Simulation of Gauge-Higgs models using the worm algorithm Y. Delgado , C. Gattringer, A. Schmidt Karl-Franzens-Universität Graz Sarajevo, February 2013 Y. Delgado (KFU) Worm algorithm Excited QCD 2013 1 / 19
Motivation: Sign problem of QCD Expectation value of observables: � O � = 1 � D [ φ, φ, U ] O [ U, φ, φ ] e − S [ φ,φ,U ] Z Use Monte-Carlo method. Generate configurations with probability: 1 Z e − S [ φ,φ,U ] At finite density: e − S ( µ ) is complex for µ > 0 e − S ( µ ) = | e − S ( µ ) | e iθ = ⇒ sign problem 1 0.5 f(x) 0 -0.5 -1 -2 -1 0 1 2 x Y. Delgado (KFU) Worm algorithm Excited QCD 2013 2 / 19
Motivation: Sign problem of QCD Way out: Taylor expansion in terms of µ/T (not exact). Complex Langevin (exact). [G. Aarts, F .A. James] Rewrite the partion sum using new variables: Dual representation (exact). P . de Forcrand (2010) (review LQCD and sign problem ) Y. Delgado (KFU) Worm algorithm Excited QCD 2013 3 / 19
Motivation: Sign problem of QCD Way out: Taylor expansion in terms of µ/T (not exact). Complex Langevin (exact). [G. Aarts, F .A. James] Rewrite the partion sum using new variables: Dual representation (exact). Test new methods using full QCD is too complicated!! Use “toy models”: SU(3) spin model (last excited QCD) [Y. D., C. Gattringer (2012)] Relativistic Bose gas [C. Gattringer, T. Kloiber (2013)] Z 3 gauge-Higgs model [C. Gattringer, A. Schmidt (2012)] U(1) gauge-Higgs model (this talk) [Y .D., C. Gattringer, A. Schmidt (2013) P . de Forcrand (2010) (review LQCD and sign problem ) Y. Delgado (KFU) Worm algorithm Excited QCD 2013 3 / 19
The U(1) Gauge-Higgs model In the continuum � d 4 x {− φ ( x ) ∗ [ ∂ ν + iA ν ( x )][ ∂ ν + iA ν ( x )] φ ( x ) S = d 4 x 1 � + [ m 2 − µ 2 ] | φ ( x ) | 2 � + iµN + 4 F µν F µν On the lattice − β � � U x,νρ + U ∗ � � S G = x,νρ 2 x ν<ρ | φ x | 2 − � � e − µδ ν 4 φ ∗ ν + e µδ ν 4 φ ∗ x U ∗ � � = S H κ x U x,ν φ x +ˆ ν,ν φ x +ˆ ν x − ˆ x x,ν � � φ (x) � � ∈ φ x C e iA ν ∈ U (1) , A ν ∈ [ − π, π ] a = U (x) U x,ν µρ ν,ρ U ∗ ρ,ν U ∗ = U x,νρ U x,ν U x +ˆ U (x) µ x +ˆ x,ρ Y. Delgado (KFU) Worm algorithm Excited QCD 2013 4 / 19
Dual representation-1 Rewrite terms of partition sum: A single nearest neighbor term: ( e − µδ ν 4 ) l x,ν e e − µδν 4 φ ∗ ν = � ( U x,ν ) l x,ν ( φ ∗ x U x,ν φ x +ˆ x ) l x,ν ( φ x +ˆ ν ) l x,ν l x,ν ! l x,ν Y. Delgado (KFU) Worm algorithm Excited QCD 2013 5 / 19
Dual representation-1 Rewrite terms of partition sum: A single nearest neighbor term: ( e − µδ ν 4 ) l x,ν e e − µδν 4 φ ∗ ν = � ( U x,ν ) l x,ν ( φ ∗ x U x,ν φ x +ˆ x ) l x,ν ( φ x +ˆ ν ) l x,ν l x,ν ! l x,ν A single plaquette term: β p x,νρ � p x,νρ � e βU x,ν U x +ˆ ν,ρ U ∗ ρ,ν U ∗ x,ρ = ν,ρ U ∗ ρ,ν U ∗ � U x,ν U x +ˆ x +ˆ x +ˆ x,ρ p x,νρ ! p x,νρ Y. Delgado (KFU) Worm algorithm Excited QCD 2013 5 / 19
Dual representation-1 Rewrite terms of partition sum: A single nearest neighbor term: ( e − µδ ν 4 ) l x,ν e e − µδν 4 φ ∗ ν = � ( U x,ν ) l x,ν ( φ ∗ x U x,ν φ x +ˆ x ) l x,ν ( φ x +ˆ ν ) l x,ν l x,ν ! l x,ν A single plaquette term: β p x,νρ � p x,νρ � e βU x,ν U x +ˆ ν,ρ U ∗ ρ,ν U ∗ x,ρ = ν,ρ U ∗ ρ,ν U ∗ � U x,ν U x +ˆ x +ˆ x +ˆ x,ρ p x,νρ ! p x,νρ Partition sum: �� � ( e − µδ ν 4 ) l x,ν β p x,νρ � � dU x,ν dφ x dφ ∗ x F ( U, φ, φ ∗ , l x,ν , p x,νρ , κ ) Z = l x,ν ! p x,νρ ! { p,l } x,νρ Y. Delgado (KFU) Worm algorithm Excited QCD 2013 5 / 19
Dual representation-2 Integrate out U(1) fields → new degrees of freedom: Links: l x,ν ∈ Z Plaquettes: p xνρ ∈ Z New partion sum: � Z ∝ W [ p, l ] C S [ l ] C L [ p, l ] { p,l } W [ p, l ] : positive weight factor (sign problem solved). C S [ l ] : site constraint → matter loops. C L [ p, l ] : link constraint → gauge surfaces. Y. Delgado (KFU) Worm algorithm Excited QCD 2013 6 / 19
Site constraint Site constraint (matter loops): � 4 � � � C S [ l ]= δ [ l x,ν − l x − ˆ ν,ν ] x ν =1 ν 2 ν 1 Y. Delgado (KFU) Worm algorithm Excited QCD 2013 7 / 19
Link constraint Link constraint (gauge surfaces): 4 � � � � � � C L [ p, l ]= [ p x,νρ − p x − ˆ ρ,νρ ] − [ p x,ρν − p x − ˆ ρ,ρν ] + l x,ν δ x ν =1 ρ : ν<ρ ρ : ν>ρ + + + ν 2 ν 1 Y. Delgado (KFU) Worm algorithm Excited QCD 2013 8 / 19
MC simulation We used two algorithms: Local Metropolis update Worm algorithm Advantages of the worm algorithm: Most suitable algorithm (for constrained variables). Local updates of the configurations. Smaller autocorrelation time in critical regions. N. Prokof’ev and B. Svistunov (2001), Y. Deng, T. M. Garoni and A. D. Sokal (2007). Y. Delgado (KFU) Worm algorithm Excited QCD 2013 9 / 19
Local Metropolis Update Plaquette update: + − ν 2 ν 1 Cube update: + + + ν 3 + ν 2 + ν + 1 Y. Delgado (KFU) Worm algorithm Excited QCD 2013 10 / 19
Elements of the WA Take smallest unit of the local update: + − ν 2 ν 1 Relax the constraints in 2 elements → segments − − + + − − + + + + − − ρ ν Y. Delgado (KFU) Worm algorithm Excited QCD 2013 11 / 19
Updating scheme One link is inserted at a random position of the lattice L 0 . 1 ν 3 ν 2 ν 1 Y. Delgado (KFU) Worm algorithm Excited QCD 2013 12 / 19
Updating scheme One link is inserted at a random position of the lattice L 0 . 1 The worm may insert a new segment at L v , healing the constraints at this 2 position and then move to one of other three links of the segment. ν 3 + + ν + = 2 ν 1 Y. Delgado (KFU) Worm algorithm Excited QCD 2013 12 / 19
Updating scheme One link is inserted at a random position of the lattice L 0 . 1 The worm may insert a new segment at L v , healing the constraints at this 2 position and then move to one of other three links of the segment. ν 3 + = + + ν 2 ν 1 Y. Delgado (KFU) Worm algorithm Excited QCD 2013 12 / 19
Updating scheme One link is inserted at a random position of the lattice L 0 . 1 The worm may insert a new segment at L v , healing the constraints at this 2 position and then move to one of other three links of the segment. ν 3 = + + + ν 2 ν 1 Y. Delgado (KFU) Worm algorithm Excited QCD 2013 12 / 19
Updating scheme One link is inserted at a random position of the lattice L 0 . 1 The worm may insert a new segment at L v , healing the constraints at this 2 position and then move to one of other three links of the segment. + ν 3 + = + + ν 2 ν 1 + Y. Delgado (KFU) Worm algorithm Excited QCD 2013 12 / 19
Updating scheme One link is inserted at a random position of the lattice L 0 . 1 The worm may insert a new segment at L v , healing the constraints at this 2 position and then move to one of other three links of the segment. The worm ends modifying the link occupation number at L v . 3 ν 3 + = + + ν 2 ν 1 + + Y. Delgado (KFU) Worm algorithm Excited QCD 2013 12 / 19
WA vs. LMA Verify correctness of WA ( µ = 0 case). χ U κ = 5; λ = 1 <U> κ = 8; λ = 1 κ = 9; λ = 1 3 0.6 2 0.4 1 0.2 SWA LMA conventional 0 0 1.2 β 1.2 β 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Y. Delgado (KFU) Worm algorithm Excited QCD 2013 13 / 19
WA vs. LMA: κ = 5 , β = 0 . 65 , V = 8 4 Close to the 1st order transition � U � function of plaquettes. �| φ | 2 � function of links. τ ’ int cubes: 25% 1000 100 10 1 <U> LMA 2 <U> SWA 2 | φ | | φ | LMA SWA Y. Delgado (KFU) Worm algorithm Excited QCD 2013 14 / 19
WA vs. LMA: κ = 8 , β = 1 . 1 , V = 8 4 Configurations dominated by closed surfaces (links are expensive). � U � function of plaquettes. �| φ | 2 � function of links. τ ’ int cubes: 99% 1000 100 10 1 <U> LMA 2 <U> SWA 2 | φ | | φ | LMA SWA Y. Delgado (KFU) Worm algorithm Excited QCD 2013 15 / 19
Scalar electrodynamics with two flavors Conventional action on the lattice: � � µ,ν U ⋆ ν,µ U ⋆ = − β S G Re U x,µ U x +ˆ x +ˆ x,ν x µ<ν � � κ 1 | φ 1 x | 2 + κ 2 | φ 2 � x | 2 S H = x � � � �� ⋆ U x,µ φ 1 ⋆ U ⋆ e δ µ 4 µ 1 φ 1 µ + e − δ µ 4 µ 1 φ 1 � µ,µ φ 1 − x x + � x x − � x − � µ x µ � � � �� ⋆ U ⋆ ⋆ U x − � � e δ µ 4 µ 2 φ 2 µ + e − δ µ 4 µ 2 φ 2 x,µ φ 2 µ,µ φ 2 − x + � x − � x x µ x µ Dual form of the partition sum: � W ( p, l 1 , l 2 ) C L ( p, l 1 , l 2 ) C S ( l 1 , l 2 ) Z = { p,l 1 ,l 2 } Y. Delgado (KFU) Worm algorithm Excited QCD 2013 16 / 19
An admissible configuration time space Chemical potential favors flux forward in time. Y. Delgado (KFU) Worm algorithm Excited QCD 2013 17 / 19
Recommend
More recommend
