A.G., Tsukuba, 16 July 2003 1

New advances in numerical simulations of theta-vacuum systems

• V. Azcoiti, V. Laliena Zaragoza U.• G. Di Carlo INFN-Gran Sasso• A.G. L’Aquila U.- Gran Sasso

hep-lat 0203017, 0210004, 0305005, 0305022

A.G., Tsukuba, 16 July 2003 2

QFT + topological termsQFT + topological termsa special case of complex action

0S S S S i Q Q

the complex part of the action has a simple form:• SU(N) with theta-term• CPN-1 models• Spin systems coupled to imaginary magnetic field• .....

A.G., Tsukuba, 16 July 2003 3

( ) ( )V

i nV

n V

Z P n e

n

nxV

density of topological charge 1,1nx

( )( ) V nVf xVP n e P.d.F. of the order parameter

PRL89 (2002) 141601, hep-lat/0203017

from real action simulations at ih

use saddle point approx. get f(x)

get q use multiprecision codeto compute Z

A.G., Tsukuba, 16 July 2003 4

the method has been tested in several cases:• 2D compact U(1) with theta-term• 1D AF Ising model coupled to imaginary magnetic field• 2D CP3

there are two possible sources of systematic effects:• saddle point approx.• a fit of simulation data is needed to have an analytical expression for f(x)

need for independent results

A.G., Tsukuba, 16 July 2003 5

2 12

1

( ) ( ) ( ) cos ( )2

n

n

nV VVyi n

V V V nn V n V y

Z P n e G n G y z

nnyV

cos2

z

2

2

( )( )

( )

n

n

n

n

Vyn V n

y

VyV n

y

y G y zy z

G y z

( ) ( ) tan2

q iy z

( ) ( ) tanh2hq ih y z cosh [1, )

2hz

cos 0,12

z z = 0

z = 1

0

from y(z<1) we get q

A.G., Tsukuba, 16 July 2003 6

z

y(z)

1 real actioncomplex action

0( 1) 2y z 00

( )dqd

( )( 0) limtan( / 2)

qy z

related to

• • • • .......

( )q

0( 0) (q c y z z I CP ) 1 0( ) 0 1 ( 0) ( )q y z z II CP 2( ) ( 0) ( )q y z z CP

real action

complexaction

ih

A.G., Tsukuba, 16 July 2003 7

to extract information on CP symmetry at define:

/ 2( ) ( )y z y e z is a “free” parameter

and the effective exponent2 log y

y

( ) ( ) ( 0) 1q y

( 0)y gives information on ( )q

A.G., Tsukuba, 16 July 2003 8

1D Ising model (data)

CP symmetric model

forbidden region

A.G., Tsukuba, 16 July 2003 9

in many models turns out to have a very mild dependence on y( )y

• extrapolate seems a reasonable possibility (assume no phase transitions at real except at most at )• the extrapolation is “easier” for asymptotically free models where from real action simulations we can measure up to very small values of y• results have to be independent of • possible systematic effects

( 0)y

( )ih ( )y

A.G., Tsukuba, 16 July 2003 10

how to extract from simulationsih ( )q

we need y(z) for z<1

use the variable and plot vs y/ 2( ) ( )y z y e z y

this function is again a very smooth one but a small region near the origin can not be determined by direct measurementfrom available data + the fit of the effective exponent we can extract for any y

( )y y ( )y( )y y

from , using an iterative procedure, we can extract y(z) and .( )y y q

PLB563 (2003) 117, hep-lat/0305005

A.G., Tsukuba, 16 July 2003 11

y y

1 / 20( )y y e z

00 0( ) 1y y z z

/ 20( )n ny y e z

...

y (simulation data)

y(from data fit)

0

A.G., Tsukuba, 16 July 2003 12

1D Ising model:exact and simulation

a model without SSB:

(1 cos )VVZ A

sin( )1 cos

Aq iA

results for the order parameterresults for the order parameter

A.G., Tsukuba, 16 July 2003 13

2D CP2D CP33 model model0.4 100 100V

q

A.G., Tsukuba, 16 July 2003 14

2D CP2D CP99

• asymptotically free • instantons solutions• -vacua• well defined topological charge• + good asymptotic scaling

, ,9 2g n n n n n nS z z z z

12 PQ F

zn 10 components scalar field 1n nz z U(1) gauge field , , 1n n

UP=eiFp plaquette PF ,n

A.G., Tsukuba, 16 July 2003 15

2D CP2D CP99 model model0.8

100 100V

0.8

200 200V

0.85

results using P.d.F and theextrapolation procedure

q

A.G., Tsukuba, 16 July 2003 16

perturbative scalingperturbative scaling

2/1 22 NLa e

non perturbative scalingnon perturbative scaling

200 200V

200 200V

0.80

0.80 0.85

0.85

0.90

0.90 q

q

A.G., Tsukuba, 16 July 2003 17

Conclusions

• we propose two different methods to simulate theta-vacuum models (tested on analitically solvable models);• both use real action (imaginary theta) simulations as input;• the two methods can be affected by (different) systematic effects but agree each other;• CP9 results scale well with perturbative calculation and indicate SSB of CP symmetry at