Proposal for the Quantum Simulation of the Model on Optical Lattices^{†}^{†}thanks: We are indebted to D. Banerjee, L. Fallani, C.V. Kraus, M. Punk, E. Rico and S. Sachdev for helpful communication. This work was supported by the Schweizerischer Nationalfonds, the European Research Council by means of the European Union’s Seventh Framework Programme (FP7/20072013)/ERC grant agreement 339220, the Mexican Consejo Nacional de Ciencia y Tecnología (CONACYT) through projects CB2010/155905 and CB2013/222812, and by DGAPAUNAM, grant IN107915. Work in Innsbruck is partially supported by the ERC Synergy Grant UQUAM, SIQS, and the SFB FoQuS (FWF Project No. F4016N23). C.L. was partially supported by NSERC.
Abstract
The 2d 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 socalled Dtheory, which provides a smooth access to the continuum limit. In that framework, we propose an experimental setup for the quantum simulation of the model. It is based on ultracold AlkalineEarth 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 vacua and of the phase diagram at finite chemical potentials, since it does not suffer from any sign problem.
Proposal for the Quantum Simulation of the Model on Optical Lattices^{†}^{†}thanks: We are indebted to D. Banerjee, L. Fallani, C.V. Kraus, M. Punk, E. Rico and S. Sachdev for helpful communication. This work was supported by the Schweizerischer Nationalfonds, the European Research Council by means of the European Union’s Seventh Framework Programme (FP7/20072013)/ERC grant agreement 339220, the Mexican Consejo Nacional de Ciencia y Tecnología (CONACYT) through projects CB2010/155905 and CB2013/222812, and by DGAPAUNAM, grant IN107915. Work in Innsbruck is partially supported by the ERC Synergy Grant UQUAM, SIQS, and the SFB FoQuS (FWF Project No. F4016N23). C.L. was partially supported by NSERC.
Catherine Laflamme, Wynne Evans, Marcello Dalmonte, Urs Gerber, Héctor MejíaDíaz, Wolfgang Bietenholz^{†}^{†}thanks: Speaker. , UweJens Wiese and Peter Zoller
Institute for Theoretical Physics, University of Innsbruck, A6020, Innsbruck, Austria
Institute for Quantum Optics and Quantum Information
Austrian Academy of Sciences, A6020 Innsbruck, Austria
Albert Einstein Center for Fundamental Physics, Institute for Theoretical Physics
Universität Bern, Sidlerstrasse 5, CH3012 Bern, Switzerland
Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México
A.P. 70543, C.P. 04510 Distrito Federal, Mexico
Instituto de Física y Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo
Edificio C3, Apdo. Postal 282, C.P. 58040, Morelia, Michoacán, Mexico
Email: wolbi@nucleares.unam.mx
\abstract@cs
1 Motivation
Lattice 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 if the Euclidean action is complex; in this approach, the phase factor is naturally incorporated (for recent reviews, see Refs. [1]). A prominent longterm goal of this concept is the exploration of the QCD phase diagram at finite baryon density, or a finite vacuum angle . In fact, within the Standard 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 model, by means of ultracold Alkaline Earth Atoms (AEAs) trapped in an optical lattice. Unlike previous suggestions for quantum simulations of lattice field theory, our proposal involves an automatic extrapolation to the continuum limit, taking advantage of asymptotic freedom.
2 models
The 2d models [2] are popular toy models for QCD. They can be considered as complex analogues of the O spin models, with a covariant derivative, as we see from the action in a continuous Euclidean plane,
(2.1) 
where is the topological charge, and .
As a remarkable property, there is a local U(1) symmetry, in addition to the global SU symmetry. In an alternative notation, we can write the fields as Hermitian projection matrices,
(2.2) 
The case corresponds to the O(3) model. For higher , all 2d models have topological sectors too (in contrast to the higher 2d O models). Therefore it is natural to include a term, as it has been done in eq. (2). As further properties in common with QCD, all the 2d models are asymptotically free, and they display a dynamically generated mass gap.
2.1 Dtheory formulation
In Dtheory, asymptotically free models are formulated in a space with an additional dimension; in the weak coupling extrapolation towards the continuum limit, this additional direction is suppressed by dimensional reduction [3].
In particular, the Dtheory formulation of 2d models starts with 2d layers, where SU quantum spins are located on a “ladder”, i.e. on a lattice with [4]. Hence each layer contains a set of long quantum spin chains. These layers are embedded in a 3d space, which includes an additional direction. The Hamiltonian can be written as
(2.3) 
where are spin operators in the (anti)fundamental representation of SU, such that (with the SU structure constants ). Here we deal with a positive coupling constant, , which corresponds to an antiferromagnetic system.
For and , Ref. [5] pointed out that (in the limit of zero temperature and infinite volume) this system undergoes spontaneous symmetry breaking , which generates NambuGoldstone bosons. They are accommodated in the coset space of the complex projection, , such that the low energy effective description coincides with the model.
In the notation (2), the Dtheory continuum action takes the form
(2.4) 
where is the spin wave velocity, and the spin stiffness.
If we return to a finite lattice on the spatial layers, asymptotic freedom implies that the (spatial) correlation length (the inverse mass of the quasi NambuGoldstone bosons) diverges exponentially when becomes large,
(2.5) 
where we assume . This divergence leads to dimensional reduction; ironically, as grows, it becomes negligible, since . Thus we recover the 2d or model, with [4] (since the integrand of the expression for is constant in ).
In view of the prospects to implement the SU quantum spin system experimentally [6], to be discussed in the next section, it is important to explore this exponential grow explicitly.
Figure 1 shows simulation results, which were obtained with a loop cluster algorithm [7] at and . The boundary conditions are open in the (short) direction, as in the experiment, and periodic in the long directions (where it hardly matters). We obtain , and observe that is sufficient for the dimensional reduction to set in, essentially. In fact, such a number of coupled quantum spin chains is experimentally realistic.
In 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 correlation length is enhanced. Numerical data for this setting, with of such defect sites are also shown in Figure 1. Again there is clear evidence that grows exponentially in (as long as finite size effects are small), in accordance with relation (2.5), which reflects asymptotic freedom. Dimensional reduction sets is even earlier in this case.
3 Experimental setup
The goal is to implement the Hamiltonian (2.1) by ultracold fermionic AEAs on the sites of an optical lattice. The latter is formed by the nodes of superimposed standing laser waves, as sketched in Figure 2, with a spacing in the m magnitude.
The temperature is of the order of nK, which keeps the atoms in their electronic ground state. For fermionic AEAs, the electron and nuclear spins decouple 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 symmetric, where and is the nuclear spin [8], and the total spin is conserved. This can be implemented up to , e.g. with Sr atoms [8].
Next we rewrite the spin operators in terms of fermionic bilinears, composed of and ,
(3.1) 
where label the states, and are generalised GellMann matrices, . Now the Hamiltonian is split into a hopping term and a potential,
(3.2) 
where the operator annihilates the nuclear spin level at site , is the hopping parameter, the onsite interaction, the occupation number, and is the energy offset between two staggered sublattices, which we denote as and . This system is illustrated in Figure 3, and the caption explains the relation between the operators and .
The initial state should be prepared with one AEA on each site of sublattice , which corresponds to one fermion; on each site of sublattice there are AEAs, corresponding to fermions, or one hole.
The hopping parameter fixes the coupling constant , to be tuned by varying the energy offset . At strong coupling, , the system is essentially in the eigenstates of , with virtual tunnelling due to the SU exchange terms in . A hopping parameter expansion up to O reproduces the form of in eq. (2.1) [9], with
(3.3) 
The preparation of the initial state proceeds as follows: first one fills each site with AEAs, where each level is occupied once^{1}^{1}1A nonuniform occupation of the Zeeman states may also be intended, since it captures the model at finite density. (this can be achieved by optical pumping [10]). Then each site is split adiabatically into a doublewell, forming the sublattices and , cf. Figure 3. The barrier is tuned to match the quantum dynamics according to the Hamiltonian ; this has already been realized for bosonic alkaline atoms [11]. Based on experimental experience, we expect this to be feasible up to large and ; according to our results in Figure 1, this is sufficient for dimensional reduction to set in. The results for the correlation length can be confronted with experimental measurements by means of Bragg spectroscopy or noise correlation [12].
4 Phase transition at and false vacuum decay
An odd number implies , where a first order phase transition and the spontaneous breaking of the C (charge conjugation) symmetry is expected [13]; Refs. [4] provide numerical evidence for this scenario.
In the experiment, a C transformation corresponds to a shift in the direction by one lattice spacing. If () belongs to sublattice (), then this shift transforms , and . The order parameter for C symmetry [14],
(4.1) 
detects dimerisation. At , C invariance holds, and . It breaks at , where there are two degenerate ground states with , which we denote as . These ground states may be distinguished by bonds between nearestneighbour sites, which can be set in two ways, as shown in Figure 4. For ultracold atoms, the singlets that contribute to can be measured by spin changing collisions [12, 15].
At last we consider a dynamical process for a single spin chain of even length : starting from total dimerisation, one turns on the hopping parameter adiabatically. We describe the gradual modification by a parameter , which increases from to , such that the dynamics is driven by the Hamiltonian
(4.2) 
Numerical results for the evolution of the first two energy eigenvalues, and , are shown in Figure 5 on the left. Their evaluation also reveals the time dependent dimerisation . The latter is shown in the right panel of Figure 5, along with a dimerisation map in the state . The evolution turns it into a false vacuum, which performs an (incomplete) decay towards the true vacuum , such that decreases in an oscillatory manner. For a similar study of the real time dynamics of coupled bosonic spin chains, we refer to Ref. [16].
This time evolution has been computed by the exact diagonalisation of at . It corresponds to the real time evolution of a false vacuum in the model, which cannot be obtained with classical Monte Carlo simulations, due to the sign problem. The experimental setup described here should enable a quantum simulation, which can be compared to the results in Figure 5, and which can be extended to large .
5 Summary
We have described a proposal for the quantum simulation of models by ultracold AEAs trapped in an optical lattice. They represent a model of SU quantum spins, with , where is the nuclear spin. This system corresponds to the 3d Dtheory formulation of the model, where dimensional reduction leads directly to the continuum limit of the 2d model, thanks to asymptotic freedom. Our results for the correlation length at show that for a realistic system size, dimensional reduction leads to the 2d continuum model.
Experimental tools for the ground state preparation in such systems, and also for its adiabatic modification, do already exist [8, 10, 11, 12, 15]. We discussed the dynamics of C symmetry restoration, which corresponds to a real time evolution in the model. For small systems it was evaluated by 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 [quantph]  [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 [quantph].

[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. CaballeroBenítez and R. Paredes, Phys. Rev. A85 (2012) 023605.