Demystifying the Strange Metal in High-temperature. Superconductors: Composite Excitations

Demystifying the Strange Metal in High-temperature Superconductors: Composite Excitations Thanks to: T.-P. Choy, R. G. Leigh, S. Chakraborty, S. Hong PRL, 99, 46404 (2007); PRB, 77, 14512 (2008); ibid, 77, 104524 (2008) ); ibid, 79,245120 (2009); RMP, 82, 1719 (2010)..., DMR/NSF

doped Mott insulator energy

doped Mott insulator energy =hole =2e mode bound

doped Mott insulator energy =hole =2e mode strange metal bound pseudogap

Strong Coupling composite or bound states not in UV theory QCD vulcanization

Strong Coupling composite or bound states not in UV theory QCD vulcanization emergent (collective) low-energy physics

AF 0 Mott insulator ρ at x

AF 0 Mott insulator ρ at LaSrCuO x

AF 0 Mott insulator ρ at x

How does Fermi Liquid Theory Breakdown? $6,000,000 question? AF 0 Mott insulator ρ at x

knee-jerk explanation 5

knee-jerk explanation quantum criticality 5

knee-jerk explanation quantum criticality Standard story 5

General Result 1.) one critical length scale 2.) charge carriers are critical 3.) charge conservation

Vector potential current

Vector potential current Charge conservation: d A = 1

scaling

PP, CC, PRL, vol. 95, 107002 (2005) σ(t ) T (d 2)/z E p z dynamical exponent

PP, CC, PRL, vol. 95, 107002 (2005) σ(t ) T (d 2)/z d=3 E p z dynamical exponent only if z<0

One-parameter Scaling breaks down new length scale new degree of freedom

NFL? P MI-BG = (0,D/6) P = (0,0) P MI-SF = (1,0 FL How to break Fermi liquid theory in d=2+1?

Polchinski, Shankar, others p 1 p = k + l, p 2 1.) e- charge carriers 2.) Fermi surface dt d 2 k 1 dl 1 d 2 k 2 dl 2 d 2 k 3 dl 3 d 2 k 4 dl 4 V (k 1, k 2, k 3, k 4 ) ψ σ(p 1 )ψ σ (p 3 )ψ σ (p 2 )ψ σ (p 4 )δ 3 (p 1 + p 2 p 3 p 4 ). No relevant short-range 4-Fermi terms in d 2

So what does one add to break this correspondence? Interacting system Free system low-energy: one-to-one correspondence

So what does one add to break this correspondence? Free system low-energy: one-to-one correspondence Interacting system new degrees of freedom!

So what does one add to break this correspondence? Free system low-energy: one-to-one correspondence new degrees of freedom!

So what does one add to break this correspondence? UV-IR mixing Free system low-energy: one-to-one correspondence new degrees of freedom!

So what does one add to break this correspondence? UV-IR mixing Free system low-energy: one-to-one correspondence new degrees of freedom! dynamically generated

classical Mottness is forbidden

classical Mottness

atomic limit: x holes density of states 1-x 2x 1-x PES IPES G σ (ω, k) = 1 2 (1 + x) + 1 2 (1 x) ω µ + U 2 ω µ U 2

atomic limit total weight=1+x= # of ways electrons can be added in lower band intensity of lower band=# of electrons the band can hold no problems yet!

quantum Mottness: U finite U t c iσ n i σ µ U c iσ (1 n i σ )

quantum Mottness: U finite U t c iσ n i σ µ U c iσ (1 n i σ ) double occupancy in ground state!!

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

Harris & Lange, 1967 α = t U c iσ c jσ > 0 ij 1 + x + α(t/u, x)

Harris & Lange, 1967 α = t U c iσ c jσ > 0 ij 1 + x + α(t/u, x) the rest of this state lives at high energy

Harris & Lange, 1967 α = t U c iσ c jσ > 0 ij 1 + x + α(t/u, x) the rest of this state lives at high energy Intensity>1+x New non-electron degrees of freedom

Harris & Lange, 1967 α = t U c iσ c jσ > 0 ij 1 + x + α(t/u, x) the rest of this state lives at high energy Intensity>1+x New non-electron degrees of freedom Can this system be a Fermi liquid?

what are the low-energy fermionic excitations? 2 1 x α 1 + x + α

what are the low-energy fermionic excitations? 2 re-definition: x = x + α 1 x α 1 + x + α

what are the low-energy fermionic excitations? 2 re-definition: x = x + α 1 x α 1 + x + α get rid of dynamical mixing (CPH, PRB 2010)

what are the low-energy fermionic excitations? 2 re-definition: x = x + α 2(x + α) 1 x α x α 1 + x + α get rid of dynamical mixing (CPH, PRB 2010)

what are the low-energy fermionic excitations? 2 re-definition: x = x + α 2(x + α) 1 x α x α 1 + x + α n qp < n e FLT breaks down get rid of dynamical mixing (CPH, PRB 2010)

2(x + α) 1 x α 1 + x + α

2(x + α) 1 x α new degrees of freedom 1 + x + α number of fermionic qp < number of bare electrons=> FL theory breaks down

2(x + α) 1 x α 1 + x + α new degrees of freedom # of addition states per e per spin>1 number of fermionic qp < number of bare electrons=> FL theory breaks down

2(x + α) 1 x α 1 + x + α new degrees of freedom # of addition states per e per spin>1 number of fermionic qp < number of bare electrons=> FL theory breaks down ways to add a particle but not an electron (gapped spectrum)

Same Physics at half-filling No proof exists? Mottness is ill-defined

A Critique of Two Metals R. B. Laughlin idea is either missing or improperly understood. Another indicator that something is deeply wrong is the inability of anyone to describe the elementary excitation spectrum of the Mott insulator precisely even as pure phenomenology. Nowhere can one find a quantitative band structure of the elementary particle whose spectrum becomes gapped. Nowhere can one find precise information about the particle whose gapless spectrum causes the paramagnetism. Nowhere can one find information about the interactions among these particles or of their potential bound state spectroscopies. Nowhere can one find precise definitions of Mott insulator terminology. The upper and lower Hubbard bands, for example, are vague analogues of the valence and conduction bands of a semiconductor, except that they coexist and mix with soft magnetic excitations no one knows how to describe very well.

Church of weak coupling A Critique of Two Metals R. B. Laughlin idea is either missing or improperly understood. Another indicator that something is deeply wrong is the inability of anyone to describe the elementary excitation spectrum of the Mott insulator precisely even as pure phenomenology. Nowhere can one find a quantitative band structure of the elementary particle whose spectrum becomes gapped. Nowhere can one find precise information about the particle whose gapless spectrum causes the paramagnetism. Nowhere can one find information about the interactions among these particles or of their potential bound state spectroscopies. Nowhere can one find precise definitions of Mott insulator terminology. The upper and lower Hubbard bands, for example, are vague analogues of the valence and conduction bands of a semiconductor, except that they coexist and mix with soft magnetic excitations no one knows how to describe very well.

= 0.6eV > dimerization (Mott, 1976) σ(ω) Ω -1 cm -1 VO 2 T=360 K T=295 K transfer of spectral weight to Recall, ev = 10 4 K high energies beyond any ordering scale cm -1

= 0.6eV > dimerization (Mott, 1976) σ(ω) Ω -1 cm -1 VO 2 T=360 K T=295 K transfer of spectral weight to Recall, ev = 10 4 K high energies beyond any ordering scale cm -1 M. M. Qazilbash, K. S. Burch, D. Whisler, D. Shrekenhamer, B. G. Chae, H. T. Kim, and D. N. Basov PRB 74, 205118 (2006)

Fermi-liquid analogy L FL (ω k ) ψ k 2 Mott Problem? L MI = (ω E LHB (k)) η k 2 + (ω E UHB (k)) η k 2 N N PES IPES

composite excitation: bound state half-filling: Mott gap doping: SWT, pseudogap? charge 2e boson

How?

perturbative interlude: standard approach H Hubb ( ) block diagonal in `double occupancy H tj ( ) each blue fermion blocks 2 states=> 2x sum rule

Transform Operators = e S e S = particle states bare fermions fictive fermions that conserve `double occupancy

What is P 0 P 0 P 0 P 0

} What is P 0 P 0 hole P P t t 0 0 + ( ) + ( ) U U } dynamical spectral weight transfer >2x

What is lost if we ignore the dynamical spectral weight transfer? [H, ] = 0 new conserved quantity Local SU(2) symmetry at half-filling

What is lost if we ignore the dynamical spectral weight transfer? [H, ] = 0 new conserved quantity Local SU(2) symmetry at half-filling Not in original model (oops)

Ψ = c c c c S = Ψ Ψ local SU(2) Ψ = hψ h = α β β α α 2 + β 2 = 1 S > Ψ h hψ = S

Operator Transform t U j (1 n i σ )a iσ + t U V σa i σ b i+ g ij n j σ a jσ + n i σ (1 n j σ )a jσ + (1 n j σ ) a jσ a iσ a jσ a iσ a i σ

Operator Transform t U j (1 n i σ )a iσ + t U V σa i σ b i+ g ij n j σ a jσ + n i σ (1 n j σ )a jσ + (1 n j σ ) a jσ a iσ a jσ a iσ a i σ Is there an easier approach?

Operator Transform t U j (1 n i σ )a iσ + t U V σa i σ b i+ g ij n j σ a jσ + n i σ (1 n j σ )a jσ + (1 n j σ ) a jσ a iσ a jσ a iσ a i σ Is there an easier approach? Collective Phenomena

yes

EffectiveTheories: S(φ) at half-filling Integrate φ=φ L +φ H Out high Energy fields Low-energy theory of M I

Half-filling U N(ω) ε Integrate out both

"I'm not into this detail stuff. I'm more concepty."

"I'm not into this detail stuff. I'm more concepty." New Concept

Key idea: similar to Bohm/Pines Extend the Hilbert space: Associate with U-scale new Fermionic oscillators N(ω) U D U 2 i D U i 2 D i D i D i D i ε

D i Fermionic constraint one per site (fermionic) transforms as a boson

D i Fermionic constraint one per site (fermionic) transforms as a boson `supersymmetry

D i Fermionic constraint one per site (fermionic) transforms as a boson `supersymmetry δ(d i θc i c i ) Grassmann

D i Fermionic constraint one per site (fermionic) transforms as a boson `supersymmetry δ(d i θc i c i ) Grassmann θϕ constraint field

Dual Theory solve constraint ϕ (Q ϕ = 2e) integrate over heavy fields UV limit d 2 θ θθl Hubb = i,σ c i,σċi,σ + H Hubb, Exact low-energy theory (IR limit)

Dual Theory solve constraint ϕ (Q ϕ = 2e) integrate over heavy fields UV limit d 2 θ θθl Hubb = i,σ c i,σċi,σ + H Hubb, Exact low-energy theory (IR limit) L hf UV = d 2 θ id Ḋ i D D U 2 (D D D D ) + t 2 D θb + t 2 θb D + h.c. + s θϕ (D θc c ) + s θ ϕ ( D θc c ) + h.c.

dynamics of ϕ

Exact low-energy Lagrangian L = #L bare (electrons) + #L bare (bosons) +f(ω)l int (c, ϕ) + f(ω)l int (c, ϕ) Ψ Ψ Ψ Ψ quadratic form: composite or bound excitations of ϕ c iσ

Exact low-energy Lagrangian L = #L bare (electrons) + #L bare (bosons) f(ω) = 0 +f(ω)l int (c, ϕ) + f(ω)l int (c, ϕ) dispersion Ψ Ψ quadratic form: Ψ Ψ of propagating light modes composite or bound excitations of ϕ c iσ

composite excitations determine spectral density γ ( k) p γ ( k) p U tε(k) p (ω) = U U + tε(k) p (ω) = U 2ω 1 + 2ω/U + 2ω 1 2ω/U. = U 4dt γ = 0 (0, 0) γ = 0 (π, π) each momentum has SD at two distinct energies

H(ϕ, ϕ, c, c ) 1.) no derivative couplings with respect to bosonic field 2.) no bare dynamics consistent with exact argument bosonic field can be treated as spatially uniform A(k,ω) bosonic field contains all the Mott physics (numerically verified that spatial variation of bosonic field does nothing) spin part irrelevant

hole-doping?

Extend the Hilbert space: Associate with U-scale a new Fermionic oscillator N(ω) U D_i ε UD i D i

charge 2e boson field (non-propagating) Non-projective

charge 2e boson field (non-propagating) Non-projective ϕ ι is an emergent degree of freedom Not made out of the elemental fields

Spectral function Bosonic field ~ independent of space

Electron spectral function t 2 /U~60meV

Electron spectral function $!"*#!,$!+!! -!!"*!!")#!") t 2 /U~60meV!"(#!"(!"'#!!"!#!"$!"$#!"%!"%#!"&!"&# +

Electron spectral function $!"*#!,$!+!! -!!"*!!")#!") t 2 /U~60meV!"(#!"(!"'#!!"!#!"$!"$#!"%!"%#!"&!"&# + Conserved charge: Q = i c i c i + 2 i ϕ i ϕ i

Graf, et al. PRL vol. 98, 67004 (2007). Spin-charge Two bands!! separation?

What is the electron operator?

What is the electron operator? UV

What is the electron operator? UV IR + σ s U ϕ i c i σ

What is the electron operator? UV IR + σ s U ϕ i c i σ P 0 P 0 =

} What is the electron operator? UV IR + σ s U ϕ i c i σ hole P 0 P 0 = t P 0 P 0 ( ) U t + + ( ) U

} What is the electron operator? UV IR + σ s U ϕ i c i σ hole P 0 P 0 = t P 0 P 0 ( ) U t + + ( ) U s=t

Origin of two bands Two charge e excitations c iσ ϕ i c i σ ϕ i is confined (no kinetic energy) New bound state Pseudogap

two types of charges free bound

direct evidence

direct evidence charge carrier density:

direct evidence charge carrier density: 100 La 2-x Sr x CuO 4 Ono, et al., Phys. Rev. B 75, 024515 (2007) 1/n eff 10 x= 0.01 0.02 0.03 (a) 0.04 0.05 0 200 400 600 800 1000 Temperature (K) n Hall (x, T ) = n 0 (x) + n 1 (x) exp( (x)/t ), PRL, vol. 97, 247003 (2006).

Our Theory 2.5 2 0.4 x = 0.05 x = 0.10 x = 0.15 x = 0.2 n Hall (T) 1.5 1 0.5 n 0 (x) 0.3 0.2 0.1 n 0 = x 0 0 0.05 0.1 0.15 0.2 x 0 0 50 100 150 T!1 (ev!1 ) exponential T-dependence

Our Theory gap 2.5 2 0.4 x = 0.05 x = 0.10 x = 0.15 x = 0.2 0.35 0.3 0.25 Theory LSCO LSCO n Hall (T) 1.5 1 0.5 n 0 (x) 0.3 0.2 0.1 n 0 = x 0 0 0.05 0.1 0.15 0.2 x!(x) (ev) 0.2 0.15 0.1 0.05 0 0 50 100 150 T!1 (ev!1 ) exponential T-dependence 0 0 0.05 0.1 0.15 0.2 0.25 0.3 x no model-dependent free parameters: just t/u

strange metal: breakup (deconfinement) of bound states 800 700 600 T * LSCO T m LSCO T m LSCO Theory T * (K) 500 400 300 200 100 0.05 0.1 0.15 0.2 x QCP:

T-linear resistivity New scenario as x increases E B µ E

Experiments

µ L =Area/# of sites Λ µ N(ω)dω L=# of single-particle addition states (qp) n h = # of ways electrons can be added= # of holes created by the dopants

µ L =Area/# of sites Λ µ N(ω)dω L=# of single-particle addition states (qp) n h = # of ways electrons can be added= # of holes created by the dopants Equivalent in Fermi liquid

Experimental Prediction: L/n h > 1 L/n h = 1 More states at lower temper ature E B = 0 E B > 0 ρ T

J. Y. Lin (arxiv:1009.2560) More addition states in PG: new charge e states

J. Y. Lin (arxiv:1009.2560) J. Lee, P. Abbamonte (here) More addition states in PG: new charge e states

J. Y. Lin (arxiv:1009.2560) J. Lee, P. Abbamonte (here) More addition states in PG: new charge e states Probe by NMR

Predictions: QO oscillations: V hp V ep = 2(x + α) x x + α Superfluid density: optical conductivity: ρ s x + α n eff (x + α) > x Thanks to R. G. Leigh and Ting- Pong Choy, Shiladitya Chakraborty, Seungmin Hong and DMR-NSF

Transformed electron operator 2x +terms in m=1 sector >2x

Transformed electron operator 2x +terms in m=1 sector Projection >2x

What is the meaning of? organizes degrees of freedom at low energy: creates new charge e states

What is the meaning of? Double occupancy bound to a hole organizes degrees of freedom at low energy: creates new charge e states