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

Review of Duality O(2) global symmetry: Lattice Lagrangian must be a function φ ( x ) → φ ( x ) + ˜ φ of finite differences angle constant shift L ∝ f ( ∂ µ φ ) = f ( ∂ µ φ + 2 π n µ ( x )) X S O (2) model = cos( ∂ µ φ ( x )) x,µ integrate out φ ( x ) Y Y X e − L ( x ) = C k µ e ik µ ( x ) ∂ µ φ ( x ) ⇒ | {z } x x,µ k µ ( x ) Fourier expansion e − L ( x ) = Y Y X C k µ δ ( ∂ µ k µ ) constraint x x,µ k µ ( x )

The vacuum is composed of closed loops of k-fluxes k-flux What are these fluxes? e i P x,µ A µ k µ ( x ) ∂ µ φ → ∂ µ φ + A µ ⇒ H dx µ A µ e i For unit flux: Integral over the worldline: PARTICLE WORLDLINE!

Charge- 2 sector TIME SPACE

Charge- 2 sector x 1 x 2 Take time large — Low temperature Z = Tr Q =2 e − β H = Tr e − E Q X n β | n, Q ih n, Q | n,Q =2 2-PARTICLE Z dx 1 dx 2 | h x 1 , x 2 | 0 , 2 i | 2 PROBABILITY Z = Tr | 0 , 2 ih 0 , 2 | = DENSITY Energy Charge position of the particle at t = 0 ≡ β Time is a cylinder, so any timeslice is equally good!

The Dual Wavefunction Meothod Make a histogram of particle distances to obtain the probability density Notice that every timeslice of a single configuration is a a datapoint! HAS ALL INFORMATION ABOUT THE 2-PART. 
 P ( x 1 , x 2 ) = | Ψ Q =2 ( x 1 , x 2 ) | 2 INTERACTIONS

Scattering phases in 1+1 D − 1 1 2 M ∂ x 1 Ψ + V ( | x 1 − x 2 | ) Ψ = E Ψ 2 M ∂ x 1 Ψ − Ψ ( x 1 , x 2 ) = ψ ( x 1 − x 2 ) e iK CM X CM Bosonic statistics: ψ ( x ) = ψ ( − x ) − 1 For x large V(x)=0 M ∂ 2 x ψ = E ψ e ik | x | + e − ik | x | +2 i δ = 2 η cos( k | x | − δ ) Unity relative coefficient: current conservation

The wave-function in 1+1 D ψ = N cos( k | x | − δ ) 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 ) ⇒ e i δ ( k ) = e ikL 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 p k 2 + m 2 W ( k ) = 2 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 1 Z n 2 = 1 d 2 x ( ∂ µ n ) 2 S = 2 g 2 X S lat = J n ( x + µ ) · n ( x ) x, ν 1 D spin chain l 3 ( x ) —angular momentum at x L 3 = Conserved charge-total angular momentum: X l 3 ( x ) x

The O(3) model in 1+1 D • Asymptotically free • Has dynamical mass gap • Excitations in the IR: massive triplet 2-particle scattering i = 1 , 2 , 3 ( 1 x 1=0+1+2 j = 1 , 2 , 3 Decomposes into a singlet, triplet and quintet channel

The O(3) model in 1+1 D State labeled by two quantum numbers: l, m total ang. mom. ang. mom. proj. 
 CHARGE Charge 2 sector: m = 2 Since m = − l, − ( l − 1) , . . . , l − 1 , l ⇒ Charge 2 sector: l = 2 , m = 2 A QUINTET — or (iso-)spin 2

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)

2.5 2-particle int. energy T/m = 0.045 2.0 = 0.023 = 0.011 = 0.0045 1.5 this knows something ∆ µ Charge 2 plateau Q about the phase-shifts 1.0 0.5 CHARGE CONDENSATION Charge 1 plateau MASSGAP METHOD 0.0 0.9 1.0 1.1 1.2 1.3 1.4 1.5 µ / m Knows about the wave-function which knows everything about the particle interactions THE WAVE-FUNCTION METHOD

THE CHARGE CONDENSATION METHOD p m 2 + k 2 = 2 m + ∆ µ - Two particle energy W ( k ) = 2 Change L and measure ∆ µ and get k Apply Luscher formula e i δ ( k ) = e ikL 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 ) Vary L to change k, and get δ ( k ) AND YOU’RE DONE!!!

The wave-function 0.30 L=20 k / m = 0.5593(16) = 0.3976(9) 0.25 L=30 = 0.3093(4) = 0.2544(4) L=50 0.20 = 0.2159(5) 0.15 Ψ L=60 0.10 L=40 0.05 0.00 0 2 4 6 8 10 12 14 Δ x m

exact J = 1.3 CC 1.4 CC The Dual Wave-Function 
 1.5 CC Method 1.3 DWF - π /4 1.4 DWF The theoretical curve 1.5 DWF δ (k) (Zamolodchikov 1987) 1.4 LW 1.5 LW 1.54 LW -3 π /8 The Charge Condensation Method - π /2 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 k / m M. Luscher and U. Wolf, Nucl. Phys. B339, 222 (1990)

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.