Duality, Finite density and Particle Interactions Tin Sulejmanpasic - - PowerPoint PPT Presentation

Duality, Finite density and Particle Interactions

Tin Sulejmanpasic North Carolina State University

In cooperation with:

• Christof Gattringer, Thomas Kloibler (Graz University)
• Falk Bruckmann (Regensburg University)

Overview

• The duality and worldlines
• The charge-

2 ensemble

• The Dual Wave-Function Method
• The Charge Condensation Method
• The O(3) lattice simulations at non-zero density, finite

volume and small temperature

O(2) global symmetry:

φ(x) → φ(x) + ˜ φ

angle constant shift Lattice Lagrangian must be a function

• f finite differences

L ∝ f(∂µφ) = f(∂µφ + 2πnµ(x))

SO(2) model = X

x,µ

cos(∂µφ(x))

integrate out φ(x)

Y

x

e−L(x) = Y

x,µ

X

kµ(x)

Ckµeikµ(x)∂µφ(x) | {z }

Fourier expansion

⇒

Y

x

e−L(x) = Y

x,µ

X

kµ(x)

Ckµδ(∂µkµ)

constraint

Review of Duality

The vacuum is composed

• f closed loops of k-fluxes

k-flux

What are these fluxes? ∂µφ → ∂µφ + Aµ ⇒ ei P

x,µ Aµkµ(x)

ei

H dxµAµ

For unit flux: Integral over the worldline: PARTICLE WORLDLINE!

Charge- 2 sector

TIME SPACE

Charge- 2 sector

Z = TrQ=2e−βH = Tr X

n,Q=2

e−EQ

n β|n, Qihn, Q|

Z = Tr|0, 2ih0, 2| = Z dx1dx2 |hx1, x2|0, 2i|2

Take time large — Low temperature

x1 x2

Time is a cylinder, so any timeslice is equally good!

Energy Charge position of the particle at t = 0 ≡ β 2-PARTICLE PROBABILITY DENSITY

The Dual Wavefunction Meothod

Make a histogram

• f particle distances

to obtain the probability density Notice that every timeslice

• f a single configuration is a

a datapoint!

P(x1, x2) = |ΨQ=2(x1, x2)|2

HAS ALL INFORMATION ABOUT THE 2-PART.
 INTERACTIONS

Scattering phases in 1+1 D

− 1 2M ∂x1Ψ − 1 2M ∂x1Ψ + V (|x1 − x2|)Ψ = EΨ Ψ(x1, x2) = ψ(x1 − x2)eiKCMXCM ψ(x) = ψ(−x)

Bosonic statistics: For x large V(x)=0

− 1 M ∂2

xψ = Eψ

Unity relative coefficient: current conservation

eik|x| + e−ik|x|+2iδ = 2η cos(k|x| − δ)

ψ = N cos(k|x| − δ)

The wave-function in 1+1 D

But we can find the wave-function from the data!
 This is the essence of the Dual Wave-Function Method If we can control the relative momentum k, we can get the phase-shift as a function of momentum k

But what is k and how do we control it?

1/k—typical separation between the two particle =typical size of the 2-particle system=L

The Luscher formula in 1+1 D

ψ(x + L) = ψ(x) ⇒ eiδ(k) = eikL

So if k as a function of L is known then so is the phase-shift But since for short range potentials the two particle energy is given by

W(k) = 2 p k2 + m2

So knowing W is equivalent to knowing k!

Finding W is the basis for the second method: 
 The Charge Condensation Method

The O(3) model in 2D

S = 1 2g2 Z d2x (∂µn)2 n2 = 1

Slat = J X

x,ν

n(x + µ) · n(x)

1 D spin chain Conserved charge-total angular momentum:

L3 = X

x

l3(x) l3(x) —angular momentum at x

• Asymptotically free
• Has dynamical mass gap
• Excitations in the IR: massive triplet

The O(3) model in 1+1 D

i = 1, 2, 3 j = 1, 2, 3

(

1 x 1=0+1+2 Decomposes into a singlet, triplet and quintet channel

2-particle scattering

State labeled by two quantum numbers: l, m total ang. mom.

• ang. mom. proj.


CHARGE Charge 2 sector: m = 2

⇒

Charge 2 sector: l = 2 , m = 2

The O(3) model in 1+1 D

A QUINTET — or (iso-)spin 2

m = −l, −(l − 1), . . . , l − 1, l Since

The Lattice setup

• Put chemical potential for the angular momentum
• Put the system at finite volume
• Reduce the temperature

This enables us to control

• Control the charge precisely (change chem. pot.)
• Control the energy of each charged sector precisely 


(change L)

0.0 0.5 1.0 1.5 2.0 2.5 0.9 1.0 1.1 1.2 1.3 1.4 1.5

Q µ / m

T/m = 0.045 = 0.023 = 0.011 = 0.0045

MASSGAP Charge 1 plateau Charge 2 plateau

2-particle int. energy Knows about the wave-function which knows everything about the particle interactions this knows something about the phase-shifts CHARGE CONDENSATION METHOD THE WAVE-FUNCTION METHOD ∆µ

THE CHARGE CONDENSATION METHOD

• Two particle energy

W(k) = 2 p m2 + k2 = 2m + ∆µ Change L and measure ∆µ and get k Apply Luscher formula eiδ(k) = eikL and get δ(k)

AND YOU’RE DONE!!!

The wave-function method

From the dual fluxes measure the probability that the two particles are at some separation Get the relative probability distribution Take the sqrt to get the wave-function Fit the wave function to the form ψ = N cos(k|x| − δ)

AND YOU’RE DONE!!!

Vary L to change k, and get δ(k)

0.00 0.05 0.10 0.15 0.20 0.25 0.30 2 4 6 8 10 12 14

Ψ Δx m

k / m = 0.5593(16) = 0.3976(9) = 0.3093(4) = 0.2544(4) = 0.2159(5)

L=20 L=30 L=40 L=50 L=60

The wave-function

• π/2
• 3π/8
• π/4

0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60

δ(k) k / m

exact J = 1.3 CC 1.4 CC 1.5 CC 1.3 DWF 1.4 DWF 1.5 DWF 1.4 LW 1.5 LW 1.54 LW

The Charge Condensation Method

• M. Luscher and U. Wolf, Nucl. Phys. B339, 222 (1990)

The theoretical curve (Zamolodchikov 1987) The Dual Wave-Function 
 Method

Conclusion

• We have developed the two methods for determining the

scattering phase-shifts: The Dual Wave-Function Method and the Charge Condensation Method

• We have tested them on the O(3) non-linear sigma model where

analytic results are known

• The agreement is extremely good, especially for the Dual-Wave-

Function method, due to the large statistics

• The Charge Condensation Method can in principle be applied to any

system for calculating the scattering length

• The Dual Wave-Function method can in principle be applied to a

system with with any dimensionality as long as the dual variable representation exists, for calculating all scattering data of a multi-particle system.