1/2 x =0 counts sign changes
40 Counting sign changes? (x) = { 0 x<0 1 x>0 1/2 x =0 counts sign changes n =2 X k (<G(k,! = 0)) Luttinger Theorem for electrons
41 How do functions change sign? E>" p " p E E<" p
42 How do functions change sign? E>" p " p E E<" p divergence
43
44 Is there another way?
45
46 Yes
47 zero-crossing " p E DetG = 0
48 zero-crossing " p E DetG = 0 no divergence is necessary
49 closer look at Luttinger s theorem n =2 X k (<G(k,! = 0))
50 closer look at Luttinger s theorem n =2 X k (<G(k,! = 0)) divergences+ zeros
51 closer look at Luttinger s theorem n =2 X k (<G(k,! = 0)) divergences+ zeros how can zeros affect the particle count?
52 what are zeros?
53 what are zeros? Im G=0 µ =0
54 what are zeros? Im G=0 Z 1 µ =0 ReG(0,p)= 1 ( ) d!! Kramers -Kronig
55 what are zeros? Im G=0 Z 1 µ =0 ReG(0,p)= 1 ( ) d!! Kramers -Kronig = below gap+above gap
56 what are zeros? Im G=0 Z 1 µ =0 ReG(0,p)= 1 ( ) d!! Kramers -Kronig = below gap+above gap = 0
57 what are zeros? Im G=0 Z 1 µ =0 ReG(0,p)= 1 ( ) d!! Kramers -Kronig = below gap+above gap = 0 DetG(k,! = 0) =0 (single band)
58 what are zeros? Im G=0 Z 1 µ =0 ReG(0,p)= 1 ( ) d!! Kramers -Kronig = below gap+above gap = 0 DetG(k,! = 0) =0 (single band) strongly correlated gapped systems
59 NiO insulates d 8? Mott mechanism (not Slater) t
60 NiO insulates d 8? perhaps this costs energy Mott mechanism (not Slater) t
61 NiO insulates d 8? perhaps this costs energy Mott mechanism (not Slater) t U t
62 NiO insulates d 8? perhaps this costs energy Mott mechanism (not Slater) t U t µ =0
63 NiO insulates d 8? perhaps this costs energy Mott mechanism (not Slater) t U t µ =0 no change in size of Brillouin zone
64 NiO insulates d 8? perhaps this costs energy Mott mechanism (not Slater) t U t µ =0 no change in size of Brillouin zone DetReG(0,p)=0
65 NiO insulates d 8? perhaps this costs energy Mott mechanism (not Slater) t U t µ =0 no change in size of Brillouin zone DetReG(0,p)=0 =Mottness
66 U/t = 10 1 interactions dominate: Strong Coupling Physics
67 simple problem: n=1 SU(2) U
68 simple problem: n=1 SU(2) µ U
69 simple problem: n=1 SU(2) µ U U 2
70 simple problem: n=1 SU(2) U 2 µ U U 2
71 simple problem: n=1 SU(2) U 2 µ U U 2 G = 1! + U/2 + 1! U/2
72 simple problem: n=1 SU(2) U 2 µ U U 2 G = 1! + U/2 + 1! U/2 =0 if! =0
73 simple problem: n=1 SU(2) U 2 µ U U 2 G = 1! + U/2 + 1! U/2 n =2 (0) = 1 =0 if! =0
74 Fermi Liquids Mott Insulators Luttinger s theorem
75
76 Is this famous theorem from 1960 correct?
77 A model with zeros but Luttinger fails
78 A model with zeros but Luttinger fails e t t t t t t t t
79 A model with zeros but Luttinger fails e t t t t t t t t
80 A model with zeros but Luttinger fails e t t t t t t t t no hopping=> no propagation (zeros)
81 A model with zeros but Luttinger fails e N flavors of e- t t t t t t t t no hopping=> no propagation (zeros)
82 SU(N) H = U 2 (n 1 + n N ) 2 2E 25 N =
83 SU(N) for N=3: no particle-hole symmetry
84 SU(N) for N=3: no particle-hole symmetry
85 SU(N) for N=3: no particle-hole symmetry lim µ(t ) T!0
86 G (! = 0) = # 2n N N
87 G (! = 0) = # 2n N N
88 Luttinger s theorem n = N (2n N)
89 Luttinger s theorem n = N (2n N) { 0, 1, 1/2
90 Luttinger s theorem n = N (2n N) n =2 N =3 { 0, 1, 1/2
91 Luttinger s theorem n = N (2n N) n =2 N =3 { 0, 1, 1/2 2=3
92 Luttinger s theorem n = N (2n N) n =2 N =3 { 0, 1, 1/2 2=3
93 Luttinger s theorem n = N (2n N) n =2 N =3 { 0, 1, 1/2 even 2=3
94 Luttinger s theorem n = N (2n N) n =2 N =3 { 0, 1, 1/2 even 2=3 odd
95 Luttinger s theorem n = N (2n N) n =2 N =3 { 0, 1, 1/2 even 2=3 odd no solution
96 does the degeneracy matter? e t t t t t t t t t =0 +
97 No
98 No as long as SU(N) symmetry is intact ab =Tr c ac b (0 + ) =1/N diag(1, 1, 1,...) mixed state pure state
99 Problem DetG =0 G = 1 E " p =0??
100 Problem DetG =0 G = 1 E " p =0?? G = 1 E " p 1
101 Problem DetG =0 G = 1 E " p =0?? G = 1 E " p 1 G=0
102 if is infinite
103 if is infinite lifetime of a particle vanishes
104 if is infinite lifetime of a particle = < p vanishes
105 if is infinite lifetime of a particle = < p vanishes no particle
106 what went wrong?
107 what went wrong? I[G] = Z d! G
108 what went wrong? I[G] = Z d! G if!1
109 what went wrong? I[G] = Z d! G if!1 integral does not exist
110 what went wrong? I[G] = Z d! G if!1 integral does not exist No Luttinger theorem!
111 zeros and particles are incompatible
112 Luttinger s theorem
113
114 are zeros important?
115 photoemission
116 photoemission particles energy E = " p
117 photoemission particles energy E = " p p 2
118 photoemission particles energy E = " p p 2
119 photoemission particles energy E = " p p 2 two crossings: closed surface of excitations
120 what is seen experimentally? Fermi arcs: no double crossings (PDJ,JCC,ZXS) seen not seen E F k
121 Problem poles ReG>0 Re G<0
122 Problem poles ReG>0 Re G<0
123 Problem How to account for the sign change without poles? poles ReG>0 Re G<0
124 Problem How to account for the sign change without poles? poles ReG>0 Re G<0 Only option: DetG=0! (zeros )
125 Problem How to account for the sign change without poles? poles Re G< ReG> Only option: DetG=0! (zeros )
126 experimental data (LSCO) `Luttinger count k F 1 x FS Bi2212 zeros do not affect the particle density
127 experimental data (LSCO) `Luttinger count k F 1 x FS Bi2212 zeros do not affect the particle density each hole = a single k-state
128 how to count particles? some charged stuff has no particle interpretation
129
130 what is the extra stuff?
131
132 scale invariance
133 new fixed point (scale invariance)?
134 new fixed point (scale invariance)
135 new fixed point (scale invariance) unparticles (IR) (H. Georgi)
136 what is scale invariance? invariance on all length scales
137 f(x) =x 2
138 f(x) =x 2 f(x/ )=(x/ ) 2 scale change
139 f(x) =x 2 f(x/ )=(x/ ) 2 scale change scale invariance
140 f(x) =x 2 f(x/ )=(x/ ) 2 scale change scale invariance f(x) =x 2 2 g( ) 1{
141 f(x) =x 2 f(x/ )=(x/ ) 2 scale change scale invariance f(x) =x 2 2 g( ) 1{ g( )= 2
142 what s the underlying theory?
143 free field theory L = µ x! x/ (x)! (x) L! 2 L
144 free field theory L = µ x! x/ (x)! (x) L! 2 L scale invariant
145 massive free theory mass L = µ + m 2 2
146 massive free theory mass L = µ + m 2 2 x! x/ (x)! (x)
147 massive free theory mass L = µ + m 2 2 x! x/ (x)! (x) m 2 2
148 massive free theory mass L = µ + m µ x! x/ (x)! (x) m 2 2
149 massive free theory mass L = µ + m µ x! x/ (x)! (x) m 2 2
150 massive free theory mass L = µ + m µ x! x/ (x)! (x) m 2 2 no scale invariance
151 mass sets a scale
152
153 unparticles from a massive theory?
154 L µ (x, m)@ µ (x, m)+m 2 2 (x, m)
155 Z 1 L µ (x, m)@ µ (x, m)+m 2 2 (x, m) 0 dm 2
156 Z 1 L µ (x, m)@ µ (x, m)+m 2 2 (x, m) 0 dm 2 theory with all possible mass!
157 Z 1 L µ (x, m)@ µ (x, m)+m 2 2 (x, m) 0 dm 2 theory with all possible mass!! (x, m 2 / 2 ) x! x/ m 2 / 2! m 2
158 Z 1 L µ (x, m)@ µ (x, m)+m 2 2 (x, m) 0 dm 2 theory with all possible mass!! (x, m 2 / 2 ) x! x/ m 2 / 2! m 2 L! 4 L scale invariance is restored!!
159 Z 1 L µ (x, m)@ µ (x, m)+m 2 2 (x, m) 0 dm 2 theory with all possible mass!! (x, m 2 / 2 ) x! x/ m 2 / 2! m 2 L! 4 L scale invariance is restored!! not particles
160 unparticles Z 1 L µ (x, m)@ µ (x, m)+m 2 2 (x, m) 0 dm 2 theory with all possible mass!! (x, m 2 / 2 ) x! x/ m 2 / 2! m 2 L! 4 L scale invariance is restored!! not particles
161 propagator Z 1 0 dm 2 m 2 p 2 i m 2 + i 1 / p 2
162 propagator Z 1 0 dm 2 m 2 p 2 i m 2 + i 1 / p 2 d U 2
163 n massless particles G / (E " p ) n 2
164 n massless particles G / (E " p ) n 2 G / (E " p ) d U 2
165 n massless particles G / (E " p ) n 2 G / (E " p ) d U 2 unparticles=fractional number of massless particles
166 n massless particles G / (E " p ) n 2 G / (E " p ) d U 2 unparticles=fractional number of massless particles d U > 2 zeros
167 n massless particles G / (E " p ) n 2 G / (E " p ) d U 2 unparticles=fractional number of massless particles d U > 2 zeros no simple sign change
168 no particle interpretation mass-distribution U = Z B(m 2 ) (m 2 )dm 2 unparticle field (non-canonical) particle field
169 d U?
170 what really is the summation over mass?
171 what really is the summation over mass? mass=energy
172 high energy (UV) QFT low energy (IR) related to sum over mass
173 high energy (UV) QFT low energy (IR) dg(e) = (g(e)) dlne locality in energy related to sum over mass
174 implement E-scaling with an extra dimension high energy (UV) QFT low energy (IR) dg(e) = (g(e)) dlne locality in energy related to sum over mass
175 implement E-scaling with an extra dimension QFT dg(e) dlne = (g(e)) locality in energy related to sum over mass
176 gauge-gravity duality (Maldacena, 1997) implement E-scaling with an extra dimension UV IR gravity QF T QFT dg(e) dlne = (g(e)) locality in energy related to sum over mass
177 gauge-gravity duality (Maldacena, 1997) implement E-scaling with an extra dimension IR gravity UV no particles QFT (conserved currents) QF T dg(e) dlne = (g(e)) locality in energy related to sum over mass
178 gauge-gravity duality (Maldacena, 1997) implement E-scaling with an extra dimension IR gravity UV no particles QFT (conserved currents) QF T dg(e) dlne = (g(e)) locality in energy related to sum over mass unparticles?
179 what s the geometry?
180 what s the geometry? dg(e) dlne = (g(e)) =0 scale invariance (continuous)
181 what s the geometry? dg(e) dlne = (g(e)) =0 scale invariance (continuous) E! E x µ! x µ /
182 what s the geometry? dg(e) dlne = (g(e)) =0 scale invariance (continuous) E! E x µ! x µ / solve Einstein equations
183 what s the geometry? dg(e) dlne = (g(e)) =0 scale invariance (continuous) E! E x µ! x µ / solve Einstein equations ds 2 = u L 2 µ dx µ dx + L E 2 de 2 anti de-sitter space
184 mass=extra dimension fixed by metric
185 can something more exact be done?
186 can something more exact be done? Z 1 L µ (x, m)@ µ (x, m)+m 2 2 (x, m) 0 m 2 dm 2
187 can something more exact be done? Z 1 L µ (x, m)@ µ (x, m)+m 2 2 (x, m) 0 m 2 dm 2 m = z 1
188 can something more exact be done? Z 1 L µ (x, m)@ µ (x, m)+m 2 2 (x, m) 0 m 2 dm 2 m = z 1 L = Z 1 0 dz 2R2 z 5+2 apple 1 2 z 2 R 2 µ (@ µ )(@ )+ 2 2R 2
189 can something more exact be done? Z 1 L µ (x, m)@ µ (x, m)+m 2 2 (x, m) 0 m 2 dm 2 m = z 1 L = Z 1 0 dz 2R2 z 5+2 apple 1 2 z 2 R 2 µ (@ µ )(@ )+ 2 2R 2 can be absorbed with AdS metric
190 generating functional for unparticles action on AdS 5+2 S = 1 2 Z d 4+2 xdz p a + 2 R 2
191 generating functional for unparticles action on AdS 5+2 S = 1 2 Z d 4+2 xdz p a + 2 R 2 ds 2 = L2 z 2 µ dx µ dx + dz 2 p g =(R/z) 5+2
192 generating functional for unparticles action on AdS 5+2 S = 1 2 Z d 4+2 xdz p a + 2 R 2 ds 2 = L2 z 2 µ dx µ dx + dz 2 p g =(R/z) 5+2 unparticle lives in d =4+2 apple 0
193 scaling dimension is fixed m = 1 z
194 scaling dimension is fixed m = 1 z m 2 AdS = d U (d U d) R 2
195 scaling dimension is fixed m = 1 z 1=d m 2 AdS = d U (d U d) U (d U d) R 2
196 scaling dimension is fixed m = 1 z 1=d m 2 AdS = d U (d U d) U (d U d) R 2 d U = d 2 + p d2 +4 2
197 G U (p) / p 2(d U d/2) G U (0) = 0
198 unparticle (AdS) propagator has zeros! G U (p) / p 2(d U d/2) G U (0) = 0
199 interchanging unparticles fractional (d_u) number of massless particles
200 interchanging unparticles fractional (d_u) number of massless particles
201 interchanging unparticles fractional (d_u) number of massless particles
202 interchanging unparticles fractional (d_u) number of massless particles
203 interchanging unparticles fractional (d_u) number of massless particles e i d U 6= 1, 0
204 interchanging unparticles fractional (d_u) number of massless particles e i d U 6= 1, 0 fractional statistics in d=2+1
205 d U = d 2 + p d > d 2 e i d U 6= e i d U time-reversal symmetry breaking from unparticle (zeros=fermi arcs) matter
206 d ln g d ln = 4d U d>0 fermions T c unparticles with unparticle propagator BCS g tendency towards pairing (any instability which establishes a gap)
207 TRSB
208 TRSB
209 TRSB unparticles
210 TRSB unparticles particles
211 breaking of scale invariance TRSB unparticles particles
212 emergent gravity High T_c unparticles variable mass
213 emergent gravity High T_c unparticles variable mass fractional statistics in d=2+1
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