Abstract
Topological quantum phases cannot be characterized by Ginzburg–Landau type order parameters, and are instead described by nonlocal topological invariants. Experimental platforms capable of realizing such exotic states now include synthetic manybody systems such as ultracold atoms or photons. Unique tools available in these systems enable a new characterization of strongly correlated manybody states. Here we propose a general scheme for detecting topological order using interferometric measurements of elementary excitations. The key ingredient is the use of mobile impurities that bind to quasiparticles of a host manybody system. Specifically, we show how fractional charges can be probed in the bulk of fractional quantum Hall systems. We demonstrate that combining Ramsey interference with Bloch oscillations can be used to measure Chern numbers characterizing the dispersion of individual quasiparticles, which gives a direct probe of their fractional charges. Possible extensions of our method to other manybody systems, such as spin liquids, are conceivable.
Introduction
Manybody systems with spontaneous symmetry breaking can be described by Ginzburg–Landau theories, formulated in terms of local order parameters. This powerful approach provides a universal description of systems with very different microscopic Hamiltonians but with similar type of symmetry breaking, such as superfluids and ferromagnets. The integer and fractional quantum Hall effects^{1,2,3} in contrast are examples of quantum phases of matter, for which no local order parameters exist. Instead, these systems are described by nonlocal topological invariants^{4}. The fractional charges of elementary excitations^{2,3}, the manybody Chern number (ref. 5) and, in the case of quantum spin liquids, fractional quantum Hall systems and fractional Chern insulators^{6,7,8,9,10,11,12,13}, the groundstate degeneracy on a torus^{14}, constitute important examples of topological order parameters.
Probing—and, in some cases, even defining—the nonlocal order parameters of topological systems with strong correlations represents a major experimental and theoretical challenge. Many indicators used in the theoretical description of such systems, such as the entanglement entropy^{15,16} and spectrum^{17}, are difficult to probe directly in current experiments, although first steps in this direction have been undertaken^{18,19,20,21}. Previously, it has been shown that edge excitations can be used to detect topological orders by measuring their fractional charges^{4,22,23,24} and statistics^{4}.
Motivated by the coherent control of mesoscopic quantum systems achievable in recent experiments, we explore new ways of detecting topological order. Our method is ideally suited to systems of ultracold atoms, which recently emerged as a new promising platform for realizing and probing various topological states of matter. The ability to perform interferometric measurements in such systems is one of their key technical advantages in comparison with other experimental setups. Cold atoms provide a versatile toolbox, allowing to engineer not only singleparticle properties of Hamiltonians, such as the shape of optical lattice potentials, but also the interactions between particles^{25,26,27,28}. Recently, the Chern number has been measured in transport experiments^{29} and the celebrated Haldane model has been realized^{30} in systems of weakly interacting ultracold atoms. An experimental realization of the fractional quantum Hall effect in such systems^{6,7,31,32,33,34,35,36} should be within reach with the currently available tools. In addition, direct and fully coherent control over individual atoms has been demonstrated in experiments with ultracold quantum gases, see, for example, refs 37, 38.
Here we present a concept of using impurity atoms as coherent probes of the topological invariants of strongly correlated manybody systems of host atoms. Our approach allows one to measure topological order parameters directly in the bulk of the system, without the need of relying on the bulkedge correspondence. The main idea is to map out the topology characterizing the effective bandstructure of elementary quasiparticle (qp) excitations. As will be pointed out, it is intimately related to the topological order of the groundstate. In particular, we show how the Chern numbers of the effective qp bandstructures can be measured by combining Bloch oscillations with a Ramsey interferometric sequence. We point out that they are directly related to the corresponding (fractional) charges for arbitrary Abelian quantum Hall states, and show how the Chern number of the manybody groundstate can be derived. Our scheme extends earlier ideas^{39,40,41,42}, which have been developed to measure topological invariants of essentially noninteracting particles (ultracold atoms in particular), to the realm of strongly correlated quantum manybody systems. More generally, our interferometric method paves the way for a detailed investigation of qp properties, including, possibly, their braiding statistics.
Results
Interferometric detection of manybody Chern numbers
The key idea of our approach is to measure the Chern number of the effective qp bandstructure using a generalization of the interferometric technique developed for noninteracting systems in refs 39, 40. First, let us briefly summarize the main idea of the interferometric protocol for a weakly interacting Bose–Einstein condensate loaded in a twodimensional (2D) Bloch band in a system with an effective magnetic field. Because of their experimental relevance for ultracold atoms we will discuss lattice systems. The first ingredient to measure the corresponding Chern number is a direct detection of the geometric Zak phase ^{39} for a given value of the quasimomentum k_{y}.
To this end, the condensate is moved to (k_{x}, k_{y}), where a π/2 Ramsey pulse is used to prepare a superposition of two internal states σ=↑, ↓. The initial wavefunction thus reads . Next, sufficiently weak opposite forces F(↓)=−F(↑) are applied such that the two components undergo Bloch oscillations (Fig. 1b). When the force is applied along the x axis for a time Δt=G_{x}/2F, where G_{x} denotes the width of the magnetic Brillouin zone (BZ) containing an integer number of magnetic flux quanta, the two components pick up a relative geometric phase (ref. 43) and the wavefunction reads . By recombining the two states using a second π/2 Ramsey pulse, the Zak phase can be read out. This measurement can be repeated for different values of k_{y} (ref. 40), and the winding of the Zak phase across the magnetic BZ in the k_{y} direction (size G_{y}) gives the Chern number^{44},
For more details, including the discussion of dynamical phases and gauge dependence of the Zak phase, the reader is referred to refs 39, 40, 41, 42.
Now we extend the interferometric protocol to strongly correlated systems. The key idea is to apply the exact same sequence as described above to a single qp excitation in the host system. To obtain fully coherent control of the qp, it is coupled to a mobile impurity. The resulting composite object (the impurity bound to the topological excitation) will be called a topological polaron (TP), which is at the heart of our scheme. (We call this bound state a TP because, as will be shown, it inherits the topological properties of the qp excitation.) We assume that the impurity has an internal degree of freedom σ, the only role of which is to enable Ramsey interferometry as described above. By introducing two pseudospin components σ=↑, ↓ a measurement of their relative phase is sufficient, that is, one of the pseudospin states effectively acts as a reference for the second. By applying opposite forces σ^{z}Fe_{x} directly to the two impurity components for the same time Δt=G_{x}/2F as in the noninteracting case (where G_{x} is defined for host particles), a topological invariant can be measured, which will be identified as the Chern number of the effective TP bandstructure (for more details see Methods section). Note that now k_{x}, k_{y}〉 stands for the full manybody wavefunction describing the TP at quasimomentum (k_{x}, k_{y}), and in this sense true manybody Zak phases are measured.
Now our main results are summarized. We will establish that the Chern number of the TP bandstructure is directly related to the fractional charges of the qps. We will use Chern–Simons effective field theory, valid at low energies, for the description of Abelian fractional quantum Hall states and their excitations. In addition, we introduce a numerically exact technique for calculating Zak phases of TPs for small lattice systems relevant for experiments. As a concrete example, Laughlintype fractional Chern insulator states at filling fraction ν=1/m (defined as a ratio of particle density to flux density) will be considered. For that case, we show that the TP Chern number, discussed in the protocol above, is given by the inverse of the fractional qp charge e*=e/m (e is the charge of particles in the host manybody system),
Therefore, a measurement of the TP Chern number directly yields the fractional qp charge.
The result in equation (2), moreover, indicates a direct relation to the manybody Chern number (ref. 5) of the incompressible ν=1/m Laughlin groundstate, . (By a fractional Chern number , with q, p integers, we refer to a situation where p degenerate states on a torus share a total Chern number q, giving rise to a fractionally quantized Hall conductivity (ref. 5). q is an integer topological invariant characterizing the vector bundle defined by the set of degenerate states on a torus.) Similar relations are true for any Abelian quantum Hall state, where e/e* needs to be expressed in terms of the Kmatrix. In that case there exist J=1, ..., n different qp types, with fractional charges and qp Chern numbers . The Chern number of the incompressible fractional quantum Hall state is given by (see Methods section).
TPs and qp Chern numbers
Our discussion of TPs will be organized as follows. In this section we focus on the physical picture of TPs in fractional quantum Hall systems. We discuss the origin of equation (2), assuming that an impurity particle is strongly bound to a qp. Therefore, the impurity current (where μ=x, y, t and σ=↑, ↓) is equivalent to the qp current, which is conserved because of a global U(1) symmetry. Detailed discussion of the binding mechanism between the impurity and a qp will be presented in the next subsection.
As in the theoretical proposal of ref. 40 and the experimental demonstration of ref. 39, we assume that the twocomponent TP can be controlled by the force F acting on the impurity. (In experiments a spindependent force F could be realized by a magnetic field gradient.) We model this by coupling the TP current to an external field that acts differently on the two internal states of the impurity but does not affect host particles. We describe this field by the potential B_{μ}(σ) so that the external force acting on the impurity is given by F_{i}(σ)=−q(∂_{t}B_{i}+∂_{i}B_{0}), where q denotes the charge of the impurity associated with the field B_{μ}. We emphasize that the impurity is not affected directly by the gauge field creating a fractional quantum Hall state for host particles. However, by binding a qp of the surrounding fractional quantum Hall state it acquires topologically nontrivial dynamics. This can be understood as arising from the particle–vortex duality of the fractional quantum Hall effect, which implies that the moving qp bound to the impurity sees surrounding host particles as a source of effective flux.
To avoid complications arising because of an underlying lattice, we first employ an effective field theory to capture the universal topological properties of our system (in particular, Chern numbers). It is valid when all relevant length scales arising from gauge fields and interactions are large compared with the lattice constant. Using this Chern–Simons description of the fractional quantum Hall effect one can show (see Methods section for details) that the effective Lagrangian describing the TP is given by (throughout the paper we set ħ=1)
Here A_{μ} denotes the external gauge field corresponding to the homogeneous magnetic field b_{z} seen by the host particles in the manybody quantum Hall system, ∂_{x}A_{y}−∂_{y}A_{x}=b_{z}. In the case of ultracold atoms, b_{z} corresponds to an artificial gauge field^{45}. The second term in the Lagrangian (3) describes the emergent coupling of the TP to the gauge field A_{μ}, albeit with a fractional A_{μ} charge e*=e/m. Hence, according to equation (3) the TP sees an effective magnetic field . Note that we consider a situation where the host system has a global U(1) symmetry, giving rise to conserved charges (or particle numbers in the context of ultracold atoms).
We emphasize again that the emergent coupling of impurity particles to the gauge field A_{μ} arises only through binding of a qp excitation. For Laughlin states, a qp acquires an Aharonov–Bohm phase 2π when going around a single host particle. Hence, the effective flux density seen by the qp is equal to the density of host particles, which is 1/m times the magnetic flux density of A_{μ}.
Now the Chern number of the TP can easily be calculated. To this end we note that, in a homogeneous magnetic field b_{0}, the Berry curvature is constant, . Similar to the manner in which we described the interferometric protocol above, we defined the TP Chern number by integrating the Berry curvature over the entire magnetic BZ of the host manybody system. Its size is G_{x} × G_{y}=2πeb_{z}, where b_{z} is the magnetic field seen by host particles. Because the Berry curvature seen by the TP is we obtain
as claimed in equation (2).
Note that because of gauge invariance has to be an integer^{46,47}, as long as the TP groundstate is nondegenerate for all k (see also Methods section). Therefore, when the qp charge is not the inverse of an integer, the band of TP groundstates needs to have degeneracies. When e*/e=p/q, where p and q are relative prime, we expect p degenerate bands sharing a total Chern number of . We find numerical signatures for this degeneracy, and a specific example is discussed in the Supplementary Note 1.
Alternatively, the qp charge e*/e of the TP could be obtained directly from an interferometric measurement of the Aharonov–Bohm phase in the magnetic field b_{z}. Here A denotes the area encircled by the TP and Φ_{0} is the magnetic flux quantum. Because obtaining control over the impurity momentum is technically less challenging than achieving full spatial resolution, the interferometric measurement of the Chern number should be easier to implement.
Microscopic description of TPs
Now we provide a microscopic description of TPs in correlated manybody systems interacting with a single impurity particle. To this end, we investigate concrete models of interacting fermions on a lattice, whose groundstates are integer and fractional Chern insulators^{6,7,8,9,10,11,12}. We develop an exact numerical method to solve this problem and compare our results to an approximate strong coupling theory of TPs, which we also introduce below. Like before, we concentrate on states in the Laughlin universality class, with a single qp type. Although the field theory employed so far assumed a continuous system in a homogeneous magnetic field, it should describe correctly the universal topological properties of the lattice systems, which will be investigated below.
We consider a topologically ordered phase in a manybody system (referred to as the host system), described by the groundstate of a Hamiltonian . For probing qp excitations, we introduce a mobile impurity with two internal states, described by . Here denotes the kinetic energy and the position operator of the impurity. The two internal states ↑ and ↓ of the impurity experience opposite forces ±F, as described by the Pauli matrix . To bind qps to the impurity, a local interaction with the manybody system is introduced. Its concrete form can differ from model to model; however, for simplicity we will assume throughout that it is independent of the internal state of the impurity. Thus, our system is described by the Hamiltonian . Local interactions between particles with multiple internal states are routinely realized in systems of ultracold atoms.
In equilibrium, that is, for F=0, the groundstate describes a TP and can be labelled by its quasimomentum q and its internal state σ=↑, ↓. The external force F couples to the quasimomentum q of the TP. This drives Bloch oscillations where the quasimomentum changes according to , where σ^{z}=1 (σ^{z}=−1) for σ=↑ (σ=↓), respectively). By applying the scheme described in ref. 39 to the states the geometric Zak phases characterizing the TP bandstructure can be measured (Fig. 1). As discussed in the Methods section, the corresponding Chern number is obtained by integrating the Berry curvature seen by the TP over the magnetic BZ of the host manybody system.
Strong coupling theory
Before turning our attention to concrete models, we introduce the strong coupling theory of TPs, which is inspired by Landau’s and Pekar’s treatment of the polaron problem in polar crystals^{48,49}. There are several important energy scales for describing TPs. First, qp excitations are characterized by the bandwidth J_{qp} of their effective dispersion and by the energy required for their creation, which is set by the bulk excitation gap Δ_{0}. Second, the impurity is characterized by the effective hopping J_{I} and the coupling strength to the host particles V. Our description of TPs requires the following hierarchy of scales. The qp gap Δ_{0}, which is larger than J_{qp}, should be larger than J_{I} and V, that is, . The impurity–host particle interaction strength V should be chosen such that the impurity binds precisely one qp. We also need J_{qp} to be smaller than J_{I}, that is, so that the total momentum of the TP, which is effectively shared by the impurity and the qp, resides predominantly in the qp. Another way of understanding this requirement is that the impurity should be fast compared with the qp and thus follow its dynamics adiabatically.
Next, we introduce a frame where the total momentum of the TP is conserved explicitly. To this end, we restrict ourselves to a single qp and approximate , with being the effective dispersion of qps and the qp state with momentum k. We apply the unitary transformation introduced by Lee, Low and Pines (LLP)^{50}, where is the impurity momentum operator and is qp position operator conjugate to its momentum operator . The transformed Hamiltonian reads
as will be explained now.
The kinetic part of the impurity Hamiltonian commutes with the impurity momentum, , and remains unchanged. denotes the interaction Hamiltonian for a qp localized in the origin of the new polaron frame, that is, . Here we assumed that (within the single qp approximation) depends only on the relative distance between impurity and qp, and we used that . Finally, because the qp momentum is shifted by an amount in the last line, .
Under the strong coupling conditions outlined above, we can make a product ansatz for the TP wavefunction, , where the impurity follows the qp adiabatically. In the polaron frame the impurity sees a quasistatic potential created by the qp and its wavefunction is determined by the strong coupling impurity Hamiltonian
This leads to a modification of the effective qp dispersion in equation (5), which we approximate by . We proceed by eliminating the last term in the first line of equation (5) by a timedependent gauge transformation . In the resulting effective Hamiltonian the force F couples to the conserved momentum k and the qp Hamiltonian in the strong coupling theory thus reads
Next, we discuss how the topological invariant measured by the TP relates to the Chern number of the effective qp bandstructure using the strong coupling wavefunction. During the protocol described above (see also Fig. 1) the force is applied and the TP wavefunction follows its groundstate adiabatically. Thereby, it picks up a phase containing a geometric part of 2πν_{TP}, which is measured by the Ramsey sequence (dynamical phases are discussed in refs 39, 40). Owing to the product form of the strong coupling wavefunction we find two contributions, ν_{TP}=ν_{qp}+ν_{I}.
The first contribution is picked up by the qp wavefunction, . When the path in momentum space described by the TP in the interferometer encloses the (magnetic) BZ, the qp invariant is related to the TP Chern number defined above,
The second contribution ν_{I} is picked up by the impurity part of the wavefunction. In the adiabatic limit of small F it is . This term corresponds to a geometric phase because it does not vanish in the limit when F→0 and needs to be considered in general. It measures the displacement of the impurity wavefunction relative to the qp located in the origin of the polaron frame (recall that in the interaction Hamiltonian) and can become relevant in lattice systems. However, when a closed loop is considered as in the interferometric sequence we discuss, the impurity invariant vanishes, ν_{I}=0.
TPs in Chern insulators
Now we turn our attention to a concrete model of interacting particles on a lattice, described by the Hofstadter–Hubbard Hamiltonian
The first term is the celebrated Hofstadter model^{51}, and it describes free particles hopping between the sites (m, n) of a square lattice in a magnetic field (using Landau gauge), where α denotes the magnetic flux density per plaquette (in units of the flux quantum) and t is the hopping amplitude. The second term describes nearest neighbour interactions of strength U between the particles. Here we consider fermions for concreteness, , but a similar Hofstadter–Hubbard model has also been discussed for bosons with contact interactions^{6,7}. For sufficiently small values of α, the groundstates of equation (9) show the integer and fractional quantum Hall effect depending on the filling fraction ν.
To study TPs we consider the Hofstadter–Hubbard model equation (9) at filling ν=1/m on a torus. We choose the number of flux quanta in the host manybody system N_{φ}=Nm+1 such that the groundstate of contains one quasihole (qh) excitation. Next, we add a single impurity, described by , hopping between the sites (located at positions r_{m,n}) of the same 2D lattice,
To bind the impurity to the qh, its interaction with the surrounding fermions is modelled by a repulsive contact potential, . In the following we consider an impurity with only a single internal state for simplicity.
The model proposed above (equations (9) and (10)) can be implemented with ultracold atoms. In refs 52, 53 the Hofstadter Hamiltonian was realized for bosons, and interactions can be introduced by using deep optical lattices. The impurity could be realized by adding a second atomic species, with different metastable internal states.
In the Methods section we present a formalism for calculating the full manybody TP wavefunction exactly for a given total momentum q, based on the LLP polaron transformation. Here we use this approach to obtain both the dispersion relation and the Chern number of TPs.
In Fig. 2a our results are shown for an integer Chern insulator, with and without fermion–fermion interactions U. We consider the case α=1/4 and use Landau gauge where the size of the magnetic unit cell is (a_{x}=4a) × a, with a denoting the lattice constant. We calculate the Chern number of the TP from the winding of the manybody Zak phase, ^{44}, see Methods section for details. For comparison, the result of a simple strong coupling analysis is shown, where the impurity is bound to a free hole. Although the distribution of Berry curvature differs, the strong coupling theory predicts correctly the TP Chern number, .
In Fig. 2b we repeat our calculations for a fractional Chern insulator at ν=1/3, corresponding to a 1/3 Laughlintype state. (We checked numerically that the incompressible groundstate of has a fractional Chern number and the expected threefold groundstate degeneracy on a torus.) The Chern number of the TP, which is determined as the winding of the Zak phase of the TP in Fig. 2b, has the value . This is expected from the fieldtheoretical arguments given in the beginning (equation (2)).
An important quantity for estimating the robustness of our protocol is the energy gap of the TP state. To get some analytical understanding, we first consider the continuum limit α→0 when the system enters the quantum Hall regime. Here the impurity (mass M_{I}) interacts with fermions (mass M_{F}) in a fillingν Laughlin state through a local contact interaction . Within the strong coupling theory introduced above, we find a finite TPbinding energy given by
where is the magnetic length and the cyclotron frequency.
In Fig. 3 we calculate the TP spectra numerically for small lattice systems and find lowenergy states where the impurity is bound to the qh forming the TP. We observe that the TP gap Δ_{TP} shows large finitesize effects. The TPbinding energy oscillates at a frequency given by the system size, but at the discrete momenta allowed by the finite quantization volume—which are indicated in the figures by circles—we find the largest values of the TP gap. In the case of integer Chern insulators, this TP gap constitutes a sizable fraction of the cyclotron gap . Note that ω_{c} provides an upper bound for the TP gap, Δ_{TP}<ω_{c}, because at this energy unbound excitons can be created in the bulk. In the case of fractional Chern insulators in contrast, the TP gap is only a small fraction of the cyclotron energy, but so is the bulk gap Δ_{LN} of the fractional Chern insulator, Δ_{LN}<ω_{c} (ref. 7). In this case Δ_{LN} similarly provides an upper bound for the TP gap as the cyclotron energy in the integer Chern insulator case, Δ_{TP}<Δ_{LN}.
To form a TP for mapping out the topological order of the host manybody system, we wish to realize the strong coupling limit described above. Thus, for impurity dynamics to be fast compared with qp tunnelling, a weak qp dispersion is desirable. This can be achieved in lattice systems by including longrange hoppings for the host particles, see ref. 54. We simulated TPs in systems with longrange tunnellings of this kind (see also Supplementary Note 1) and verified that the TP Chern number is a robust feature of the groundstates.
Discussion
By coupling a mobile impurity to the topological qp excitations of an incompressible manybody groundstate of host particles, a TP can be formed. Using internal degrees of freedom of the impurity, fully coherent control can be gained over individual qp excitations of the host manybody system. We demonstrated that TPs can be used to measure the topological invariants characterizing such qp excitations. In particular, we developed an interferometric measurement scheme for qp Chern numbers and showed that these are directly related to the fractional charges of topological excitations in two dimensions. To this end, we generalized schemes developed earlier for noninteracting systems and showed by explicit calculations that our scheme can be applied to integer and fractional quantum Hall systems and Chern insulators.
In systems of ultracold atoms, impurities can be realized, for example, by introducing another atomic species and their internal degree of freedom corresponds to different hyperfine states. To neglect the effects of interactions between TPs, the density of impurity atoms needs to be well below one atom per cyclotron orbit. We expect that, in order to realize TPs in an experiment, a main challenge will be their preparation. Because TPs can carry fractional charges as in the case of fractional Chern insulators, their preparation requires nonlocal operations in general. Therefore, we suggest to build impurity atoms into the state as defects already before preparing the incompressible manybody state of host atoms. One concrete approach would be to start from this system and cool it down into its groundstate. If the densities of majority and impurity atoms are suitably chosen, this leads to the formation of TPs. Alternative approaches have also been discussed in the literature^{55,56,57,58}.
The method presented in this article can provide a powerful experimental tool for detecting topological order in interacting manybody systems. It is ideally suited for cold atom experiments, which offer a rich toolbox with precise and fully coherent control over individual atoms. The concept of TPs as experimental probes for topological order can be generalized to other systems, however, including, for example, qp excitations on topological superconductors, or systems with symmetryprotected topological orders. Another interesting direction would be to probe the (nonAbelian) braiding statistics of anyons by coupling them to impurities and forming TPs. In this case, one can envision interferometric sequences designed to probe braiding statistics, which work in realspace rather than momentum space, as considered in this work.
We expect that our idea—which is to witness topological order in a groundstate by probing the topology characterizing the effective bandstructure of its qp excitations—may also be of broader theoretical interest. It provides a direct route how the concepts developed for noninteracting systems can be generalized to correlated manybody systems: we can define a set of topological invariants for a manybody state by considering singleparticle topological invariants of all qp excitations. These may include Chern numbers, invariants or quantized Zak phases. The approach may be useful, for example, in the study of fractional topological insulators^{59} or quantum spin liquid states^{14}. In the case of chiral quantum spin liquids the analogy to our work is particularly close. On a meanfield level the chiral quantum spin liquid can be described by an integer Chern insulator formed by spinons with a unity Chern number (for example, ref. 60). Therefore, it would be interesting to study the binding of a spinon to a mobile impurity and apply the interferometric method introduced in our paper to detect the spinon Chern number.
Methods
Chern number of TPs
In the main text we generalized the interferometric protocol developed for the measurement of Chern numbers of noninteracting particles^{39,40} to a single qp excitation in a strongly correlated manybody system. The coupling to an impurity particle was necessary for adapting the interferometric protocol. In this way a topological invariant of the TP was defined which, as we will now argue, is the Chern number of the TP.
Consider a qp state , which is characterized by its quasimomentum (k_{x}, k_{y}). Let us assume that the state is nondegenerate for all quasimomenta. The corresponding Chern number is then defined by the quantized Hall response to an external force F. In the case of TPs the external force F acts directly on the impurity and couples to the quasimomentum according to . Using the Kubo formula, Thouless et al.^{46} have shown that the Chern number can be defined as an integral of the TP Berry curvature over the (magnetic) BZ,
The Berry curvature is defined through the Bloch wavefunction constructed from . The magnetic BZ is defined by the periodicity of the qp Hamiltonian, including gaugedependent vector potentials^{61}. The periodicity of the TP wavefunction , where G is a reciprocal lattice vector, guarantees the integer quantization of the Chern number^{46,47}.
The magnetic BZ of the TP is determined by the microscopic details of the model for both the impurity as well as the host manybody system. Let us assume that the impurity either lives in the continuum (such that effectively the impurity lattice constant a→0 vanishes) or that the unit cell of the impurity lattice fits into the magnetic unit cell of the host manybody system an integer number of times in a commensurable way. In either case, the magnetic BZ of the TP is then equal to the magnetic BZ of the host manybody system. This explains why, in our interferometric protocol, the TP Chern number needs to be defined as the winding of the Zak phase of the TP over the magnetic BZ of the host manybody system, k_{y}→k_{y}+G_{y}.
Field theory of TPs
Here we discuss a field theory description of TPs, allowing us to derive the topological invariants characterizing their effective bandstructure. We consider a more general situation than discussed in the main text, and allow quantum Hall states with an arbitrary Abelian topological order. The generalization of our interferometric protocol to this case is discussed in the end.
Our starting point is a fieldtheoretical description of the topologically ordered host manybody system. We consider an incompressible groundstate with an Abelian topological order, described by a Chern–Simons theory of level n. Such theories are believed to classify all Abelian topological orders and are relevant, for example, for the hierarchical description of the fractional quantum Hall effect^{62} or multilayer fractional quantum Hall systems. They are characterized by the symmetric integer nbyn matrix and the charge vector . The Lagrangian is^{4}
Here are the auxiliary compact U(1) gauge fields from which the conserved current J_{μ} of the manybody system can be derived, . As usual, μ, ν, ...=t, x, y denote temporal and spatial coordinates.
The first two terms of the Lagrangian equation (13) describe the response of the manybody system to the external U(1) gauge field A_{μ}. Here e denotes the A_{μ} charge of the indistinguishable host particles constituting the manybody system. From the Euler–Lagrange equations the quantized Hall response is obtained, , where the manybody Chern number is given by
The third term in equation (13) describes the conserved currents j_{Iμ} of the I=1...n different qps (in the bulk of the system). The integers denote the number of qps that are bound together, in particular, for elementary qp (qh) excitations. Edge terms and kinetic energy corrections will be ignored in the Lagrangian equation (13) in the following. The qps carry fractional charges (ref. 4). Thus, we note that the Chern number of the incompressible groundstate is given by the sum of the fractional charges of elementary topological excitations, .
When the impurity (in a given internal state) binds qps of type I, their currents can be directly related to the TP current j_{μ} by . This is demonstrated by a microscopic calculation in the main text. Using this expression and integrating out the auxiliary U(1) gauge fields in the Lagrangian equation (13) (ref. 63), we obtain
where . We also included a coupling qB_{μ}j_{μ} of the impurity to an additional external field B_{μ}. The first term in equation (15) is a Chern–Simons term for the external gauge field A_{μ}. The second term describes the braiding statistics of the TP, which coincides with the expected qp statistics, see, for example, ref. 64. The statistical phase e^{iθ} picked up when interchanging two TPs adiabatically is given by (ref. 4). The last term, most important to our discussion, corresponds to an effective gauge field seen by the TP, where the A_{μ} charge of the TP is given by .
By applying the interferometric protocol introduced in the main text to different flavours J=1, ..., n of TPs, all the Chern numbers (or, equivalently, all the fractional charges)
of elementary topological excitations can be measured. To this end a single qp (or qh) of flavour J is bound to the impurity. The last equation is then derived as equation (4) in the main text. When the TP Chern numbers of all qps are known, the Chern number of the incompressible groundstate can be derived from equation (14),
Here (−1) needs to be inserted if the elementary qh excitation of flavour J is used, with and (+1) for elementary qps with .
In the main text we discuss the case of ν=1/m Laughlin states where n=t=1 and K=m. In this case there exists one qh type, with fractional charge e*=−e/m. According to equation (16) the Chern number of the TP consisting of an impurity bound to a hole is given by . We confirm this by a microscopic calculation for a fractional Chern insulator in Fig. 3. Then, equation (17) predicts a fractional Chern number of the incompressible Laughlin state, in agreement with the established result by Niu et al.^{5}
Exact polaron transformation
To calculate the topological invariants characterizing TPs exactly, we need its full wavefunction for any given value of the total TP quasimomentum q. Here we consider lattice systems for which we develop a method allowing to calculate using exact numerical methods. Our approach is based on the LLP unitary transformation^{50} introduced in the context of conventional polaron physics, which makes the conservation of the polaron momentum explicit. The effect of the external force F acting on the impurity is also discussed in this framework.
Our starting point is the impuritycentred LLP transformation
To define the impurity position operator and the fermion momentum operator , a gauge choice is made. We introduce the magnetic unit cell of size a_{x} × a_{y} for the fermion Hamiltonian , see equation (9), which contains an integer number of flux quanta. Using the Landau gauge as in equation (9) and assuming α=r/s with r, s integers, we have a_{x}=a and a_{y}=sa. Next, we label sites within the unit cell by an integer μ and define the impurity position operator
Here the integers j_{x,y} label unit cells, and (j, μ) is merely an alternative way of parametrizing the site indices (m, n) that were used previously in the definition of the model. Hence, we see that represents only the position of the unit cell, but not the positions of individual sites within one cell. Similarly, the fermion momentum operator is
where we introduced operators in momentum space . The wavevector k takes quantized values k=2π(i_{x}/L_{x}, i_{y}/L_{y})^{T} for integers i_{x}=1, ..., L_{x}/a_{x} and i_{y}=1, ...., L_{y}/a_{y} and with L_{x,y} denoting system size in x and y directions. Note that, although the impurity lattice has a smaller period of a, we have chosen the larger magnetic unit cell of the fermion model in equation (19). This is necessary to distinguish between inequivalent sites μ within one magnetic unit cell for both the fermions and the impurity.
We proceed by applying the LLP transformation defined above to the Hamiltonian (equations (9) and (10)). The new effective Hamiltonian in the polaron frame reads . First we note that the manybody (fermion) Hamiltonian trivially commutes with the LLP transformation because of its translational invariance by multiples of one magnetic unit cell. The potential term in equation (10) also commutes with the LLP transformation, as can easily be checked.
To transform the kinetic energy of the impurity, we introduce the singleparticle band Hamiltonian defined in the magnetic unit cell of the fermions. This allows us to write in the absence of the force F. The momentum operators are defined as in the case of fermions discussed above. The impurity position operator is the infinitesimal generator of displacements in quasimomentum space. Therefore, it holds
as can also be checked directly from the definition of the impurity momentum operators . Hence, after the LLP transformation we obtain the effective impurity Hamiltonian
Finally, we apply the LLP transformation to the impurity–fermion interaction . To keep the discussion general, we consider a density–density interaction of the form
where V_{μ,ν}(r) denotes an arbitrary potential. Because the LLP transformation displaces the manybody system (recall that is the infinitesimal generator of fermion translations), it holds
Using this relation and restricting ourselves to the subspace of one impurity, we obtain
This corresponds to a static potential for the manybody fermion system, centred around the central unit cell (where r_{j}=0). For the local interaction assumed in the main text it holds V_{μ,ν}(r)=Vδ_{r,0}δ_{μ,ν}.
For a single impurity, the resulting Hamiltonian in the polaron frame (written in second quantization for notational convenience) is
Here we used the relation to make the conservation of the TP momentum apparent, . Equation (26) demonstrates that the force F directly couples to the TP momentum. Using exact diagonalization techniques we solved the TP band Hamiltonian equation (26) for different values of k (Figs 2 and 3). This allows us to extract the TP Chern number for systems with a small number of fermions.
Fractional TP Chern numbers
Now we discuss TPs with qp charge e*/e=p/q, which is not the inverse of an integer, p≠1, assuming p, q are relative prime. In this case, the topological field theory of the TP presented above predicts a Chern number
Because the TP Chern number needs to be an integer because of gauge invariance, the groundstate manifold has to be at least pfold degenerate. In that case all states share a total Chern number .
To understand the origin of this degeneracy we investigate the ν=1/3 Laughlin state. To this end we bind p=2 qhs to the impurity, which realizes the simplest nontrivial fraction e*/e=−2/3 because each of the qhs has charge e*/e=−1/3. Every qh sees the original host particles as a source of magnetic flux. In a meanfield theory this leads to a homogeneous effective magnetic field seen by the impurity.
Our reasoning so far relies on the effective lowenergy topological field theory, which does not take into account lattice effects. Let us now treat the lattice as a perturbation, which is justified when the magnetic flux per plaquette is sufficiently small. The lattice defines the magnetic unit cell of the TP, the size of which, , is equal to that of the original magnetic unit cell of the host particles. The size of the magnetic unit cell in the effective field , on the other hand, is larger by a factor of 3/2, (Supplementary Fig. 1). To construct groundstates in the full lattice model from the groundstate in the effective continuum model, we choose the smallest commensurable unit cell, with a size .
In reciprocal space, we thus arrive at a description of the system in a reduced zone with size 1/3 of the original magnetic BZ. Because it has half the size of the effective magnetic BZ, it contains two degenerate groundstates. (As a result of the lattice, small gaps may open in some places in the BZ.) We can thus understand the twofold degeneracy as a consequence of the incommensurability of the original magnetic unit cell and the effective magnetic unit cell seen by the impurity.
We used our numerics to verify the existence of a twofold degenerate TP band in small systems (Supplementary Fig. 2a). The corresponding Berry curvature is compatible with a total Chern number shared by the two degenerate states (Supplementary Fig. 2b). Details of our calculations are discussed in Supplementary Note 1.
Data availability
The data that support the findings of this study are available from the corresponding author upon request.
Additional information
How to cite this article: Grusdt, F. et al. Interferometric measurements of manybody topological invariants using mobile impurities. Nat. Commun. 7:11994 doi: 10.1038/ncomms11994 (2016).
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Acknowledgements
We acknowledge useful discussions with M. Knap, M. Hafezi, F. Letscher, W. Phillips, J. Sau, J. Taylor, I. Bloch, E. Mueller, M. Greiner, F. Gerbier, J. Dalibard and C. Weitenberg. F.G. acknowledges support by the Excellence Initiative (DFG/GSC 266), the Marion Köser Stiftung and the Moore foundation. E.D. and F.G. acknowledge support from HarvardMIT CUA, NSF Grant No. DMR1308435, AFOSR Quantum Simulation MURI, the AROMURI on Atomtronics and the AROMURI Quism programme. E.D. acknowledges support from the Humboldt foundation, Dr Max Rössler, the Walter Haefner Foundation and the ETH Foundation. D.A. acknowledges support by the Alfred Sloan Foundation. N.Y.Y. acknowledges support from the Miller Institute for Basic Research in Science.
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Affiliations
Department of Physics and Research Center OPTIMAS, University of Kaiserslautern, Kaiserslautern 67663, Germany
 F. Grusdt
 & M. Fleischhauer
Graduate School Materials Science in Mainz, GottliebDaimlerStrasse 47, 67663 Kaiserslautern, Germany
 F. Grusdt
Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
 F. Grusdt
 & E. Demler
Department of Physics, University of California, Berkeley, California 94720, USA
 N. Y. Yao
Department of Theoretical Physics, University of Geneva, 24 quai ErnestAnsermet, 1207 Geneva, Switzerland
 D. Abanin
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Contributions
All authors contributed substantially to the writing of the manuscript. F.G. performed the calculations. F.G. and E.D. conceived the method. N.Y.Y., D.A. and M.F. contributed to clarifying the initial idea of the method.
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The authors declare no competing financial interests.
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Correspondence to F. Grusdt.
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