1. A.G., Tsukuba, 16 July 20031 New advances in numerical simulations of theta-vacuum systems V. Azcoiti, V. LalienaZaragoza U. G. Di CarloINFN-Gran Sasso A.G.L’Aquila U.- Gran Sasso hep-lat 0203017,0210004, 0305005, 0305022

2. A.G., Tsukuba, 16 July 20032 QFT + topological terms a special case of complex action the complex part of the action has a simple form: SU(N) with theta-term CP N-1 models Spin systems coupled to imaginary magnetic field.....

3. A.G., Tsukuba, 16 July 20033 density of topological charge P.d.F. of the order parameter PRL89 (2002) 141601, hep-lat/0203017 from real action simulations at use saddle point approx. get f(x) get use multiprecision code to compute use multiprecision code to compute

4. A.G., Tsukuba, 16 July 20034 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 CP 3 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

5. A.G., Tsukuba, 16 July 20035 z = 0 z = 1 from y(z<1) we get

6. A.G., Tsukuba, 16 July 20036 z y(z) 1 real actioncomplex action related to....... real action complex action

7. A.G., Tsukuba, 16 July 20037 to extract information on CP symmetry at define: is a “free” parameter and the effective exponent gives information on

8. A.G., Tsukuba, 16 July 20038 1D Ising model (data) CP symmetric model forbidden region

9. A.G., Tsukuba, 16 July 20039 in many models turns out to have a very mild dependence on 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

10. A.G., Tsukuba, 16 July 200310 how to extract from simulations we need y(z) for z<1 use the variable and plot vs y this function is again a very smooth one but a small region near the origin can not be determined by direct measurement from available data + the fit of the effective exponent we can extract for any y from, using an iterative procedure, we can extract y(z) and. PLB563 (2003) 117, hep-lat/0305005

11. A.G., Tsukuba, 16 July 200311... (simulation data) (from data fit)

12. A.G., Tsukuba, 16 July 200312 1D Ising model: exact and simulation a model without SSB: results for the order parameter

13. A.G., Tsukuba, 16 July 200313 2D CP 3 model

14. A.G., Tsukuba, 16 July 200314 2D CP 9 asymptotically free instantons solutions -vacua well defined topological charge + good asymptotic scaling z n 10 components scalar field U(1) gauge field U P =e iFp plaquette

15. A.G., Tsukuba, 16 July 200315 2D CP 9 model results using P.d.F and the extrapolation procedure

16. A.G., Tsukuba, 16 July 200316 perturbative scaling non perturbative scaling

17. A.G., Tsukuba, 16 July 200317 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; CP 9 results scale well with perturbative calculation and indicate SSB of CP symmetry at