In recent years there has been a renewed effort toward describing high energy scattering processes in terms of the intrinsic correlations present in the final state at asymptotic spatial distances. In quantum field theories (QFTs), these properties can be properly studied in terms of time-integrated correlation functions of conserved current operators living on a celestial sphere—these correspond to the theoretical realizations of detectors used in real high energy experiments. This class of correlation functions has been studied to great depth in conformal field theories (CFTs) [1–11], providing a wealth of techniques and physics insights. These developments have been translated to quantum chromodynamics (QCD), establishing a remarkable connection between theory [12–19], phenomenology [20–32], and experiment [33,34].¹

See also [35,35–37] for related developments in N=4 Super Yang Mills (SYM).

In contexts closer to QCD, the most common realization of this program is in terms of the N-point energy correlators (ENCs), which measure the correlations between asymptotic energy flows [16,38–40]. ENCs can be defined in terms of the light-transformed energy momentum tensor (EMT), Tμν, E(n→)=limr→∞rd-2∫0∞dtniT0i(t,rn→),(1)where E denotes the LRO corresponding to the local EMT operator, depending only on the unit vector n→ labeling a point on the celestial d-1 dimensional sphere. Using Eq. (1), ENCs are then correlation functions of the generic form ⟨Ω|K†E(n→1)E(n→2)⋯E(n→N)K|Ω⟩⟨Ω|K†K|Ω⟩,(2)where |Ω⟩ denotes an equilibrium state of the theory, e.g., the vacuum, and the operator K generates a local excitation. In this work we also will consider charge correlations, obtained by replacing T0i in Eq. (1) with an appropriate conserved current operator Ji.

Correlation functions of the form of Eq. (2) have been studied, in QCD, to high perturbative precision in certain kinematical limits for N≤3 [15,23,41,42]. Accessing the properties of ENCs in full kinematics, for large values of N and beyond the perturbative region, is a challenging task,²

See Refs. [22,43–47] for studies on nonperturbative effects on ENCs.

In this work, we argue that real time quantum simulations can provide such a new direction, both in the context of gauge theories and for CFTs, serving as a laboratory to test theoretical predictions. While this program could be realized in the near-to-mid future, taking into account the development of quantum technologies and devices for the study of QFTs [48–53], here we will illustrate our approach in the Schwinger model [54], QED in 1+1-d, making use of tensor network methods [55]. These are ideally suited for quantum simulations of lower dimensional systems and can be naturally mapped quantum computer based approaches. In the A0=0 gauge, this theory is defined by the Hamiltonian H=∫dxE2(x)2+ψ¯(x)(-iγ1∂1+gγ1A1(x)+m)ψ(x),(3)with ψ a two component, single flavor massive fermionic field. Due to dimensionality, the properties of correlations of LROs in this model are quite distinct from what is seen in 3+1-d. Firstly, in 1+1-d the celestial sphere consists of a set of two disconnected points; as a result there is, for example, no notion of a light-ray operator product expansion (OPE), which plays a big role in the study of ENCs, even though there is a local OPE. Further, since there are only two spatial asymptotic positions where one can place detectors, it only makes sense to consider single and two body correlations functions of the form C1,kμν(q)=∫dtdxeiqαxα⟨Jμ(t,x)Ek({R,L})Jν(0)⟩,C2,kμν(q)=∫dtdxeiqαxα⟨Jμ(t,x)Ek(L)Ek(R)Jν(0)⟩.(4)Here the electromagnetic currents Jμ=ψ¯γμψ act on the vacuum state, generating a local injection of momentum q, which puts the system out of equilibrium, following Eq. (4), while the R, L denotes the insertion of the detector at asymptotic positive (R) or negative (L) spatial locations.³

We note that to obtain the projection that would contribute to physical processes, one should contract C1/2 with the appropriate polarization vectors; however in the Schwinger model the gauge field is strictly nonpropagating, and thus such a contraction is vanishing. Secondly, the correlators should be normalized by the expectation value of the current insertions without measurement, as in Eq. (4).

To study the EEC in the Schwinger model one could use the fact that the theory, in the charge zero sector, is dual to a quantum Sine-Gordon model [57] via a bosonization map. In particular, in the limit of massless fermions, the dual theory is that of free massive bosons, with both being connected by the identity [57,58] ψ¯γμψ=14πϵμν∂νϕ,(5)where ϕ is a bosonic field with mass mϕ=g/π. Combining this with Eqs. (4) and that the off-diagonal component for the bosonic EMT reads T0i=∂0ϕ∂iϕ, one can obtain for example C1,1μν=limy→∞16π2∫0∞dsϵμαϵνβ×∂xα∂x0β∂y10∂y21⟨ϕ(x)ϕ(y1)ϕ(y2)ϕ(x0)⟩y1=y2=y.(6)Thus, computing the ENC boils down to the computation of 2(N+1)-body Wightman functions of bosonic fields; in the free limit the evaluation of such expectation values is immediate through Wick’s theorem (for the creation/annihilation operators). However for massive fermions, the dual theory is interacting (gϕ∝m), and the computation of the above correlation is nontrivial, see, e.g., [59,60]. As a result, even in this simple model, the analytical computation of ENCs beyond perturbation theory is far from obvious.

Alternatively, one can study this model on the lattice, where the above mentioned real-time simulation methods can be directly applied. To that end, we discretize space into Ns/2 sites, with a physical lattice spacing 2a. Following the Kogut-Susskind prescription [61,62] for fermions,⁴

See Ref. [63] for further details following the same conventions.

Using the latticized Schwinger model and the appropriate representations of the detector operators, we can then study the correlation functions of the form of Eq. (2) numerically. To that end, we implement the protocol illustrated in Fig. 1. First, we prepare the ground state of the Schwinger model at finite m and g; this generates the state |Ω⟩ in Eq. (2). Second, we insert an expanding electric field⁶

The form of the quench is not particularly important, as long as it leads to an increase in energy density and it is localized in space. It is simple to check that this is indeed achieved by this quench [65,66], see also [67–71] for related studies.

In Fig. 2 we show the evaluation of C2,klat(Δx,t,t).⁷

Note that here we take t=t′ and we do not perform the time integrals. The reasoning for this is twofold. First, the EEC in two dimensions is a number, and thus having the time differential correlator provides more information. Secondly, taking different times for the two operator insertions makes the simulation technically more costly; since we do not compute the full EEC looking at the equal time correlator seems sufficient.

In Fig. 2 (top) we show the late time behavior of C2,klat(Δx,t,t), for two values of spatial separation of the detectors and two values of m/g. At times t<6a, the correlator exactly vanishes. This is in agreement with the naive expectation from the right hand side illustration in Fig. 1, where a finite amount of time is required for the signal to propagate from the initial localized perturbation to the detectors. Comparing the starting time for the variations of the energy correlator for the two values of Δx, one concludes that the momentum spread happens close to the (lattice) speed of light, i.e., Δ(Δx)/δt∼O(0.5–1) where we roughly estimate the time interval by comparing the first peaks of the dark green and red data sets. The oscillatory-like behavior seen after the first peaks is sensitive to the lattice edge (due to the small gap with respect to the spatial infinity where the detectors are placed), and thus the extracted results there are not physical. To overcome these issues, one should not only ensure that Δx∼Nsa but also that |xdetector|<Nsa/2, with xdetector the spatial position of a detector; nonetheless the realization of such a limit is numerically hard to achieve due to the need to time evolve the full quantum state on a larger lattice.

A similar exercise to that shown on the top panel of Fig. 2 was also carried out for the charge correlator. Remarkably, we found for several parameters sets that if one normalizes the curves as in Fig. 2, i.e., by dividing all points in each data set by the maximum value in the considered time domain, that the energy and charge correlators have a very close behavior. This qualitative observation agrees with the following simple picture: 1) the evolution of the correlators is constrained by conservation of energy and conservation of total charge⁸

We work in a charge zero sector with no net momentum.

On the bottom of Fig. 2, we show the results for the initial value of the energy and charge correlator, i.e., the correlator value for times earlier than the observation of the first peak, for several values of k. The odd values are not shown since for the those cases the extracted value is exactly vanishing. More, we observed that for the odd values of k the behavior in time of the energy and charge correlation functions is very close to what is seen for k=1. These statements do not hold for even k, where we observe that the charge correlator acquires a finite initial value, which does not depend on k, while the energy correlator acquires a positive value that grows with k.⁹

Note that in the figure the evolution with k has the opposite trend. This is because we normalize each point by ⟨H2k⟩quench, which grows considerably faster than the evolution of the bare correlator.

In this manuscript we have provided a first discussion on the study of LROs’ correlations from real-time lattice simulations. This new approach can be potentially used to explore nonperturbative features of ENCs in gauge theories and complements existing perturbative methods. Of course, the realization of such a program is tightly dependent on the development of large scale quantum computers, which are not available at the moment. This is particularly important for the extraction of LROs’ correlations functions since the measurements must be performed at large spatial separations. Combined with the need to take continuum limits and the large number of degrees of freedom necessary to implement gauge theories in quantum devices, such an endeavor is extremely resource intensive. Hopefully continuum and large volume extrapolations may be possible in the mid to long term future, using more advanced quantum devices. Perhaps more interesting for near term implementations is the study of 2+1-d field theories where the behavior near critical points of LROs can be theoretically described; a natural example is the three-dimensional critical Ising model, which can be more easily realized in quantum devices and (large) tensor networks, when compared to gauge theories. We leave such a study for future work, see Refs. [77,78] for related discussions. From the point of view of quantum simulation of QFTs, the study of LROs is of critical importance since they provide a proper way to introduce the notion of detector, a necessary element in the study of high energy physics using lower dimensional models, see, e.g., [48].