(Quasi-) Nambu-Goldstone Fermion in Hot QCD Plasma and Bose-Fermi Cold Atom System

Transcription

1 (Quasi-) Nambu-Goldstone Fermion in Hot QCD Plasma and Bose-Fermi Cold Atom System Daisuke Satow (RIKEN/BNL) Collaborators: Jean-Paul Blaizot (Saclay CEA, France) Yoshimasa Hidaka (RIKEN, Japan)

2 Supersymmetry (SUSY) Symmetry related to interchanging of a boson and a fermion = b f b f 2

3 Supersymmetry (SUSY) Q: supercharge b f b f SUSY: [Q, H]=0 Supercharge operator: annihilate one fermion and create one boson (and its inverse process) 3

4 Spontaneous SUSY breaking V. V. Lebedev and A. V. Smilga, Nucl. Phys. B 318, 669 (1989) (medium effect) SUSY breaking nf nb E E 4

5 NG fermion which is related to SUSY breaking Generally, spontaneous symmetry breaking generates zero energy excitation (Nambu-Goldstone (NG) mode). symmetry breaking NG mode We expect that SUSY breaking order parameter also generates NG mode. 5

6 NG fermion which is related to SUSY breaking Nambu-Goldstone theorem: ik µ (fermion ver.) Broken symmetry d 4 xe ik (x y) TJ µ (x)o(y) = {Q, O} J µ : supercurrent Q=J 0 : supercharge NG mode Order parameter When the order parameter is finite, the propagator in the left-hand side has a pole at k 0. 6

7 NG fermion which is related to SUSY breaking ik µ Broken symmetry d 4 xe ik (x y) TJ µ (x)o(y) = {Q, O} NG mode If we set O=Q, NG mode appears in <QQ>. Order parameter is energy-momentum tensor (T µν γν) in the present case. SUSY is always broken when ρ is finite. { } Order parameter (µ=0) T µν =diag(ρ,p,p,p), ε, SUSY is fermionic symmetry, so the NG mode is fermion (Goldstino). Rare fermionic zero mode 7

8 Quasi-goldstino in hot QED/QCD V. V. Lebedev and A. V. Smilga, Annals Phys. 202, 229 (1990) = q g Both of the quark and the gluon are regarded as massless at high T, so there is SUSY approximately if we neglect the interaction. 8

9 Quasi-goldstino in hot QED/QCD Actually, in weak coupling regime, we established the existence of the (quasi) goldstino in QED/QCD. Y. Hidaka, D. S., and T. Kunihiro, Nucl. Phys. A 876, 93 (2012) D. S., PRD 87, (2013). dispersion relation Reω p/3 Damping rate Imω=ζq+ζg=O(g 2 T) Residue g 2 144π 2 QED g 2 (4 + N f ) 2 48π 2 QCD The dispersion relation is linear (Type-I NG mode). 9

10 Why Cold Atom System? Lattice Structure (optical lattice) Hubbard model Tunableness of interaction strength (laser intensity, magnetic field strength: Feshbach resonance) Cold atom system can be used as experiment station of many-body system whose experiment is difficult. Wess-Zumino model: Y. Yu, and K. Yang, PRL 105, (2010) Dense QCD: K. Maeda, G. Baym and T. Hatsuda, PRL 103, (2009) Relativistic QED: Kapit and Mueller, PRA 83, (2011) 10

11 Motivation of this research If we simulate the SUSY with the cold atom system, we can observe the goldstino experimentally!! 11

12 T. Shi, Y. Yu, and C. P. Sun, PRA 81, (R) (2010) SUSY in Cold Atom System Trap two kinds of fermion (f, F) and their bound state (boson: b) on optical lattice. b f F 12

13 T. Shi, Y. Yu, and C. P. Sun, PRA 81, (R) (2010) SUSY in Cold Atom System H α = t α a α i aj α µ α a α i ai α ij i ( ) f b = (tf =tb) (µf =µb) How to tune t: M. Snoek, S. Vandoren, and H. T. C. Stoof, PRA 74, (2006) 13

14 SUSY in Cold Atom System T. Shi, Y. Yu, and C. P. Sun, PRA 81, (R) (2010) U bb 2 i n b i ( n b i 1 ) + U bf i n b i nf i, = (Ubb =Ubf) When tf =tb, Ubb =Ubf, µf =µb, Q =bf commutes with the Hamiltonian. F decouples. 14

15 NG fermion which is related to SUSY breaking ik µ Broken symmetry d 4 xe ik (x y) TJ µ (x)o(y) = {Q, O} Q =bf Q =b f NG mode Order parameter If we set O=Q, NG mode appears in <QQ >. Order parameter is density (<{Q, Q }>=ρ) in this case. SUSY is always broken when ρ is finite. 15

16 Setup d=2 No BEC. Continuum limit a << (kf) -1, (T) -1 (a=1 unit, for simplicity.) Δµ=µf -µb 0 Explicit SUSY breaking 16

17 Analysis with Resummed Perturbation Calculate the propagator (spectrum) of the Goldstino. Since Q =bf, the propagator at the one-loop order is <Q (x)q(0)>= b f The Hamiltonian has SUSY, so the bare dispersion relations of the fermion and the boson are the same, and thus the loop integral diverges at ω-δµ, p 0. (pinch singularity) (Δµ=µf -µb) (fermion term)= d 2 k n F (ɛ f k ) (2π) 2 ω µ t(2k p + p 2 ), 2 17

18 Analysis with Resummed Perturbation T. Shi, Y. Yu, and C. P. Sun, PRA 81, (R) (2010) (1) the density correction to the dispersion relations. fermion: Uρb Uρf + 2Uρb boson: the pinch singularity is regularized. d 2 k n F (ɛ f k ) (2π) 2 ω ( µ + t(2k p + p 2 )+Uρ), (fermion term)= U -1 18

19 Analysis with Resummed Perturbation T. Shi, Y. Yu, and C. P. Sun, PRA 81, (R) (2010) All ring diagrams contributes at the same order. U -1 U -1 U U -1 =U -1 We need to sum up infinite ring diagrams. 19

20 Analysis with Resummed Perturbation T. Shi, Y. Yu, and C. P. Sun, PRA 81, (R) (2010) (2) Random Phase Approximation U The self-consistent equation can be rewritten in explicit form. 20

21 Expansion in terms of energy, momentum Calculate the propagator of the Goldstino by using (1), (2). We are interested in low energy-momentum region. Expand in terms of energy, momentum p <<kf, Uρ/(kf t) 21

22 Spectral properties of Goldstino Dispersion Relation ω Δµ -αp 2 Strength 1 (maximum value allowed by sum rule) Type-II NG mode (while it is Type-I in relativistic system). α 1 ρ ( 4πt 2 ρ 2 f Uρ (T=0 case) ) t(ρ f ρ b ) The damping rate is expected to be zero at p=0 because of NG theorem. Damping rate when p is finite can not be evaluated since we neglected collision effect. 22

23 Results at T 0 17 U/t=0.1, f =0.5, b = /t T/t The coefficient increases when T increases. 23

24 Analogy with Magnon in Ferromagnet Goldstino Conserved Charge: Q, Q, ρ Magnon in ferromagnet Conserved Charge: m +, m -, m z Broken order parameter Broken order parameter Q m + Q m - b Q 2 =Q 2 =0 f up m +2 =m -2 =0 down m ± =m x ± im y 24

25 Analogy with Magnon in Ferromagnet Derivation of spectrum of magnon P. M. Chaikin, T. C. Lubensky, Principles of condensed matter physics. Expand the free energy density in terms of spatial inhomogeneity. f = ρ s 2M 2 0 ρ (( m x ) 2 +( m y ) 2 ) ρs: spin wave stiffness constant M0:spontaneous magnetization 25

26 Analogy with Magnon in Ferromagnet df =-hidmi h effective magnetic field external magnetic field (x, y) h a = ρ s (z) hz =h M m a, Equation of Motion of m i t m i = ih j [m i,m j ] = (h m) i, (H-himi) [mi, mj]=iεijkmk Linearization (mi: small) t m x = ρ s M 0 2 m y + hm y, t m y = ρ s M 0 2 m x hm x, 26

27 Analogy with Magnon in Ferromagnet rewrite using m ± instead of m x, m y. t m ± = i ( ) ρs 2 + h M 0 m ±. ( ) Dispersion relation: ω = ± ( ) ρs p 2 + h M 0 27

28 Analogy with Magnon in Ferromagnet Generally, expectation values of commutators among conserved charges are essentially important to predict whether the NG mode is type-i or II. NII=rank<[Qa, Qb]>/2 NI=NBS -2NII H. Watanabe and H. Murayama, PRL 108, (2012) Y. Hidaka, PRL 110, (2013) ( <[m ±, m z ]>=0 ) <[m +, m - ]>=2m0 rank<[qa, Qb]>/2=1 NII =1 NI =0 NBS =2 28

29 Analogy with Magnon in Ferromagnet The Case of Goldstino Expectation values of (anti-) commutators have the same structure as those in ferromagnet. <[Q, ρ]>=0 <{Q, Q }>=ρ Q, Q m+, m- ρ mz Therefore we understand the reason why the both spectrums have the same form. 29

30 Analogy with Magnon in Ferromagnet ( β ) ( ) Δµ h ω = ± ρ p2 + µ β = αρ = 4πt2 ρ 2 f Uρ t(ρ f ρ b ) It corresponds to the spin wave stiffness constant in ferromagnet. As that of magnon, the damping rate has the form of Dp 4. (D can be calculated by using Kubo formula) The momentum dependence of the energy and the damping rate is model-independent! 30

31 Summary We obtained the expression of dispersion relation and the strength of the goldstino (at weak coupling, continuum limit). We understand the similarity between the goldstino and the magnon in ferromagnet, by using the fact that the (anti-)commutation relations have the same structure. We obtained the momentum dependence of the excitation energy and the damping rate in modelindependent way. 31