Toolbox for Abelian lattice gauge theories with synthetic matter
Fundamental forces of Nature are described by field theories, also known as gauge theories, based on a local gauge invariance. The simplest of them is quantum electrodynamics (QED), which is an example of an Abelian gauge theory. Such theories describe the dynamics of massless photons and their coupling to matter. However, in two spatial dimension (2D) they are known to exhibit gapped phases at low temperature. In the realm of quantum spin systems, it remains a subject of considerable debate if their low energy physics can be described by emergent gauge degrees of freedom. Here we present a class of simple two-dimensional models that admit a low energy description in terms of an Abelian gauge theory. We find rich phase diagrams for these models comprising exotic deconfined phases and gapless phases - a rare example for 2D Abelian gauge theories. The counter-intuitive presence of gapless phases in 2D results from the emergence of additional symmetry in the models. Moreover, we propose schemes to realize our model with current experiments using ultracold bosonic atoms in optical lattices
pacs:67.85.Lm, 03.75.Lm, 73.43.-f
The invariance of a system under the action of a local symmetry is one of the central ideas in modern physics. It is the building block of gauge invariant theories that provide the foundations of our current understanding of fundamental forces of nature. The simplest of them, quantum electrodynamics, describes the dynamics of gauge bosons (photons) and their coupling to matter (electrons). Gauge theories can also emerge as a low energy description of low-dimensional quantum spin system (1); (2); (3). Quantum spin liquids (4), exotic states of matter that do not break any symmetry down to zero temperature, provide the obvious example but the phenomenon is more general at least at a mean-field level. Quantum fluctuations, however, most of the time conspire against emerging gauge theories and strongly bind the gauge bosons together, producing a low energy sector dominated by standard spin fluctuations (5); (6); (7); (8). For this reason the validity of such emergent gauge theory descriptions still remains a subject of considerable debate (3); (9); (10); (11); (12); (13).
In the present paper we take a fresh look on the emergence of gauge theories in the context of bosonic Hamiltonians in two dimensions. We focus on Hamiltonians that can be realized in experiments with ultra-cold atoms, and describe in detail how to design the corresponding experiments. In particular we analyze a system made by two-species of bosons hopping on a two dimensional square lattice. Our main result shows that the emerging gauge theory description of the system naturally accounts for the appearance of an exotic gapless dipolar liquid phase.
In particular we show how to design the tunneling geometry of one of the two bosonic species (referred to as the auxiliary particles), such that the other species is forced to behave, at sufficiently low energies, as an effective gauge boson. We also describe the regime in which we can ensure that the gauge bosons remains mass-less even after including the quantum fluctuations. The construction we propose is very flexible, and provides a complete toolbox for generating “exotic” low energy gauge theories. The emerging gauge theories are always Abelian, but depending on the setup can have a discrete or continuous symmetry.
Their Hamiltonians, however, are different from those used to describe high-energy “standard” gauge theories and, as a consequence, their phase diagrams are richer. Standard gauge theories with discrete gauge invariance (e.g. ) show both confined and deconfined phases (6); (7); (5); (8). Both are gapped, and the extent of the latter in the parameter space vanishes as tends to infinity. As a result, the standard gauge theory exhibits only the gapped confined phase, as was originally pointed out by Polyakov (14); (15). Our “exotic” gauge theories exhibit, in addition, exotic gapped deconfined phases with dipolar and kink excitations and, most remarkably, a gapless dipolar liquid phase.
Taking a complementary perspective we can identify the proposed bosonic system as a very flexible toolbox to generate, at low energies, exotic gauge theories in cold atom experiments. Most of the seminal proposals in this direction have focused on implementing microscopic models of gauge theories, (16); (17); (18); (20); (21); (19); (22); (23); (24); (25); (26). Instead, here we focus on engineering emerging gauge theories rather than microscopic gauge theories. As a result we believe that our paper provides a plausible novel and original proposal to perform 2D quantum simulations of a gauge theories with ultra-cold atoms.
Ii The model
We consider a 2D lattice and identify its sites by Latin letters and links by the pairs , where denotes the site they originate from and the direction they point to. It is understood that the lattice direction is given by . With a slight abuse of notation we also identify the nearest neighbor of site in the direction as site . On each site of the lattice we have two species of bosons: auxiliary a-bosons and b-bosons. The creation and annihilation operators for the two species are respectively , and , . The number operator for b-bosons is defined as . We assume that a-bosons are hardcore, and thus we can have at most one a-boson per site. We also define the difference of the number operators of b-bosons on neighboring sites, , and associate it to an operator living on the link that connects the two sites.
Our aim is to investigate the low energy physics of the Hamiltonian
where h.c. stands for Hermitian conjugate. This is a generalization of the Bose-Hubbard Hamiltonian, in which the tunneling amplitudes of a-bosons are link dependent, , while those for b-bosons are uniform. More importantly, the phases of the tunneling amplitudes of a-bosons are modulated by the occupation of b-bosons on neighboring sites as illustrated in the right-hand panel of Fig.1a. The strength of the modulation, , depends on the direction. It vanishes along the direction , while (with being a positive integer). The lattice configurations for both bosons are shown in Fig.1a. The b-bosons are assumed to be non-interacting. The a-bosons interact strongly with being the dominant energy scale, so that in effect they are considered to be hard core bosons.
In the present paper we show that the low energy sector of Hamiltonian (1), is described by an exotic gauge theory for b-bosons filling . Note that similar models have been considered recently in (27); (28); (29); (30); (31); (32); (33); (34); (35); (36); (37). We discuss the details of the physical implementations of the Hamiltonian (1) to the later sections. Let us here just mention that the above Hamiltonian can be engineered experimentally in an ultra-cold atoms set up. In that context the a-bosons tunneling inhomogeneity can be realized by trapping them in optical super-lattices. The large filling of b-bosons is obtained by allowing the b-bosons to form extended tubes (38) along a direction perpendicular to the lattice. Such species dependent traps have been demonstrated experimentally by using state-selective optical lattices (39); (40); (41); (42); (43); (44). The last ingredient is the possibility to tune both interaction among a-bosons and b-bosons, at the same time ensuring that b-bosons do not interact. This requires an accurate choice of the species representing the bosons. For example one could choose K-Cs mixture that has the necessary hyperfine structure accompanied by a rich landscape of Feshbach resonances (45). One may use the inter-species resonance around G (45) where the a-bosons interact strongly (a) whereas b-bosons are essentially noninteracting (a). Still, since we assume large occupation of b-bosons per site, , one should tune the magnetic field to reduce b-bosons interaction to an almost exact zero of the Feshbach resonance.
We present in the next Section III our main result – the dynamical gauge theory and ’exotic’ phases emerging from the Hamiltonian Eq.(1). In Section IV, we discuss the possible use of our setup as a toolbox for certain kind of gauge theories. In Section V we provide two different detailed schemes that use periodic shaking mechanisms of the optical lattice and allow to generate the Hamiltonian, Eq. (1), as the effective Hamiltonian after time-averaging.
Iii Emerging lattice gauge theory
As an example of a possible gauge theory we consider a-bosons hopping on a dimerized lattice as presented in the left-hand panel of Fig. 1b. The considered tunneling amplitudes are if is odd and for even . The tunneling of auxiliary bosons along y-direction is given by . We are interested in the limit where sites and are dimerized for odd , i.e . The a-bosons are in the insulating phase when tunnelings between the dimer links vanish, i.e . The insulating phase is represented by each dimerized link containing exactly one a-boson as in Fig. 1b – we assume here “half-filling” for the a-bosons – so that in the low energy sector the term assures their hard-core nature. This gives our “zero-order” Hamiltonian:
The ground state of this model, with energy denoted as , is a Mott insulator of the a-bosons in the dimer states localized on the odd horizontal links, and an arbitrary state of the b-bosons. Being independent of the state of the b-species, it is thus highly degenerate. We denote the projector on the manifold of the degenerate ground states as .
Now, we take into account the b-bosons tunneling, as well as -tunneling () within the perturbation theory. We will assume that the horizontal even bonds are strictly zero, . The perturbation consists of the diagonal and non-diagonal terms
where the diagonal part (i.e. the part, which acts as a block matrix on the ground state manifold), and the non-diagonal part (i.e. the part that transforms the ground states outside the ground state manifold) are, respectively:
The off-diagonal part mixes different dimer links, creating an effective potential for b-bosons due to phase modulation of the inter-dimer tunnelings. We could also take into account the coupling between dimer rows due to small nonzero , but at second order of the perturbation theory, this leads to an uninteresting constant term only. Let us write the effective Hamiltonian to second order of the perturbation theory
It is interesting to notice that at this level acts as a kinetic term for b-bosons, whereas acts as a potential term .
Let us first discuss this potential term that is a genuinely second order term in (5). It depends on a linear combination of occupations around the shaded plaquettes (see Fig. 1b), containing the sites for odd . It is given by the sum of the shaded plaquettes ,
where the plaquette operators are introduced as
and plaquette strength . The operator has eigenvalues .
At this moment, in the effective Hamiltonian is expressed in terms of the original creation and annihilation operators for the b-bosons, whereas the second order part is expressed by the plaquette operators for the shaded plaquettes. It is thus interesting to express in terms of the conjugated lattice gaige theory operators. To this aim we follow the standard procedure (7), and introduce the ladder operators, , to construct a algebra. fulfills the following commutation relations with the operator defined on the same plaquette ,
while commuting with defined on different plaquettes, .
Next we can try to express the tunneling Hamiltonian of the b-bosons, , in terms of the ladder operators, where we have introduced an coupling constant (defined later in terms of tunneling amplitude) to make better contact with the standard notation used in the context of gauge theories. Generic examples of such expressions are:
where the final expressions are valid only for a large b-boson filling, . In this limit may be expressed fully in terms of plaquette ladder operators.
We also carry out the transformations: . The resulting Hamiltonian reads in dimensionless units, i.e. after dividing by :
and coupling strength . In (9), the first, second and third terms lead to first, third, and second tunneling expression in (11), respectively. Note that the sums in the above Hamiltonian run over the shaded plaquettes. The interactions induced by and all vertical bonds occur between the shaded plaquettes within each shade column independently. It is only the tunnelings that induce effective coupling between the shaded columns. This important asymmetry between horizontal and vertical direction is an key feature of our model, which in fact leads to its exotic properties.
The Hamiltonian (10) describes a spin model defined on the dual lattice whose sites are at the centers of plaquettes of the original lattice. Therefore, by performing a duality transformation, we can go back to the original lattice and write the spin model as a gauge theory. Since originally (11) involves only every second plaquette, the lattice, where the gauge theory is defined, is a coarse-grained version of the original lattice made by grouping together plaquettes inside the squares as shown in Fig. 1(b). As a result, from now on, the plaquette index denotes the entire plaquettes group. We reverse the standard duality between gauge and spin systems (46); (6); (7); (1); (48); (47) and define the electric-field operator on the links of the coarse grained lattice. Similarly . By construction, the electric-field operators obey the Gauss law: . The relation between and can be inverted in several ways, one possibility is . Under the same duality transformation, the plaquette magnetic field is transformed to the standard curl of the Wilson line, where the operator has the same form as the operator, but is defined on the links of the lattice. Since also has the same form as , and fulfill the commutation relations (III) when acting on the same link and commute on different links. After the duality transformation, the Hamiltonian (10) becomes,
where the sums run now over the rectangular, coarse-grained plaquettes (see Fig. 1b).
Expressing the low energy theory as a gauge theory gives us a better way to describe the phase diagram, and allows us to provide a precise prescription on how to measure the gauge field correlations in actual experiments. As an example, the -field configurations can be obtained directly through counting the particle number at each sites. In the context of ultracold atoms, those can be easily available through single-site measurements (49); (50); (51) or by measuring the -momentum particle density of a time-of flight image of a plaquette. Such correlations are important and perhaps the unique way to probe the phases of the model that as we will discuss in detail include a Coulomb phase, where the gauge fields correlations decay as a power law of their separation, or the gapped spin-liquid phases where the gauge field correlations decay exponentially. Moreover, magnetic field correlations can clearly distinguish between the spin-liquid phases of the present work and standard superfluid or
Mott-insulating phases, where for .
Weak coupling phases, . When , only the first term in Eq. (12) survives, and the ground state is given by a state with no magnetic charge, i.e, . In the presence of a small non-zero coupling strength , the first excited states consist of frustrating two ’s from the same column in order to fulfill charge conservation. Obviously, the phase is gapped, since the excitations have a finite energy. As we shall see below, the nature of low energy excitations undergoes crossover from the magnetic charge to magnetic dipole excitations as grows.
We start by constructing two families of delocalized excitations. The first family contains two frustrated plaquettes in the same column having magnetic charge of unit as shown in Fig.2(a). An example of the magnetic charge state is given by where so that the charges are from the same column and is the state with no magnetic charge. Such magnetic charge states are created and de-localized by the action of the first row of the kinetic part of the Hamiltonian in Eq.(11). It is thus natural to consider both in the same column , and study the energy of the maximally de-localized state of a column , where is the lattice size. Corresponding excitation energy is given by with for . The other family of states contains the magnetic dipole state oriented along :
In particular we can again consider the zero momentum state that has an excitation energy . Similarly, one could construct magnetic dipoles oriented along -direction with the opposite sign in the exponent. As opposed to magnetic charges, the dipoles delocalize by tunneling along both lattice directions due to the last two terms in the Hamiltonian (12) as pictorially sketched in Fig. 2(b).
Comparing and , we see that for we have for a weak coupling, and the lower energy excitations are magnetic charge states; the ground state is here similar to the deconfined phase of the corresponding standard gauge theories. This can be seen by measuring the expectation value of the Wilson loop (67),
where is a closed loop on the lattice, shown by the red tine in Fig.3. The deconfined phase is characterized by where stands for the perimeter of the loop . This means that the expectation value of Wilson loop in the coarse-grained lattice decays to zero exponentially fast with the perimeter of the loop.
We now focus on the scenario where dipoles have lower energy than magnetic charges, which occurs for . Here we can write the ground state for small as . As a consequence, the expectation value of the rectangular Wilson loop (see Fig. 2(c)) () of width and height is . We call this regime of the gapped deconfined phase dipole-deconfined since the first excited state consists of delocalized magnetic dipoles. A footprint of this phase is that the Wilson loop expectation value only decays exponentially fast in the horizontal width of the enclosed area.
Here we point out that as one goes from to , the ground state remains deconfined though the low-energy excitations change from being charge-like to dipole-like.
Gapless Dipolar Liquid. In order to investigate the presence of a gapless phase, we focus on the nature of the system in the limit of with constant. In this limit, we derive (details in Appendix A) an equivalence of the partition function between Hamiltonian in Eq. (12) and dipolar Sine-Gordon model in Euclidean space-time (with renaming the co-ordinates as , and ): . The dipolar action reads,
where is the fugacity of dipole excitations, denotes charge excitation field and the integral . Additionally we introduce the symbolic differentials action on a function as, where . Unlike the original sine-Gordon model (52), the kinetic part of the field operators have quadratic and quartic components. Moreover, the cosine potential contains a derivative of the fields along -direction, originating from the presence of dipolar excitations of the underlying model (52). We solve Eq.(14) variationally by using Gibbs-Bogoliubov-Feynman inequality. Our variational trial Gaussian action is given by expanding the cosine, , where is a variational parameter. The Gibbs-Bogoliubov-Feynman inequality then states that . When , the propagator for the kinetic energy is quartic along the plane, whereas for non-zero , the effective kinetic part of the action becomes quadratic along plane. The resulting free energy is expressed as,
where we have introduced a short-distance cutoff along direction corresponding to the lattice spacing and only considered the dependent terms. We find the optimum free energy by minimizing with respect to . For each fugacity , there exists a critical strength such that for , the optimum free energy is obtained for , whereas for , optimal is non-zero.
We term the phase with as a dipolar liquid phase with exotic correlation functions. For example, the charge-charge correlation function is given by when , where is a function of coupling strength . The correlation function has an intermediate character between a power-law decay and an exponential decay. Moreover, the dipolar correlation function , has a three dimensional character. As an example, we find that, when and , where are functions of coupling strength. More examples of such correlation functions are shown in Appendix B.
One dimensional Bose Liquid.
In the one dimensional limit with , the charge-charge correlation functions are similar to a dimensional gapless xy model, , and the corresponding dipolar correlations also show one dimensional character. To measure such correlations experimentally, we connect the charge correlation to the operators . The scaling of these correlation functions, in principle, can be obtained from the experimental analysis of the visibility of interference fringes, similar to the one already performed in (53).
Strong coupling phases, : We start with the limit in Eq. (12). In this limit the low energy sector is highly degenerate. Indeed any choice of constant in the direction, minimizes (12) at . This means that in a lattice we have degenerate ground states. We can label these states by the value of the constant operator of the plaquettes in a given column and see that the ground state becomes
We can construct a column Fourier basis . In particular we can see that the ladder operator for the is given by . Under periodic boundary conditions, our magnetic field operator satisfies . As a result, the ground state becomes non-degenerate and is given by, . The first excited state is again made of a manifold of states where one of the column state is changed from to , where and are two different eigenvalues of the operator. These states have a gap of order over the ground state manifold. Interestingly these excitations are localized in the direction, since once more their hopping only arises at order of the perturbation theory. Nevertheless there is no energy cost related to separating two of them in the direction so that they are energetically deconfined. The same holds in the -direction, so that these domain walls are deconfined in both direction. Next one can show that, in the periodic QED limit (7); (6), the system is equivalent to a dimensional xy-model, which has a gapped phase for (effective high temperature phase of the classical Coulomb gas). We present in Fig. 2(b) a summary of the qualitative phase diagram.
Iv A toolbox for generating low energies gauge theories
In the previous section we have provided a specific atomic set-up whose low energy is described by a class of emerging gauge theories, all Abelian, from to displaying exotic deconfined phases. This already provides a great deal of flexibility, since typically, changing the gauge group requires important changes in the implementation. Here we want to explain that the setup we propose can be easily adapted to generate an even larger set of theories. For example, here we have focused on the specific case in which the auxiliary boson are in a Mott insulating phase and are integrated out from the low energy dynamics, but we could consider a different regime in which the auxiliary particle behave as fully dynamical charged matter fields. Furthermore we have chosen the dimerized configuration of auxiliary particles sketched in Fig. 1, but this can be generalized to, for example, configurations in which there is only one auxiliary particle every plaquette, giving rise to a different low energy theory. The bosons could also be trapped in different lattice geometries (for example triangular or honeycomb lattices rather than the square lattice considered here), once more giving rise to different low energy theories, appearing at different orders in perturbation theory. Finally, the auxiliary bosons can be substituted by auxiliary fermions, so to introduce a new energy scale, the Fermi energy, that could drastically modify the low-energy physics. These are just few of the possible extensions we are currently characterizing. As a result, the set-up we are considering here constitutes a very flexible toolbox to generate gauge theories at low energy.
V Cold atoms implementation
Having described already the main results of the theory emerging from (1) let us discuss the details for the derivation of the effective Hamiltonian. Recall that we consider two species of particles, auxiliary, a-bosons and b-bosons. The assumed optical lattices potentials for these particles are
where parameter controls the relative heights of the super-lattice along -direction for a-bosons. The optical lattice depths are denoted by . We assume a tight trap (with frequency ) for the a-particles. Here is a typical optical laser wavelength. The masses of the a- and b-bosons are and respectively. The b-atoms are trapped in a square lattice with the lattice constant . In the third orthogonal direction b-bosons feel an elongated trap (compare the experimental setup in (38)). Such a scheme allows to reach the regime of high boson fillings, namely: . The superlattice potential enables dimerized tunnelings of auxiliary particles as discussed above (similar scheme has been implemented in (59)).
Let us now discuss two possible implementations of shaking procedures that lead to the effective Hamiltonian Eq. (1).
v.1 Scheme A: Shaking tunnelings and interactions
We consider a standard tight-binding model in the lattice potential given above with the Hamiltonian:
where we have added the inter-species interaction with strength and the b-boson density is denoted by to better differentiate from the a-boson density . Comparing (19) with (1) we see that the phase modulation of the a-species tunneling amplitude is missing in (19). To generate the effective Hamiltonian, (1) we add to periodically modulated (with frequency ) terms of the form
where is a shaken component of the tunnelings and is non-zero only along the direction . denotes the strength of the inter-species interaction modulation.
The harmonic shaking of the tunnelings may be realized by appropriately periodically modulating the depth of the optical lattices in the direction:
where is the shaking frequency and is the amplitude modulation strength. The modulation of lattice depth not only induces the time dependence in tunneling but also induces periodic modulation in single-particle onsite energies. However, since the tunneling rates depend exponentially on the lattice depth the main effect of the lattice modulation is on tunneling rates. The shaking frequency, while large, should not be resonant with the energy difference between the and bands (60).
Observe that not only the tunnelings but also interactions between species are assumed to be modulated. The latter can be performed with the help of magnetic Feshbach resonance (see below for discussion of possible choice of atomic species).
The Hamiltonian thus becomes . To perform time-averaging over fast oscillations we apply the unitary transformation: which transfers the time-dependence of the total Hamiltonian into the tunneling amplitudes yielding . Using Jacobi-Anger identity the Hamiltonian may be expressed as
with the tunneling amplitudes taking quite complicated form
with, let us recall, being the difference of population operators for tubes separated by . While for each tube , . The tunneling of auxiliary particles is thus modulated dynamically both in amplitude (61); (33) and in phase by the presence of b-bosons.
The expressions (V.1) simplify considerably assuming that the shaken tunneling component value is chosen such that
Then for one obtains the following approximate expressions:
Since the tunneling shaking is assumed along direction only we have , . This approximation even for yields the error less than ten percent. Within this approximation the b-boson tunneling remains unchanged. Furthermore, we assume that , so contains fast oscillating terms only and may be dropped altogether (as it averages to zero over the period of the perturbation). Then we arrive at Eq. (1).
v.2 Scheme B: Quasi-resonant lattice shaking
The previous proposal allows us to make a weak modulation of tunneling phases. To reach the regime of strong phase modulated tunnelings, we introduce another shaking scheme. We assume a standard lattice shaking potential (62); (38) represented by the Hamiltonian,
where observe the additional phase difference in shaking for a-bosons. This is easily accomplished since we assume different lattices for both species anyway. As before we assume to be much larger than but additionally we assume this frequency to be resonantly adjusted to the interspecies interaction strength (or the latter to be modified by Feshbach resonance) with the condition:
with being an integer.
To average over the fast oscillations one has now to take this resonant condition into account. Define
We transform the Hamiltonian to the rotating frame with the help of the unitary transformation taking the form
where the first term takes care of the lattice shaking with frequency and strength .
Again, in the limit of fast compared to other frequency (e.g. tunneling) scales, we carry out the time-averaging procedure, as for the first scheme, resulting in the effective Hamiltonian:
where and . The averaging procedure is valid provided is much larger than the tunnelings and .
In the limit of strong shaking strength, , and taking , we may approximate Bessel functions as
provided . As a result we obtain the Hamiltonian of the form given by Eq. (1) with modified, small , as well as renormalized . For the tunneling of the b-bosons, we adjust the shaking parameter such that .
The effect of asymptotic approximation of the Bessel function is that the number fluctuations become constrained by the shaking strength. Let us also note that such a strong shaking can induce heating due to the coupling to higher bands (60); (68) and the corresponding losses. Those effects as well as possible choices of particular parameter values are discussed in Appendix C.
v.3 Necessary properties of atomic species
The simulation of the gauge Hamiltonian requires noninteracting b-bosons with large mean occupation per site , assured through appropriate lattice arrangement (with b-bosons confined to tubes perpendicular to the plane. Thus one has to choose almost non-interacting atomic species e.g. using the zero crossing scattering length around Feshbach resonance. One experimentally available possibility is K in the hyperfine states and Cs in the hyperfine state (45). The role of auxiliary particles is taken by Cesium atoms which are hardcore bosons due to strong interactions. The magnetic field is tuned to a zero crossing of the Potassium atoms which is around Gauss (63); (65); (64). This range of magnetic field is also suitable due to the presence of Feshbach resonance in K-Cs interaction at Gauss. Thus a time-periodic inter-species interaction around Gauss can be used to generate oscillating inter-species interaction (as required for the shaking scheme A). Moreover, one can of course additionally control the scattering length using optical Feshbach resonances (66), especially for generating the time-periodic force. Use of optical Feshbach resonance can potentially allow for a utilization of a broader range of available ultracold atomic species.
Shaking scheme B, on the other hand is more versatile with respect to atomic species as no time modulation of interaction is necessary. The necessary condition is that the interspecies interactions, are strong (again possible for the exemplary species discussed above close to the Feshbach resonance). Then adjusting the shaking frequency one can easily realize the resonant condition <