Different symmetry realizations in relativistic coupled Bose systems

at finite temperature and densities

Collaborators: R.L.S. Farias and R.O. Ramos

Rodrigo Vartuli

Department of Theoretical Physics,Department of Theoretical Physics, UERJII Latin American Workshop on

High Energy Phenomenology

5th December 2007São Miguel das Missões, RS,

Brazil

Outline• Motivation

• Study of symmetry breaking (SB) and symmetry restoration (SR)

• In multi-scalar field theories at finite T and are looking for the phenomena

* symmetry nonrestoration (SR) * inverse symmetry breaking (ISB)

• How a nonzero charge affects the phase structure of a multi-scalar field theory?

• Work in progress and future applications

1- Motivation1- MotivationThe larger is the temperature, the larger is the The larger is the temperature, the larger is the

symmetrysymmetrythe smaller is the temperature, the lesser is the the smaller is the temperature, the lesser is the

symmetry:symmetry:

Symmetry Breaking/Restoration in O(N) Scalar Models

Boundness:

+ unbroken

- broken0

Relativistic case:

The potential V( ) for –m2, N=1 (Z2)

Let´s heat it up!!

Thermal Mass at high-T and N

M ²(N=2)

For ALL single field models:

Higher order corrections do NOT alter this pattern !!

(1974)

O(N)xO(N) Relativistic Models

Boundness: λ>0 OR: λ<0!!

Thermal masses to one loop

Critical Temperatures at high-T and N=2

or i

M² (N=2)

Both m² < 0:SR in the ψ sector

SNR in the sector

Transition patterns

M ²

ISB

Transition patterns

Both m² > 0: sector:

unbroken sector : ISB

Temperature effects in multiscalar field models Temperature effects in multiscalar field models can change the symmetry aspects in unexpected ways:can change the symmetry aspects in unexpected ways:

e.g. in the O(N)xO(N) example, it shows the possibilities e.g. in the O(N)xO(N) example, it shows the possibilities of phenomena like inverse symmetry breaking (ISB)of phenomena like inverse symmetry breaking (ISB)

and symmetry nonrestoration (SNR)and symmetry nonrestoration (SNR)

But be careful:

Question: Can we trust perturbative methods Question: Can we trust perturbative methods at high temperatures ?at high temperatures ?

NO ! (but these phenomena appear too in nonperturbative

approaches) THEY ARE NOT DUE BROKEN OF PERTURBATION THEORY

~ O( T ) ~ O( T . T/m )

Perturbation theory breaks down for Perturbation theory breaks down for temperatures temperatures l T/m > 1

2 2

Requires nonperturbation methods: Requires nonperturbation methods: daisy and superdaisy resum, daisy and superdaisy resum,

Cornwall-Jackiw-Tomboulis method, Cornwall-Jackiw-Tomboulis method, RG, RG,

large-N, large-N, epsilon-expansion,epsilon-expansion,

gap-equations solutions, lattice, etcgap-equations solutions, lattice, etc

Nonperturbative methods are quite discordant aboutthe occurrence or not of ISB/SNR phenomena:

NONO

PLB 151, 260 (1985), PLB 157, 287 (1985), PLB 151, 260 (1985), PLB 157, 287 (1985), Z. Phys. C48, 505 (1990)Z. Phys. C48, 505 (1990)

Large-N expansion Large-N expansion

Gaussian eff potential Gaussian eff potential PRD37, 413 (1988), Z. Phys. C43, 581 (1989) PRD37, 413 (1988), Z. Phys. C43, 581 (1989)

Chiral lagrangian method Chiral lagrangian method

Monte Carlo simulations Monte Carlo simulations

PRD59PRD59,025008 (1999),025008 (1999)

Bimonte et al NPB515, 345 (1998),Bimonte et al NPB515, 345 (1998), PRL81, 750 (1998)PRL81, 750 (1998)

Large-N expansion Large-N expansion

Gap equations solutions Gap equations solutions PLB366, 248 (1996), PLB388, 776 (1996), PLB366, 248 (1996), PLB388, 776 (1996),

NPB476, 255 (1996) NPB476, 255 (1996)

Renormalization Group Renormalization Group

Monte Carlo simulations Monte Carlo simulations

PRD54, 2944 (1999), PLB367, 119 (1997)PRD54, 2944 (1999), PLB367, 119 (1997)

Bimonte et al NPB559, 103 (1999),Bimonte et al NPB559, 103 (1999),Jansen and Laine PLB435, 166 (1998)Jansen and Laine PLB435, 166 (1998)

YESYES

PLB403, 309 (1997) PLB403, 309 (1997)

Optimized PT (delta-exp) Optimized PT (delta-exp) M.B. Pinto and ROR, PRD61, 125016 (2000)M.B. Pinto and ROR, PRD61, 125016 (2000)

Conclusions for O(N)xO(N) relativistic:ISB/SNR are here to stay!! Applications?

Cosmology, eg, Monopoles/Domain Walls

What happens in real condensed matter systems ?

((potassiumpotassium sodium tartrate tetrahydrate) sodium tartrate tetrahydrate)Liquid crystalsLiquid crystals

(SmC*)(SmC*)Reentrant phaseReentrant phase383K < T < 393K383K < T < 393K

Manganites: (Pr,Ca,Sr)MnO ,Manganites: (Pr,Ca,Sr)MnO ,ferromagnetic reentrant phase above the Curie ferromagnetic reentrant phase above the Curie

temperature (colossal magnetoresistence)temperature (colossal magnetoresistence)

Inverse melting (~ ISB) liquid Inverse melting (~ ISB) liquid crystal: crystal: He3,He4, He3,He4,

binary metallic alloys (Ti, Nb, Zr, Ta) binary metallic alloys (Ti, Nb, Zr, Ta) bcc to amorphous at high T bcc to amorphous at high T

Etc, etc, etc ….Etc, etc, etc ….

33

Review: cond-mat/0502033Review: cond-mat/0502033

Phase structure and the effective potential at fixed

chargeWe start with the grand partition function

Where H is the ordinary Hamiltonian and

Using the standard manipulations like Legendre transformations … we get

Phase structure and the effective potential at fixed

chargeZ is evaluated in a systematic way

where

or

where

Phase structure and the effective potential at fixed

chargeUsing imaginary time formalism

The renormalized effective potential in the high density and temperatures is given by

where

Neglecting the zero point contribution similar made in PRD 44, 2480 (1991)

or

Phase structure and the effective potential at fixed

chargeThe phase structure depends on the minima of the effective potential

We have two minima:

for unbroken symmetry

for broken symmetry

and

Phase structure and the effective potential at fixed

chargeMinimizing the effective potential with respect to µ

In the high density limit µ >> m

Now we will show numerical results for broken

and unbroken phase of the theory with one complex scalar field

Working at high density µ >> m and high temperature T

0mfor !42

1 2422 mU

Numerical Results (broken phase)

PRD 44, 2480 (1991)Small charge - Symmetry restored

Charge increase - Symmetry never restored (SNR)

Numerical Results (unbroken case)

Unbroken case 02 m Ordinary η=0 have no Symmetry breaking

But at high T and µ

Remember that

0mfor !42

1 2422 mU

Numerical Results (unbroken case)

PRD 44, 2480 (1991)Broken symmetry at high T

(ISB)

In Preparation

& For one complex scalar field we showvery interesting results like(PRD 44, 2480 (1991))

Symmetry non restorationInverse symmetry breaking

& We are extending these calculations for two complex scalar fields

Future applications

* In collaboration with L.A. da Silva R.L.S. Farias and R.O. Ramos

Nonequilibrium dynamics of multi-scalar field Theories

• Markovian and

• Non-Markovian evolutions for the fields…

See poster: Langevin Simulations with Colored

Noise and Non-Markovian Dissipation