Classical and Quantum Simulation of Gauge …blogs.bu.edu/ppcm/files/2014/05/Wiese.pdfOutline...
Transcript of Classical and Quantum Simulation of Gauge …blogs.bu.edu/ppcm/files/2014/05/Wiese.pdfOutline...
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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)
![Page 2: Classical and Quantum Simulation of Gauge …blogs.bu.edu/ppcm/files/2014/05/Wiese.pdfOutline Quantum Simulation in Condensed Matter Physics Wilson’s Lattice Gauge Theory versus](https://reader035.fdocuments.us/reader035/viewer/2022070803/5f03374d7e708231d4081c4e/html5/thumbnails/2.jpg)
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
![Page 3: Classical and Quantum Simulation of Gauge …blogs.bu.edu/ppcm/files/2014/05/Wiese.pdfOutline Quantum Simulation in Condensed Matter Physics Wilson’s Lattice Gauge Theory versus](https://reader035.fdocuments.us/reader035/viewer/2022070803/5f03374d7e708231d4081c4e/html5/thumbnails/3.jpg)
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
![Page 4: Classical and Quantum Simulation of Gauge …blogs.bu.edu/ppcm/files/2014/05/Wiese.pdfOutline Quantum Simulation in Condensed Matter Physics Wilson’s Lattice Gauge Theory versus](https://reader035.fdocuments.us/reader035/viewer/2022070803/5f03374d7e708231d4081c4e/html5/thumbnails/4.jpg)
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.
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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.
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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
![Page 7: Classical and Quantum Simulation of Gauge …blogs.bu.edu/ppcm/files/2014/05/Wiese.pdfOutline Quantum Simulation in Condensed Matter Physics Wilson’s Lattice Gauge Theory versus](https://reader035.fdocuments.us/reader035/viewer/2022070803/5f03374d7e708231d4081c4e/html5/thumbnails/7.jpg)
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
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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
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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
![Page 10: Classical and Quantum Simulation of Gauge …blogs.bu.edu/ppcm/files/2014/05/Wiese.pdfOutline Quantum Simulation in Condensed Matter Physics Wilson’s Lattice Gauge Theory versus](https://reader035.fdocuments.us/reader035/viewer/2022070803/5f03374d7e708231d4081c4e/html5/thumbnails/10.jpg)
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.
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Probability Distribution of the Order Parameters: L2 = 242
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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
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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
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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.
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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.
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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.
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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 .
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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
![Page 19: Classical and Quantum Simulation of Gauge …blogs.bu.edu/ppcm/files/2014/05/Wiese.pdfOutline Quantum Simulation in Condensed Matter Physics Wilson’s Lattice Gauge Theory versus](https://reader035.fdocuments.us/reader035/viewer/2022070803/5f03374d7e708231d4081c4e/html5/thumbnails/19.jpg)
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.
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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
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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
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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)
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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
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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
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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
)
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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
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Can a “rishon abacus” implemented with ultra-cold atoms beused as a quantum simulator?
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Optical lattice with ultra-cold alkaline-earth atoms(87Sr or 173Yb) with color encoded in nuclear spin
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D. Banerjee, M. Bogli, M. Dalmonte, E. Rico, P. Stebler, UJW, P. Zoller,Phys. Rev. Lett. 110 (2013) 125303
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Restoration of chiral symmetry at baryon density nB ≥ 12
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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.
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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
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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.