Is Strongly Correlated Electron Matter Full of Unparticles? Kiaran dave Charlie Kane Brandon Langley
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1 Is Strongly Correlated Electron Matter Full of Unparticles? Kiaran dave Charlie Kane Brandon Langley
2 Correlated Matter
3 Correlated Matter What is carrying the current?
4
5 current-carrying excitations?
6 current-carrying excitations? e = e q (q odd)
7 fractional quantum Hall effect current-carrying excitations? e = e q (q odd)
8 fractional quantum Hall effect current-carrying excitations?
9 fractional quantum Hall effect current-carrying excitations? e =?
10 Correlated Electron Matter What is carrying the current? UV IR
11 Correlated Electron Matter What is carrying the current? e e e UV e e e e e e e e IR
12 Correlated Electron Matter What is carrying the current? e e e UV e e e e e e e e IR
13 Correlated Electron Matter What is carrying the current? e e e UV e e e e e e e e not particles IR
14 can the charge density be given a particle interpretation?
15 if not?
16 if not? charge stuff= particles + other stuff
17 if not? charge stuff= particles + other stuff unparticles!
18 counting particles
19 counting particles 1
20 counting particles 2 1
21 counting particles 2 3 1
22 counting particles
23 counting particles is there a more efficient way?
24 µ
25 n = µ N(ω)dω µ
26 n = µ N(ω)dω µ IR UV
27 n = µ N(ω)dω µ IR UV can n be deduced entirely from the IR(lowenergy) scale?
28 Luttinger s `theorem
29 Each particle has an energy
30 Each particle has an energy particles energy E = ε p
31 Each particle has an energy particles energy E = ε p Green function (propagator) 1 G(E) = E ε p particle=infinity
32 Each particle has an energy particles particle count is deducible from Green function energy E = ε p Green function (propagator) 1 G(E) = E ε p particle=infinity
33 Luttinger s Theorem Green function (propagator) G(E) = G(E) 1 E ε p ε p E
34 Luttinger s Theorem Green function (propagator) G(E) = G(E) 1 E ε p E>ε p ε p E
35 Luttinger s Theorem Green function (propagator) G(E) = G(E) 1 E ε p E>ε p ε p E E<ε p
36 Luttinger s Theorem Green function (propagator) G(E) = G(E) 1 E ε p E>ε p ε p E E<ε p particle density= number of sign changes of G
37 Counting sign changes?
38 Counting sign changes? Θ(x)
39 Counting sign changes? Θ(x) = { 0 x<0
40 Counting sign changes? Θ(x) = { 0 x<0 1 x>0
41 Counting sign changes? Θ(x) = { 0 x<0 1 x>0 1/2 x =0
42 Counting sign changes? Θ(x) = { 0 x<0 1 x>0 1/2 x =0 counts sign changes
43 Counting sign changes? Θ(x) = { 0 x<0 1 x>0 1/2 x =0 counts sign changes n =2 p Θ(G(E =0, p)) Luttinger Theorem for electrons
44 How do functions change sign? E>ε p ε p E E<ε p
45 How do functions change sign? E>ε p ε p E E<ε p divergence
46
47 Is there another way?
48
49 Yes
50 zero-crossing ε p E
51 zero-crossing ε p E no divergence is necessary
52 closer look at Luttinger s theorem n =2 p Θ(G(E =0, p))
53 closer look at Luttinger s theorem n =2 p Θ(G(E =0, p)) divergences+ zeros
54 closer look at Luttinger s theorem n =2 p Θ(G(E =0, p)) divergences+ zeros how can zeros affect the particle count?
55
56 Is this famous theorem from 1960 correct?
57 A model with zeros but Luttinger fails
58 A model with zeros but Luttinger fails e t t t t t t t t
59 A model with zeros but Luttinger fails e t t t t t t t t
60 A model with zeros but Luttinger fails e t t t t t t t t no hopping=> no propagation (zeros)
61 A model with zeros but Luttinger fails N flavors e t t t t t of e- t t t no hopping=> no propagation (zeros)
62 e spin SU(2)
63 e spin SU(2) generalization N flavors of spin SU(N)
64 e spin SU(2) generalization N =5 N flavors of spin SU(N)
65 e spin SU(2) generalization N =5 N flavors of spin SU(N)
66 e spin SU(2) generalization 2E 25 N = N flavors of spin SU(N)
67 e spin SU(2) generalization 2E N flavors of spin SU(N) 25 N = H = U 2 (n 1 + n N ) 2
68 SU(2) U
69 SU(2) U G = # ω + U/2 + # ω U/2
70 SU(N) for N=3: no particle-hole symmetry
71 SU(N) for N=3: no particle-hole symmetry lim µ(t ) T 0
72 G αβ (ω = 0) = δ αβ K(n + 1) K(n) 2n N N
73 G αβ (ω = 0) = δ αβ K(n + }> 1) K(n) 0 2n N N
74 G αβ (ω = 0) = δ αβ K(n + }> 1) K(n) 0 2n N N
75 < 0 G αβ (ω = 0) = δ αβ K(n + }> 1) K(n) 0 2n N N
76 < 0 > 0 G αβ (ω = 0) = δ αβ K(n + }> 1) K(n) 0 2n N N
77 a zero must exist < 0 > 0 G αβ (ω = 0) = δ αβ K(n + }> 1) K(n) 0 2n N N
78 Luttinger s theorem n = NΘ(2n N)
79 Luttinger s theorem n = NΘ(2n N) { 0, 1, 1/2
80 Luttinger s theorem n = NΘ(2n N) n =2 N =3 { 0, 1, 1/2
81 Luttinger s theorem n = NΘ(2n N) n =2 N =3 { 0, 1, 1/2 2=3
82 Luttinger s theorem n = NΘ(2n N) n =2 N =3 { 0, 1, 1/2 2=3
83 Luttinger s theorem n = NΘ(2n N) n =2 N =3 { 0, 1, 1/2 even 2=3
84 Luttinger s theorem n = NΘ(2n N) n =2 N =3 { 0, 1, 1/2 even 2=3 odd
85 Luttinger s theorem n = NΘ(2n N) n =2 N =3 { 0, 1, 1/2 even 2=3 odd no solution
86 Luttinger s theorem
87
88 Problem
89 Problem G=0
90 Problem G=0 G = 1 E ε p =0??
91 Problem G=0 G = 1 E ε p =0?? G αβ (ω = 0) = δ αβ K(n + 1) K(n) 2n N N
92 Problem G=0 G = 1 E ε p =0?? G αβ (ω = 0) = δ αβ K(n + 1) K(n) 2n N N must diverge
93 G = 1 E ε p Σ G=0
94 self energy diverges G = 1 E ε p Σ G=0
95 if Σ is infinite
96 if Σ is infinite lifetime of a particle vanishes
97 if Σ is infinite lifetime of a particle Σ < p vanishes
98 if Σ is infinite lifetime of a particle Σ < p vanishes no particle
99 zeros and particles are incompatible
100 zeros and particles are incompatible Luttinger theorem is wrong!
101 how to count particles?
102 how to count particles? some charged stuff has no particle interpretation
103
104 what is the extra stuff?
105 Properties all particles share?
106 Properties all particles share? charge fixed mass!
107 Properties all particles share? charge fixed mass! conserved (gauge invariance)
108 Properties all particles share? charge fixed mass! conserved (gauge not conserved invariance)
109 mass sets a scale
110 mass sets a scale mass breaks scale invariance
111 what is scale invariance? invariance on all length scales
112 f(x) =x 2
113 f(x) =x 2 f(x/λ) =(x/λ) 2 scale change
114 f(x) =x 2 f(x/λ) =(x/λ) 2 scale change scale invariance
115 f(x) =x 2 f(x/λ) =(x/λ) 2 scale change scale invariance f(x) =x 2 λ 2 g(λ) 1{
116 f(x) =x 2 f(x/λ) =(x/λ) 2 scale change scale invariance f(x) =x 2 λ 2 g(λ) 1{ g(λ) =λ 2
117 f(x) =ax 2 + bx 3 x x/λ
118 f(x) =ax 2 + bx 3 x x/λ f(x) Λ 2 (ax 2 + bx/λ)
119 f(x) =ax 2 + bx 3 x x/λ f(x) Λ 2 (ax 2 + bx/λ) not scale invariant
120 free field theory L = 1 2 µ φ µ φ x x/λ φ(x) φ(x) L Λ 2 L
121 free field theory L = 1 2 µ φ µ φ x x/λ φ(x) φ(x) L Λ 2 L scale invariant
122 massive free theory mass L = 1 2 µ φ µ φ + m 2 φ 2
123 massive free theory mass L = 1 2 µ φ µ φ + m 2 φ 2 x x/λ φ(x) φ(x)
124 massive free theory mass L = 1 2 µ φ µ φ + m 2 φ 2 x x/λ φ(x) φ(x) m 2 φ 2
125 massive free theory mass L = 1 2 µ φ µ φ + m 2 φ 2 Λ µ φ µ φ x x/λ φ(x) φ(x) m 2 φ 2
126 massive free theory mass L = 1 2 µ φ µ φ + m 2 φ 2 Λ µ φ µ φ x x/λ φ(x) φ(x) m 2 φ 2
127 massive free theory mass L = 1 2 µ φ µ φ + m 2 φ 2 Λ µ φ µ φ x x/λ φ(x) φ(x) m 2 φ 2 no scale invariance
128 how should we think about field theory? QFT
129 how should we think about field theory? g 7 g 6 g 5 g 4 g 3 g 2 g 1 QFT
130 how should we think about field theory? g7 g6 g5 g4 g3 g2 g1 QFT
131 g7 g6 g5 g4 g3 g2 g1 QFT
132 what if m depends on the scale? g7 g6 g5 g4 g3 g2 g1 QFT
133 what if m depends on the scale? QFT
134 what if m depends on the scale? Lmi+7 Lmi+6 Lmi+5 Lmi+4 Lmi+3 Lmi+2 Lmi+1 Lmi QFT
135 sum over all mass L = i L(m i )
136 L = µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m)
137 L = µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m) 0 dm 2
138 L = µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m) 0 dm 2 theory with all possible mass!
139 L = µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m) 0 dm 2 theory with all possible mass! φ φ(x, m 2 /Λ 2 ) x x/λ m 2 /Λ 2 m 2
140 L = µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m) 0 dm 2 theory with all possible mass! φ φ(x, m 2 /Λ 2 ) x x/λ m 2 /Λ 2 m 2 L Λ 4 L scale invariance is restored!!
141 L = µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m) 0 dm 2 theory with all possible mass! φ φ(x, m 2 /Λ 2 ) x x/λ m 2 /Λ 2 m 2 L Λ 4 L scale invariance is restored!! not particles
142 unparticles L = µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m) 0 dm 2 theory with all possible mass! φ φ(x, m 2 /Λ 2 ) x x/λ m 2 /Λ 2 m 2 L Λ 4 L scale invariance is restored!! not particles
143 scale-invariant stuff is weird particles energy E = ε p Green function (propagator) 1 G(E) = E ε p particle=infinity
144 scale-invariant stuff is weird
145 scale-invariant stuff is weird unparticles
146 scale-invariant stuff is weird unparticles G (E ε p ) d U 2 d U > 2 unparticles=zero propagator
147 n massless particles G (E ε p ) n 2 n>2 unparticles=fractional number of massless particles
148 n massless particles G (E ε p ) n 2 n>2 n<2 unparticles=fractional number of massless particles unparticles=luttinger liquids
149 n massless particles G (E ε p ) n 2 n>2 n<2 unparticles=fractional number of massless particles unparticles=luttinger liquids common ingredient:scale invariance
150 Where do we expect to find zeros? UV IR
151 Where do we expect to find zeros? e e e UV e e e e e e e e IR
152 Where do we expect to find zeros? e e e UV e e e e e e e e IR
153 Where do we expect to find zeros? e e e UV e e e e e e e e not particles IR
154 G(p, ω) = 0 UV no physical quark or electron states G(p L,ω = 0) = 0 IR G BCS (ω, p) = ω + v( p p F ) ω 2 2 v 2 ( p p F ) 2 G 1+1QCD = ik m 2 g 2 /π +2k + k + g 2 k /πλ i
155 where else do we expect to see zeros?
156 where else do we expect to see zeros? strongly correlated systems
157 where else do we expect to see zeros? strongly correlated systems Mott insulators
158 NiO insulates d 8? What is a Mott Insulator?
159 NiO insulates d 8? What is a Mott Insulator? EMPTY STATES= METAL
160 NiO insulates d 8? What is a Mott Insulator? EMPTY STATES= METAL band theory fails!
161 NiO insulates d 8? perhaps this costs energy What is a Mott Insulator? EMPTY STATES= METAL band theory fails!
162 Mott Problem: NiO (Band theory failure) (N rooms N occupants) t U t
163
164
165 U/t = 10 1 interactions dominate: Strong Coupling Physics
166 photoemission
167 photoemission particles energy E = ε p
168 photoemission particles energy E = ε p p 2
169 photoemission particles energy E = ε p p 2
170 photoemission particles energy E = ε p p 2 two crossings: closed surface of excitations
171 what is seen experimentally? Fermi arcs: no double crossings (PDJ,JCC,ZXS)
172 what is seen experimentally? Fermi arcs: no double crossings (PDJ,JCC,ZXS) seen infinities not seen zeros E F k
173 zeros do not affect the particle density
174 zeros do not affect the particle density no Luttinger theorem
175 zeros do not affect the particle density experimental data no Luttinger theorem
176 where do these zeros come from?
177 classical (atomic) limit
178 warm-up no hopping: x holes density of states 1+x 1 x } }
179 warm-up no hopping: x holes density of states 1+x 1 x } 1-x 2x }
180 warm-up no hopping: x holes density of states 1+x 1 x } 1-x 2x }
181 warm-up no hopping: x holes density of states 1+x 1 x } 1-x 2x } spectral weight: x- dependent G = 1+x ω + µ + U/2 + 1 x ω + µ U/2
182 quantum Mottness: U finite U t c iσ n i σ µ U c iσ (1 n i σ )
183 quantum Mottness: U finite U t c iσ n i σ µ U c iσ (1 n i σ ) double occupancy in ground state!!
184 quantum Mottness: U finite U t c iσ n i σ µ U c iσ (1 n i σ ) double occupancy in ground state!! W PES > 1 + x
185 beyond the atomic limit: any real system density of states 1 + x + α(t/u, x) α = t U c iσ c jσ > 0 ij 1 x α
186 beyond the atomic limit: any real system density of states 1 + x + α(t/u, x) α = t U c iσ c jσ > 0 ij 1 x α Harris & Lange, 1967 dynamical spectral weight transfer
187 beyond the atomic limit: any real system density of states 1 + x + α(t/u, x) α = t U c iσ c jσ > 0 ij 1 x α Harris & Lange, 1967 dynamical spectral weight transfer Intensity>1+x
188 beyond the atomic limit: any real system density of states 1 + x + α(t/u, x) α = t U c iσ c jσ > 0 ij 1 x α Harris & Lange, 1967 dynamical spectral weight transfer Intensity>1+x # of charge e states
189 beyond the atomic limit: any real system density of states 1 + x + α(t/u, x) α = t U c iσ c jσ > 0 ij 1 x α Harris & Lange, 1967 dynamical spectral weight transfer Intensity>1+x # of charge e states # of electron states in lower band
190 beyond the atomic limit: any real system density of states 1 + x + α(t/u, x) α = t U c iσ c jσ > 0 ij 1 x α Harris & Lange, 1967 dynamical spectral weight transfer Intensity>1+x # of charge e states # of electron states in lower band not exhausted by counting electrons alone?
191 Wilsonian analysis density of states d(uhb)
192 Wilsonian analysis density of states 1 + x + α(t/u, x) charge 2e stuff d(uhb)
193 Wilsonian analysis density of states 1 + x + α(t/u, x) charge 2e stuff d(uhb) zeros n< conserved charge
194 do particles +unparticles=infinities+zeros?
195 what really is the summation over mass? Lmi+7 Lmi+6 Lmi+5 Lmi+4 Lmi+3 Lmi+2 Lmi+1 Lmi
196 mass=energy
197 high energy (UV) QFT low energy (IR) related to sum over mass
198 high energy (UV) QFT low energy (IR) dg(e) = β(g(e)) dlne locality in energy related to sum over mass
199 implement E-scaling with an extra dimension QFT low energy (IR) dg(e) = β(g(e)) dlne locality in energy high energy (UV) related to sum over mass
200 implement E-scaling with an extra dimension QFT dg(e) dlne = β(g(e)) locality in energy related to sum over mass
201 gauge-gravity duality (Maldacena, 1997) implement E-scaling with an extra dimension UV IR gravity QF T QFT dg(e) dlne = β(g(e)) locality in energy related to sum over mass
202 what s the geometry?
203 what s the geometry? dg(e) dlne = β(g(e)) =0 scale invariance (continuous)
204 what s the geometry? dg(e) dlne = β(g(e)) E λe x µ x µ /λ =0 scale invariance (continuous)
205 what s the geometry? dg(e) dlne = β(g(e)) E λe x µ x µ /λ =0 scale invariance (continuous) solve Einstein equations
206 what s the geometry? dg(e) dlne = β(g(e)) E λe x µ x µ /λ =0 scale invariance (continuous) solve Einstein equations ds 2 = u L 2 ηµν dx µ dx ν + L E 2 de 2 anti de-sitter space
207 unparticles L = µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m) 0 dm 2
208 unparticles L = µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m) 0 dm 2 classical gravity in a d+1 curved spacetime
209 unparticles L = µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m) 0 dm 2 classical gravity in a d+1 curved spacetime gauge theory in AdS 4
210 High T_c unparticles variable mass UPt_3
211 emergent gravity High T_c unparticles variable mass UPt_3
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