The 2d CP(N 1) models share a number of features with QCD, like asymptotic freedom, a dynamically generated mass gap and topological sectors. They have been formulated and analysed successfully in the framework of the so-called D-theory, which provides a smooth access to the continuum limit. In that framework, we propose an experimental set-up for the quantum simulation of the CP(2) model. It is based on ultra-cold Alkaline-Earth Atoms (AEAs) located on the sites of an optical lattice, where the nuclear spins represent the relevant degrees of freedom. We present numerical results for the correlation length and for the real time decay of a false vacuum, to be compared with such a future experiment. The latter could also enable the exploration of q-vacua and of the phase diagram at finite chemical potentials, since it does not suffer from any

sign problem

Bietenholz, Wolfgang

Dalmonte, Marcello

Evans, Wynne

Gerber, Urs

Laflamme, Catherine

Mejía-Díaz, Héctor

Wiese, Uwe-Jens

Zoller, Peter

Bern Open Repository and Information System (BORIS)

PoS(LATTICE 2015)311Proposal for the Quantum Simulation of the CP(2)Model on Optical Lattices∗Catherine Laflamme a,b, Wynne Evans c, Marcello Dalmonte a,b, Urs Gerber d,e,Héctor Mejía-Díaz d, Wolfgang Bietenholz† d, Uwe-Jens Wiese c and Peter Zoller a,ba Institute for Theoretical Physics, University of Innsbruck, A-6020, Innsbruck, Austriab Institute for Quantum Optics and Quantum InformationAustrian Academy of Sciences, A-6020 Innsbruck, Austriac Albert Einstein Center for Fundamental Physics, Institute for Theoretical PhysicsUniversität Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerlandd Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de MéxicoA.P. 70-543, C.P. 04510 Distrito Federal, Mexicoe Instituto de Física y Matemáticas, Universidad Michoacana de San Nicolás de HidalgoEdificio C-3, Apdo. Postal 2-82, C.P. 58040, Morelia, Michoacán, MexicoE-mail: wolbi@nucleares.unam.mxThe 2d CP(N−1) models share a number of features with QCD, like asymptotic freedom, a dy-namically generated mass gap and topological sectors. They have been formulated and analysedsuccessfully in the framework of the so-called D-theory, which provides a smooth access to thecontinuum limit. In that framework, we propose an experimental set-up for the quantum simula-tion of the CP(2) model. It is based on ultra-cold Alkaline-Earth Atoms (AEAs) located on thesites of an optical lattice, where the nuclear spins represent the relevant degrees of freedom. Wepresent numerical results for the correlation length and for the real time decay of a false vacuum,to be compared with such a future experiment. The latter could also enable the exploration ofθ -vacua and of the phase diagram at finite chemical potentials, since it does not suffer from anysign problem.The 33rd International Symposium on Lattice Field Theory14 -18 July 2015Kobe International Conference Center, Kobe, Japan∗We are indebted to D. Banerjee, L. Fallani, C.V. Kraus, M. Punk, E. Rico and S. Sachdev for helpful commu-nication. This work was supported by the Schweizerischer Nationalfonds, the European Research Council by meansof the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement 339220, the MexicanConsejo Nacional de Ciencia y Tecnología (CONACYT) through projects CB-2010/155905 and CB-2013/222812, andby DGAPA-UNAM, grant IN107915. Work in Innsbruck is partially supported by the ERC Synergy Grant UQUAM,SIQS, and the SFB FoQuS (FWF Project No. F4016-N23). C.L. was partially supported by NSERC.†Speaker.c© Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). http://pos.sissa.it/PoS(LATTICE 2015)311Quantum Simulation of the CP(2) Model Wolfgang Bietenholz1. MotivationLattice simulations of models in quantum field theory employing a quantum system — i.e.analog quantum computing — could overcome the notorious sign problem that usually occurs ifthe Euclidean action is complex; in this approach, the phase factor is naturally incorporated (forrecent reviews, see Refs. [1]). A prominent long-term goal of this concept is the exploration ofthe QCD phase diagram at finite baryon density, or a finite vacuum angle θ . In fact, within theStandard Model of particle physics, this is one of the major issues that remain mysterious; so far,the sign problem has prevented reliable numerical studies.As a step towards that goal, we present a proposal for the quantum simulation of the 2d CP(2)model, by means of ultra-cold Alkaline Earth Atoms (AEAs) trapped in an optical lattice. Un-like previous suggestions for quantum simulations of lattice field theory, our proposal involves anautomatic extrapolation to the continuum limit, taking advantage of asymptotic freedom.2. CP(N−1)modelsThe 2d CP(N− 1) models [2] are popular toy models for QCD. They can be considered ascomplex analogues of the O(N) spin models, with a covariant derivative, as we see from the actionin a continuous Euclidean plane,S[~z] =∫d2x (Dµ~z)† ·Dµ~z− iθQ[~z] , ~z(x)∈CN , |~z(x)|= 1, Dµ = ∂µ+ 12(∂µ~z† ·~z−~z† ·∂µ~z) ,(2.1)where Q ∈ Z is the topological charge, and N = 2,3,4 . . . .As a remarkable property, there is a local U(1) symmetry, in addition to the global SU(N)symmetry. In an alternative notation, we can write the fields as N×N Hermitian projection matri-ces,P(x) = |~z(x)〉〈~z(x)| , TrP(x) = 1, P(x) = P(x)2 = P(x)† . (2.2)The case N = 2 corresponds to the O(3) model. For higher N, all 2d CP(N − 1) models havetopological sectors too (in contrast to the higher 2d O(N) models). Therefore it is natural to includea θ -term, as it has been done in eq. (2.1). As further properties in common with QCD, all the 2dCP(N−1) models are asymptotically free, and they display a dynamically generated mass gap.2.1 D-theory formulationIn D-theory, asymptotically free models are formulated in a space with an additional dimen-sion; in the weak coupling extrapolation towards the continuum limit, this additional direction issuppressed by dimensional reduction [3].In particular, the D-theory formulation of 2d CP(N− 1) models starts with 2d layers, whereSU(N) quantum spins are located on a “ladder”, i.e. on a L× L′ lattice with L L′ [4]. Henceeach layer contains a set of L′ long quantum spin chains. These layers are embedded in a 3d space,which includes an additional β -direction. The Hamiltonian can be written asH =−J ∑〈xy〉N2−1∑a=1T ax Ta∗y , (2.3)2PoS(LATTICE 2015)311Quantum Simulation of the CP(2) Model Wolfgang Bietenholzwhere T ax (Ta∗y ) are spin operators in the (anti-)fundamental representation of SU(N), such that[T ax ,Tby ] = iδxy f abcT cx (with the SU(N) structure constants f abc). Here we deal with a positivecoupling constant, J > 0, which corresponds to an anti-ferromagnetic system.For N = 3 and 4, Ref. [5] pointed out that (in the limit of zero temperature and infinite volume)this system undergoes spontaneous symmetry breaking SU(N)→U(N−1), which generates 2(N−1)Nambu-Goldstone bosons. They are accommodated in the coset space of the complex projection,CP(N− 1) = SU(N)/U(N− 1), such that the low energy effective description coincides with theCP(N−1) model.In the notation (2.2), the D-theory continuum action takes the formS[P] =1g2∫ β0dx3∫d2x Tr[ 2∑i=1∂iP∂iP+1c2∂3P∂3P]− iθ Q[P] ,iθQ[P] =1pi∫ β0dx3∫d2x Tr [P∂1P∂3P] , g2 =cρs L′, (2.4)where c is the spin wave velocity, and ρs the spin stiffness.If we return to a finite L×L′ lattice on the spatial layers, asymptotic freedom implies that the(spatial) correlation length ξ (the inverse mass of the quasi Nambu-Goldstone bosons) divergesexponentially when L′ becomes large,ξ ∝ econst. L′, (2.5)where we assume L′  L, β . This divergence leads to dimensional reduction; ironically, as L′grows, it becomes negligible, since ξ  L′. Thus we recover the 2d CP(2) or CP(3) model, withθ = L′pi [4] (since the integrand of the expression for Q[P] is constant in x2).In view of the prospects to implement the SU(N) quantum spin system experimentally [6], tobe discussed in the next section, it is important to explore this exponential grow explicitly. Figure 1 10 100 1000 0  5  10  15  20ξL′no defects0.1 % defectsFigure 1: The correlation length ξ as a function of L′, for an SU(3) quantum spin model on a β ×L×L′lattice, at L = 1500 and βJ = 1000. The red crosses (blue asterisks) refer to the setting when all sitesare occupied (with 0.1% of random distributed empty sites). Error bars are included, but hardly visible.The results for ξ and for the second moment correlation length ξ2 agree up to tiny differences, which arenot visible either. With or without defect sites, ξ grows exponentially in L′, which confirms eq. (2.5) andtherefore asymptotic freedom; we further see that a moderate L′ ≈ 10 is sufficient for dimensional reduction.1 shows simulation results, which were obtained with a loop cluster algorithm [7] at L = 1500 andβJ = 1000. The boundary conditions are open in the (short) L′-direction, as in the experiment, andperiodic in the long directions (where it hardly matters). We obtain const.= 0.270784, and observethat L′ ≈ 10 is sufficient for the dimensional reduction to set in, essentially. In fact, such a numberof coupled quantum spin chains is experimentally realistic.3PoS(LATTICE 2015)311Quantum Simulation of the CP(2) Model Wolfgang BietenholzIn an experiment it may happen that a few sites in the optical lattice remain unoccupied.Frequent repetition of this experiment restores translation invariance statistically, but the correlationlength is enhanced. Numerical data for this setting, with 0.1% of such defect sites are also shown inFigure 1. Again there is clear evidence that ξ grows exponentially in L′ (as long as finite size effectsare small), in accordance with relation (2.5), which reflects asymptotic freedom. Dimensionalreduction sets is even earlier in this case.3. Experimental set-upThe goal is to implement the Hamiltonian (2.3) by ultra-cold fermionic AEAs on the sites of anoptical lattice. The latter is formed by the nodes of superimposed standing laser waves, as sketchedin Figure 2, with a spacing in the µm magnitude. The temperature is of the order of nK, whichFigure 2: Illustration of optical lattices, where the sites correspond to the nodes of standing laser waves.keeps the atoms in their electronic ground state. For fermionic AEAs, the electron and nuclear spinsdecouple almost completely in the ground state manifold, which excludes spin changing collisions.Moreover, in an external magnetic field, the interactions between the Zeeman states are SU(N)symmetric, where N ≤ 2I+1 and I is the nuclear spin [8], and the total spin is conserved. This canbe implemented up to N = 10, e.g. with 87Sr atoms [8].Next we re-write the spin operators in terms of fermionic bilinears, composed of dx and d†x ,T ax = d†xmλamm′dxm′ , −T a ∗x =−d†xmλ a ∗mm′dxm′ , (3.1)where m, m′= 1 . . .N label the states, and λ a are generalised Gell-Mann matrices, Tr [λ a,λ b] = δ ab.Now the Hamiltonian is split into a hopping term and a potential,H = Ht +HU , Ht =−t ∑〈xy〉,m(c†xmcym+ c†ymcxm), HU =U2 ∑xnx(nx−1)+V ∑x∈Anx , (3.2)where the operator cxm annihilates the nuclear spin level m ∈ {−I . . . I} at site x, t is the hoppingparameter, U the on-site interaction, nx = ∑m c†xmcxm the occupation number, and V is the energyoffset between two staggered sub-lattices, which we denote as A and B. This system is illustrated inFigure 3, and the caption explains the relation between the operators cxm and dxm. The initial stateshould be prepared with one AEA on each site of sub-lattice A, which corresponds to one fermion;on each site of sub-lattice B there are N−1 AEAs, corresponding to N−1 fermions, or one hole.The hopping parameter t fixes the coupling constant J, to be tuned by varying the energyoffset V . At strong coupling, t  U,V , the system is essentially in the eigenstates of HU , withvirtual tunnelling due to the SU(N) exchange terms in Ht . A hopping parameter expansion up toO((t/U)2) reproduces the form of H in eq. (2.3) [9], withJ =t2U[−V +U(N−3)] [V −U(N−1)] . (3.3)4PoS(LATTICE 2015)311Quantum Simulation of the CP(2) Model Wolfgang BietenholzLL’↔Figure 3: An illustration of the experimental set-up: a) SU(N) spins on a bipartite L′×L lattice, L L′. b)The N ≤ 2I+ 1 hyperfine states of an AEA, in an external magnetic field which induces Zeeman splitting.c) On sub-lattice A the SU(N) quantum spins are in the fundamental representation, and we identify cx =d†x . On sub-lattice B they are in the anti-fundamental representation, and cx = dx. Hence the exchange ofthe staggered sub-lattices A↔ B corresponds to a particle-hole transformation. d) The nearest-neighbourinteractions between T ax∈A and −T by∈B.The preparation of the initial state proceeds as follows: first one fills each site with N AEAs,where each level m is occupied once1 (this can be achieved by optical pumping [10]). Then each siteis split adiabatically into a double-well, forming the sub-lattices A and B, cf. Figure 3. The barrieris tuned to match the quantum dynamics according to the Hamiltonian H; this has already beenrealized for bosonic alkaline atoms [11]. Based on experimental experience, we expect this to befeasible up to large L = O(1000) and L′ ≈ 12; according to our results in Figure 1, this is sufficientfor dimensional reduction to set in. The results for the correlation length ξ can be confronted withexperimental measurements by means of Bragg spectroscopy or noise correlation [12].4. Phase transition at θ = pi and false vacuum decayAn odd number L′ implies θ = pi , where a first order phase transition and the spontaneousbreaking of the C (charge conjugation) symmetry is expected [13]; Refs. [4] provide numericalevidence for this scenario.In the experiment, a C transformation corresponds to a shift in the L-direction by one latticespacing. If x (x+ 1ˆ) belongs to sub-lattice A (B), then this shift transforms T ax → −T a ∗x+1ˆ, and−T ax → T a ∗x+1ˆ. The order parameter for C symmetry [14],D = ∑x∈A〈T ax T a ∗x+1ˆ−T ax T a ∗x−1ˆ〉 , (4.1)1A non-uniform occupation of the Zeeman states may also be intended, since it captures the CP(N− 1) model atfinite density.5PoS(LATTICE 2015)311Quantum Simulation of the CP(2) Model Wolfgang Bietenholzdetects dimerisation. At θ = 0, C invariance holds, and D = 0. It breaks at θ = pi , where thereare two degenerate ground states with ±D, which we denote as |±〉. These ground states may bedistinguished by bonds between nearest-neighbour sites, which can be set in two ways, as shown inFigure 4. For ultra-cold atoms, the singlets that contribute to D can be measured by spin changingcollisions [12, 15].Figure 4: An illustration of the two degenerate ground states at θ = pi . In this case, there is dimerisation (inan ambiguous way), and C symmetry is broken.At last we consider a dynamical process for a single spin chain of even length L: starting fromtotal dimerisation, one turns on the hopping parameter t adiabatically. We describe the gradualmodification by a parameter τ , which increases from 0 to 1, such that the dynamics is driven by theHamiltonianH(τ) = τH− (1− τ) J ∑x∈AT ax Ta ∗x+1ˆ . (4.2)Numerical results for the evolution of the first two energy eigenvalues, E0 and E1, are shown inFigure 5 on the left. Their evaluation also reveals the time dependent dimerisation D(t). The latteris shown in the right panel of Figure 5, along with a dimerisation map in the state |−〉. The evolutionturns it into a false vacuum, which performs an (incomplete) decay towards the true vacuum |+〉,such that D(t) decreases in an oscillatory manner. For a similar study of the real time dynamics ofcoupled bosonic spin chains, we refer to Ref. [16].0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Figure 5: The time evolution of the first two energy eigenvalues, E0/J and E1/J (on the left), and of thedimerisation D(t) (on the right), if one starts from total dimerisation and switches on the hopping parametert adiabatically. Then the Hamiltonian (4.2) drives the dynamics, turning the initial vacuum state |−〉 into afalse vacuum, which decays such that D(t) decreases and oscillates, as shown on the right.This time evolution has been computed by the exact diagonalisation of H(τ) at L = 14. Itcorresponds to the real time evolution of a false vacuum in the CP(2) model, which cannot beobtained with classical Monte Carlo simulations, due to the sign problem. The experimental set-updescribed here should enable a quantum simulation, which can be compared to the results in Figure5, and which can be extended to large L.6PoS(LATTICE 2015)311Quantum Simulation of the CP(2) Model Wolfgang Bietenholz5. SummaryWe have described a proposal for the quantum simulation of CP(N − 1) models by ultra-cold AEAs trapped in an optical lattice. They represent a model of SU(N) quantum spins, withN ≤ 2I+1, where I is the nuclear spin. This system corresponds to the 3d D-theory formulation ofthe CP(N−1) model, where dimensional reduction leads directly to the continuum limit of the 2dmodel, thanks to asymptotic freedom. Our results for the correlation length at N = 3 show that fora realistic system size, dimensional reduction leads to the 2d continuum CP(2) model.Experimental tools for the ground state preparation in such systems, and also for its adiabaticmodification, do already exist [8,10–12,15]. We discussed the dynamics of C symmetry restoration,which corresponds to a real time evolution in the CP(2) model. For small systems it was evaluatedby the diagonalisation of the Hamiltonian; for larger systems it can be measured by the experiment,which acts as an analog quantum computer.References[1] U.-J. Wiese, Annalen Phys. 525 (2013) 777.E. Zohar, J.I. Cirac and B. Reznik, arXiv:1503.02312 [quant-ph][2] A. D’Adda, M. Lüscher and P. Di Vecchia, Nucl. Phys. B146 (1978) 63.[3] S. Chandrasekharan and U.-J. Wiese, Nucl. Phys. B492 (1997) 455.R. Brower, S. Chandrasekharan and U.-J. Wiese, Phys. Rev. D60 (1999) 094502.R. Brower, S. Chandrasekharan, S. Riederer and U.-J. Wiese, Nucl. Phys. B693 (2004) 149.[4] B.B. Beard, M. Pepe, S. Riederer and U.-J. Wiese, Phys. Rev. Lett. 94 (2005) 010603.S. Riederer, Ph.D. thesis, Universität Bern, 2006.[5] K. Harada, N. Kawashima and M. Troyer, Phys. Rev. Lett. 90 (2003) 117203.[6] C. Laflamme et al., arXiv:1507.06788 [quant-ph].[7] H.G. Evertz, G. Lana and M. Marcu, Phys. Rev. Lett. 70 (1993) 875.U.-J. Wiese and H.-P. Ying, Z. Phys. B93 (1994) 147.[8] A.V. Gorshkov et al., Nature Phys. 6 (2010) 289.[9] A. Auerbach, Interacting Electrons and Quantum Magnetism (Springer, 1994).[10] G. Pagano et al., Nature Phys. 10 (2014) 198. F. Scazza et al., Nature Phys. 10 (2014) 779.[11] S. Nascimbène et al., Phys. Rev. Lett. 108 (2012) 205301.[12] I. Bloch, J. Dalibard and S. Nascimbène, Nature Phys. 8 (2012) 267.[13] N. Seiberg, Phys. Rev. Lett. 53 (1984) 637.[14] N. Read and S. Sachdev, Phys. Rev. Lett. 62 (1989) 1694.[15] B. Paredes and I. Bloch, Phys. Rev. A77 (2008) 023603.[16] S.F. Caballero-Benítez and R. Paredes, Phys. Rev. A85 (2012) 023605.7

Proposal for the Quantum Simulation of the \mathbb CP(2) Model on Optical Lattices

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Proposal for the Quantum Simulation of the \mathbb CP(2) Model on Optical Lattices

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