Group Structure of Gauge Theories
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Returning user. Request Username Can't sign in? The upper bound shows that N should scale as which is somewhat bad since it considers a very general setting. If we restrict ourselves to the observation of intensive quantities we expect this scaling to be much better after completion of this work we became aware of a later work which proves this behavior [ 70 ].
However, there are observables in lattice gauge theories, e. Wilson loops, which do not fulfill this requirement and thus need to be bounded by more general estimates like the ones given above. With this general scheme for the digital construction of three-dimensional lattice gauge theories at hand, we can turn to the implementation of some concrete examples with ultracold atoms.
Typical gauge groups of interest are compact e.
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U 1 , for which the link Hilbert spaces are infinite. A truncation of this Hilbert space is therefore required to make the quantum simulation feasible.
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Previous proposals have performed this truncation in the representation basis [ 13 — 15 ]. This procedure, however, spoils unitarity of the group element operators U and prevents the use of isometries see section 3. Thus, the Hilbert space of the gauge field should be truncated using group element states instead. A truncation of U 1 in this sense is given by the finite groups which converge to U 1 in the limit.
The digital quantum simulation of gauge theories has been studied in [ 55 , 56 ].
We summarize below their main features, and then we build on these to tackle the simulation of simple non-abelian gauge models with dihedral symmetry given by the group D N. Lattice gauge theories with a finite abelian gauge group play an important role as they approximate compact QED [ 71 ].
Since the Hilbert space of the gauge field is reduced to dimension N if the gauge group is considered, ultracold atoms can be used to represent these gauge degrees of freedom. These N states are labeled by and we define unitary operators P and Q on them:. Since the group is abelian, its representations are one dimensional and we need to consider a single fermionic species, , on the vertices.
We can now define the Hamiltonian of lattice gauge theory with fermionic matter:. Possible implementations for [ 55 ] and [ 56 ] with isometries have been discussed in two space dimensions. These proposals can be readily generalized to three dimensions following the scheme presented in the previous section. The matter content is represented by a fermionic atomic species whereas the gauge fields can be represented by a second atomic species with the appropriate ground state manifold, e. Furthermore, auxiliary atoms must be trapped in the center of each second cube.
These species are confined to the desired lattice geometry by suitable optical lattices and their interactions are realized by ultracold atomic scattering. Since the type of interactions appearing in two and three dimensions are the same, the implementation in three dimensions follows closely the steps explained in [ 55 , 56 ] and the reader should refer to the original references for more details.
Here we just report the bounds on the Trotter error that can be computed following the discussion in section 3. In three dimensions and for the gauge group , we obtain the first order formula see 44 :. Note that these formulas give a more accurate bound with respect to the original analysis in [ 55 , 56 ]. We now turn our attention to a simple example—using the above procedure for the quantum simulation of a D 3 lattice gauge theory.
It is the smallest non-abelian group, and therefore provides both a simple and a non-trivial example. In the previous works we have constructed examples. Unfortunately, such a series does not exist for non-abelian groups of interest such as SU 2. Therefore, our interest in D 3 as an example is not due to its physical properties or using it to approximate another group but rather as the simplest non-abelian example.
D 3 gauge theory was discussed in the context of doubled lattice Chern—Simons theory in [ 73 ]. The dihedral group D N can be characterized by a set of rotations R in a two-dimensional plane and reflections S along a certain axis:. It is thus useful to write the states of the gauge field Hilbert space as states living in the tensor product of an N -dimensional Hilbert space and a two-dimensional one,.
In the implementation, such a product Hilbert space can be realized by using two atoms with the appropriate hyperfine structure. This allows us to write down the Hamiltonians.
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The last part, the electric Hamiltonian, takes its simplest form if the states in are expressed in the usual group element states but the states of in , the conjugate basis to defined by , see the appendix for details :. Generally, the coefficients in 52 can be chosen in a way that respects the combined large- N and continuum limit as in the case. However, we do not have this option for D N as explained above; nevertheless, it is convenient to identify the second term of 52 with the electric energy of a lattice gauge theory see above , and fix the coefficients accordingly.
Group Structure of Gauge Theories
Our implementation scheme is in principle applicable to all dihedral groups but we focus here on the simplest case D 3 isomorphic to the group of permutations S 3. We first discuss the system we will use as a platform to perform the quantum simulation. For the simulation of the matter fields it is crucial to use fermionic atoms to obtain the correct commutation relations. A natural, minimal choice for the two fermionic d. These atoms must be trapped by a superlattice that allows one to modulate the depth of the minima to account for the staggering and the tunneling rate between nearest neighbors to switch tunneling on and off in the different steps of the Trotter sequence.
To simulate the gauge field and auxiliary Hilbert spaces, we will exploit the product structure as mentioned above:. In total, we need four different atomic species: two atoms trapped at the middle of each link, and two extra atoms that must be addressed independently of the previous two in the middle of each second cube. For the links, we identify:. Every state of the Hilbert space on the link can be obtained as a tensor product of the two multiplets, e. It is useful to introduce unitary operators P 3 , Q 3 and P 2 , Q 2 acting, respectively, on the three-level and two-level atoms.
They are defined as:. The operators P 3 , Q 3 fulfill the algebra whereas the operators P 2 , Q 2 fulfill the algebra. The link and auxiliary atoms must be trapped in the desired positions by arranging suitable optical potentials. The individual minima must contain exactly one atom and must be deep and well separated so that the dynamics is frozen no tunneling, no interactions between nearest neighbors. When requested, the lattices must undergo a rigid translation so that specific pairs of atoms can overlap and interact via two-body scattering.
The resulting setup—for convenience projected to two dimensions—is depicted in figure 5. Figure 5. The simulating system consists of one atomic species on the vertices representing the matter red and two for both the gauge fields blue on the links and the controls green located at the center of every second cube projected into two dimensions for better visualization. The simulated degrees of freedom are encoded in the hyperfine structure of the atoms, i. The alternating occupation of vertices with fermionic atoms shall illustrate the staggered fermion picture, in which this configuration corresponds to the non-interacting vacuum see Dirac sea in the continuum.
The empty green circles indicate the need to move the auxiliary atoms between even and odd cubes. All interactions between the constituents of the simulating system from above are in the form of two-body scattering. As will become clear in the following, we need to impose specific constraints on the scattering. First we want interactions that are diagonal in m F and do not change the internal level of the atoms. This can be achieved by lifting the degeneracy of the hyperfine multiplets such that transitions changing m F will cost energy.
A possible way to do this is by introducing a uniform magnetic field which adds the following correction to the Hamiltonian Zeeman shift :. The energy splitting has to be different for different atomic species to avoid resonant exchanges, therefore we need to choose species with different Lande factors.
Another possible approach to realize the different energy splittings is to address each species individually, for example exploiting the AC Stark effect. Second, at some point we need to modulate the interaction strengths depending on the internal level of the atoms. This can be achieved for example by spatially separating the different m F levels via a magnetic field gradient.
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The different m F levels will experience forces pointing in different directions and reach different equilibrium positions within the same potential well.