classical and quantum simulation of gauge …blogs.bu.edu/ppcm/files/2014/05/wiese.pdfoutline...

Classical and Quantum Simulation of Gauge Theoriesin Particle and Condensed Matter Physics

Uwe-Jens Wiese

Albert Einstein Center for Fundamental PhysicsInstitute for Theoretical Physics, Bern University

Field Theoretic Computer Simulationsfor Particle Physics and Condensed Matter

Boston University, May 9, 2014

Collaboration:Debasish Banerjee, Michael Bogli, Pascal Stebler, Philippe Widmer

(Bern), Christoph Hofmann (Colima), Fu-Jiun Jiang (Taipei),Mark Kon (BU), Marcello Dalmonte, Peter Zoller (Innsbruck),

Enrique Rico Ortega (Strasbourg), Markus Muller (Madrid)

Outline

Quantum Simulation in Condensed Matter Physics

Wilson’s Lattice Gauge Theory versus Quantum Link Models

The (2 + 1)-d U(1) Quantum Link and Quantum Dimer ModelsMasquerading as Deconfined Quantum Criticality

Atomic Quantum Simulator for U(1) Gauge Theory Coupled toFermionic Matter

Atomic Quantum Simulator for U(N) and SU(N) Non-AbelianGauge Theories

Conclusions

Ultra-cold atoms in optical lattices as analog quantumsimulators

Superfluid-Mott insulator transition in the bosonic Hubbardmodel

M. Greiner, O. Mandel, T. Esslinger, T. Hansch, I. Bloch,Nature 415 (2002) 39.

Digital quantum simulation of Kitaev’s toric code(a Z(2) lattice gauge theory) with trapped ions

• Precisely controllable many-body quantum device, which canexecute a prescribed sequence of quantum gate operations.• State of simulated system encoded as quantum information.• Dynamics is represented by a sequence of quantum gates,following a stroboscopic Trotter decomposition.

A. Y. Kitaev, Ann. Phys. 303 (2003) 2.B. P. Lanyon et al., Science 334 (2011) 6052.

Outline






Conclusions

Hamiltonian formulation of Wilson’s U(1) lattice gauge theory

U = exp(iϕ), U† = exp(−iϕ) ∈ U(1)

Electric field operator E

E = −i∂ϕ, [E ,U] = U, [E ,U†] = −U†, [U,U†] = 0

Generator of U(1) gauge transformations

Gx =∑

i

(Ex−i ,i − Ex ,i ), [H,Gx ] = 0

U(1) gauge invariant Hamiltonian

H =g2

2

∑

x ,i

E 2x ,i −

1

2g2

∑

�

(U� + U†�),

U� = Ux ,iUx+i ,jU†x+j ,i

U†x ,j

operates in an infinite-dimensional Hilbert space per link

U(1) quantum link model

U = S1 + iS2 = S+, U† = S1− iS2 = S−

r rx x + i

Ex,i

Ux,i

Electric flux operator E

E = S3, [E ,U] = U, [E ,U†] = −U†, [U,U†] = 2E

Generator of U(1) gauge transformations

Gx =∑

i

(Ex−i ,i − Ex ,i ), [H,Gx ] = 0

Gauge invariant Hamiltonian for S = 12

H = −J∑

�

(U� + U†�)

H = J

H = 0

defines a gauge theory with a 2-d Hilbert space per link.

D. Horn, Phys. Lett. B100 (1981) 149D. S. Rokhsar, S. A. Kivelson, Phys. Rev. Lett. 61 (1988) 2376P. Orland, D. Rohrlich, Nucl. Phys. B338 (1990) 647S. Chandrasekharan, UJW, Nucl. Phys. B492 (1997) 455R. Brower, S. Chandrasekharan, UJW, Phys. Rev. D60 (1999) 094502

Outline






Conclusions

Hamiltonian with Rokhsar-Kivelson term

H = −J[∑

�

(U� + U†�)− λ∑

�

(U� + U†�)2

]

Phase diagram

1

λ0−1

T

Deconfined

λc

l

M

M

M

MB

A

B

A

MB

MA

ll

l l

l

l

l

l

Confined

l

l

ll

D. Banerjee, F.-J. Jiang, P. Widmer, UJW, JSTAT (2013) P12010.

Probability Distribution of the Order Parameters: L2 = 242

Energy density of charge-anti-charge pair Q = ±2: L2 = 722

0

20

40

60

0 20 40 60

(c)

0

20

40

60(a)

0 20 40 60

(d)c

-1

-0.8

-0.6

-0.4

-0.2

(b)d

-1

-0.8

-0.6

-0.4

-0.2

0

Rokhsar and Kivelson’s Quantum Dimer ModelConfining columnar phase:

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

Confining plaquette phase:

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

Deconfined staggered phase:

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

Hamiltonian with Rokhsar-Kivelson term

H = −J[∑

�

(U� + U†�)− λ∑

�

(U� + U†�)2

]

Probability distribution of order parameters: L2 = 242

A soft mode arises near λ ≈ −0.2 and extends all the way tothe RK-point λ = 1.D. Banerjee, M. Bogli, C. P. Hofmann, F.-J. Jiang, P. Widmer, UJW,in preparation.

Soft mode leads to an almost perfect rotor spectrum: L2 = 82

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

-1 -0.5 0 0.5 1

Ei−E

0

λ

E1, E2E3

E4E5, E6

E7E8

The small splitting E4 − E3 originates from an explicit breaking ofthe emergent SO(2) symmetry.

External charges and confining potential: L2 = 1202, βJ = 100

8

10

12

14

16

18

20

22

10 15 20 25 30 35 40 45V

(r)

r

λ=-0.5λ=0.5λ=0.8

String tension:σ(λ = −0.5) = 0.370(1)J/a,σ(λ = 0.5) = 0.057(1)J/a,σ(λ = 0.8) = 0.029(1)J/aThe soft mode almost deconfines. Since for λ 6= 1 it is not exactlymassless, it can not quite be identified as a dual photon.

Energy density of charge-anti-charge pair Q = ±2: L2 = 1442

0

20

40

60

80

100

120

140

0 20 40 60 80 100 120 140-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

The string separates into 8 strands of fractionalized flux 14 .

Outline






Conclusions

Hamiltonian for staggered fermions and U(1) quantum links

H = −t∑

x

[ψ†xUx ,x+1ψx+1 + h.c.

]+m

∑

x

(−1)xψ†xψx+g2

2

∑

x

E 2x ,x+1

Bosonic rishon representation of the quantum links

Ux ,x+1 = bxb†x+1, Ex ,x+1 =

1

2

(b†x+1bx+1 − b†xbx

)

Microscopic Hubbard model Hamiltonian

H =∑

x

hBx ,x+1 +∑

x

hFx ,x+1 + m∑

x

(−1)xnFx + U∑

x

G 2x

= −tB∑

x odd

b1x†b1x+1 − tB

∑

x even

b2x†b2x+1 − tF

∑

x

ψ†xψx+1 + h.c.

+∑

x ,α,β

nαxUαβnβx +

∑

x ,α

(−1)xUαnαx

D. Banerjee, M. Dalmonte, M. Muller, E. Rico, P. Stebler, UJW,P. Zoller, Phys. Rev. Lett. 109 (2012) 175302.

Optical lattice with Bose-Fermi mixture of ultra-cold atoms

b)

F

a)b1

b2

tF

2U

tB

2U

2U2 (U + m)

2

✓U +

g2

4� t2B

2U

◆

tFtB

1 2 3 4 x

U↵�

F

b1

b2

From string breaking to false vacuum decay

c)

d)

Q

Q

Q

Q qq

fermion:

|0i|1i

a)

b)

link:

S=1/2

|1/2i| � 1/2i

S=1

|0i| � 1i

| + 1i

†xUx,x+1 x+1x x + 1x x + 1 x x + 1

†xb� †

x+1b�x x+1

Quantum simulation of the real-time evolution of stringbreaking and of coherent vacuum oscillations

0 10 20 30t τ

-12

-6

0

E

0 30 60 90t τ

0

2

4

E

e) f)

Outline






Conclusions

U(N) quantum link operators

U ij = S ij1 +iS ij

2 , Uij† = S ij

1−iSij2 , i , j ∈ {1, 2, . . . ,N}, [U ij , (U†)kl ] 6= 0

SU(N)L × SU(N)R gauge transformations of a quantum link

[La, Lb] = ifabcLc , [Ra,Rb] = ifabcR

c , a, b, c ∈ {1, 2, . . . ,N2 − 1}

[La,Rb] = [La,E ] = [Ra,E ] = 0

Infinitesimal gauge transformations of a quantum link

[La,U] = −λaU, [Ra,U] = Uλa, [E ,U] = U

Algebraic structures of different quantum link models

U(N) : U ij , La, Ra,E , 2N2+2(N2−1)+1 = 4N2−1 SU(2N) generators

SO(N) : O ij , La, Ra, N2+2N(N − 1)

2= N(2N−1) SO(2N) generators

Sp(N) : U ij , La, Ra, 4N2+2N(2N+1) = 2N(4N+1) Sp(2N) generators

R. Brower, S. Chandrasekharan, UJW, Phys. Rev. D60 (1999) 094502

Low-energy effective action of a quantum link model

S [Gµ] =

∫ β

0dx5

∫d4x

1

2e2

(Tr GµνGµν +

1

c2Tr ∂5Gµ∂5Gµ

), G5 = 0

undergoes dimensional reduction from 4 + 1 to 4 dimensions

S [Gµ]→∫

d4x1

2g2Tr GµνGµν ,

1

g2=

β

e2,

1

m∼ exp

(24π2β

11Ne2

)

Fermionic rishons at the two ends of a link

{c ix , c j†y } = δxyδij , {c ix , c jy} = {c i†x , c j†y } = 0

Rishon representation of link algebra

r rx y

cix cjy

Uij

U ijxy = c ixc

j†y , L

axy = c i†x λ

aijc

jx , R

axy = c i†y λ

aijc

jy , Exy =

1

2(c i†y c iy−c i†x c ix)

Can a “rishon abacus” implemented with ultra-cold atoms beused as a quantum simulator?

Optical lattice with ultra-cold alkaline-earth atoms(87Sr or 173Yb) with color encoded in nuclear spin

x x + 1x � 1

�� ++� +

a)

U Uc)

�3/2 �1/2 1/2 3/2

| "i | #i | "i | #i

( (

e)

b)

�t

�t

�t

+t

t tt

ix

cix,� ci

x,+

U ijx,x+1 i

x

d)

D. Banerjee, M. Bogli, M. Dalmonte, E. Rico, P. Stebler, UJW, P. Zoller,Phys. Rev. Lett. 110 (2013) 125303

Restoration of chiral symmetry at baryon density nB ≥ 12

1e-05

0.0001

0.001

0.01

0.1

1

10

0 2 4 6 8 10 12 14

∆E

= E

B+ -

EB

-

L

∆E with constant Baryon density nB

nB = 0nB = 1/2

nB = 1

Other proposals for digital quantum simulators for Abelian andnon-Abelian quantum link modelsM. Muller, I. Lesanovsky, H. Weimer, H. P. Buchler, P. Zoller,Phys. Rev. Lett. 102 (2009) 170502; Nat. Phys. 6 (2010) 382.L. Tagliacozzo, A. Celi, P. Orland, M. Lewenstein,Nature Communications 4 (2013) 2615.L. Tagliacozzo, A. Celi, A. Zamora, M. Lewenstein,Ann. Phys. 330 (2013) 160.

Other proposals for analog quantum simulators for Abelian andnon-Abelian gauge theories with and without matter

H. P. Buchler, M. Hermele, S. D. Huber, M. P. A. Fisher, P. Zoller,Phys. Rev. Lett. 95 (2005) 040402.E. Kapit, E. Mueller, Phys. Rev. A83 (2011) 033625.E. Zohar, B. Reznik, Phys. Rev. Lett. 107 (2011) 275301.E. Zohar, J. Cirac, B. Reznik, Phys. Rev. Lett. 109 (2012) 125302;Phys. Rev. Lett. 110 (2013) 055302; Phys. Rev. Lett. 110 (2013) 125304.

Review on quantum simulators for lattice gauge theories

UJW, Annalen der Physik 525 (2013) 777, arXiv:1305.1602.

Outline






Conclusions

Conclusions

• Quantum link models implemented with ultra-cold atoms serve asquantum simulators for Abelian and non-Abelian gauge theories, whichcan be validated in efficient classical cluster algorithm simulations.

• Such simulations have revealed a soft mode in the quantum dimermodel. Can this play the role of “phonons” in high-Tc materials?

• Quantum simulator constructions have already been presented for theU(1) quantum link model as well as for U(N) and SU(N) quantum linkmodels with fermionic matter.

• This would allow the quantum simulation of the real-time evolution ofstring breaking as well as the quantum simulation of “nuclear” physicsand dense “quark” matter, at least in qualitative toy models for QCD.

• Accessible effects may include chiral symmetry restoration, baryonsuperfluidity, or color superconductivity at high baryon density, as well asthe quantum simulation of “nuclear” collisions.

• The path towards quantum simulation of QCD will be a long one.

However, with a lot of interesting physics along the way.

classical and quantum simulation of gauge …blogs.bu.edu/ppcm/files/2014/05/wiese.pdfoutline...

Documents