Spontaneous symmetry breaking in open systems

Spontaneous symmetry breaking in open systems

Yoshimasa Hidaka

RIKEN

Collaboration with Yuki Minami

1509.05042[cond-mat.stat-mech], 1712.xxxx

Open systems

System

Environment

CC BY-SA 2.0

Environment: Air

System: flock of birds QGP

Environment: QGP

System: Heavy quarks

heavy quarks

Symmetry breaking in open systems Synchronization

Driven dissipative condensate

Kuramoto model

Metronome, fireflies, … diehl

Driving force and dissipative causes a condensate.

figure is taken from Diehl's website

Diehl, Micheli, Kantian, Kraus, Büchler, Zoller, Nature Physics 4, 878 (2008);

Kraus, Diehl, Micheli, Kantian, Büchler, Zoller, Phys. Rev. A 78, 042307 (2008).

Questions

Hamiltonian systems Continuum

symmetry @ _µ J ^µ = 0

Open systems

@ _µ J ^µ 6= 0 because of friction

What is the symmetry?

Is there any symmetry breaking?

Does a NG mode appear?

Nambu-Goldstone theorem

For Lorentz invariant vacuum

Spontaneous breaking of global symmetry

# of NG modes

Dispersion relation:

N _{N G} = N _BS

# of broken symmetries

Nambu(’60), Goldstone(61), Nambu Jona-Lasinio(’61),

Goldstone, Salam, Weinberg(’62).

CC by-sa Aney

CC by-sa Roger McLassus

CC by Zouavman Le Zouave CC BY-SA 2.0

8 is described by the quadratic Hamiltonian [13]

H0 = HC +HX +HC X, (4) where the parts of the Hamiltonian involving only photons and excitons, respectively, take the same form, which is given by (here the index ↵ labels cavity photons, ↵ = C, and excitons,

↵ = X, respectively)² H↵ =

Z dq (2⇡)²

X!_↵(q)a^†_↵, (q)a↵, (q). (5)

Field operators a^†↵, (q) and a↵, (q) create or destroy a photon or exciton (note that both are bosonic excitations) with in-plane momentum q and polarization (there are two polarization states of the exciton which are coupled to the cavity mode [13]).

For simplicity, we neglect polarization e↵ects leading to an ef- fective spin-orbit coupling [13]. Due to the confinement in the transverse (z) direction, i.e., along the cavity axis, the motion of photons in this direction is quantized as qz,n = ⇡n/lz, where n is a positive integer, and lz is the length of the cavity. In writing the Hamiltonian (5), we are assuming that only the lowest trans- verse mode is populated, which leads to a quadratic dispersion as a function of the in-plane momentum q = |q| = q

q²_x +q²_y:

!_C(q) = cq

q²_z,1 +q² = !⁰_C + q²

2mC +O(q⁴). (6) Here, c is the speed of sound, !⁰_C =cqz,1, and the e↵ective mass of the photon is given by m_C = q_z,1/c. Typically, the value of the photon mass is orders of magnitude smaller than the mass of the exciton, so that the dispersion of the latter appears to be flat on the scale of Fig.1(b).

Upon absorption of a photon by the semiconductor, an exciton is generated. This process (and the reverse process of the emis- sion of a photon upon radiative decay of an exciton) is described by

HC X = ⌦_R

Z dq (2⇡)²

X ⇣a^†_X, (q)aC, (q) + H.c.⌘

, (7) where ⌦R is the rate of the coherent interconversion of photons into excitons and vice versa. The quadratic Hamiltonian (4) can be diagonalized by introducing new modes — the lower and upper exciton-polaritons, LP, (q) and UP, (q) respectively, which are linear combinations of photon and exciton modes. The dispersion of lower and upper polaritons is depicted in Fig. 1 (b). In the regime of strong light-matter coupling, which is reached when ⌦R is larger than both the rate at which pho- tons are lost from the cavity due to mirror imperfections and the

2In Ref. [149,150], a di↵erent model for excitons is used: they are assumed to be localized by disorder, and interactions are included by imposing a hard- core constraint.

exciton lower polariton

upper polariton

photon relaxation

!

q

emission excitation

Bragg mirror

Bragg mirror

photon

e h

exciton

(a)

(b)

Figure 1. (a) Schematic of two Bragg mirrors forming a microcavity, in which a quantum well (QW) is embedded. In the regime of strong light-matter interaction, the cavity photon and the exciton hybridize and form new eigenmodes, which are called exciton-polaritons. (b) Energy dispersion of the upper and lower polariton branches as a function of in- plane momentum q. In the experimental scheme illustrated in this figure (cf. Ref. [12]), the incident laser is tuned to highly excited states of the quantum well. These undergo relaxation via emission of phonons and scattering from polaritons, and accumulate at the bottom of the lower polariton branch. In the course of the relaxation process, coherence is quickly lost, and the e↵ective pumping of lower polaritons is incoher- ent.

non-radiative decay rate of excitons, it is appropriate to think of exciton-polaritons as the elementary excitations of the system.

In experiments, it is often sufficient to consider only lower polaritons in a specific spin state, and to approximate the disper- sion as parabolic [13]. Interactions between exciton-polaritons originate from various physical mechanisms, with a dominant contribution stemming from the screened Coulomb interactions between electrons and holes forming the excitons. Again, in the low-energy scattering regime, this leads to an e↵ective contact interaction between lower polaritons. As a result, the low-energy description of lower polaritons takes the form (in the following we drop the subscript indices in _LP, ) [13]

HLP = Z

dx

"

†(x) !⁰_LP r² 2mLP

!

(x) + uc †(x)² (x)²

# . (8) While this Hamiltonian is quite generic and arises also, e.g., in cold bosonic atoms in the absence of an external potential, the peculiarity of exciton-polaritons is that they are excitations with relatively short lifetime. In turn, this necessitates continuous re-

chiral symm.

spin symm.

U(1) symm.

translation symm.

U(1) 1-form symm. rotation symm. U(1) symm.

Gapless modes in nature

superfluid phonon

spin waves

photon surface waves diffusive modes

Nambu-Goldstone modes

pion

Need generalization

NG theorem OK

We focus on open systems

Classification of

Nambu-Goldstone modes

in Hamiltonian system

Exception of NG theorem

N _BS 6= N ^{N G and} ! 6= k

Schafer, Son, Stephanov, Toublan, and Verbaarschot (’01) Miransky, Shovkovy (’02)

Dispersion :

N _BS = 3, N _NG = 2

NG modes in Kaon condensed CFL phase

SU (2) _I ⇥ U(1) ^Y ! U(1) ^em

! / k ^{and !} / k ²

exist NG modes with

Dispersion :

Magnon

N _BS = dim(G/H) = 2 N _NG = 1

! / k ²

spin rotation SO(3) ! SO(2)

Internal symmetry breaking

N _BS = dim(G/H)

# of flat direction

G H

Symmetry group

This does work in nonrelativistic system

at zero and finite temperature

Intuitive example

for type-B NG modes

Pendulum with a spinning top

Rotation symmetry is explicitly 
 broken by a weak gravity

Rotation along with z axis is unbroken.

Rotation along with x or y is broken.

The number of broken

symmetry is two.

Pendulum has two oscillation motions

if the top is not spinning.

Intuitive example

for type-B NG modes

If the top is spinning,

the only one rotation motion (Precession) exists.

In this case, _{L x , L _y } ^P = L _z 6= 0

Intuitive example

for type-B NG modes

Type-A Type-B

Classification of NG modes

Harmonic oscillation Precession

N _A = N _BS rank h[iQ ^a , Q _b ] i N _B = 1

2 rank h[iQ ^a , Q _b ] i Ex. ) superfluid phonon Ex. ) magnon

Watanabe, Murayama (’12), YH (’12)

N _NG = N _BS 1

2 hi[Q ^a , Q _b ] i

cf. Takahashi, Nitta (’14), Beekman (’14)

Type-A Type-B

Dispersion relation

! ⇠ p

g _⇠ ^{p k 2} ! ⇠ g ^{⇠ k 2}

gravity

N _BS N _type-I N _type-II ¹

2 rank h[Q ^a , Q _b ] i N _BS N _{N G}

Spin wave in ferromanget

SO(3)→SO(2) 2 0 1 1 2

NG modes in Kaon

condensed CFL

SU(2)xSU(1)

→U(1)

3 1 1 1 3

Spinor BEC


SO(3)xU(1)→U(1) 3 1 1 1 3

nonrelativistic massive CP

model

U(1)xR

→R

2 0 1 1 2

Examples of Type-B NG modes

N _BS ¹

2 rank h[Q ^a , Q _b ] i

N _BS N _NG = 1

2 rank h[Q ^a , Q _b ] i N _type-A + 2N _type-B = N _BS

N _type-A N _{type-B N type-A + 2N type-B}

At finite temperature

The interaction with thermal particles modifies the dispersion relation

! = ak ibk ²

! = a ⁰ k ² ib ⁰ k ⁴

Type-A:

Type-B:

Hayata, YH (’14)

CC BY-SA 2.0

Open system

8 is described by the quadratic Hamiltonian [13]

H

= H

+ H

+ H

, (4) where the parts of the Hamiltonian involving only photons and excitons, respectively, take the same form, which is given by (here the index ↵ labels cavity photons, ↵ = C, and excitons,

↵ = X, respectively)

H

=

Z d q (2⇡)

X !

(q)a

(q)a

(q). (5)

Field operators a

(q) and a

(q) create or destroy a photon or exciton (note that both are bosonic excitations) with in-plane momentum q and polarization (there are two polarization states of the exciton which are coupled to the cavity mode [13]).

For simplicity, we neglect polarization e↵ects leading to an ef- fective spin-orbit coupling [13]. Due to the confinement in the transverse (z) direction, i.e., along the cavity axis, the motion of photons in this direction is quantized as q

= ⇡ n/l

, where n is a positive integer, and l

is the length of the cavity. In writing the Hamiltonian (5), we are assuming that only the lowest trans- verse mode is populated, which leads to a quadratic dispersion as a function of the in-plane momentum q = |q| = q

q

+ q

:

!

(q) = c q

q

+ q

= !

+ q

2m

+ O(q

). (6) Here, c is the speed of sound, !

= cq

, and the e↵ective mass of the photon is given by m

= q

/ c. Typically, the value of the photon mass is orders of magnitude smaller than the mass of the exciton, so that the dispersion of the latter appears to be flat on the scale of Fig. 1 (b).

Upon absorption of a photon by the semiconductor, an exciton is generated. This process (and the reverse process of the emis- sion of a photon upon radiative decay of an exciton) is described by

H

= ⌦

Z d q (2⇡)

X ⇣ a

(q)a

(q) + H.c. ⌘

, (7)

where ⌦

is the rate of the coherent interconversion of photons into excitons and vice versa. The quadratic Hamiltonian (4) can be diagonalized by introducing new modes — the lower and upper exciton-polaritons,

(q) and

(q) respectively, which are linear combinations of photon and exciton modes. The dispersion of lower and upper polaritons is depicted in Fig. 1 (b). In the regime of strong light-matter coupling, which is reached when ⌦

is larger than both the rate at which pho- tons are lost from the cavity due to mirror imperfections and the

2

In Ref. [149, 150], a di↵erent model for excitons is used: they are assumed to be localized by disorder, and interactions are included by imposing a hard- core constraint.

exciton lower polariton

upper polariton

photon

relaxation

!

q

emission excit ation

Bra g g mi rro r

Bra g g mi rro r

photon

e h

exciton

(a)

(b)

Figure 1. (a) Schematic of two Bragg mirrors forming a microcavity, in which a quantum well (QW) is embedded. In the regime of strong light-matter interaction, the cavity photon and the exciton hybridize and form new eigenmodes, which are called exciton-polaritons. (b) Energy dispersion of the upper and lower polariton branches as a function of in- plane momentum q. In the experimental scheme illustrated in this figure (cf. Ref. [12]), the incident laser is tuned to highly excited states of the quantum well. These undergo relaxation via emission of phonons and scattering from polaritons, and accumulate at the bottom of the lower polariton branch. In the course of the relaxation process, coherence is quickly lost, and the e↵ective pumping of lower polaritons is incoher- ent.

non-radiative decay rate of excitons, it is appropriate to think of exciton-polaritons as the elementary excitations of the system.

In experiments, it is often sufficient to consider only lower polaritons in a specific spin state, and to approximate the disper- sion as parabolic [13]. Interactions between exciton-polaritons originate from various physical mechanisms, with a dominant contribution stemming from the screened Coulomb interactions between electrons and holes forming the excitons. Again, in the low-energy scattering regime, this leads to an e↵ective contact interaction between lower polaritons. As a result, the low-energy description of lower polaritons takes the form (in the following we drop the subscript indices in

) [13]

H

= Z

d x

"

†

(x) !

r

2m

!

(x) + u

(x)

(x)

#

. (8)

While this Hamiltonian is quite generic and arises also, e.g., in

cold bosonic atoms in the absence of an external potential, the

peculiarity of exciton-polaritons is that they are excitations with

relatively short lifetime. In turn, this necessitates continuous re-

Ex2) Vicsek model

T. Vicsek, et al., PRL (1995).

x _i (t + t) = x _i (t) + v _i t

✓ _i (t + t) = h✓ ⁱ (t) i ^r + ⇠ _i

velocity

angle of velocity

v _i = v ₀ (cos ✓ _i , sin ✓ _i )

noise average

angle

✓

r

r

Some cells on fish skin

An example of SSB in open systems

B. Szabo, et al., Phys. Rev. E 74, 061908 (2006)

Model of active matter: Vicseck model, Active hydrodynamics, ….

T. Vicsek, et al., PRL (1995). J. Toner, and Y. Tu, PRL (1995).

high density

low density

@ _t v + (v · r)v = ↵v v ² v rP + D ^L r(r · v) + D ^l (v · r) ² v + f

@⇢ + r · (⇢v) = 0

Ex.) NG mode in Active hydrodynamics

J. Toner, and Y. Tu, PRE (1998)

noise

nonconserved term

! = ck ! = i k ^{2 NG modes}

diffusive propagating

O(3) ! O(2)

Steady state solution:

v = (v ₀ + v _x , v _y , v _z )

Fluctuation:

v ² = ↵/ ⌘ v ₀ ²

Symmetry breaking:

Field theoretical model

Can we discuss symmetry breaking

without ordinary conservation law?

Ex) Symmetry of Brownian motion

d

dt u(t) = u(t) + ⇠(t) d

dt x(t) = u(t)

L = x ⇥ u

Angular momentum d

dt hL(t)i = hx ⇥ u(t)i 6= 0 Langevin equation

h⇠ ⁱ (t)⇠ _j (t ⁰ ) i = 2 ^ij T (t t ⁰ )

not conserved

Langevin equation

d

dt u(t) = u(t) + ⇠(t)

Fokker-Planck equation

@ _t P (t, u) = @

@u _i

⇣ T @

@u _i + u _i ⌘

P (t, u)

Path integral Martin-Siggia-Rose formalism

Z =

Z

D Due ^{iS[ ,u]}

Dynamic action: ^{iS = Z dt h i i ⇣ d} _dt ^{u i + u i ⌘ T 2} i

i

i ! R ^{ij j} u _i ! R ^ij u _j

O(3) symmetry

with ^R ik R _kj = _jk

Symmetry of Dynamic action

Noether charge

L _MSR = ⇥ u L = x ⇥ u

L _MSR 6= L

iS =

Z

dt h

i _i ⇣ d

dt u _i + u _i ⌘

T ² _i i

Open quantum system

1 2

time

complex

Schwinger-Keldysh Path integral

cf. for review, Sieberer, Buchhold, Diehl, 1512.00637

Z = Z

D ¹ D ² exp h

iS[ ₁ ] iS[ ₂ ] + iS ₁₂ [ ₁ , ₂ ] i

: Interaction with environment

S[ ₁ ] S[ ₂ ]

: forward evolution

: backward evolution

S ₁₂ [ ₁ , ₂ ]

Example

Lindblad equation

@ _t ⇢ = i[H, ⇢] + ⇣

L⇢L ^† 1

2 (L ^† L⇢ + ⇢L ^† L) ⌘ fluctuation and dissipation

Action

S[ ] = 1

2 (@ _µ ) ² 1

2 m ^{2 2}

4!

4

iS ₁₂ [ ₁ , ₂ ] = L( ₁ )L ^† ( ₂ )

2

⇣ L( ₁ )L ^† ( ₁ ) + L( ₂ )L ^† ( ₂ ) ⌘

R/A basis

S[ ₁ ] S[ ₂ ] = Z

d ⁴ x _A ⇣

@ _{µ R} ² m ² _R

3!

3 R

⌘ + Z

d ⁴ x

24

3 A R

R = 1

2 ( ₁ + ₂ ) _A = _{1 2}

classical field fluctuation

classical Equation of motion

iS ₁₂ [ _R , _A ] = _A @ _{0 R} A 2

2 A + · · ·

For example,

Symmetry of Open quantum system

1 2

time

S[ ₁ ], S[ ₂ ]

Q ₁ , Q ₂ :Symmetry generators:

Q _R = Q ₁ + Q ₂ 2

Q _A = Q ₁ Q ₂

Suppose S ₁₂ [ ₁ , ₂ ] 2

are invariant.

is invariant under

complex

We also define

Schwinger-Keldysh Path integral

cf. for review, Sieberer, Buchhold, Diehl, 1512.00637

Z = Z

D ¹ D ² exp h

iS[ ₁ ] iS[ ₂ ] + iS ₁₂ [ ₁ , ₂ ] i

Ex1) SU(2)xU(1) model Type-A ^{V ( )}

Linear analysis

NG type-A mode

! ² i ! + k ² = 0

diffusion mode (@ ₀ ² + @ ₀ r ² )⇡ _a = 0

! = i

2 ± i 2

p 2 4k ² ⇠ i

k ² , i + i

k ²

Spontaneous symmetry breaking

Minami, YH (’15)

' _R/A two component complex field

' _R = (⇡ ₁ + i⇡ ₂ , v + h + i⇡ ₃ )

iS = Z

d ⁴ x h

† A ( @ ₀ ² + r ² @ ₀ )' _R 2 |' ^R | ² ' _R A' ^† _A ' _A i

+ · · ·

Ex2)

V ( )

with chemical potential μ

✓ @ ₀ ² @ ₀ + r ² 2µ@ ₀

2µ@ ₀ @ ₀ ² @ ₀ + r ²

◆ ✓ ⇡ ₁

⇡ ₂

◆

= 0,

SU(2)xU(1)model Type-B

! = k ²

4µ ² + ² ( ±2µ i )

Spontaneous symmetry breaking

Minami, YH (’15)

iS = Z

d ⁴ x h

† A ( (@ ₀ + iµ) ² + r ² @ ₀ )' _R 2 |' ^R | ² ' _R A' ^† _A ' _A i

+ · · ·

quadratic dispersion

h[iQ ¹ A , Q ² _R ] i 6= 0 Type-B

Ex3)

with complex potential

SU(2)xU(1)model Type-B

Spontaneous symmetry breaking

Minami, YH (’17)

iS = Z

d ⁴ x ⇣

i' ^† _A (( @ ₀ ² + r ² ( + 2iµ)@ ₀ m ² _r im ² _i )' _R 2( _r + i _i )(' ^† _R ' _R )' _R ) A' ^† _A ' _A ⌘

+ · · ·

(! ₀ ² 2µ! ₀ m ² _r 2 _r v ² + i( ! ₀ m ² _i 2 _i v ² ))v = 0 ' _R = (0, ve ^{i! 0 t} )

Assuming

Gap equation

Symmetric phase Broken phase

v = 0

v 6= 0, ! ₀ 6= 0

Ex3)

with complex potential

We still have quadratic dispersion

SU(2)xU(1)model Type-B

Spontaneous symmetry breaking

Minami, YH (’17)

! = k ²

4(µ ! ₀ ) ² + ² ( ±2(µ ! ₀ ) i )

✓ @ ₀ ² @ ₀ + r ² 2(µ ! ₀ )@ ₀

2(µ ! ₀ )@ ₀ @ ₀ ² @ ₀ + r ²

◆ ✓ ⇡ ₁

⇡ ₂

◆

= 0,

Linear analysis

Similarly, we find

! = i k ²

+ 2(µ ! ₀ ) _i / _r

Diffusive mode for type-A

Minami, YH (’17)

[G _⇡ ¹ (k)] ^↵ = iC ^{µ; ↵} k _µ + C ^{µ⌫; ↵} k _µ k _⌫ + · · ·

Inverse propagator and dispersion

i Z

d ^D x h[iQ ^↵ _R , L ¹² (x)]j _A ^µ (0) i ^⇡c

k lim !0

@

@k _⌫ i Z

d ^D xe ^{ik ⇢ x ⇢} h[iQ ^↵ R , L ¹² (x)]j _A ^µ (0) i ^⇡c

C ^{µ; ↵} = h[iQ ^↵ _R , j _A ^µ (0)] i

C ^{µ⌫; ↵} = i

Z

d ^D x hj _R ^↵µ (x)j _A ^⌫ (0) i ^⇡c

Hamiltonian system Open system

Our result is too general

Need to impose symmetry of S 12

Ex)’Standard’ Fokker-Plank eq.

Type-A mode

! = ik ² Diffusive

Type-B mode

! = ak ² ik ^{2 0}

Next step: classification

N _B = 1

2 rank h[iQ ^↵ R , Q _A ] i

N _A = N _BS rank h[iQ ^↵ R , Q _A ] i

Spontaneous breaking of


symmetry of Dynamic action

Type-A mode

! = ik ² Diffusive

Type-B mode

! = ak ² ik ^{2 0}

Two-type of diffusive NG modes

Next step: classification

Summary

Type Dispersion

Conserved charge Examples

Re Im

A

k k ^{2 Q A, Q R} Superfluid, etc.

0 k ^{2 Q A} flock of birds,

Exciton-polariton condensates

B 


<[Q

Q

]> 0

k ² k ^{4 Q A, Q R} Ferromagnet

k ² k ^{2 Q A}

Spinor BEC in open quantum system?

Magnetotactic bacteria?

What is the condition satisfying this table?

Magnetotactic bacteria Collective motion of

Possible active matter 


with type B modes?