**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

q^{2}_{x} +q^{2}_{y}:

!_{C}(q) = cq

q^{2}_{z,1} +q^{2} = !^{0}_{C} + q^{2}

2mC +O(q^{4}). (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} r^{2}
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)

Y### →U(1)

em### 3 1 1 1 3

### Spinor BEC

### SO(3)xU(1)→U(1) 3 1 1 1 3

### nonrelativistic *massive CP*

^{1}

### model

### U(1)xR

^{3}

### →R

^{2}

### 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

_{0}

### = H

_{C}

### + H

_{X}

### + H

_{C 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 d q (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 q

z,n### = ⇡ n/l

_{z}

### , where n is a positive integer, and l

z### 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

^{2}

_{x}

### + q

^{2}

_{y}

### :

### !

_{C}

### (q) = c q

### q

^{2}

_{z,1}

### + q

^{2}

### = !

_{C}

^{0}

### + q

^{2}

### 2m

C### + O(q

^{4}

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

_{C}

^{0}

### = cq

_{z,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

### H

_{C X}

### = ⌦

_{R}

### Z d q (2⇡)

^{2}

### X ⇣ a

^{†}

_{X,}

### (q)a

C,### (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

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

_{LP,}

### ) [13]

### H

_{LP}

### = Z

### d x

### "

†

### (x) !

^{0}

_{LP}

### r

^{2}

### 2m

_{LP}

### !

### (x) + u

c †### (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-

**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}

^{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.

**k**

**k**

^{2}^{Q}

^{A, }

^{Q}

^{R }

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

**k**

^{2}^{Q}

^{A}

### Exciton-polariton condensates

### B **
**

### <[Q

A,### Q

R### ]> 0

**k** ^{2} **k** ^{4} ^{Q} ^{A, } ^{Q} ^{R } Ferromagnet

**k**

^{2}**k**

^{4}^{Q}

^{A, }

^{Q}

^{R }

**k** ^{2} **k** ^{2} ^{Q} ^{A}

**k**

^{2}**k**

^{2}^{Q}

^{A}