Introduction
Dual Representation of the Scalar Field
Dual Representation of Non-Abelian Fields
Conclusions & Outlook
Dual Methods for Lattice Field Theories at
Non-Zero Density
Thomas Kloiber
Supervisor: Christof Gattringer
November 27, 2013
Introduction
Dual Representation of the Scalar Field
Dual Representation of Non-Abelian Fields
Conclusions & Outlook
Disclaimer
No new (physical) results since the Jena talk.
It’s more about formalism and techniques, not so
much about (new) physics.
Second half may contain some technicalities.
Introduction
Dual Representation of the Scalar Field
Dual Representation of Non-Abelian Fields
1 Introduction
2 Dual Representation of the Scalar Field
3 Dual Representation of Non-Abelian Fields
4 Conclusions & Outlook
Conclusions & Outlook
Introduction
Dual Representation of the Scalar Field
Dual Representation of Non-Abelian Fields
Conclusions & Outlook
Main Motivation
Our ignorance of the QCD phase diagram
Introduction
Dual Representation of the Scalar Field
Dual Representation of Non-Abelian Fields
Conclusions & Outlook
. . . then let’s calculate it on the lattice . . .
Expectation value of some Observable O
⟨O⟩ =
1
−SG [U]
det[D] O[U]
∫ D[U] e
Z
Importance sampling in lattice MC simulation
Probability weight ∝ det[D] ∼ Fermion determinant
Density driven by chemical potential µ
µ = 0 Ô⇒ det[D] ∈ R Ô⇒ probabilistic interpretation
"
µ ≠ 0 Ô⇒ det[D] ∈ C Ô⇒ no probabilistic interpretation
%
Introduction
Dual Representation of the Scalar Field
Dual Representation of Non-Abelian Fields
Conclusions & Outlook
Sign Problem
No access to non-zero chemical potential and therefore
non-vanishing density with MC simulations.
Only temperatue axis of QCD phase diagram well explored.
[Aoki et. al., Nature 443 (2006)]
In other models
Chemical potential introduces particle / antiparticle asymmetry
⇓
Action not invariant under charge conjugation
⇓
Complex Action Problem
Introduction
Dual Representation of the Scalar Field
Dual Representation of Non-Abelian Fields
Conclusions & Outlook
”Solutions” I
Reweighting
Compute averages of observable O over configurations C
⟨O⟩ =
∑C e −W [C] O[C] ⟨O⟩
=
⟨1⟩
∑C e −W [C]
⟨O⟩ =
with
W = WR + i W I
∑C e −WR (O e −iWI ) ⟨O e −iWI ⟩R
≡
⟨e −iWI ⟩R
∑C e −WR (e −iWI )
Problems, when average phase ⟨e −iWI ⟩R ≈ 0
Ð→ Severe sign problem
%
Introduction
Dual Representation of the Scalar Field
Dual Representation of Non-Abelian Fields
Conclusions & Outlook
”Solutions” II
Imaginary chemical potential
µ = 0 : γ5 -hermiticity holds
γ5 Dγ5 = D †
and leads to a real Fermion determinant det[D] ∈ R .
µ ≠ 0 : γ5 -hermiticity breaks down
γ5 D(µ)γ5 = D † (−µ) .
µ = i η : γ5 -hermiticity is restored Ð→ analytical continuation.
Introduction
Dual Representation of the Scalar Field
Dual Representation of Non-Abelian Fields
”Solutions” III
Series expansions
O(µ) ∼ ∑ cn (µ = 0) (
n
µ n
)
T
Thermodynamic limit: phase transitions ⇐⇒ singularities
Stochastic quantization
No sign problem, but numerically very demanding.
[Sexty, PoS LATTICE 2013]
Conclusions & Outlook
Introduction
Dual Representation of the Scalar Field
Dual Representation of Non-Abelian Fields
Conclusions & Outlook
”Solutions” IV – Our Approach
Dual representations (a.k.a. flux representation, world-line
approach, . . . )
Partition function Z ∈ R
Z = ∫ D[φ] P[φ] with P = e −S ∼ probability of configuration φ
µ ≠ 0 ⇒ complex action ⇒ P ∈ C
Find degrees of freedom l where
Z = ∑ W[l] with W ∈ R+ ∼ probability of configuration l
l
Introduction
Dual Representation of the Scalar Field
Dual Representation of Non-Abelian Fields
Conclusions & Outlook
Dual Representations
• In principle complete and exact solution to the sign problem.
• Model dependent.
• “Historical” approach: Kramers-Wannier-Duality,
Flux-Tube-Models, . . .
• “Modern” approach:
[Endres, Phys. Rev. D75 (2007)]
[Chandrasekharan, PoS LATTICE 2011]
[Gattringer, Delgado Mercado, Schmidt, . . . ]
• It’s (at best) a long way to QCD . . .
Introduction
Dual Representation of the Scalar Field
Dual Representation of Non-Abelian Fields
Conclusions & Outlook
φ4 Model on the Lattice
Continuum action
S = ∫ d 4 x [∣∂ν φ∣2 + (m2 − µ2 ) ∣φ∣2 + λ ∣φ∣4 + µ (φ∗ ∂4 φ − ∂4 φ∗ φ)]
Lattice action
4
S = ∑ (− ∑ (e µ δν,4 φx φ∗x+ν̂ + e −µ δν,4 φ∗x φx+ν̂ ) + κ ∣φx ∣2 + λ ∣φx ∣4 )
x
ν=1
with kinetic, mass and self-interaction terms.
Chemical potential µ couples only in 4-direction (time).
→ Complex action problem for µ ≠ 0.
Introduction
Dual Representation of the Scalar Field
Dual Representation of Non-Abelian Fields
Conclusions & Outlook
Partition function in dual representation
Z= ∑
∏
{k,m} x,ν
1
(∣kx,ν ∣ + mx,ν )! mx,ν !
∏ δ (∑[kx,ν − kx−ν̂,ν ])
x
ν
× ∏ e µkx,4 W (∑ [∣kx,ν ∣ + ∣kx−ν̂,ν ∣ + 2(mx,ν + mx−ν̂,ν )])
x
ν
• New degrees of freedom (dual variables, fluxes)
kx,ν ∈ Z (constrained) and mx,ν ∈ N0 (unconstrained)
• Weight factors W(nx ) (numerical integrals)
• Complex action problem is gone
Z = ∑ P[k, m]
{k,m}
• Exact rewriting
with
P[k, m] ∈ R+ ∼ probability weight
Introduction
Dual Representation of the Scalar Field
Dual Representation of Non-Abelian Fields
Conclusions & Outlook
Numerical Simulation
constraint ⇐⇒ flux conservation at each lattice site (closed loops)
→ Generalization of the Prokof’ev-Svistunov worm algorithm
• Standard Metropolis update sweep for the m-variables
• Worm update for the restricted k-variables
Introduction
Dual Representation of the Scalar Field
Dual Representation of Non-Abelian Fields
Conclusions & Outlook
Physical Results (for Example)
Phase diagram and relativistic Bose condensation
3
Vertical sections below and above µcrit .
κ = 7.44 (χ'|Φ|2 data)
2.5
0.25
κ=9
2
0.2
1.5
<n>
T/µcrit,T=0
κ = 7.44 (χ'n data)
pseudo Silver Blaze
behavior
1
strong
µ-dependence
0.5
0
0.5
0.1
0.05
Silver Blaze region
0
0.15
0
1
µ/µcrit,T=0
1.5
2
0
0.05
0.1
0.15
0.2
0.25
0.3
T/µcrit,T=0
0.35
0.4
0.45
Introduction
Dual Representation of the Scalar Field
Dual Representation of Non-Abelian Fields
Conclusions & Outlook
2-Point Functions – General considerations
Temporal correlator
C (t) ≡ ⟨φa φ∗b ⟩∣
p fixed
∝ e −E t with E 2 = p 2 + m2 and t = ∣a4 − b4 ∣
Free (λ ≡ 0) zero-momentum (p ≡ 0) correlator
+∞
dp4
C̃ (t) ∝ ∫
−∞
e ip4 t
p42 + m2 − µ2 + i2µp4
Im p4
i(m-μ)
C̃ (t) ∝ e −(m−µ)t . . . forward propagation
Re p4
C̃ (t) ∝ e (m+µ)t . . . backward propagation
-i(m+μ)
Introduction
Dual Representation of the Scalar Field
Dual Representation of Non-Abelian Fields
Conclusions & Outlook
2-Point Functions
Correlators
⟨φa φ∗b ⟩ =
1
1
−S
∗
Za,b
∫ D[φ] e φa φb ≡
Z
Z
cannot be expressed as partial derivatives of ln Z .
Dual representation of Za,b
∑
{k,m}
∏ δ (∑ [ . . . ]− δx,a + δx,b ) W (∑ [ . . . ]+ δx,a + δx,b )
x
ν
ν
→ modified arguments of
• Kronecker deltas and
• Weightfactors at lattice sites a and b.
⎛ ⎞
...
⎝ ⎠
Introduction
Dual Representation of the Scalar Field
Dual Representation of Non-Abelian Fields
New Constraint for Za,b
site
constraint
x =a
∑ν (kx,ν − kx−ν̂,ν ) = 1
source of flux
x =b
∑ν (kx,ν − kx−ν̂,ν ) = −1
sink of flux
else
∑ν (kx,ν − kx−ν̂,ν ) = 0
local flux conservation
b
a
Conclusions & Outlook
Introduction
Dual Representation of the Scalar Field
Dual Representation of Non-Abelian Fields
Conclusions & Outlook
Numerical Simulation
We enlarge the ensemble [Korzec et. al., Comp. Phys. Comm. 182 (2011)]
Z ≡ ∑ Za,b ∼ closed loops (a = b) + open lines (a ≠ b)
a,b
Monte Carlo simulation again with a generalization of the
Prokof’ev-Svistunov worm algorithm but with open ends.
Dual zero momentum temporal correlator
C (t) ∝ ⟨δt , a4 −b4 ⟩Z ∝ e −m t
Introduction
Dual Representation of the Scalar Field
Dual Representation of Non-Abelian Fields
Conclusions & Outlook
100
µ
C(t)/C(0)
10-1
µcrit = 0.170(1)
0
mred ≡ E± ± µ
10-2
E+=0.1687(3)
10-3
E-=0.1684(3)
10-4
0
10
20
30
40
50
60
70
80
90
100
t
Introduction
Dual Representation of the Scalar Field
Dual Representation of Non-Abelian Fields
Conclusions & Outlook
100
E-=0.2295(66)
E+=0.0196(3)
mred=0.1696(3)
µ
10-1
0
C(t)/C(0)
E+=0.0685(2)
mred=0.1685(2)
0.05
0.1
10-2
E+=0.1183(2)
mred=0.1683(2)
0.15
E+=0.1687(3)
E-=0.2665(11)
0.18
0.2
10-3
0.3
E-=0.2178(3)
E-=0.1684(3)
10-4
0
10
20
30
40
50
60
70
80
90
100
t
Introduction
Dual Representation of the Scalar Field
Dual Representation of Non-Abelian Fields
Conclusions & Outlook
Correlators in the condensed phase
• Massive mode φσ
lim ⟨φσx φσy ⟩ ∝ e −E ∣x−y ∣
∣x−y ∣→∞
• Massless (Goldstone) mode φπ
lim ⟨φπx φπy ⟩ ∝
∣x−y ∣→∞
Im Φ
U(1)
Φπ
Φσ
Re Φ
1
∣x − y ∣n
→ external source to break symmetry
“in a controlled way”.
S → S + ∑(Jx∗ φx + Jx φ∗x )
x
Introduction
Dual Representation of the Scalar Field
Dual Representation of Non-Abelian Fields
Conclusions & Outlook
The Non-Abelian Problem
• Fields cannot be arranged according to our needs.
• Fields only appear within invariant functions (e. g., the trace).
• Integrals over dim > 1 group manifolds are more involved.
• ...
Historical approach: Character expansion
Invariant functions ∼ sum over all irreducible representations
f (U) = f (V −1 UV ) = ∑ χr (U) fr
r
[Wade Cherrington - “spin foams”]
Ð→ artificial sign problem!
Introduction
Dual Representation of the Scalar Field
Dual Representation of Non-Abelian Fields
Conclusions & Outlook
E.g., SU(2) Spin system with external field
Z = ∫ D[U] ∏ e β Tr[Ux Ux+µ̂ ]
†
+ h Tr[Ux ]
Ux ∈ SU(2)
x,µ
First try – “Naive expansion”
kx,µ
∏∫
x
SU(2)
†
])
dUx ∏ (Tr[Ux Ux+µ̂
Ð→ split off group center
µ
kx,µ
†
Σ
Σ
∏ ∫ SU(2) d Ũx ((−1) + 1 ) ∏ (Tr[Ũx Ũx+µ̂ ])
x
µ
{−1,1}
Constraint Ð→ Σ ≡ ∑ (kx,µ + kx−µ̂,µ ) = 2 N0
∀x
µ
kx,µ
†
Problem: (Tr[Ũx Ũx+
µ̂ ])
leads to terribly complicated expressions . . .
Introduction
Dual Representation of the Scalar Field
Dual Representation of Non-Abelian Fields
Conclusions & Outlook
A few tries later
Consider the trace as an “inner product” in the space of SU(2)
matrix elements
4
†
Tr[Ux Ux+µ̂
] ≡ ∑(Ux )i (U†x+µ̂ )i ≡ Ax,µ + Ax,µ + Ax,µ + Ax,µ
(1)
(2)
(3)
(4)
i=1
(i)
Ax,µ ∝ products of SU(2) matrix elements
∏e
x,µ
†
β Tr[Ux Ux+
] + h Tr[Ux ]
µ̂
(i)
↝
∏e
x,µ,i
↝
(i)
(Ax,µ )lx,µ
(i)
Ax,µ
∏ ∑
x,µ,i l (i)
x,µ
(i)
lx,µ !
Factorization Ð→ fields at different sites can be integrated out
independently.
Introduction
Dual Representation of the Scalar Field
Dual Representation of Non-Abelian Fields
Conclusions & Outlook
General SU(2) element
Ux = (
cos θx e iχx
sin θx e −iηx
−sin θx e iηx
)
cos θx e −iχx
and Haar measure
π/2
∫
SU(2)
dU ∝ ∫
2π
sin θ cos θ dθ ∫
0
0
2π
dχ ∫
dη
0
Ð→ leads to integrals of the form
2π
∫
0
dx e i n x ∝ δ(n)
π/2
and
∫
0
dx (sin x)n1 (cos x)n2 ∝
1+n2
1
Γ( 1+n
2 ) Γ( 2 )
Γ( 2+n21 +n2 )
Introduction
Dual Representation of the Scalar Field
Dual Representation of Non-Abelian Fields
Conclusions & Outlook
Partition function in dual representation
(1)
(1)
(2)
(2)
⎛
⎞
β ∣kx,ν ∣+2mx,ν +∣kx,ν ∣+2mx,ν
Z= ∑
∏ (1)
(1)
(1)
(2)
(2)
(2)
{k,m,p,q} ⎝ x,ν (∣kx,ν ∣ + mx,ν )! mx,ν ! (∣kx,ν ∣ + mx,ν )! mx,ν ! ⎠
(∏
x
h∣qx ∣+2px
) ∏ Γx ∏ δx
(∣qx ∣ + px )! px !
x
x
• New degrees of freedom (dual variables)
(1)
(2)
kx,ν , kx,ν , qx ∈ Z
and
(1)
(2)
mx,ν , mx,ν , px ∈ N0
• Weight factors Γx (ratios of gamma functions)
• Constraints
(1)
(1)
(2)
(2)
δx ≡ δ (∑[kx,ν − kx−ν̂,ν ] + qx ) δ (∑[kx,ν − kx−ν̂,ν ])
ν
ν
Introduction
Dual Representation of the Scalar Field
Dual Representation of Non-Abelian Fields
Conclusions & Outlook
Numerical check
Trace and susceptibility of nearest neighbor terms for different external fields
(colors) compared to conventional simulations (black crosses).
1.8
0.003
1.6
0.0025
1.4
0.002
1
χTr[Ux Ux+µ]
<Tr[Ux Ux+µ]>
1.2
0.8
0.6
0.0015
0.001
0.4
0.2
0.0005
0
-0.2
0
0.2
0.4
0.6
0.8
1
β
1.2
1.4
1.6
1.8
2
0
0
0.2
0.4
0.6
0.8
1
β
1.2
1.4
1.6
1.8
2
Introduction
Dual Representation of the Scalar Field
Dual Representation of Non-Abelian Fields
Conclusions & Outlook
Conclusions
• Dual representations (where applicable) give a complete
and exact solution to the sign problem for a given model.
• Essentially all abelian, scalar and U(N) symmetric vector
models can be mapped to a dual representation
Ð→ access to finite density.
• In this talk: phase diagram, condensation studies and
spectroscopy calculations at non-zero chemical potential
within the φ4 -model.
• Non-Abelian case: SU(2)-spin model successfully
reformulated.
Introduction
Dual Representation of the Scalar Field
Dual Representation of Non-Abelian Fields
Conclusions & Outlook
Outlook
Find dual representations of other non-abelian models.
A gauge theory would be nice . . .