Spontaneous supersymmetry breaking on the latticegies/Conf/SIFT2012/jena12_wenger.pdfOverview SUSY breaking and sign problem N = 2 supersymmetric QM N = 1 Wess-Zumino model Spontaneous

Overview SUSY breaking and sign problem N = 2 supersymmetric QM N = 1 Wess-Zumino model

Spontaneous supersymmetry breakingon the lattice

David Baumgartner, Kyle Steinhauer and Urs Wenger

Albert Einstein Center for Fundamental PhysicsUniversity of Bern

Workshop on Strongly-Interacting Field Theories29 November – 1 December 2012

Jena (Germany)


Outline

Spontaneous supersymmetry breaking:Witten index and sign problem

New approach for simulating fermions on the lattice:Loop formulation for Majorana Wilson fermionsSolution of the sign problem

Two examples:N = 2 supersymmetric QMN = 1 Wess-Zumino model in d = 2


Supersymmetry and its breaking

Unbroken supersymmetry:Vanishing ground state energyDegenerate mass spectrum

Broken supersymmetry:No supersymmetric ground stateParticle masses not degenerateEmergence of Goldstino mode


Spontaneous SUSY breaking (SSB) and the Witten index

Witten index provides a necessary but not sufficientcondition for SSB:

W ≡ limβ→∞

Tr(−1)F exp(−βH) ⇒{

= 0 SSB may occur$= 0 no SSB

Index counts the difference between the number ofbosonic and fermionic zero energy states:

W ≡ limβ→∞

[TrB exp(−βH) − TrF exp(−βH)

]= nB − nF

Index is equivalent to partition function with periodic b.c.:

W =

∫ ∞

−∞Dφ det [/D(φ)] e−SB[φ] = Zp

⇒ Determinant (or Pfaffian) must be indefinite for SSB.


Example: N = 2 SUSY QM

Consider the Lagrangian for N = 2 supersymmetricquantum mechanics

L =12

(dφ

dt

)2+

12P

′(φ)2 + ψ

(ddt + P′′(φ)

)ψ ,

real commuting bosonic ’coordinate’ φ,complex anticommuting fermionic ’coordinate’ ψ,superpotential P(φ)

Two supersymmetries in terms of Majorana fields ψ1,2:

δAφ = ψ1εA, δBφ = ψ2εB ,δAψ1 = dφ

dt εA, δBψ1 = −iP′εB ,δAψ2 = iP′εA, δBψ2 = dφ

dt εB.


Example: N = 2 SUSY QM

Integrating out the fermion fields yields (indefinite)determinant

∫Dψ Dψ exp

(−ψD(φ)ψ

)= detD(φ)

The (regulated) fermion determinant can be calculatedexactly:

det[∂t + P′′(φ)

∂t +m

]= sinh

∫ T

0

P′′(φ)

2 dt =⇒ Z0 − Z1

If under some symmetry φ ↔ φ̃ of SB(φ) we have∫ T

0

P′′(φ̃)

2 dt = −∫ T

0

P′′(φ)

2 dt =⇒ Z0 = Z1 and SSB


Spontaneous SUSY breaking (SSB) and the sign problem

On the lattice we find with Wilson type fermions

det [∇∗ + P′′(φ)] =∏

t[1 + P′′(φt )] − 1 .

For odd potentials, e.g. P(φ) = m2

2λ φ + 13λφ3 we have

det [∇∗ + P′′] =∏

t[1 + 2λφt ] − 1

no longer positive... ⇒ sign problem!

Every supersymmetric model which allows SSB must havea sign problem:

SUSY QM with odd potential,N = 16 Yang-Mills quantum mechanics [Catterall, Wiseman ’07],N = 1 Wess-Zumino model in 2D [Catterall ’03; Wipf, Wozar ’11],. . .


Solution of the sign problem

We propose a (novel) approach circumventing theseproblems:⇒ fermion loop formulation.

Alternative way of simulating fermions on the lattice:based on the exact hopping expansion of the fermionaction,eliminates critical slowing down,allows simulations directly in the massless limit,

⇒ solves the fermion sign problem.

Applicable to Wilson fermions in theO(N) Gross-Neveu model in d = 2 dimensions,Schwinger model in the strong coupling limit in d = 2 and 3,SUSY QM,N = 1 and 2 supersymmetric Wess-Zumino model,supersymmetric matrix QM.


Exact hopping expansion for Wilson fermions

Fermionic part of N = 2 SUSY QM,

L = ψ(∂t + P′′(φ))ψ .

Using Wilson lattice discretisation yields backwardderivative

∇∗ψ(t) = ψ(t) − ψ(t − a)

and eliminates fermion doubling in 1D.

Using the nilpotency of Grassmann elements we expandthe Boltzmann factor

∫DψDψ

∏

t

(1 −M(t)ψ(t)ψ(t)

) ∏

t

(1 + ψ(t)ψ(t − a)

)

where M(t) = 1 + P ′′(φ(t)).


Exact hopping expansion for Wilson fermions

At each site t , the fields ψ and ψ must be exactly paired togive a contribution to the path integral:

∫DψDψ

∏

t

(−M(t)ψ(t)ψ(t)

)m(t) ∏

t

(ψ(t)ψ(t − a)

)nf (t)

with occupation numbersm(t) = 0, 1 for monomers,nf (t) = 0, 1 for fermion bonds (or dimers),

satisfying the constraint

m(t) +12 (nf (t) + nf (t − a)) = 1.

Only closed paths survive the integration.


Exact hopping expansion for scalar fields

Analogous treatment for the bosonic field [Prokof’ev, Svistunov ’01]:(∂tφ)2 → φt+aφt ,expand hopping term exp{−φt+aφt} to all orders:∫

Dφ∏

t

∑

nb(t)

1nb(t)!

(φtφt+a)nb(t) exp (−V (φt )) M(φt )

m(t)

with bosonic bond occupation numbers nb(t) = 0, 1, 2, . . .

Integrating out φ(t) yields bosonic site weights

Q(N) =

∫dφ φN exp (−V (φ))

where N includes powers from M(φ).


Loop formulation of SUSY QM

Loop representation in terms of fermionic monomers anddimers and bosonic bonds.

Especially simple for supersymmetric QM:

{nf = 0,m = 1} ⇒ no fermion, bosonic vacuum{nf = 1,m = 0} ⇒ fermion present, fermionic vacuum

!!! ! ! !!!! ! ! !0 1 2 Lt − 1

Partition functions sum over all bond configurations:

Zp = Z0 − Z1 ⇒ Witten indexZa = Z0 + Z1 ⇒ finite temperature


Exact calculation using a transfer matrix

Loop representation allows the construction of a transfermatrix.

Each state on the link is characterised by the bondoccupation numbers, i.e. |nf ,nb〉,

!!!t

!!! !!!t

Tn′;n takes the system from state |nf ,nb〉 to state |n′f ,n′b〉.

Tn′;n block diagonalises into T nf=0n′b;nb

and T nf=1n′b;nb

.



Loop representation allows the construction of a transfermatrix.

Each state on the link is characterised by the bondoccupation numbers, i.e. |nf ,nb〉,

!!!t T 0

2;4

!!! !!!t T 1

0;4

Tn′;n takes the system from state |nf ,nb〉 to state |n′f ,n′b〉.

Tn′;n block diagonalises into T nf=0n′b;nb

and T nf=1n′b;nb

.



Specifically,

T 1n′,n =

Q(n′ + n)√n′! n!

with Q(N) =

∫dφ φN exp (−V (φ))

and T 0 analogously.

Exact lattice partition functions given by Znf = Tr[(Tnf )Lt

],

Witten index by W ≡ Zp = Z0 − Z1.

Mass gaps from ratios of eigenvalues of T 0 and T 1.

Exact results for other observables (n-point functions, etc.)


Witten index for unbroken supersymmetry, P = 12mφ2 + 1

4gφ4

0 0.01 0.02 0.03 0.04 0.05a/L

0.9975

0.998

0.9985

0.999

0.9995

1

Z p/Za

mL = 30mL = 10mL = 5

g/m2 = 1.0


Witten index for broken supersymmetry, P = − 12λm2φ + 1

3λφ3

0 0.01 0.02 0.03 0.04 0.05a/L

-1

-0.5

0

Z p/Za

mL = 0.1mL = 0.2mL = 0.5mL = 0.8mL = 1.0mL = 2.0mL = 5.0mL = 8.0mL =10.0mL =12.0mL =16.0mL =20.0mL =30.0

/m3/2=1.0


Mass gaps for unbroken SUSY

Z0 Z1

0



Z0 Z1

0

""

""# m0F0



Z0 Z1

0 $m0B0



Z0 Z1

0

%%

%%

%%

%%&

m0F1



Z0 Z1

0 $

m0B1



Z0 Z1

0

'''''''''''''(

m0F2



Z0 Z1

0 $

m0B2



Z0 Z1

0


Mass gaps for broken SUSY

Z0 Z1

E0 = 0.1685



Z0 Z1

E0 = 0.1685

)m0F0



Z0 Z1

E0 = 0.1685

**

*+m0F1



Z0 Z1

E0 = 0.1685

$m0B0



Z0 Z1

E0 = 0.1685

$m1B0



Z0 Z1

E0 = 0.1685

%%

%%

%%%&

m0F2



Z0 Z1

E0 = 0.1685

$

m0B1



Z0 Z1

E0 = 0.1685

$

m1B1



Z0 Z1

E0 = 0.1685


Mass gaps for unbroken SUSY, Q1-exact action (twisted SUSY)

Z0 Z1


Ward identity for unbroken SUSY

Ward identity 〈δO〉 = 〈OδS〉:O = ψxφx → W = 〈ψxψy 〉 + 〈(∆− + P′)xφy 〉

0 0.2 0.4 0.6 0.8 1t / L

-0.1

-0.05

0

0.05

0.1

W(t)

aPBC

Lt = 50Lt = 100Lt = 200Lt = 300Lt = 600

fg = 1 µL = 4


Ward identity for unbroken SUSY


0 0.2 0.4 0.6 0.8 1t / L

-0.1

-0.05

0

0.05

0.1

W(t)

PBC

Lt = 50Lt = 100Lt = 200Lt = 300Lt = 600

fg = 1 µL = 4


Ward identity for unbroken SUSY, continuum limit


0 0.1 0.2 0.3 0.4a µ

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

W(L

t/2)

µL = 3 for aPBCµL = 4 for aPBCµL = 5 for aPBCµL = 7 for aPBCµL = 10 for aPBCµL = 3 for PBCµL = 4 for PBCµL = 5 for PBCµL = 7 for PBCµL = 10 for PBC

fg = 1


Ward identity for unbroken SUSY, Q-exact action


0 0.2 0.4 0.6 0.8 1t / L

-0.01

0

0.01

0.02

0.03

0.04

0.05

W(t)

Lt = 30 for PBCLt = 60 for PBCLt = 90 for PBCLt = 120 for PBCLt = 30 for aPBCLt = 60 for aPBCLt = 90 for aPBCLt = 120 for aPBC

fg = 1 µL = 10


Ward identity for unbroken SUSY, Q-exact action, continuum limit


0 0.1 0.2 0.3 0.4a µ

0

0.02

0.04

0.06

0.08

0.1

W(L

t/2)

all µL for PBCµL = 3 for aPBCµL = 4 for aPBCµL = 5 for aPBCµL = 10 for aPBC

fg = 1


MC simulation of fermion loops: proof of concept

0 0.1 0.2 0.3 0.4 0.5 0.6am

1

1.2

1.4

1.6

1.8

2

mX/m

mF, mL=17.0mB, mL=17.0mF, mL=31.0mB, mL=31.0mF, mL=10.0mB, mL=10.0

continuum limit, g/m2=1.0, Q-exact action


MC simulation of fermion loops: proof of concept


N = 1 Wess-Zumino model

The N = 1 Wess-Zumino model in 2D:

L =12 (∂µφ)2 +

12P

′(φ)2 +12ψ (∂/ + P′′(φ)) ψ

with ψ a two component Majorana field,and φ real bosonic field,superpotential, e.g. P(φ) = −m2

4g φ + 13gφ3.

Symmetries:Supersymmetry

δφ = εψ

δψ = (∂/φ − P′)ε

δψ = 0

ZZ (2) chiral symmetry

φ → −φ

ψ → γ5ψ

ψ → −ψγ5


Classical potential

The scalar potential is a standard φ4 theory:

12P

′(φ)2 = −12m2

2 φ2 +12g

2φ4 + const

ZZ (2) symmetry may bebroken.

-g/2m 0 +g/2m0

m4/g2

P’( )2/2

Witten index is vanishing (W = 0) with one bosonic andone fermionic ground state.

Integrating out Majorana fermions yields indefinite Pfaffian.


ZZ (2) and supersymmetry breaking

For large mg the ZZ (2) symmetry is spontaneously broken,

φ = ±m/2g selects a definite ground state:

φ = +m/2g =⇒ bosonicφ = −m/2g =⇒ fermionic

⇒ SUSY unbroken

For small mg the ZZ (2) symmetry is unbroken,φ = 0 allows both bosonic and fermionic ground state:

⇒ SUSY broken

Tunneling between the two ground states:

⇒ Goldstino mode, PfMpp = 0


N = 1 Wess-Zumino model on the lattice

Using Wilson lattice discretisation for the fermionic part

L =12ξTC(γµ∂̃µ − 1

2∂∗∂ + P′′(φ))ξ ,

and expanding the Boltzmann factor,∫

Dξ∏

x

(−M(x)ξT (x)Cξ(x)

)m(x) ∏

x,µ

(ξT (x)CΓ(µ)ξ(x + µ̂)

)bµ(x)

where M(x) = 2 + P ′′(φ).m(x) = 0, 1 and bµ(x) = 0, 1 satisfy the constraint

m(x) +12

∑

µ

bµ(x) = 1.

Only closed, non-intersecting paths survive the integration.


Fermion loop formulation

Loop representation in terms of fermionic monomers anddimers and bosonic fields φ.

Partition function becomes a sum over all non-oriented,self-avoiding fermion loops +

ZL =

∫Dφ

∑

{$}∈L

ω[+, φ], L ∈ L00 ∪ L10 ∪ L01 ∪ L11

= ZL00 + ZL10 + ZL01 + ZL11

Represents system with unspecified fermionic b.c. [Wolff ’07].

Simulate fermions by enlarging the configuration space byone open fermionic string [Wenger ’08].


Reconstructing the fermionic boundary conditions

The open fermionic string corresponds to the insertion of aMajorana fermion pair {ξT (x)C, ξ(y)} at position x and y :

It samples the relative weights between ZL00 ,ZL10 ,ZL01 ,ZL11 .Reconstruct the Witten index a posteriori

W ≡ Zpp = ZL00 − ZL10 − ZL01 − ZL11 ,

or a system at finite temperature

Zpa = ZL00 − ZL10 + ZL01 + ZL11 .


ZZ (2) broken phase

ZZ (2) broken phase with 〈|φ|〉 - 2 at m/g = 4

0 5×105 1×106

MC time

-4

-2

0

2

4|

⇒ tunneling suppressed for L→ ∞ or m/g → ∞


ZZ (2) broken phase


-4 -2 0 2 40

1000

2000

3000

4000

5000

6000

7000

Z00Z10Z01Z11Z

|

〈φ〉 - −2 : Z00 - Z10 - Z01 - Z11 ⇒ Zpp - −Zpa〈φ〉 - +2 : Z00 - 1,Z10 - Z01 - Z11 - 0 ⇒ Zpp - Zpa


ZZ (2) broken phase


-4 -2 0 2 40

1000

2000

3000

4000

5000

6000

7000

Z00Z10Z01Z11Z

|

supersymmetry restored (at least in the limit L→ ∞)


ZZ (2) symmetric phase

ZZ (2) symmetric phase with 〈φ〉 - 0 at m/g = 0.16

2×105 3×105

MC time

-4

-2

0

2

4|

⇒ ZZ (2) exact only in the limit a→ 0


ZZ (2) symmetric phase

ZZ (2) symmetric phase with 〈φ〉 - 0 at m/g = 0.16

-4 -2 0 2 40

5000

10000

15000

20000

Z00Z10Z10Z11Z

|

〈φ〉 - 0 ⇒ Z00 - Z10 + Z01 + Z11 ⇒ Zpp - 0supersymmetry broken


Scanning for the transition

0 0.05 0.1 0.15 0.2a m

-0.5

0

0.5

1Z pp

/Z

L= 8

a g = 0.03125



0 0.05 0.1 0.15 0.2a m

-0.5

0

0.5

1Z pp

/Z

L= 8L=16

a g = 0.03125



0 0.05 0.1 0.15 0.2a m

-0.5

0

0.5

1Z pp

/Z

L= 8L=16L=32

a g = 0.03125



0 0.05 0.1 0.15 0.2a m

-0.5

0

0.5

1Z pp

/Z

L= 8L=16L=32L=64

a g = 0.03125



0 0.05 0.1 0.15 0.2a m

-0.5

0

0.5

1Z pp

/Z

L= 8L=16L=32L=64L=128

a g = 0.03125


Supersymmetry breaking phase transition

0 0.05 0.1 0.15 0.2a m

-0.5

0

0.5

1Z pp

/ZL= 8L=16L=32L=64L=128L=256

a g = 0.03125

bosonic vacuum preferred (ZZ (2)χ broken at a $= 0)amc - 0.05 for L→ ∞



0 0.2 0.4 0.6 0.8 1a m

0

0.2

0.4

0.6

0.8

1Z pp

/Zpa

gL=2gL=4gL=6gL=8

Witten index W ∝ Zpp/Zpa as an order parameter



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1a m

0

0.05

0.1

0.15

0.2P’

ag=0.25ag=0.125ag=0.0625

Ward identity 〈P ′〉 as an order parameter,volumes gL = 2, 4, 6, 8


ZZ (2) breaking phase transition

0 0.05 0.1 0.15 0.2a m

0

0.2

0.4

0.6

0.8

1

<s> pa

L= 8L=16L=32L=64L=128L=256

a g = 0.03125



0.05 0.1 0.15 0.2am

0

0.1

0.2

0.3

0.4Is

ing

L=16L=32L=64L=128L=256

0 100 200 300 400L

2

4

6

8

max

’Ising susceptibility’ χm with m = 1/V∑

x sign[φx ]



0.06 0.08 0.1 0.12 0.14am

0.1

0.2

0.3

0.4

0.5

0.6

0.7Bi

nder

cum

ulan

tL=16L=32L=64L=128L=256

Binder cumulant



0 0.5 1 1.51/gL

0.04

0.06

0.08

0.1

0.12

0.14am

c

IsingBinder cumulant

Thermodynamic limit


ZZ (2)/SUSY breaking phase transition

0 0.5 1 1.5 2 2.5 31/gL

0

0.05

0.1

0.15

0.2

amc

Thermodynamic limit


ZZ (2)/SUSY breaking phase transition

0 0.05 0.1 0.15 0.2 0.25ag

1

1.5

2

2.5

3f c-1

= m

c/gmc

bare

mcR

Continuum limit: 1-loop renormalised critical coupling


Boson mass spectrum

Boson field correlator in the SUSY symmetric/ZZ (2) brokenphase:

0 10 20 301.8

1.805

1.81

1.815

1.82

Cag=0.25, am=0.71, 64x128

0 10 20 301e-06

1e-05

0.0001

0.001

0.01

Cco

nn

0 10 20 30t/a

0.420.440.460.480.5

0.520.54

mef

f

perfect fit with e−MBt plus constant shift


Boson mass spectrum

Boson field correlator in the SUSY broken/ZZ (2) symmetricphase, ZZ (2) odd state φ:

0 10 20 30 40

0.050.1

0.150.2

Cag=0.25, am=0.20, 64x128

0 10 20 30 400.0001

0.001

0.01

Cco

nn

0 10 20 30 40t/a

0.080.1

0.120.140.160.180.2

mef

f

perfect fit with A0e−M(0)B t + A1e−M

(1)B t


Boson mass spectrum

Boson field correlator in the SUSY broken/ZZ (2) symmetricphase, ZZ (2) even state φ:

0 10 20 30 400.22

0.2250.23

0.2350.24

0.2450.25

Cag=0.25, am=0.20, 64x128

0 10 20 30 40

0.0001

0.01

Cco

nn

0 10 20 30 40t/a

0.20.40.60.8

1

mef

f

perfect fit with e−M(2)B t plus constant shift


Fermion mass spectrum

Fermion field correlator in the SUSY symmetric/ZZ (2) brokenphase (spans O(107) due to fermion loop algorithm):

0 10 20 300

0.5

1

1.5

2

Cag=0.25, am=0.71, 64x128

0 10 20 301e-06

0.0001

0.01

1

C

0 5 10 15 20 25 30t/a

0

0.5

1

1.5

2

mef

f

perfect fit with e−MF t



Fermion field correlator in the SUSY broken/ZZ (2) symmetricphase, ZZ (2) even/odd state ψ and ψφ:

0 10 20 30 40 50

0.60.70.80.9

1

Cag=0.25, am=0.20, 64x128

0 10 20 30 40 500

0.02

0.04

0.06

C

0 5 10 15 20 25 30t/a

00.20.40.60.8

1

mef

f

Z(2) even state

Z(2) odd state

good fits with A0e−M(0)F t + A1e−M

(1)F t and e−M

(2)F t


Boson mass spectrum

0 0.5 1 1.5 2 2.5 3am

0

0.5

1

1.5

2aM

B

MB bosonic sector

MB(2) bosonic sector

ag=0.25, gL=16

0 0.1 0.2 0.3 0.4 0.5

MB bosonic sector


~(mc2-m2)3/4

~(m-mc)1/2


Boson mass spectrum

0 0.5 1 1.5 2 2.5 3am

0

0.5

1

1.5

2aM

B

MB bosonic sector


ag=0.25, gL=16

0 0.1 0.2 0.3 0.4 0.5

MB bosonic sector


MB fermionic sector

MB(2) fermionic sector

~(mc2-m2)3/4

~(m-mc)1/2



0 0.5 1 1.5 2 2.5 3am

0

0.5

1

1.5

2aM

F

MF bosonic sector

MF(2) bosonic sector

ag=0.25, gL=16

0 0.1 0.2 0.3 0.4 0.5



0 0.5 1 1.5 2 2.5 3am

0

0.5

1

1.5

2aM

F

MF bosonic sector


ag=0.25, gL=16

0 0.1 0.2 0.3 0.4 0.5

Goldstino amplitudeGoldstino mass



0 0.5 1 1.5 2 2.5 3am

0

0.5

1

1.5

2aM

F

MF bosonic sector


MF fermionic sector

MF(2) fermionic sector

ag=0.25, gL=16

0 0.1 0.2 0.3 0.4 0.5


Combined mass spectrum

0 0.5 1 1.5 2 2.5 3am

0

0.5

1

1.5

2

aMF

MF bosonic sector


ag=0.25, gL=16

0 0.1 0.2 0.3 0.4 0.5

0 0.5 1 1.5 2 2.5 3am

0

0.5

1

1.5

2

aMB

MB bosonic sector


ag=0.25, gL=16

0 0.1 0.2 0.3 0.4 0.5

MB bosonic sector


~(mc2-m2)3/4

~(m-mc)1/2


Conclusions

Reformulation of Majorana fermions in terms of fermionloops:

separation into bosonic and fermionic sector,avoids the fermion sign problem,allows simulations without critical slowing down.

Transfer matrix yields exact results for SUSY QM.

Determination of the critical coupling for the ZZ (2)/SUSYbreaking phase transition for the N = 1 WZ model.

Determination of the particle mass spectrum above andbelow the transition.

Complete non-perturbative description of aspontaneous supersymmetry breaking phase transition.

Documents

Spontaneous supersymmetry breaking on the latticegies/Conf/SIFT2012/jena12_wenger.pdfOverview SUSY breaking and sign problem N = 2 supersymmetric QM N = 1 Wess-Zumino model Spontaneous