Lecture on “Spontaneous Symmetry Breaking” · 2012. 12. 10. · 12/7/12 B. Rosenow, Lecture on...

12/7/12 B. Rosenow, Lecture on “Spontaneous Symmetry Breaking”

Lecture on “Spontaneous Symmetry Breaking”

1

Bernd Rosenow Universität Leipzig


Outline

• Ising model

• ordered state: energy vs. entropy

• numerical simulation

• thermodynamic limit

• continuous symmetry and Goldstone modes


Symmetry of physical laws - structures in the physical world

• fundamental laws of nature have high degree of symmetry

• symmetry under translations, rotations, reflections, time reversal, charge

conjugation

• the world we live in has many structures which do not obey the symmetries of

the physical laws

The underlying mechanism is called spontaneous symmetry breaking

source: wikipedia


Ising Model

• simple model in which spontaneous symmetry breaking occurs

• model for a ferromagnet on a lattice

• at each lattice site has classical variable Si = ±1,

• can also describe lattice gas, activity of neurons, etc.

• coupling between spins favors parallel alignment

S. Kobe, Braz. J. Phys. 2000


Symmetry of Ising model

• Ising model has important symmetry (time reversal symmetry)

• when changing the signs of spins Si → - Si for all i, and h → -h, the energy is

unchanged

• H[h, {Si}] = H(-h, {-Si}] and for h=0 H[h=0, {Si}] = H[h=0, {-Si}]


Ground states of Ising model

• two fully polarized ferromagnetic states with Si =1 8i and Si = -1 8i do not

share time reversal symmetry of Hamiltonian

• fully polarized states have lowest possible energy

• E0 = - N J (one bond per spin on square lattice)

• why could these states important at low temperature?


Reminder: free energy

equilibrium state is determined by minimum of free energy

F = E - T S

E energy of system

S entroy of system

entropy measures number Wm of microscopic realizations {Si} for magnetization m


Existence of ordered phase

completely disordered paramagnetic state with m ¼ 0 ) two possibilities Si = ± 1

per spin

) S = kB log 2N = N kB log 2

paramagnet E = 0, S = N kB log 2 ) Fpara ' - N kB T log 2

ferromagnet E = - N J, S = 0 ) Fferro ' - N J

there is transition temperature Tc where Fpara = Fferro ) Tc ' J/kB log 2

T > Tc paramagnet lower free energy

T < Tc ferromagnet lower free energy

two degenerate ferromagnetic states with m=± 1, which one (or both?) is chosen?


Ising model simulation

simulation http://itp.tugraz.at/MML/isingxy/ Siegfried Gürtler, Hans Gerd Evertz

high temperature T >> Tc random magnet

low temperature T << Tc ordered state with m=1 (m=-1) stable

low temperatue T << Tc disordered state orders, domain walls


Symmetry breaking from dynamics

• T << Tc ) either state m=1 or m=-1 “spontaneously” realized

• time ¿ to switch between them?

• switching needs domain wall across system

• domain wall breaks n bonds ) ¢ E = 2 J n

• at each lattice point, three directions to go ) entropy S = n kB log 3

domain wall free energy ¢ Fwall = 2 n J - n kB T log 3

for activated dynamics (“Arrhenius law”)


Symmetry breaking due to slow dynamics

domain wall free energy ¢ Fwall = 2 n J - n kB T log 3

for activated dynamics (“Arrhenius law”)

microscopic time scale ¿0 ¼ 10-12 s

to switch m=1 to m=-1, need domain wall through whole system ) n ¸ N1/2

for T = Tc/2 find ¯¢Fwall = N1/2 log 3

¿ = 4.3 1017 s (age of the universe) ) N1/2 ' log(4.3 1029) ' 68

ground state of system with ¸ 5000 spins at T = 0.5 Tc will never switch

) spontaneous symmetry breaking


“Impossibility” of a phase transition

free energy density

time reversal symmetry of H ) f(h,T) = f(-h,T)

due to m(h) = - m(-h), for zero external field h=0 we find m(0) = - m(0) ´ 0

what is wrong with the argument?

had to assume


Non-analyticity of free energy

• assumption correct for systems with finite N

• all fN(h,T) are analytic functions of h (polynomials in e±¯h)

• assumption fails in the limit N → 1 as convergence of fN (h,T) to f(h,T) is only

pointwise and not uniform

• limit function f(h,T) can be non-analytic

f(h,T) develops discontinuity in


as an example, assume that

f(h) = f(h=0) - ms |h| + O(h¾) with ¾ >1

Consequences of derivative discontinuity

as h → 0


Magnetization and system size

one sees that the spontaneous

magnetization is indeed given by

key observation: the limits i) N → 1 and ii) h → 0 do not commute


Magnetization and temperature

ms is reduced to zero when

T is raised towards Tc

magnetization jumps across

branch cut of free energy in

(h,T)-plane

end of branch cut


Ergodicity breaking

average magnetization of spin Si

at h=0 clearly h Si i = 0

use of limh → 0+ after the thermodynamic limit implies

average over a restricted ensemble of configurations

{Si}, excluding configurations with - ms

system does not sample a representative region of full phase space

) ergodicity breaking


Breaking of a continuous symmetry

consider now a classical spins

there are infinitely many ground states with

the same free energy (t < 0)

there is no energy cost for

changing the direction of

other example: crystall lattice breaks translational symmetry of

free space


Goldstone mode

• no energy cost for changing direction of in whole system

• small energy cost for changing direction over distance L

• can show that ¢ f / 1/L2

• more generally, Á(x) is function of position

• energy ²(k) = ½s k2 vanishes for k → 0

• “soft” excitation is called Goldstone mode, is necessary consequence of breaking

a continous symmetry spontaneously (note that ²(k) / k possible as well)

Goldstone mode responsible for restoring broken symmetry for T > Tc


Higgs particle

theory of electroweak interaction considers quantum fields

Lagrangian

for reasons of renormalizability, mass term for electrons and gauge fieldcannot be added directly

solution: add complex field which undergoes spontaneous symmetry breaking

spontaneous symmetry breaking for t<0 adds mass terms to gauge field and electrons


Conslusions

• spontaneous symmetry breaking due to large free energy barrier

between configurations with ms and - ms

• mathematically, symmetry breaking due to non-analyticity of free

energy in the thermodynamic limit N → 1

• need h → 0 N → 1 as limits do not commute

• in case of broken continuous symmetry, low energy fluctuations

(Goldstone boson) try to restore symmetry

Lecture on “Spontaneous Symmetry Breaking” · 2012. 12. 10. · 12/7/12 B. Rosenow, Lecture on...

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