BEYOND THE TOMONAGA-LUTTINGER MODEL: Dynamical critical behaviour - - PowerPoint PPT Presentation

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BEYOND THE TOMONAGA-LUTTINGER MODEL: Dynamical critical behaviour of the xxz spin chain

with RG Pereira, J Sirker, JS Caux, JM Maillet, R Hagemans and SR White

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1ST LECTURE-small q

• Introduction
• Band curvature effects at small q
• Effective Hamiltonian – exact coupling

constants

• Exact results for high energy tail of S(q,ω)
• Behaviour near peak of S(q,ω)
• Bethe ansatz and Density Matrix

Renormalization group (DMRG) results

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OUTLINE-2nd LECTURE

• “X-ray edge” field theory methods
• Finite size spectrum and Bethe Ansatz
• Long time behaviour of self-correlation

function

• Comparison to Dynamical Density Matrix

Renormalization Group results

• Open questions

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Introduction

> < = = < < Δ − ⋅ Δ + ⋅ + ⋅ =

∫ ∑ ∑

∞ ∞ − = − − + + + + =

| ) ( ) ( | 1 ) , ( , , 1 | | ], [

' 1 ' , ) ' ( 1 1 1 1 1 1 z j z j t i N j j j j iq N S z j z j z j y j y j x j N j x j

S t S e dt e N q S S S h h hS S S S S S S H

ω

ω r r

ω

Field theory works only here, at low ω??

q 2π 2kF

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Jordan-Wigner transformation:

) ( ) ( ] .) . ( [

1 1 1 2 1

j e j e c n n c h c c H

L j ik R j ik j j j j j N j

F F

ψ ψ

− + + + =

+ ≈ Δ + + − = ∑

kF εF Low energy q≅2kF excitation ε k

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Linearizing the fermion dispersion relation and bosonizing gives: )] / ( [ exp ] 2 4 [ cos const ) ( ) (

/ 2 2

φ π θ π ψ φ π φ π ψ ψ ψ ψ ψ ψ ψ ψ K K i j k K K e e S

L R F x L R j ik R L j ik R R L L z j

F F

± − ∝ + ⋅ + ∂ ≈ + + + ≈

+ − + + +

With an effective free boson Hamiltonian density:

2 2

) ( 2 2 1 φ

x

v v H ∂ + Π =

• The velocity, v, and Luttinger parameter, K, can be

determined from Bethe ansatz results

• We can obtain low energy S(q,ω) and fermion Green’s

functions in terms of free boson Green’s functions

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BAND CURVATURE EFFECTS

• At small q, Luttinger liquid theory predicts

Szz(q,ω)=(K/π)<∂xφ ∂xφ>=Kqδ(ω-v|q|)

• In free fermion case this simple form is

a result of linear dispersion:

q

All particle-hole excitations of wave-vector q have same energy, ω=v|q| ω

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Including band curvature gives finite width: q ω

Δ=0

width, Δω~q2 so LL theory is almost correct as q⇒0

<Sz>=

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• What happens if we include both band

curvature and interactions (Δ≠0)?

• Does width still scale as q2?
• What is line shape?
• Could steps turn into power law divergences?
• Can we study these questions with field

theory techniques?

• What information can we extract about these

questions from Bethe ansatz solution?

• What can we learn from numerical techniques

(Density Matrix Renormalization Group)?

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EFFECTIVE HAMILTONIAN: EXACT COUPLING CONSTANTS

• Band curvature gives interactions upon

bosonizing even for Δ=0!

( )

3 2

3 2

L x L L

φ π ψ ψ ∂ ≈ ∂

+

• Interactions in lattice model given another

cubic term. Including both:

• These are most general dimension 3

interactions allowed by parity

• They destroy Lorentz invariance and

particle-hole symmetry ( ϕ→-ϕ)

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• We may derive exact identities relating η± to

derivatives of Luttinger parameter and velocity with respect to field, h:

• v(h) and K(h) can be determined to high

accuracy from numerical solution of thermodynamic Bethe ansatz equations

• Thus coupling constants, η± can be

determined essentially exactly

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BEHAVIOUR NEAR TAIL OF S

• Lowest order perturbation theory in η± gives

retarded Green’s function:

K + − − + − + − =

+ −

) ( 18 |) | ( |) | ( 24 |) | (

2 2 2 4 2 2 2 5 2

q v v q v Kq q v d d Kq q v Kq S ω ω θ η ω δ ω η ω δ

• Note that, while both corrections are suppressed

by 2 extra power of q or ω, they diverge “on mass shell” near ω~v|q|

• We can only trust this perturbation theory far

from mass shell: ω-v|q|>> q2η±

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• η+

2 term predicts a “high energy tail” for S

• We compared this to Bethe ansatz:
• Up to 2000 form factors were calculated:
• <0|Sz(q)|n>, 2-particle, … up to 8-particle

for finite length chains up to length N=600

• We extend the field theory result to finite N

by the usual conformal transformation

• Field theory predicts states at discrete

energies: ω=2πvn/N, n=1,2,3,… whereas Bethe ansatz states are scattered

• We bin BA states to make comparison:

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Bethe ansatz squared form factors

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• Red dots are BA results, line is field theory
• Note that we have no adjustable parameters

(all fixed from thermodynamic BA) and we are at strong coupling

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BEHAVIOUR NEAR PEAK

• We need to somehow sum series in η± to

get sensible result near peak

• We explicitly calculated series in η- to 4th
• rder:

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Here w≡iω-vq and the 4th order term gets contributions from all 3 diagrams: 1/[5x24]=1/144 + 1/504 + 1/280

• Tells us that no simple partial resummation

(RPA, SCBA,…) will be sufficient!

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• Series in η- should sum up to give free

fermion result, including band curvature at small q Implying: but we have no idea how to sum η+ terms This crude (η- only) approximation, seems to give correct line-width, ~q2 and height but misses line shape and power law singularities

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• Blue dots are BA results (2 particle states
• nly)
• Dashed line is field theory result ignoring η+

Δ=.25 σ=-.1 N=6000

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• We can’t get enough resolution from finite

size BA, with limited number of states kept, to study singularities

• We don’t know how to sum series in η+ to

make field theory predictions for singularities

• Singularities may also exist at q of O(1) –

we would like to study those also

q 2π 2kF

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X-Ray Edge Field Theory Methods

We can calculate S(q,ω) near ωmin (q) for any q using

• ther field theory techniques first developed for study
• f X-ray edge singularities – Pustilnik, Glazman, …

q 2π 2kF

ω q ωmin(q)

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• Consider excitations with one “deep hole” at k1 and many

particles and holes near +kF

• These appear to give lower threshold for hole Green’s

function at momentum k or of S(q,ω) at q=±kF-k1 kF εF k k1 q

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To calculate GF(k1,ω) and S(+kF-k,ω), we consider a “low energy” effective Hamiltonian, Hk, containing

• nly Fourier modes of fermions near k and + kF for

the particular value of k=k1 of interest kF εF k k1 Λ ε1

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d d d iu d H H j d e j e j e c

R R R L L L x TLL j ik L j ik R j ik j

F F

+ + + + −

+ + ∂ − + ≈ + + ≈ ) ~ ~ ( ) ( ) ( ) ( ) (

1 1

1

ψ ψ κ ψ ψ κ ε ψ ψ

• N.B. we haven’t included any terms that would allow

the “heavy hole” to decay

• Validity of this is complicated and not completely understood
• Now looks like model for a single “heavy” hole

interacting with a Tomonaga-Luttinger Liquid (see Tsukamoto, Fujii, Kawakami, PRL 100, 126403, 1998)

• Now we bosonize the fermions near the Fermi surface only:

d d K d iu d v v H

R x R L x L x x + +

∂ + ∂ + ∂ − + ∂ + Π = ) ( 2 1 ) ( ) ( 2 2 1

1 1 2 2

φ κ φ κ π ε φ

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• All parameters: ε1, u1, κL, κR, depend on our arbitrary

choice of heavy hole momentum, k1, and are renormalized by the interactions, as we integrate out all wave-vectors except for the 3 narrow bands near k1, +kF

• Nonetheless, we determine them exactly using Bethe Ansatz
• Once these parameters are known, we can calculate

the edge exponents by making a unitary transformation:

[ ]

⎭ ⎬ ⎫ ⎩ ⎨ ⎧ + − =

∫

∞ ∞ − +d

d dx K i U

L L R R

φ γ φ γ π 2 exp

• The parameters γL/R are chosen to eliminate the

marginal interactions ~∂xφL/Rd+d, leaving only irrelevant

• nes with 2 or more derivatives

1 / /

u v

R L R L

± = κ γ

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The transformed Hamiltonian is just free fields:

d iu d v v H

x x

~ ) ( ~ ) ~ ( 2 ~ 2 1

1 1 2 2

∂ − + ∂ + Π =

+ ε

φ

But the fermion d-field now contains a boson factor:

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + − = ) ~ ~ ( 2 exp ~

L L R R

K i d d φ γ φ γ π

and thus:

+ − + +

+ − d i i d

L R R

~ ] ~ 2 ~ 2 [ exp φ πν φ πν ψ p

and similarly for d+ψL where:

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ± =

±

π γ ν

L R

K K

/

1 1 4 1

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• Fortunately, the free fermion Green’s function has a very

simple form:

) ( | ) , ( ~ ) , ( ~ |

1 ) (

1 1 1

t u x e e dk e d t x d

t i t u x ik t i

− ≈ >≈ <

− Λ Λ − − − +

∫

δ

ε ε

Thus, for example:

∫

∞ ∞ − −

− +

− + − − ∝

ν ν ω ω

δ δ ω ] ) [( ] ) [( ) , (

1 1 ) (

i t u v i t u v e dt q S

t i

L

where δ→0+

• N.B. if v>u1, this vanishes for ω<ωL, as expected

for a lower edge singularity:

1

)] ( )][ ( [ ) ~ , (

− +

− +

− − ∝

ν ν

ω ω ω ω θ ω ω q q q S

L L L

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FINITE SIZE SPECTRUM AND BETHE ANSATZ

• Energy and velocity of heavy hole, ε1, u1 are determined

from Bethe ansatz: ε(k) is energy to add one hole

• i.e. we remove a single root from ground state (a

topological excitation) of momentum k

• u=dε/dk
• At half-filling (zero field) ε(k)=-v cos k

Δ ⋅ Δ − = arccos 2 1

2

π v

• Away from half-filling ε(k) appears to be modified

in a non-trivial trivial way by the interactions

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n=.2 (<Sz>=-.3), Δ=1/2

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• To get γL/R, coupling constants between heavy hole and

bosonized excitations near Fermi surface, we compare 1/N terms in finite size spectrum

• Without the heavy hole, a low energy excitation with

an excess charge, ΔN and D particles transported from left to right Fermi point has excitation energy: ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + + + Δ = Δ

− +

n n KD K N N v E

2 2

4 ) ( 2π where n± are non-negative integers – the standard Gaussian model result

• The change in ΔN and D under the unitary transformation

which eliminates the interaction with the deep hole gives the spectrum with the deep hole present:

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K D dx K D N dx K N

L R R x L x N L R R x L x N

π γ γ φ φ π π γ γ φ φ π 4 ) ( 2 1 2 1 2 ) ( 2 − − → ∂ + ∂ − ≈ + − Δ → ∂ − ∂ ≈ Δ

∫ ∫

⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + − Δ = Δ

− +

n n K D K N K N v E

L R L R 2 2

4 2 4 1 2 π γ γ π γ γ π

• The corresponding states, for arbitrary ΔN, D, in the

presence of the deep hole, can be constructed from the Bethe ansatz From Euler-MacLaurin expansion of density of rapidities in presence of the deep hole, we obtain ΔE(ΔN,D) in the above form, thus determining γL/R

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For h=0 we predict, for any q:

K L

q v q v q S

−

− − ∝

1

] sin [ ] sin [ ) ~ , ( ω ω θ ω ω

• Step function at Δ=1 evolves into square root

singularity at Δ=1 (Heisenberg model) in agreement with Mueller ansatz and exact 2-spinon result (except for log corrections which are missed in this approach)

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At q→2kF, (for h ≠0) we obtain the exponent:

) 1 ( 2 ) ( 1 K K − → + −

− +

ν ν

This is different than the well-known Tomonaga-Luttinger liquid result

K F K F F

k q v k q v k q v q S

− −

− + − − − − ∝

1 1

)] 2 ( [ )] 2 ( [ |) 2 | ( ) , ( ω ω ω θ ω

The TLL exponent, 1-K is replaced by

K K K − > − 1 ) 1 ( 2

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• The standard TLL theory approximates the dispersion

relation as linear, thus giving a higher energy threshold

• We expect that it becomes correct by some sort of

crossover at this slightly higher energy scale

q 2π 2kF

As q→2kF, the region of ω where the TLL theory fails shrinks to zero

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LONG-TIME BEHAVIOUR OF SELF-CORRELATION FUNCTION

• Our field-theory methods gives directly S(q,t) at long times
• Comparison with DMRG is most direct through G(x,t)
• It is interesting to consider G(x=0,t) at long times

∫

−

=

π π

) , ( ) , ( t q dqS t G

At long times, this is dominated by several special values of q:

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q≅0 q≅2kF q≅kF (and others)

• Only q=0 and ±2kF

particle-hole pairs are low energy excitations

• Nonetheless other

contributions usually dominate at large t!

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Consider ½-filling (h=0): hole at k=0, particle at k=kF=π/2 dominates

2 / 1 ) 2 / ( ) 2 / ( ) 2 / ' ( ) 2 / ( ) 2 / ( ) 2 / ( ) ( ) ( ) (

2

' ) , (

+ + − + ∞ ∞ − + −

− + − + − +

∫ ∫ ∫

≈ →

π ν π ν π ν π ν π ω ν ν ω π π

t e e dq t e t e dq t q G dq

ivt t m q i t i q q t q i

L L

p

π/2

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For h=0 we can evaluate ν± explicitly for lower threshold:

K + + + → =

+ − 2 3 2 2 2 / 1 1

) , ( t B t B t e B t x G

K K ivt

q=π/2 q=π q=0 For 0<Δ<1, ½<K<1, so q=π/2 contribution always dominates

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COMPARISON TO DENSITY MATRIX RENORMALIZATION GROUP RESULTS

• DMRG is a numerical, iterative, variational approach

that gives excellent accuracy for 1D quantum systems at T=0

• Recently generalizations have been found which give

real time correlation functions out to moderate times with good accuracy

• Keeping up to 400 lattice sites and 1000 states we can

get G(x,t) out to times of 30 – 60 (in units of 1/J=1) with errors of order 10-4 -10-5

• We can then Fourier transform in x

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• For 10<t<30 we compare DMRG to field theory/Bethe ansatz

predictions – checking predicted frequencies and exponents

• Once we are convinced it works we can supplement DMRG

for short to intermediate times with asymptotic results

• ut to infinite time – the time-Fourier transform can then

be performed without usual rounding of singularities due to finiteness of time interval h=0, lower edge energy and exponent from G(0,t)

• Similar comparison works well for GF

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Fourier transformed DMRG + field theory data for h=0

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CONCLUSIONS

• Luttinger-liquid theory fails to describe some critical features
• f 1D many body systems
• Limited progress can be made by including

higher dimension operators in bosonized Hamiltonian

• “X-ray edge” methods, combined with Bethe ansatz

predict new critical exponents that seem to agree with DMRG calculations

• Open questions remain regarding decay of “heavy

hole” and robustness of critical singularities, among

• ther topics