Susy Lattice Models - uni-jena.de · 1 Introduction 2 Supersymmetric Landau-Ginzburg Models 3 d = N= 2 Wess-Zumino Model 4 Potentials for WZ-Models 5 Simulations 6 Nonlinear Sigma

Susy Lattice Models

A. Wipf

Theoretisch-Physikalisches Institut, FSU Jena

with G. Bergner, T. Kastner, S. Uhlmann, C. Wozar

earlier work: Annals Phys. 303, 315, 316, arXiv:0705.2212

Heidelberg, 15th December 2007

Andreas Wipf (FSU Jena) Susy Lattice Models 1 / 32

1 Introduction

2 Supersymmetric Landau-Ginzburg Models

3 d = N = 2 Wess-Zumino Model

4 Potentials for WZ-Models

5 Simulations

6 Nonlinear Sigma Models

7 Conclusions


Introduction

Lattice Susy

susy breaking, phase diagrams, treatment of fermions

how much susy we give up on lattice: off- and on-shell

low lying masses, Ward identities, matching anomalies

fine tuning: improvement program in doubt


related earlier work:

Dondi, Nicolai: exact susy on lattice, 77Elitzur, Schwimmer: N = 2 Wess-Zumino in d=2, 82Sakai, Sakamoto: lattice susy & Nicolai mapping, 83Beccaria, Curci, D’Ambrosio: simulation & Nicolai map, 98

related recent work:

Catterall et.al: exact susy on lattice, 01 –Cohen, Kaplan, Katz, Unsal: susy with N charges, 02 –Feo, Bonini, et.al: simulations of WZ-models, 02 –Sugino: extended SYM and exact susy on lattice, 04 –Giedt, Poppitz: perturbation theory, det D, 04 –

Particle Physics:

Montvay et.al: N = 1 SYM on lattice, 01 –


d =N =2 Supersymmetric Landau-Ginzburg Models

Φ chiral superfield: complex φ, Dirac ψ, complex auxiliary F :

L(x) ∝∫

d2θd2θK (Φ, Φ) +

[∫

d2 θW (Φ) + h.c.

]

K (z , z) : potential on Kahlerian target space M,

gpq =∂2K

∂zp∂z q, 1 ≤ p, q ≤ n.

W (z) : holomorphic superpotential

K and W enter scalar potential

V = gpqWpWq, Wp = ∂W /∂zp, Wq = (Wq)∗


W not renormalized in perturbation theory (R-symmetries)

But K → Keff from matching microscopic and IR lattice datasimulations ⇒ nonperturbative renormalization K → Keff

first step: → 〈φ〉, mlight, . . . constrains

Vφ, Vφφ, Vφφ

N = 2 Wess-Zumino model: M = C, K = zz and

Lwz ∝∫

d2θd2θ ΦΦ +

[∫

d2θW (Φ) + h.c.

]

Supersymmetric O(3) model: M = CP1,W = 0 and

Lσ ∝∫

d2θd2θ log(

1 + ΦΦ)

or Calabi-Yau models


Wess-Zumino Model

Euclidean off-shell Lagrangian density for component fields

Lwz = 2∂φ∂φ+ ψMψ +1

2|W ′(φ)|2 − 1

2

∣

∣F + W ′(φ)∣

∣

2

On-shell density bounded below:

Lwz = 2∂φ∂φ+ ψMψ +1

2|W ′(φ)|2

Dirac operator with scalar and pseudoscalar Yukawa coupling

M = /∂ + W ′′(φ)P+ + W ′′(φ)P−

Here: superpotential

W (φ) =m

2φ2 +

g

3φ3


Symmetries

(2, 2) supersymmetry:

δφ ∼ ǫψ, δψ ∼ ǫ(∂φ+ F ), δF ∼ ǫ∂ψ

unbroken for continuum model → susy multiplets (Witten index)conflict with lattice regularization

R-type symmetries:

U(1)V ; ψ → e iλψ, ψ → ψe−iλ

Z2 : φ→ −(φ+ m/g), ψ → γ∗ψ, ψ → −ψγ∗

Z2 probably broken for all g 6= 0.


Potentials for WZ-Models

Classical potential for φ = (φ1 − ξ) + iφ2 with shift ξ = m/2g :

V =1

2|W ′|2 =

g2

2

{

(φ21 + φ2

2)2 + 2ξ2

(

φ22 − φ2

1

)

+ ξ4}

.

φ1

V

ξ = m2g

m4

32g2

W ′=0

W ′′ = 2g(φ1 + iφ2)


On-shell effective potential

S(φ,ψ) = kinetic + Yukawa +1

2|W ′(φ)|2

Finite one-loop on-shell result

U(1)on (ϕ) =

1

2|W ′|2 − |W

′′|28π

(

log(1− X 2) + 2Xartanh(X ))

,

where

X =|W ′′′W ′||W ′′|2 =

|W ′|2g |ϕ|2

extreme at classical minima ±ξ, where W ′ = 0. Effective mass

m2eff = m2

(

1− 1

π

g2

m2

)

⇒ gc =√πm


Off-shell effective potential

S(φ,ψ,F ) = kinetic + Yukawa − 1

2|F |2 +

1

2FW ′ +

1

2F W ′

Finite one-loop off-shell result

U(1)off (ϕ) =

1

2|W ′|2 − |W

′′|28π

(

log(1− X 2) + 2Xartanh(X ))

,

where now

X =|FW ′′′||W ′′|2 , |F |+ |W

′′′|4π

artanh

( |FW ′′′||W ′′|2

)

− |W ′| = 0

minimal at classical minima ±ξ, where W ′ = 0. Effective mass

m2eff =

m2

1 + g2

πm2

= m2

(

1− 1

π

g2

m2+ . . .

)


supersymmetry unbroken

U(1)on non-positive, complex, unstable for strong couplings

U(1)off real, stable for all g

off-shell on-shell

classical

ϕ1

U(1)

m=10, g =10 m=10, g =20

U(1)on 6= U

(1)off although Uon = Uoff


classical, on-shell, off-shell, g=15 m=10

line 1line 2line 3

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

phi1-0.4

-0.3-0.2

-0.1 0

0.1 0.2

0.3 0.4

phi2

-1

-0.5

0

0.5

1

1.5

2

2.5


Zero-momentum toy model for constant (ψ, φ,F )

classical action

S = ψMψ +1

2(FW ′ + F W ′ − F F ), M = W ′′P+ + W ′′P−

potentials via Legendre transform of w(j) in

ew(j) =

∫

d(ψ, φ,F ) e−S+jφ1

Saddle-point approximations before and after eliminating F :

U(1) =1

2|W ′|2 +

1

2log

(

1− X 2)

, X =|W ′′′F ||W ′′|2

|F | = |W ′| on-shell

|F | =(

|F |2 − |W ′′|2)

(|F | − |W ′|) off-shell


exact constraint effective potential

u(1)(ϕ) = − log

∫

e−Sδ(φ1 − ϕ) = V (ϕ, 0) − log ∆(ϕ)

∆(ϕ) = 4√

gH eH2{

gϕ2K1/4(H2) + H

(

K3/4(H2)− K1/4(H

2))

}

on- and off-shell ’masses’ (α = 2g/m2)

on-shell: m2eff = m2

(

1− α2)

off-shell: m2eff =

m2

1 + α2


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.2 0.4 0.6 0.8 1 1.2 1.4

u, g=1, m=5off-shell, g=1, m=5on-shell, g=1, m=5

-0.5

0

0.5

1

1.5

2

2.5

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

u, g=5, m=5off-shell, g=5, m=5on-shell, g=5, m=5

weak coupling

U(1)on complex, unstable

shifted minimas ⇒wave function Z 6= 1

strong coupling


Simulations

no Leibniz → susy broken for naive discretization, Z not central

fermion-doublingsusy−→ boson-doubling

if ∂ 6= −∂T ⇒ large lattice artifacts in {Qα,Qβ}Wilson fermions:no doubling, ultra-local, cheap, non-chiral, fine tuningwith improvement: S = Sst + Simpr with 〈Sst〉 ≈ 〈Simpr〉 ≫ 〈S〉Slac fermions Ds = γµ∂slac

µ + m:no doubling, chiral, antisymmetric, ’exact’, non-localone-loop renormalizable (Bergner)with improvement 〈Sst − Simpr〉 ≪ 〈S〉 ⇒ reweighting?

HMC, Fourier accelerated; direct inverter in 1d and very small latticesin 2d ; else pseudo fermions; reweighting for g ≪ m (with Frommer)so far no PHMC or RHMC needed (checked for small EV)


Nicolai improvement in 1d

Nicolai map of bosonic variables

φ −→ ξ(φ), Rn → R

n

Ssusy = 12

∑

x ξx(φ)2 +∑

x ,y ψx∂ξx

∂φyψy

exact susy: δ(1)φx = ǫψx , δ(1)ψx = 0, δ(1)ψx = −ǫξx

choice: ξx(φ) = (∂φ)x + Wx

Ssusy =1

2

∑

x

((∂φ)x + Wx)2 +

∑

x ,y

ψx (∂xy + Wxy )ψy

= Snaive +∑

x

Wx(φ) (∂φ)x

Nicolai-improvement exists also for d = N = 2-Wess-Zumino


Comparing different discretizations

discretize supersymmetric quantum mechanics:

Scont =

∫

dτ(1

2φ2 +

1

2W ′2 + ψψ + ψW ′′ψ + improvement

)

model kbos mbos(0) kferm mferm(0)

Snaivw 139.52 ± 8.45 12.23 ± 0.08 −186.25 ± 4.98 18.40 ± 0.05Sw −136.85 ± 5.22 16.68 ± 0.05 −146.10 ± 3.84 16.73 ± 0.04

Sslac −25.22 ± 6.24 16.92 ± 0.07 −33.64 ± 2.52 16.97 ± 0.03

Ssusyw −135.11 ± 7.36 16.68 ± 0.07 −138.50 ± 2.85 16.64 ± 0.03

Ssusyslac −17.97 ± 2.41 16.84 ± 0.03 −18.53 ± 0.91 16.81 ± 0.01

Table: linear interpolations for masses, mphys = 16.865 (g/m2 = 1)


PSfrag repla ements

latti e spa ing ae�e tivemass

m

naive SLAC fermions

naive Wilson bosonsnaive Wilson fermionsnaive SLAC bosonsnaive SLAC fermions

00.10.20.30.40.50.60.70.80.91

0.005 0.01 0.015 0.02 0.025

00.10.20.30.40.50.60.70.80.9112131415161718

Figure: Boson and fermion masses for Wilson fermions without improvement. andnon-supersymmetric SLAC fermions.


Back to two dimensions

continuum action

Scont =

∫(

2∂φ∂φ+1

2|W ′|2 + ψMψ

)

, M = /∂ + W ′′P+ + W ′′P−

N = 2 supersymmetry broken by naive discretization

Nicolai variables

ξx = 2(∂φ)x + Wx , Sbos =1

2

∑

x

ξxξx

Nicolai improved action

Sbos =∑

x

(

2(∂φ)x(∂φ)x +1

2|Wx |2 + Wx(∂φ)x + W ′

x(∂φ)x

)

Sferm =∑

x ,y

ψxMxyψy , Mxy = γµ∂µ,xy + WxyP+ + WxyP−


one (out of 4) susy remains ⇒ supersymmetric Ward identities , e.g.

〈SB〉 = N2 for improved models (cp. Catterall)

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

0 0.05 0.1 0.15 0.2

Sb

/ N2

a

bosonic action for unimproved models with slac and wilson derivative at g/m=0.1

WilsonSlac

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

0 0.05 0.1 0.15 0.2S

b / N

2

a

bosonic action for unimproved models with slac and wilson derivative at g/m=1.0

WilsonSLAC

susy breaking for unimproved models increases with g/m (0.1 and 1);SLAC much better for strong coupling


Unimproved models: Monte Carlo history of φ1 at g/m = 1 (7× 7)

-2

-1.5

-1

-0.5

0

0.5

1

1.5

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

phi 1

monte carlo time t

Monte Carlo history of phi1 at m/g=1.0 on a 7x7 lattice

with unimproved action using slac derivativewith unimproved action using wilson (r=1) derivative

locations of classical vacuaa

red: SLAC; green: Wilson; blue lines: classical vacuaWilson term breaks Z2 r-symmetry, so do improvement terms


histogram for ϕ1 shows Z2 symmetry for SLAC , g/m = 1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

-2 -1.5 -1 -0.5 0 0.5 1

P(p

hi1)

phi1

(symmetrized) histogram of phi1 at g/m=1.0

unimproved action using slac derivative

unimproved model with SLAC: Z2 (spontaneously) broken


constrained effective potential for weak coupling g/m = 0.1 and SLAC

-2

-1

0

1

2

3

4

5

6

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

Vef

f

phi1

effective potential for unimproved action and various lattice sizes using slac derivative at g/m=0.1

19x19classical potential

well approximated by classical potential 12 |W ′|2, perturbative regime (lit)


cep for strong coupling g/m = 1, various lattice sizes and SLAC

-1

0

1

2

3

4

5

6

-2 -1.5 -1 -0.5 0 0.5 1

Vef

f

phi1

effective potential for unimproved action and various lattice sizes using slac derivative at g/m=1.0

7x711x1115x15

minimas further out; mass extraction under way (statistics)


Nonlinear Sigma Models (with Wozar and Georg)

Here CP1 model with K = log(1 + zz), Φ ∼ (ψ, u,F ), eliminate

Lσ =ρ

2

(

∂µu∂µu − i2 ψ /∂ψ + i

2∂µψγµψ + ωµψγ

µψ + ρ (ψψ)2)

ρ = (1 + uu)−2, ωµ = i√ρ (u∂µu − u∂µu)

Simulation on curved target space? 4-fermi term: when doesHubbard-Stratonovich work? θ-term; purely bosonic sector:

LB ∝∂µu∂µu(1 + u2)2

≡ ρ ∂µu∂µu , u =

(

u1

u2

)

discretizations: well-kown ordering problem

Sa = 4β∑

〈x ,y〉

ρx + ρy

2(ux − uy )2 , Sg = 4β

∑

〈x ,y〉

√ρxρy (ux − uy)

2


0.001

0.01

0.1

1

0.01 0.1 1 10

Σ x ⟨n

(0)n

(x)⟩

/V

beta=g-2

geometric mean, L=30arithmetic mean, L=30geometric mean, L=20arithmetic mean, L=20


-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

n 3

n1

arithmetic mean, L=20, single point values

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

n 3

n1

geometric mean, L=20, single point values

S2 = SO(3)/SO(2): action Sa not SO(3) invariant. Here

Sg ≡β

2

∑

x ,µ

(nx+µ − nx)2 = −β

∑

〈x ,y〉

nxny + const


0

200

400

600

800

1000

1200

1400

1600

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

# co

nfig

s

n3

histograms of single point values

arithmetic meangeometric mean

a must for coset models M = G/H : choose G -invariant discretizationnon-coset sigma-models (path integrals on curved M)?


dependency on discretization (e.g. derivative) for n-formulationstandard action → SLAC derivative → fixpoint action (cp. Bergner)

6.2

6.3

6.4

6.5

6.6

6.7

6.8

0 0.002 0.004 0.006 0.008 0.01

g R

sqrt(z)*exp(-z), z=L/xi

SLAC-Action, Heatbath, beta=0.96, L in [21,31]Balog et al., Std-Action, L in [60,140]

Balog et al., FP-Action, L in [18,26]


Conclusions

Fourier accelerated HMC-code works fine (JenLaTT, many tests!)

Wilson and SLAC-fermions, with and without improvementWilson: strong susy and r -symmetry breaking for g/m ≥ 1

numerics for WZ difficult: extremely flat potential for g > m

related to pletoria of susy ground state of Himpr

soon masses and precise Ueff for g/m ≈ 1; discriminate

m2eff = m2

(

1− 1

π

g2

m2

)

←→ m2eff = m2

(

1 +g2

πm2

)−1

See no difficulties N = 2 −→ N = 1 (Pfaffian, sign,..)HMC-code for CPn model: work in progress; CP1 soonNicolai maps for Sigma- or even Landau-Ginzburg models?

Four dimensions: beyond local cluster → next year


Documents

Susy Lattice Models - uni-jena.de · 1 Introduction 2 Supersymmetric Landau-Ginzburg Models 3 d = N= 2 Wess-Zumino Model 4 Potentials for WZ-Models 5 Simulations 6 Nonlinear Sigma