YKIS2016@YITP (2016/6/15)

Supersymmetry breaking andNambu-Goldstone fermionsin lattice models
Hosho Katsura(Department of Physics, UTokyo)
Collaborators:
Yu Nakayama (IPMU  Rikkyo)
Noriaki Sannomiya (UTokyo)
 arXiv:1606.03947

Susy and me

1/23

 What I’ve been working on…
Condensed Matter & Statistical Physics
・ Strongly correlated systems,
・ Topological phases of matter,・ Quantum entanglement, …
VBS/CFT correspondence (2d AKLT ⇔ 1d Heisenberg)
J.Lou, S.Tanaka, H.K., N.Kawashima, PRB 84 (2011).

 I’m not a high-energy physicist, or a string theorist, …
(at least for the moment). But…

 My first paper (undergrad)“Exact supersymmetry in the relativistic hydrogen
atom in general dimensions” (arXiv:quant-ph/0410174),
H. Katsura & H. Aoki, J. Math. Phys. 47, 032301 (2006).

Today’s talk2/23

 Many-body systems with built-in supersymmetry!
• Lattice-fermion model in 1 (spatial) dimension• Spontaneous supersymmetry breaking• Gapless excitation with linear dispersion

 Edge of the workshop• My talk contains some Information (S≠0)• The model lives in (1+1)-dim. flat Spacetime• Super-weird Quantum Matter, never synthesized …, cold atoms?

Outline
1. Introduction & Motivation• Supersymmetry and lattice models• Extended Nicolai model

2. SUSY breaking in extended Nocolai model• Definition of SUSY breaking• 1) Finite chain, 2) infinite chain

3. Nambu-Goldstone fermions• 1. Variational result, 2. Numerical result• Bosonization & RG analysis
4. Summary

3/23

Supersymmetry (SUSY QM) demystified
 Algebraic structure• Supercharges (Q & Q† ) and fermion number (F)• Hamiltonian• Conserved charges
Energy Spectrum of H
• E ≧ 0 for all states• E > 0 states come in pairs• E = 0 iff a state is a singlet (cohomology)
SUSY breaking ⇔ No E=0 state

04/23

5/23

Elementary examples
 Boson-fermion system• Creation & annihilation operators (b: boson, c: fermion)• Vacuum state

Total number of B and F!

• Supercharges|vac> is a SUSY singlet.

 Lattice bosons and fermions• Lattice sites: i, j = 1,2, …, N
• Creations & annihilations(b and f are mutually commuting.)• Vacuum stateElementary examples (contd.)

6/23

 Generalization

Total number of B and F! |vac> is a SUSY singlet.

 SUSY in Bose-Fermi mixtures
Realization in cold-atom systems?
M. Snoek et al., PRL 95 (‘05); PRA 74 (‘06); G.S.Lozano et al., PRA 75 (‘07).

• Yu-Yang model (PRL 100, (‘08))
Hubbard-type model with equal hopping & equal-int. for any pair of sites.

HYY commutes with Q & Q†! ( HYY is not {Q, Q†} )Lattice models with built-in SUSY

7/23

 Fendley-Schoutens-de Boer model
PRL 90, 120402 (‘03); PRL 95, 046403 (’05).

• Supercharge

Hard-core constraint• Hamiltonian

 Nicolai model“Supersymmetry and spin systems”,
H. Nicolai, JPA 9, 1497 (‘76).

2• Supercharge
12k

32k-1 2k+1

# of E=0 states ~ exp (# of sites)
Both models tend to have massively degenerate E=0 states...

Extended Nicolai model

8/232k2N

 Definition• Setting

12k-1 2k+1
31d lattice of length N (even). PBCs are imposed (cN+1 = c1).

• New supercharge (g > 0)

Nilpotent. Comprised solely of fermions.

• Hamiltonian Symmetries• SUSY• U(1)• Translation• Reflection

Linear term in c!
NN-1
N-2

12
3

9/23

Hamiltonian (explicit expression)
42

1. Hopping term
13

2. Charge-charge int.

3. Pair hopping

Very complicated and seems intractable…
cf) Original Nicolai model (g = 0):

8

6

5

7

9

10/23Large-g limit
 Free fermions
In the large-g limit,(The (many-body) g.s. energy of Hhop) ∝ gN
• SUSY is broken (No E=0 states)• Gapless excitations
Dirac fermions in continuum limit

Nambu-Goldstone theorem?

Also the case for finite g?

Results
 SUSY breaking
1) Finite chain: SUSY is broken for any g > 0.
2) Infinite chain: SUSY is broken when g > NG fermions
Rigorous result
Existence of low-lying states with E(p) ≦ (const.) |p|.
Analytical & numerical result
Effective field theory ~ c=1 CFT with TL parameter close to 1.

Outline
1. Introduction & Motivation• Supersymmetry and lattice models• Extended Nicolai model

2. SUSY breaking in extended Nocolai model• Definition of SUSY breaking• 1) Finite chain, 2) infinite chain

3. Nambu-Goldstone fermions• 1. Variational result, 2. Numerical result• Bosonization & RG analysis
4. Summary

11/23

SUSY breakingEnergy Naïve definition
SUSY is unbroken ⇔ E=0 state exists
SUSY is broken ⇔ No E= 0 state
Subtle issue... (Witten, NPB 202 (’82))“SUSY may be broken in any finite volume yet
restored in the infinite-volume limit.”0 More precise definition• (Normalized ) ground state:• Ground-state energy density:
V= (# of sites) for lattice systems

SUSY is said to be spontaneously broken if the ground-stateenergy density (energy per site) is strictly positive.
Applies to both finite and infinite-volume systems!

12/23

SUSY breaking in finite Nicolai chains

13/23

 Theorem 1
Consider the extended Nicolai model on a finite chain oflength N. If g > 0, then SUSY is spontaneously broken.

 Proof• Local operator s.t.

well-defined when g>0• Proof by contradiction
Suppose ψ0 is an E=0 ground state.
Then we have

But this leads to ψ0 =0. Contradiction. No E=0 state!
(g.s. energy/N) > 0 for any finite N.

SUSY breaking in the infinite Nicolai chain Theorem 2Consider the extended Nicolai model on the infinite chain.
If, then SUSY is spontaneously broken.

 Proof• Lower bound for g.s. energy
Original Nicolai (g=0)

Since HNic is positive semi-definite, the g.s. energy of H
is bounded from below by• G.s. energy of Hhop (Free-fermion chain)

NOTE) The condition

may not be optimal…

14/23

Outline
1. Introduction & Motivation• Supersymmetry and lattice models• Extended Nicolai model

2. SUSY breaking in extended Nocolai model• Definition of SUSY breaking• 1) Finite chain, 2) infinite chain

3. Nambu-Goldstone fermions• 1. Variational result, 2. Numerical result• Bosonization & RG analysis
4. Summary

15/23

16/23

Nambu-Goldstone (NG) fermions?
 NG bosons in non-relativistic systems
Effective Lagrangian approach, counting rules, …
Watanabe-Murayama, PRL 108 (‘12); Hidaka, PRL 110, (’13).

 Analogy (Blaizot-Hidaka-Satow, PRA 92 (‘15))• Ferromagnet (Type B)• SUSY system

Quadratic dispersion

 Examples• Yu-Yang model (Bose-Fermi mixture)YES
SUSY spin-wave states

Quadratic dispersion?
Fermionic excitation?
Fermionic!

• Extended Nicolai modelNO!
Low-lying fermionic states with ω ≦ (const.)|p| exist.

Warm-up: Heisenberg model

17/23
Ferro (J < 0)Antiferro (J > 0)

Hamiltonian

 Ferromagnetic case
• Fully polarized ground state:• Spin wave

, Exact eigenstate!

 Antiferromagnetic case (Horsch-von der Linden, ZPB, 72 (‘88))• Neel state: not even an eigenstate!
• Bijl-Feynman ansatz
Exact ground state: ψ0Fourier component of spins:
Linear around18/23

Variational argument
 SUSY “spin waves” in extended Nicolai model• Local supercharge

22k• Fourier components
132k-1 2k+1

• Ansatz (ψ0: SUSY broken g.s.)

But,is a sum of local operators. (may not be so because)
for all k, l.

19/23

Variational argument (contd.) Useful inequality (Pitaevskii-Stringari, JLTP 85 (‘91))
Holds for any state ψ and any operators A, B.

 Upper bound for the lowest dispersionLocal!

For |p|<<1,fn(p): 1. Local, 2. fn(-p) = fn(p), 3. fn(0) = 0fd(p): 1. Local, 2. fd(-p) = fd(p), 3. fd(0) = E0
E0 > 0 if SUSY is broken!
NOTE) U(1), translation, reflection symmetries have been used.
Implicit assumption: The g.s. multiplicity is finite in the infinite-N limit.

20/23

Numerical results
 Exact diagonalization
N= 12, 14, …, 22
4 ground states
8 first excited states(Independent of g & N)

• 1st excitation energy

ΔE
F=N/2-2

N/2-1

N/2

N/2+1

N/2+2• Central charge
Finite-size scaling
Blote-Cardy-Nightingale, PRL 56 (‘86)

g2.0

4.0

6.0

8.0

c0.9705

1.008

1.020

1.025

Gapless linear dispersion!
Described by c=1 CFT!

21/23

Tomonaga-Luttinger liquid parameter
c=1 CFTs are further specified by Tomonaga-Luttinger (TL)parameters K. (Or equivalently, boson compactification radius.)
 Number fluctuation
Song, Rachel et al., PRB 59 (‘12)

A
L
Similar toCalabrese-Cardy!

K is almost independent of g and is close to 1 (free-Dirac).

22/23

Bosonization and RG• Lattice fermion  Dirac(a: lattice spacing)ψLψR• Dirac fermion  boson• Sin-Gordon Hamiltonian Free boson!
 Cos termVelocity• Scaling dim. of cos

TL parameterCos term is irrelevant. Gapless!
~ Massless Thirring modelSummary

23/23• Introduced one-parameter extension of Nicolai’s model

Lattice model with exact supersymmetry
1. Original Nicolai (g=0), 2. Free fermions (g=∞)• Spontaneous SUSY breaking
1. Finite chain: broken for any g >0.
2. Infinite chain: broken when g >• Nambu-Goldstone fermions
1. Rigorous result: E(p) ≦ (const.) |p|
2. Numerical result: E(p) = v |p|, gapless, linear
3. Field theory: gapless c=1 CFT with K close to 1