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MANY BODY PHYSICS

45 YEARS OF NUCLEAR THEORY at STONY BROOK

A Tribute To Gerald E. Brown Stony Brook

November 25, 2013

Jean-Paul Blaizot, IPhT- Saclay

MANY BODY PHYSICS

45 YEARS OF NUCLEAR THEORY at STONY BROOK

A Tribute To Gerald E. Brown Stony Brook

November 25, 2013

Jean-Paul Blaizot, IPhT- Saclay

Collective fermionic excitations in the quark-gluon plasma

and cold atom systems

THE QUARK-GLUON PLASMA

AS A MANY-BODY SYSTEM

- Ideal playground for application, and development, of many-body and quantum field theoretical techniques (and so are cold atom systems)

- Here we focus on high temperature ( ), weak interactions (small gauge coupling). [Leave aside issues related to the so-called strongly coupled QGP, AdS/CFT, etc]

- Basic degrees of freedom are quarks and gluons (quasiparticles), interacting with strength g (long range interactions)

- Various collective phenomena

HIERACHY OF SCALES (AND PHENOMENA) AT WEAK COUPLING

plasma ‘particles’ (hard)

collective excitations, screening, Landau damping (soft)

‘magnetic scale’, hydrodynamical modes (ultra-soft)

T





T

Collective features are ‘natural’ in gluon (boson) sectorBut they also appear in quark (fermion) sector





In the rest of this talk, focus on collective phenomena carrying fermionic quantum numbers

T

Collective features are ‘natural’ in gluon (boson) sectorBut they also appear in quark (fermion) sector

THE PLASMINO

Gluon (boson) Quark (fermion)

Interactions

Origin of the split dispersion relation

THE «SCHEMATIC MODEL» INTERPRETATION

Note how the smallness of the coupling can be overcome by the large number of degrees of freedom (N) to produce a large effect even when g is tiny

Nature of hard excitation changes from boson to fermion

Soft degree of freedom « oscillates». It can be described as oscillation of fermionic mean field [see JPB and E. Iancu Phys. Rept. 359 (2002)]

Turning bosons into fermions (and vice versa)

Suggestive of supersymmetry

Ultra-soft fermionic mode analog of hydrodynamic sound ? Phonino, ‘quasi-goldstino’ ?

Ultra soft modes

[Y. Hidaka, D. Satow, T. Kunihiro, arXiv1105.0423 (Yukawa theory)]

Indeed the case in simple supersymmetric model (Wess-Zumino model)

V. V. Lebedev and A. V. Smilga, Nucl. Phys. B 318, 669 (1989).In QED/QCD supersymmetry emerges in the absence of interaction.

Other effects come into play (chiral symmetry, charge symmetry)

JPB, Daisuke Satow, work in progress

conservation of energy-momentum

24 Chapter 2. Supersymmetry at finite temperature

where p is the pressure and ρ the energy density while

uµ = (1 − "v2)−1/2 (1,"v)

is the four-velocity of the fluid. In equilibrium, we take "v = 0, in otherwords, we go to the rest frame of the heat bath.

After a perturbation, fast relaxation processes take place and local ther-modynamic equilibrium is established. The local deviation from the globalaverage can be described by two parameters, the temperature ∆T (x, t) andvelocity "v(x, t). In a linear approximation, the components of the energy-momentum tensor are then given by

〈T00(x, t)〉 =(

1 + ∆T (x, t)∂

∂T

)ρ,

〈T0i(x, t)〉 = vi(x, t)(ρ+ p),

〈Tij(x, t)〉 = δij

(1 + ∆T (x, t)

∂

∂T

)p.

Since the energy-momentum tensor is conserved, ∂µTµν = 0, the param-eters v and ∆T are related by

∂0∆T∂ρ

∂T− ∂ivi(x, t)(ρ+ p) = 0,

∂0vi(x, t) (ρ+ p) − ∂i∆T∂p

∂T= 0.

By substituting v, one finds the wave equation(∂ρ

∂T∂2

0 − ∂p

∂T"∂2

)∆T (x, t) = 0,

describing the propagation of sound waves with the velocity

v2S =

∂p/∂T

∂ρ/∂T. (2.4)

Hence, as a consequence of the spontaneous breakdown of Lorentz symmetryin the thermal bath, perturbations in the energy-momentum tensor prop-agate as sound waves. Their quanta, the phonons, can be viewed as theGoldstone bosons associated with the spontaneous symmetry breaking.

Now, let us come to the supersymmetric case. Here, we have the con-served supercurrent Jµ. Consider a small, space and time dependent su-persymmetry variation characterized by a parameter ξ(x, t) which leads toa local deviation of the supercurrent from the global average, δξJµ(x, t).Since we know that by such a supersymmetry transformation the supercur-rent is transformed into the energy-momentum tensor, its variation can beexpressed as

δξJµ = −2iγνξ Tµν .

small deviations from local equilibrium (measured by local temperature T(x,t))



uµ = (1 − "v2)−1/2 (1,"v)



〈T00(x, t)〉 =(

1 + ∆T (x, t)∂

∂T

)ρ,

〈T0i(x, t)〉 = vi(x, t)(ρ+ p),


(1 + ∆T (x, t)

∂

∂T

)p.


∂0∆T∂ρ

∂T− ∂ivi(x, t)(ρ+ p) = 0,

∂0vi(x, t) (ρ+ p) − ∂i∆T∂p

∂T= 0.


∂T∂2

0 − ∂p

∂T"∂2

)∆T (x, t) = 0,


v2S =

∂p/∂T

∂ρ/∂T. (2.4)




dispersion relation ω=vs p



uµ = (1 − "v2)−1/2 (1,"v)



〈T00(x, t)〉 =(

1 + ∆T (x, t)∂

∂T

)ρ,

〈T0i(x, t)〉 = vi(x, t)(ρ+ p),


(1 + ∆T (x, t)

∂

∂T

)p.


∂0∆T∂ρ

∂T− ∂ivi(x, t)(ρ+ p) = 0,

∂0vi(x, t) (ρ+ p) − ∂i∆T∂p

∂T= 0.


∂T∂2

0 − ∂p

∂T"∂2

)∆T (x, t) = 0,


v2S =

∂p/∂T

∂ρ/∂T. (2.4)




conservation of supercurrent

2.3. Hydrodynamics of the supersymmetric plasma 25

The conservation equation, ∂µJµ = 0, then tells us that also the local vari-ation of the supercurrent must be conserved, giving

∂µ〈δξJµ(x, t)〉 = −2iγν∂µξ(x, t)〈Tµν(x, t)〉 = 0.

In terms of the components of the energy-momentum tensor, this can bewritten as

(ρ γ0∂0 + p γi∂i) ξ(x, t) = 0,

which is nothing but the Dirac equation for a massless fermion propagatingwith the velocity

vSS =p

ρ. (2.5)

Therefore, in a supersymmetric system whose supersymmetry is bro-ken spontaneously by the thermal bath, there should exist ‘supersymmetricsound’. Its quanta, interpreted as the Goldstone fermions associated withthe spontaneous symmetry breaking, would naturally be called phoninos.Of course there is no classical picture of these fermionic waves, but in theframework of thermal field theory it should be possible to verify the ex-istence of such a fermionic collective excitation. Just as we expect soundwaves to appear as poles in the correlator 〈Tµν(x)Tρσ(y)〉T , supersymmetricsound should lead to poles in the correlation function 〈Jµ(x)Jν(y)〉T .

Both of these types of waves have a very characteristic dispersion law. Inthe relativistic limit, when the temperature is much higher than the particlemasses, pressure and energy density are related by p = 1

3ρ, and therefore

v2S = vSS =

13, T $ m. (2.6)

In the non-relativistic limit of a massive theory, we have p = Tmρ, resulting

inv2S = vSS =

T

m, T % m. (2.7)

In the intermediate range, T & m, v2S and vSS differ by a few percent [9].

So far, the derivation is completely general and model independent. In aninteracting theory, the fluid is of course not perfect and one will in general getcorrections to the dispersion law and also damping. This naturally dependsstrongly on the model. In any case, sound and supersymmetric sound wavescan exist only for wavelengths greater than the mean free path so that thereis time to establish local thermodynamic equilibrium. At higher frequencies,these waves are strongly damped.





(ρ γ0∂0 + p γi∂i) ξ(x, t) = 0,


vSS =p

ρ. (2.5)



3ρ, and therefore

v2S = vSS =

13, T $ m. (2.6)


inv2S = vSS =

T

m, T % m. (2.7)



ω=vss p





(ρ γ0∂0 + p γi∂i) ξ(x, t) = 0,


vSS =p

ρ. (2.5)



3ρ, and therefore

v2S = vSS =

13, T $ m. (2.6)


inv2S = vSS =

T

m, T % m. (2.7)



V. V. Lebedev and A. V. Smilga, Nucl. Phys. B 318, 669 (1989).Y. Hidaka, D. S., and T. Kunihiro, Nucl. Phys. A 876, 93 (2012)D. S., PRD 87, 096011 (2013).

Imω=ζq+ζg=O(g2T)Reω＝p/3quasi-goldstino in hot QED/QCD

Phonon and phonino

mode equation

dispersion relation





(ρ γ0∂0 + p γi∂i) ξ(x, t) = 0,


vSS =p

ρ. (2.5)



3ρ, and therefore

v2S = vSS =

13, T $ m. (2.6)


inv2S = vSS =

T

m, T % m. (2.7)



One can study some of these issues in cold atom systems

Two kinds of fermion (f, F) and their bound state (b: boson) on an optical lattice.

T. Shi, Y. Yu, and C. P. Sun, PRA 81, 011604(R) (2010)

f Fb

Ubb =Ubfµf =µbWhen

(external field adjusted so that F decouples)

Suggested experimental setup

the hamitonian is supersymmetric with

2

projection operator method [14, 20, 21]. We also estab-lish the analogy between the goldstino and the magnon ina ferromagnet. Section III is devoted to derivation of thegeneralized kinetic equation corresponding to the RPA,and the calculation of the goldstino’s dispersion relation.We summarize this paper and give concluding remarksin Sec. IV. We give some useful properties of terms ap-pearing in the Langevin equation in Appendix A.

II. MODEL-INDEPENDENT ANALYSIS

In this section, we obtain the form of the dispersionrelation of the goldstino in a model independent way, byusing Mori’s projection operator method [14, 20, 21]. Westart by providing a short introduction on the supersym-metry.

A. Momentum dependence of goldstino’s spectrum

In this paper, we consider a system that consistsof single-component bosons and fermions. The anni-hilation operators for the boson (fermion) are denotedby b(t,x) (f(x)), and the creation operators by b†(x)(f†(x)). These operators satisfy the usual equal-timecommutation or anti-commutation relations, respectively[b(x), b†(y)] = �d(x� y) and {f(x), f†(y)} = �d(x� y),all other commutators (anticommutators) being zero.Here, [, ] and {, } denote the commutator and the anti-commutator, respectively. The operators that count thetotal numbers of bosons and fermions are given by

Nb =

Z

ddx b†(x)b(x), Nf =

Z

ddx f†(x)f(x). (2.1)

Central to our discussion is the operator that annihi-lates one boson and creates one fermion at point x. Thisis given by

q(x) ⌘ b(x)f†(x). (2.2)

The spatial integral of q(x), Q ⌘ R

ddx q(x) will be re-ferred to as the “supercharge”. Together with it conju-gate, Q†, and the fermion and boson number operators,they form a closed algebra,

�

Q,Q† = N, [N,Q] = [N,Q†] = 0, [�N,Q] = Q,(2.3)

where N ⌘ Nb+Nf and �N = (Nf�Nb)/2, and the var-ious number operators comment among themselves. Inthe present context, this algebra is referred to as the su-persymmetric algebra, with the corresponding operatorsbeing the generators of supersymmetric transformations.The non trivial transformations are those which involve

the supercharger Q:

�f(x) ⌘ �i[✏Q, f(x)] = �i✏{Q, f(x)} = �i✏b(x),(2.4)

�b†(t,x) ⌘ �i[✏Q, b†(t,x)]

= �i✏[Q, b†(t,x)] = �i✏f†(t,x), (2.5)

where ✏ is a Grassmann parameter (✏2 = 0). We empha-size that this supersymmetric transformation is di↵er-ent from that considered in the context of high-energyphysics, which is related to the spacetime symmetry,while the SUSY that we consider is not [4]. In fact, theanticommutator of the supercurrents can be expressed byusing energy-momentum tensor in models that have theSUSY in high-energy physics, while it is not the case inmodels we consider: {Q,Q†} = N .We assume that the hamiltonian is invariant under

these supersymmetric transformations. That is, H com-mutes with all the generators above

[Q,H] = 0 = [N,H] = [�N,H]. (2.6)

We are interested in situations where supersymmetryis broken. In fact only the vacuum state is supersymmet-ric, i.e., Q|0i = 0. All other states are degenerate. Forinstance, let | ni be an eigenstate of H with nb bosonsand no fermion. Since [Q,H] = 0, the state Q| ni, whichcontains one fermion and nb � 1 bosons, is also an eigen-state with the same energy.This may occur in two ways. First there is a possible

explicit symmetry breaking that occurs when the chemi-cal potentials for bosons and fermions di↵er. The grandcanonical Hamiltonian can be written as

HG = H � µfNf � µbNb

= H � µN ��µ�N,(2.7)

where µf (µb) is the fermion (boson) chemical potentialand we set µ ⌘ (µf +µb)/2, and �µ ⌘ µf �µb. By usingEq. (2.6) and the last of Eqs. (2.3), we obtain

[Q,HG] = �µQ. (2.8)

Therefore HG no longer commutes with the superchargewhenever �µ 6= 0. This explicit breaking lifts the degen-eracy of level that occurs when the chemical potentialsare equals. Consider for instance the case where �µ > 0(i.e., µf > µb). Let | 0

i be the ground state of H � µN .This is also an eigenstate of HG if �µ 6= 0. It is in thiscase degenerate with the state Q†|

0

i. When �µ 6= 0,we have HGQ†|

0

i = (E0

+�µ)Q†| 0

i.In addition to the explicit breaking, supersymmetry

may be broken by the presence of matter, that is when-ever the expectation value of the number density is nonvanishing. DS: Following to Jean-Paul’s comment,the following sentence needs to be improved: In-tuitively, it can be understood by remembering the factthat the form of the Bose distribution function is di↵erentfrom the Fermi one. In this case, we have

h{Q†, q(x)}ieq = ⇢0

. (2.9)

2






Nb =

Z


Z

ddx f†(x)f(x). (2.1)


q(x) ⌘ b(x)f†(x). (2.2)



�

Q,Q† = N, [N,Q] = [N,Q†] = 0, [�N,Q] = Q,(2.3)


the supercharger Q:

�f(x) ⌘ �i[✏Q, f(x)] = �i✏{Q, f(x)} = �i✏b(x),(2.4)

�b†(t,x) ⌘ �i[✏Q, b†(t,x)]

= �i✏[Q, b†(t,x)] = �i✏f†(t,x), (2.5)



[Q,H] = 0 = [N,H] = [�N,H]. (2.6)




= H � µN ��µ�N,(2.7)


[Q,HG] = �µQ. (2.8)



0


0

i = (E0

+�µ)Q†| 0



h{Q†, q(x)}ieq = ⇢0

. (2.9)

2






Nb =

Z


Z

ddx f†(x)f(x). (2.1)


q(x) ⌘ b(x)f†(x). (2.2)



�

Q,Q† = N, [N,Q] = [N,Q†] = 0, [�N,Q] = Q,(2.3)


the supercharger Q:

�f(x) ⌘ �i[✏Q, f(x)] = �i✏{Q, f(x)} = �i✏b(x),(2.4)

�b†(t,x) ⌘ �i[✏Q, b†(t,x)]

= �i✏[Q, b†(t,x)] = �i✏f†(t,x), (2.5)



[Q,H] = 0 = [N,H] = [�N,H]. (2.6)




= H � µN ��µ�N,(2.7)


[Q,HG] = �µQ. (2.8)



0


0

i = (E0

+�µ)Q†| 0



h{Q†, q(x)}ieq = ⇢0

. (2.9)

2






Nb =

Z


Z

ddx f†(x)f(x). (2.1)


q(x) ⌘ b(x)f†(x). (2.2)



�

Q,Q† = N, [N,Q] = [N,Q†] = 0, [�N,Q] = Q,(2.3)


the supercharger Q:

�f(x) ⌘ �i[✏Q, f(x)] = �i✏{Q, f(x)} = �i✏b(x),(2.4)

�b†(t,x) ⌘ �i[✏Q, b†(t,x)]

= �i✏[Q, b†(t,x)] = �i✏f†(t,x), (2.5)



[Q,H] = 0 = [N,H] = [�N,H]. (2.6)




= H � µN ��µ�N,(2.7)


[Q,HG] = �µQ. (2.8)



0


0

i = (E0

+�µ)Q†| 0



h{Q†, q(x)}ieq = ⇢0

. (2.9)

2






Nb =

Z


Z

ddx f†(x)f(x). (2.1)


q(x) ⌘ b(x)f†(x). (2.2)



�

Q,Q† = N, [N,Q] = [N,Q†] = 0, [�N,Q] = Q,(2.3)


the supercharger Q:

�f(x) ⌘ �i[✏Q, f(x)] = �i✏{Q, f(x)} = �i✏b(x),(2.4)

�b†(t,x) ⌘ �i[✏Q, b†(t,x)]

= �i✏[Q, b†(t,x)] = �i✏f†(t,x), (2.5)



[Q,H] = 0 = [N,H] = [�N,H]. (2.6)




= H � µN ��µ�N,(2.7)


[Q,HG] = �µQ. (2.8)



0


0

i = (E0

+�µ)Q†| 0



h{Q†, q(x)}ieq = ⇢0

. (2.9)

2






Nb =

Z


Z

ddx f†(x)f(x). (2.1)


q(x) ⌘ b(x)f†(x). (2.2)



�

Q,Q† = N, [N,Q] = [N,Q†] = 0, [�N,Q] = Q,(2.3)


the supercharger Q:

�f(x) ⌘ �i[✏Q, f(x)] = �i✏{Q, f(x)} = �i✏b(x),(2.4)

�b†(t,x) ⌘ �i[✏Q, b†(t,x)]

= �i✏[Q, b†(t,x)] = �i✏f†(t,x), (2.5)



[Q,H] = 0 = [N,H] = [�N,H]. (2.6)




= H � µN ��µ�N,(2.7)


[Q,HG] = �µQ. (2.8)



0


0

i = (E0

+�µ)Q†| 0



h{Q†, q(x)}ieq = ⇢0

. (2.9)

2






Nb =

Z


Z

ddx f†(x)f(x). (2.1)


q(x) ⌘ b(x)f†(x). (2.2)



�

Q,Q† = N, [N,Q] = [N,Q†] = 0, [�N,Q] = Q,(2.3)


the supercharger Q:

�f(x) ⌘ �i[✏Q, f(x)] = �i✏{Q, f(x)} = �i✏b(x),(2.4)

�b†(t,x) ⌘ �i[✏Q, b†(t,x)]

= �i✏[Q, b†(t,x)] = �i✏f†(t,x), (2.5)



[Q,H] = 0 = [N,H] = [�N,H]. (2.6)




= H � µN ��µ�N,(2.7)


[Q,HG] = �µQ. (2.8)



0


0

i = (E0

+�µ)Q†| 0



h{Q†, q(x)}ieq = ⇢0

. (2.9)

2






Nb =

Z


Z

ddx f†(x)f(x). (2.1)


q(x) ⌘ b(x)f†(x). (2.2)



�

Q,Q† = N, [N,Q] = [N,Q†] = 0, [�N,Q] = Q,(2.3)


the supercharger Q:

�f(x) ⌘ �i[✏Q, f(x)] = �i✏{Q, f(x)} = �i✏b(x),(2.4)

�b†(t,x) ⌘ �i[✏Q, b†(t,x)]

= �i✏[Q, b†(t,x)] = �i✏f†(t,x), (2.5)



[Q,H] = 0 = [N,H] = [�N,H]. (2.6)




= H � µN ��µ�N,(2.7)


[Q,HG] = �µQ. (2.8)



0


0

i = (E0

+�µ)Q†| 0



h{Q†, q(x)}ieq = ⇢0

. (2.9)

2






Nb =

Z


Z

ddx f†(x)f(x). (2.1)


q(x) ⌘ b(x)f†(x). (2.2)



�

Q,Q† = N, [N,Q] = [N,Q†] = 0, [�N,Q] = Q,(2.3)


the supercharger Q:

�f(x) ⌘ �i[✏Q, f(x)] = �i✏{Q, f(x)} = �i✏b(x),(2.4)

�b†(t,x) ⌘ �i[✏Q, b†(t,x)]

= �i✏[Q, b†(t,x)] = �i✏f†(t,x), (2.5)



[Q,H] = 0 = [N,H] = [�N,H]. (2.6)




= H � µN ��µ�N,(2.7)


[Q,HG] = �µQ. (2.8)



0


0

i = (E0

+�µ)Q†| 0



h{Q†, q(x)}ieq = ⇢0

. (2.9)

NOTE: supersymmetry is here only an internal symmetry

Supersymmetric algebra (Continuum limit, d=2, no BEC)

Supercharge

Algebra

Boson and fermion field operators

Supersymmetric hamiltonian

Supersymmetry and its breaking

2






Nb =

Z


Z

ddx f†(x)f(x). (2.1)


q(x) ⌘ b(x)f†(x). (2.2)



�

Q,Q† = N, [N,Q] = [N,Q†] = 0, [�N,Q] = Q,(2.3)


the supercharger Q:

�f(x) ⌘ �i[✏Q, f(x)] = �i✏{Q, f(x)} = �i✏b(x),(2.4)

�b†(t,x) ⌘ �i[✏Q, b†(t,x)]

= �i✏[Q, b†(t,x)] = �i✏f†(t,x), (2.5)



[Q,H] = 0 = [N,H] = [�N,H]. (2.6)




= H � µN ��µ�N,(2.7)


[Q,HG] = �µQ. (2.8)



0


0

i = (E0

+�µ)Q†| 0



h{Q†, q(x)}ieq = ⇢0

. (2.9)

2






Nb =

Z


Z

ddx f†(x)f(x). (2.1)


q(x) ⌘ b(x)f†(x). (2.2)



�

Q,Q† = N, [N,Q] = [N,Q†] = 0, [�N,Q] = Q,(2.3)


the supercharger Q:

�f(x) ⌘ �i[✏Q, f(x)] = �i✏{Q, f(x)} = �i✏b(x),(2.4)

�b†(t,x) ⌘ �i[✏Q, b†(t,x)]

= �i✏[Q, b†(t,x)] = �i✏f†(t,x), (2.5)



[Q,H] = 0 = [N,H] = [�N,H]. (2.6)




= H � µN ��µ�N,(2.7)


[Q,HG] = �µQ. (2.8)



0


0

i = (E0

+�µ)Q†| 0



h{Q†, q(x)}ieq = ⇢0

. (2.9)

2






Nb =

Z


Z

ddx f†(x)f(x). (2.1)


q(x) ⌘ b(x)f†(x). (2.2)



�

Q,Q† = N, [N,Q] = [N,Q†] = 0, [�N,Q] = Q,(2.3)


the supercharger Q:

�f(x) ⌘ �i[✏Q, f(x)] = �i✏{Q, f(x)} = �i✏b(x),(2.4)

�b†(t,x) ⌘ �i[✏Q, b†(t,x)]

= �i✏[Q, b†(t,x)] = �i✏f†(t,x), (2.5)



[Q,H] = 0 = [N,H] = [�N,H]. (2.6)




= H � µN ��µ�N,(2.7)


[Q,HG] = �µQ. (2.8)



0


0

i = (E0

+�µ)Q†| 0



h{Q†, q(x)}ieq = ⇢0

. (2.9)

2






Nb =

Z


Z

ddx f†(x)f(x). (2.1)


q(x) ⌘ b(x)f†(x). (2.2)



�

Q,Q† = N, [N,Q] = [N,Q†] = 0, [�N,Q] = Q,(2.3)


the supercharger Q:

�f(x) ⌘ �i[✏Q, f(x)] = �i✏{Q, f(x)} = �i✏b(x),(2.4)

�b†(t,x) ⌘ �i[✏Q, b†(t,x)]

= �i✏[Q, b†(t,x)] = �i✏f†(t,x), (2.5)



[Q,H] = 0 = [N,H] = [�N,H]. (2.6)




= H � µN ��µ�N,(2.7)


[Q,HG] = �µQ. (2.8)



0


0

i = (E0

+�µ)Q†| 0



h{Q†, q(x)}ieq = ⇢0

. (2.9)

2






Nb =

Z


Z

ddx f†(x)f(x). (2.1)


q(x) ⌘ b(x)f†(x). (2.2)



�

Q,Q† = N, [N,Q] = [N,Q†] = 0, [�N,Q] = Q,(2.3)


the supercharger Q:

�f(x) ⌘ �i[✏Q, f(x)] = �i✏{Q, f(x)} = �i✏b(x),(2.4)

�b†(t,x) ⌘ �i[✏Q, b†(t,x)]

= �i✏[Q, b†(t,x)] = �i✏f†(t,x), (2.5)



[Q,H] = 0 = [N,H] = [�N,H]. (2.6)




= H � µN ��µ�N,(2.7)


[Q,HG] = �µQ. (2.8)



0


0

i = (E0

+�µ)Q†| 0



h{Q†, q(x)}ieq = ⇢0

. (2.9)

2






Nb =

Z


Z

ddx f†(x)f(x). (2.1)


q(x) ⌘ b(x)f†(x). (2.2)



�

Q,Q† = N, [N,Q] = [N,Q†] = 0, [�N,Q] = Q,(2.3)


the supercharger Q:

�f(x) ⌘ �i[✏Q, f(x)] = �i✏{Q, f(x)} = �i✏b(x),(2.4)

�b†(t,x) ⌘ �i[✏Q, b†(t,x)]

= �i✏[Q, b†(t,x)] = �i✏f†(t,x), (2.5)



[Q,H] = 0 = [N,H] = [�N,H]. (2.6)




= H � µN ��µ�N,(2.7)


[Q,HG] = �µQ. (2.8)



0


0

i = (E0

+�µ)Q†| 0



h{Q†, q(x)}ieq = ⇢0

. (2.9)

Associated Goldstone mode «Goldstino»

Explicit breaking

Explicit symmetry breaking when

Only supersymmetric state is the vacuum

Spontaneous symmetry breaking by matter

Random Phase Approximation

f

b<Q†(x)Q(0)>=

This spectrum can be understood in terms of symmetry breaking in a model independent way

ω＝Δµ -αp2

1

I.

−ikµ

∫

d4xeik·(x−y)〈TJµ(x)O(y)〉 = 〈{Q,O(y)}〉, (1.1)

−ikµ

∫

d4xeik·(x−y)〈TJµ(x)O(y)〉 = 〈{Q,O}〉 (1.2)

ρs2M2

0

((∇mx)2 + (∇my)

2)

=ρs

2M20

∇m+∇m−,

(1.3)

In a = 1 unit,

Πf0 (ω,p) %

∫

d2k

(2π)2nF (ε

fk)

ω −∆µ− t(2k · p+ p2), (1.4)

Πb0(ω,p) %

∫

d2k

(2π)2nB(εbk)

ω −∆µ− t(2k · p− p2). (1.5)

Πf0 (ω,p) %

∫

d2k

(2π)2nF (ε

fk)

ω − (∆µ+ t(2k · p+ p2) + Uρ), (1.6)

Πb0(ω,p) %

∫

d2k

(2π)2nB(εbk)

ω − (∆µ+ t(2k · p− p2) + Uρ). (1.7)

α ≡t24πρ2fUρ2

−t(ρf − ρb)

ρ

α ≡1

ρ

(

4πt2ρ2fUρ

− t(ρf − ρb)

) (1.8)

(sp strength is unity)

[JPB, Y. Hidaka and D. Satow, wotk in progress]

Analogy with Magnon in Ferromagnet

fb

Q†

Q

Q2=Q†2=0

GoldstinoConserved Charge: Q, Q†, ρ

7

fb

Q

Q†

up down

S-

S+

Q2=Q†2=0S+2=S-2=0

<{Q, Q†}>=ρ

<[Q, ρ]>=0

<[S+, S-]>=2M0

<[S±, Sz]>=0

(∆µ=0) Charge: S+, S-, SzCharge: Q, Q†, ρ, ∆ρ

FIG. 2: Similarity between the magnon in the ferromagnet and the goldstino.

or, equivalently,

∂tqx =β

ρ∇2qy (3.36)

∂tqy = −β

ρ∇2qx, (3.37)

where we have introduced qx ≡ (q + q†)/2 and qy ≡ (q − q†)/(2i). We note that these quantities are hermite. Thedispersion relation is

ω = ±(

β

ρp2 +∆µ

)

. (3.38)

This similarity between the magnon in the ferromagnet and the goldstino results from the fact that only thecommutation relationship among the conserved charge is important in determining the spectrum of NG mode (Fig. 2).

IV. KINETIC EQUATION WHICH IS EQUIVALENT TO THE EQUATION IN RPA

We derive the generalized kinetic equation which is equivalent to the equation for the goldstino propagator Eq (6.1)in the RPA. We use the method developed in Ref. [3, 4]. The equations of motion for b and f in the continuum limitis

i∂t〈b(x)〉 = −ta2∂2〈b(x)〉 − µ′b〈b(x)〉+ jbind(x) + g0〈f(x)η(x)〉, (4.1)

i∂t〈f †(x)〉 = ta2∂2〈f †(x)〉+ µ′f 〈f †(x)〉 + jf†ind(x) + g0〈b†(x)η(x)〉, (4.2)

where jbind ≡ U〈b†bb+ f †fb〉, jf†ind ≡ −U〈b†bf †〉 , and η(x) ≡ F (x) exp(−i(k0 · x − δ0t)). By differentiating Eq. (4.1)with respect to jf†(y) and Eq. (4.2) with jb(y), we get

(i∂t + ta2∂2x + µ′

b)K(y, x) = ijbind(x)

δjf†(y)+ g0S(x, y)η(x), (4.3)

(i∂t − ta2∂2x − µ′

f )K(x, y) = iδjf†ind(x)

δjb(y)+ g0D(y, x)η(x), (4.4)

Here we have introduced the fermion-boson propagator K(x, y) ≡ 〈TCf †(x)b(y)〉, and S(x, y) ≡ 〈TCf(x)f †(y)〉 andD(x, y) ≡ 〈TCb(x)b†(y)〉. By interchanging x with y in Eq. (4.3), we get

(i∂′t + ta2∂2

y + µ′b)K(x, y) = i

jbind(y)

δjf†(x)+ g0S(y, x)η(y). (4.5)

up down

m-

m+

m+2=m-2=0

Magnon in ferromagnetConserved Charge: m+, m-, mz

1

I.

−ikµ

∫

d4xeik·(x−y)〈TJµ(x)O(y)〉 = 〈{Q,O(y)}〉, (1.1)

−ikµ

∫

d4xeik·(x−y)〈TJµ(x)O(y)〉 = 〈{Q,O}〉 (1.2)

f =ρs

2M20

((∇mx)2 + (∇my)

2)

=ρs

2M20

∇m+∇m−,(1.3)

∂tmx =ρsM0

∇2my + hmy, (1.4)

∂tmy = −ρsM0

∇2mx − hmx, (1.5)

∂tm± = ∓i

(

ρsM0

∇2 + h

)

m±. (1.6)

ω = ±

(

ρsM0

p2 + h

)

. (1.7)

f =β

ρ2∇q†∇q, (1.8)

In a = 1 unit,

Πf0 (ω,p) &

∫

d2k

(2π)2nF (ε

fk)

ω −∆µ− t(2k · p+ p2), (1.9)

Πb0(ω,p) &

∫

d2k

(2π)2nB(εbk)

ω −∆µ− t(2k · p− p2). (1.10)

Πf0 (ω,p) &

∫

d2k

(2π)2nF (ε

fk)

ω − (∆µ+ t(2k · p+ p2) + Uρ), (1.11)

Πb0(ω,p) &

∫

d2k

(2π)2nB(εbk)

ω − (∆µ+ t(2k · p− p2) + Uρ). (1.12)

α ≡t24πρ2fUρ2

−t(ρf − ρb)

ρ

α ≡1

ρ

(

4πt2ρ2fUρ

− t(ρf − ρb)

)

β = αρ =4πt2ρ2fUρ

− t(ρf − ρb)

(1.13)

Type-II NG mode (while it is Type-I in relativistic model)

H. Watanabe and H. Murayama, PRL 108, 251602 (2012); Y. Hidaka, PRL 110, 091601 (2013)JPB, Y. Hidaka, D. Satow, work in progress

SOME LEGACIES FROM GERRY


Many-body physics is fun, and beautiful

One can learn a lot by looking at, and comparing, different physical systems


Many-body physics is fun, and beautiful

many body physics - stony brook universitytonic.physics.sunysb.edu/gerrybrown/talks/blaizot.pdf ·...

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