The Lagrangian density of the charge-q Schwinger model is given as² (1) where q is a positive integer and we take m > 0. Here, F = dA (F₀₁ = ∂₀A₁ − ∂₁A₀ in components) is the field strength of the U(1) gauge field A, and we normalize it so that the Dirac quantization is given by for any closed 2-manifolds M₂. Equivalently, the U(1) gauge transformation is imposed by the invariance under (2) where the gauge transformation parameter λ is a 2π-periodic compact scalar field. When λ is well defined as -valued functions, we call it the small gauge transformation, and others are called the large gauge transformation.

We first take the space as S¹ for the canonical quantization. For the Hamiltonian formulation, the temporal gauge A₀ = 0 is convenient. When imposing this condition, we must also impose (3) which gives the Gauss law constraint: (4) The canonical momentum for A₁ is (5) Therefore the Hamiltonian density becomes (6) With the canonical variables, the Gauss law constraint (4) is rewritten as (7) Furthermore, we also impose the invariance under the large gauge transformation on the physical Hilbert space.

Because of the presence of the charge-q electric matter, the -form symmetry in the pure Maxwell theory is explicitly broken. Still, this model has -form symmetry, generated by (8) with dΛ = 0 and . This quantization of ∮Λ is important to have the single-valuedness for ψ after the transformation. For example, we can choose (9) as a nontrivial generator for . Note that qΛ = dϕ, where is a 2π-periodic scalar, and thus it is part of the large gauge transformation, which explains why this is a transformation.

The generator of is given by (10) Therefore, U(x) does not depend on x for physical states. To see this, let Ψ be a physical state that satisfies the Gauss law. Then, (11) Moreover, U(x) commutes with the Hamiltonian. The nontrivial part is the covariant derivative in the fermion kinetic term, so let us only check that part: (12) As almost everywhere, we have checked the commutation relation. In the lattice regularized formulation, we can see this more explicitly without any subtlety.

Because of in (1 + 1)D, the Hilbert space decomposes into q sectors, (13) where each Hilbert space is defined by (14) We note that, due to the topological nature of U(x), the condition does not depend on x. Here, the label k is identified in mod q, i.e., k ∼ k + q.

We can fix the label k by gauging the 1-form symmetry. Since the 1-form symmetry is , the 2-form gauge field is now subject to the constraint (15) where is an auxiliary -form gauge field. Now, the 1-form gauge transformation is given by (16) and thus the gauge-invariant combinations are given as and . Using this 1-form gauge transformation, we can set A = 0 as a gauge-fixing condition, which effectively replaces A with . Then, we obtain the replacement g² → q²g² and θ₀ → (θ₀ + 2πk)/q, where k is the label for the discrete θ term. In this way, we can extract the Hilbert space by gauging including the discrete θ term. That is, the U(1) gauge theory with charge-q matters can be identified as the discrete sum of U(1) gauge theories with charge-1 matters with rescaled coupling q²g² and with different fractionalized θ angles (θ₀ + 2πk)/q. Each sector of the decomposition has recently been called a universe [57–60].

These different universes can be connected by introducing Wilson line operators. When a charge-q_p Wilson loop is introduced in the kth universe, its string tension σ is given by (17) where E_k(θ₀) describes the ground-state energy density³ for , k = 1, …, q. When the fermion mass m² is small enough compared with the photon mass μ² = q²g²/π, the ground-state energy density can be approximated as (18) When the fermion mass m is large enough, the theory is approximately the pure Maxwell theory with theta angle θ₀ and the ground-state energy density behaves as (19) When q = 1, the Wilson loop does not change the universe, and thus any integer electric charge can be screened by the pair creation. When q > 1, this is not the case, and the Wilson loop obeys an area law for generic values of θ₀ and .

Let us point out that the string tension does not have to be positive semi-definite, and the appearance of negative string tensions is suggested by the formula (17) with Eqs. (18) or (19). However, one might think at first sight that this is counter-intuitive: the string tension in (1 + 1)D is the difference between the energy densities inside and outside the Wilson loop, and, since the outside of the loop is the ground state, we necessarily have a positive string tension. How can we create states that have lower energy density than the ground state? Here, the notion of the universe [57–60] plays an important role. Since U(x) is a topological local operator, any physical process cannot change its value as long as the 1-form symmetry is preserved. This means that any processes are closed inside one of the universes. In particular, if we have a ground state of a specific universe, then it never decays even if it has a larger energy density than another universe. Since the Wilson loop connects different universes as we have discussed, it is possible to have negative string tensions. We also comment on when the string tension changes its sign. From Eqs. (18) and (19), we see that the sign of the string tension changes at the same (q, q_p, θ₀), at least in small and large mass regimes. Therefore it would be natural to expect that the condition for having negative string tension is independent of m as long as it is nonzero.

When we take the massless limit of the fermion, m = 0, the formula (18) suggests the presence of q degenerate vacua and the deconfinement of the Wilson loops. In this limit, the charge-q Schwinger model enjoys chiral symmetry, and . This transformation may look to be instead of , but we note that the fermion parity is part of the U(1) gauge redundancy, so the proper symmetry transformation is just . The vacuum degeneracy is the consequence of its spontaneous breaking, and it is required to match the ’t Hooft anomaly between and chiral symmetry with nonzero mass gap [48–51].

Even with nonzero m, we have an ’t Hooft anomaly, and/or global inconsistency [61–67], to constrain the θ₀ dependence of low-energy physics of the charge-q Schwinger model [51]. As we can see explicitly in the formula (18) of the ground-state energy, each ground state E_k(θ₀) does not have the 2π periodicity of θ₀. Instead, it satisfies E_k(θ₀ + 2π) = E_{k − 1}(θ₀), and thus the energy spectrum has 2π periodicity thanks to the multi-branch structure of the ground states. This nicely imitates the situation of the (3 + 1)D pure Yang–Mills theory [68–70].⁴ This level crossing of the ground states is required to satisfy the anomaly and/or global inconsistency matching conditions.

Now, let us discuss what happens if we consider the theory on the interval [0, L]. As discussed in the following sections, our strategy of quantum computation for the Schwinger model is designed for the open boundary condition, not for the periodic boundary condition. We need to understand how the physics changes by the choice of the boundary condition.⁵

To discuss gauge theories with the boundaries, we have to specify the charges at the boundaries. When external charges are put on the boundaries, the Gauss law constraint is given by (20) Let us denote the corresponding Hilbert space as . The Gauss law above tells us that the 1-form symmetry generator is now completely fixed upon acting on the physical states and its value is given by (21) as we do not introduce any test charges except at the boundaries. This point is drastically different from the case with the periodic boundary condition. In the periodic boundary condition, U(x) can take q distinct values, and the Hilbert space is decomposed into q universes by the eigenvalue of U(x). In some sense, on the open interval, the projection to the single universe is automatically selected by the choice of the boundary condition.

Let us discuss how these properties affect the θ-angle periodicity, θ₀ ∼ θ₀ + 2π. For the periodic boundary condition, we can use the spatial Wilson loop operator, W(S¹) = exp (i∮A₁(x)dx), as the gauge-invariant unitary operator, and we can relate θ₀ + 2π and θ₀ by the unitary transformation of W(S¹). This ensures the 2π periodicity of the spectrum for any U(1) gauge theories with the periodic boundary condition, and the formula (18) gives an explicit realization. For the open boundary condition, however, we do not have such a gauge-invariant unitary operator. Therefore, nothing ensures that physical quantities have 2π periodicity with respect to θ₀. Indeed, we can see that 2π periodicity is completely lost, , when we consider the charge-q Schwinger model with the open boundary.

For the charge-q Schwinger model, the θ-angle periodicity can be found for a single sector for bulk local observables, and that periodicity is given by θ₀ ∼ θ₀ + 2πq. This is a natural expectation based on the fact that the 1-form symmetry group is just and we do not have other 1-form symmetries. For example, let us find the ground-state energy density: (22) Since U(x)|GS, k〉 = e^2πik/q|GS, k〉, |GS, k〉 should behave as the ground state of the kth universe if the volume is sufficiently large. When the location x is sufficiently far from both ends, i.e., 0 ≪ x ≪ L, it is given by the same formula (18) or (19) up to exponentially small corrections: (23) where μ denotes the mass gap of the theory, which is given by in the massless fermion limit, m → 0. The exponentially small terms represent the boundary effects and need not have any θ₀ periodicity. Only when we can neglect the boundary contributions can we observe the manifest 2πq periodicity, E_k(θ₀ + 2πq) = E_k(θ₀).

Here, we would like to emphasize the importance of using the local density H(x). Another popular way to compute the energy density is that we calculate the total energy first and divide it by the volume; however, the result becomes exponentially worse. In order to see this, let us compute the total energy (24) where the second term on the right-hand side represents the localized energy around the boundary. If we try to obtain the energy density as (25) then we really have to take the infinite volume limit to suppress the boundary effect, . However, since the system is generically gapped including the massless point, m = 0, we should be able to calculate E_k(θ₀) using the open boundary condition with exponentially good accuracy, and Eq. (23) realizes this physical intuition. The same comment also applies for other quantities, such as chiral condensates.

Let us put the theory on a lattice with N sites and lattice spacing a. We realize the two-component Dirac fermion ψ(x) using the staggered fermion χ_n, where n labels the lattice site (x = na). Here, χ_n is a single-component complex fermionic operator, and the Dirac fermion at x extends over two sites on the lattice: (26) The gauge field and its canonical momentum are represented by the link variables, (27) which are defined on the link between the sites n and n + 1. We will take open boundary conditions for the fields.

The lattice discretization of the Hamiltonian (6) is given by (28) where the parameters J and w are defined as (29) The θ term is realized by a constant shift of link variable L_n, so that it can be interpreted as an constant electric flux. No additional difficulty appears in the performance of the numerical simulation in the nonzero θ₀ regime in contrast to the conventional Monte Carlo approach.⁶ The field operators obey the canonical commutation relations: (30)

On a lattice, the Gauss law (7) takes the form (31) With the open boundary condition, we can solve it as (32) Note that the q dependence appears in the Gauss law. We impose a boundary condition on the electric field operator by setting L₋₁ = 0, which corresponds to the choice k = 0 in Eq. (21). Upon fixing the gauge so that U_n = 1 for all n, we have the Hamiltonian as follows: (33) Notice that this Hamiltonian only contains the fermionic operators and thus the local Hilbert space is finite dimensional.

We note that the 2π periodicity of θ₀ is lost by taking the open boundary condition. If we take the periodic boundary condition instead, then we can define the gauge-invariant unitary operator, W = ∏_nU_n, which is nothing but the spatial Wilson loop, and we have . For the periodic boundary condition, however, we necessarily have an infinite-dimensional local Hilbert space, and we cannot map to the system only with fermions.⁷ Since we would like to realize this system on the quantum computer, locally finite Hilbert space is more natural, so we take the open boundary condition in the following.

We introduce the two probe charges +q_p and −q_p on the th and th sites, respectively. Here, denotes the dimensionless quantity, . This gives a constant shift for the link variable between the th and th sites, which corresponds to the insertion of a long rectangular Wilson loop with width ℓ as shown in Fig. 1.

This is realized by introducing the position-dependent θ angle: (34) For open boundary conditions, it would be appropriate to take (35) with odd for even N and even for odd N. In this way, we equally separate the probe charges from both boundaries, so that the boundary effect is suppressed as much as possible in the simulation with finite volumes.

In the presence of the probes, the lattice Hamiltonian is (36) where θ₀ in Eq. (33) is replaced by the position-dependent one, ϑ_n, in Eq. (34). Using the Jordan–Wigner transformation, (37) we obtain the Hamiltonian in terms of spin operators, (38) where (X_n, Y_n, Z_n) stands for the Pauli matrices (σ¹, σ², σ³) at site n. Here, we have dropped irrelevant constants independent of ϑ_n in the Hamiltonian. The form of the spin Hamiltonian indicates the convenient relation, (39) which enables us to relate the q = 1 case to the general q case. We would like to emphasize that this translation to q = 1 is possible because we take the open boundary condition. When we take the periodic boundary condition, we cannot eliminate the spatial link variables completely by gauge fixing. Then, the spatial hopping term, , genuinely depends on the choice of q ≥ 1, and we cannot relate them by simple replacements of coupling constants. As we have discussed in Sect. 2.1, the results for charge q can be technically translated in the language of the q = 1 case with a fractional probe charge, but this necessarily accompanies the projection to the single universe. It requires extra modification of the large gauge invariance to keep the global information of the theory with a periodic boundary condition.

In the adiabatic state preparation, the first step is to choose an initial Hamiltonian H₀ whose ground state |GS₀〉 is unique and known. The second step is to choose a time-dependent adiabatic Hamiltonian H_A(t) such that (41) Then the adiabatic theorem claims that we can construct the ground state of the target Hamiltonian H: if the system with the Hamiltonian H_A(t) has a unique gapped ground state for any t ∈ [0, T], then the ground state |GS〉 is obtained by [72,73] (42) Here, the symbol indicates the time ordering of subsequent operators.

In the present study, we choose the initial Hamiltonian as (43) for some m₀ > 0. The ground state of H₀ is given by the Néel ordered state, (44) with Z|0〉 = +|0〉 and Z|1〉 = −|1〉, and it can be easily constructed by (45) Here, ⌊x⌋ is the floor function of x, which denotes the largest integer no greater than x.

We express the unitary evolution in Eq. (42) using the Suzuki–Trotter product formula. Considering the following decomposition of the Hamiltonian turns out to be useful [26]: (46) where, for odd N, each term is (47) Here, C is just a constant, but it is important to compute the θ₀ dependence of the ground-state energy. The summands of commute with one another, and thus we can obtain the unitary operator, , by taking the product of local unitary operations, , without care about ordering. The same is true for and H_Z.

Combining the adiabatic theorem and the product formula, we find the approximate form of the ground state: (48) Here, M ≔ T/δt is a positive integer, which should be taken to be large in practice for good approximation. This is the second-order Suzuki–Trotter approximation, and its error is estimated as for the whole operator for fixed T [74,75]. Here, , , and H_{Z, r} are obtained by the following replacements in Eq. (47): (49) where the function f(s) called the adiabatic schedule is smooth and satisfies (50) There are infinitely many choices of f(s) and any choice of f(s) works in principle as long as the setup satisfies the assumptions of the adiabatic theorem and T is sufficiently large. However, the choice of f(s) affects the accuracy of the approximation in practice. In Appendix B, we study various choices of adiabatic schedules at various values of the parameters and identify the best one among them. As a conclusion, we choose the adiabatic schedule as (51)

In this subsection, let us quickly mention the quantum circuits to obtain |GS_A〉. The universality theorem tells that one can approximate any unitary operations on multiple qubits by a combination of simple qubit gates and the controlled NOT (CNOT) gate. Indeed, our expression (48) only contains two-quibit unitary transformations, so we just have to express them. Here, we use the following three operations as elementary gates to realize the unitary evolution (48):

For notational convenience, we also introduce a circuit diagram of the S gate, which is defined by

The last equality holds up to an overall phase, which does not affect the following discussion.

When trying to generate the ground state by Eq. (48), the only nontrivial step is to realize the two-qubit operations, and , using the above elementary gates. These operations can be expressed as

Here the top and bottom lines correspond to the sites n and n + 1, respectively, and α is a real parameter.

We first demonstrate that the slope of the potential between the probes can change its sign on varying the parameters. Figure 2 shows the potential V(ℓ)/g as a function of gℓ for several values of θ₀. Here, we take q = 3, m = 0.15, and q_p = −1. We can see clear linear potentials where the string tension is positive in 0 ≤ θ₀ < π while it is negative in π < θ₀ ≤ 2π. At θ = π, the potential is consistent with zero in all gℓ and this indicates the screening property.

Next we focus on the negative string tension case and compare it with the continuum results. Here we take θ₀ = 2π as a representative. In Fig. 3, we plot the potential V(ℓ)/g against the probe distance gℓ for various values of (a, N) keeping the physical volume as ga(N − 1) = 9.6 in order to find the string tension for various values of a at a fixed physical volume. To obtain the string tension (σ), we fit the potential for each (a, N) and mass using a linear function of ℓ of V(ℓ) = σℓ + c₀. We take a fitting range of 3.0 ≤ gℓ ≤ 7.0 to remove the boundary effect on the finite lattice extent with the open boundary condition. Consequently, we find that the χ²/d.o.f. values for all the fitting processes are reasonable, namely χ²/d.o.f. ≈ 1. We summarize the obtained values for the string tension by the fitting in Table 1.

In terms of the values of σ in Table 1, we do a continuum extrapolation for m = 0.05, 0.15, and 0.25 as shown in Fig. 4. All the fit qualities, namely the χ²/d.o.f., in both the linear and quadratic extrapolations are reasonable. Now we compare the obtained values of the negative string tension with the analytic predictions given by the mass perturbation theory up to in infinite volume. The energy density of the second-order mass perturbation theory is given by⁸ [76] (75) where⁹C₊ ≃ −8.9139 and C₋ ≃ 9.7384. In the small mass cases (m = 0.05 and m = 0.15), the extrapolated values by the quadratic extrapolation in terms of a are consistent with the analytic predictions by the mass perturbation theory in the continuum limit. In the larger mass case (m = 0.25), the quadratic extrapolation starts to deviate from the mass perturbation results. One possibility for the deviation is that the approximations by the mass perturbation are no longer reliable in this regime. Indeed, at m = 0.25, we see a non-negligible difference between the and results, which may suggest the importance of higher-order terms in the mass perturbation theory. The other possibility is systematic errors in our simulation. If we include systematic errors coming from the fitting ansatz of the extrapolation, however, then the extrapolated value is consistent, so we cannot say much within this numerical setup. The main source of the large systematic error is the smallness of N in the present simulation. If we perform a simulation for large N, say N ≈ 100, ga ≈ 0.1 in the near future, then we can numerically obtain the string tension in such a large mass regime using this simulation strategy, and we may obtain a clearer signal that goes beyond the mass perturbation regime.

Now, let us confirm the transformation property under the 1-form symmetry in our numerical simulation. In the previous subsection, we observed the stability of the negative string tension, which suggests the existence of an unconventional selection rule, and it is natural to interpret this as a consequence of the 1-form symmetry. Here, we would like to see if the 1-form symmetry group is actually . To this end, we carry out simulations with q_p = 2 and compare the results with q_p = −1. Since the dynamical charge is set to q = 3, the two probe charges q_p = −1 and 2 are the same mod q = 3 and therefore in the same sector of the -form symmetry. This naturally predicts that they should have the same string tension unless an unknown selection rule exists. In numerical simulations, we take a longer adiabatic time T = 198 for q_p = 2, because we find that the adiabatic error for the q_p = 2 simulation is larger than that for q_p = −1 (see Appendix B for details). In this subsection, the mass is always set to m = 0.15.

Figure 5 shows a comparison between the potentials for q_p = −1 (black circles) and q_p = 2 (red squares) for representative values of the positive (θ₀ = 0, left panel), zero (θ₀ = π, middle panel), and negative (θ₀ = 2π, right panel) string tension cases. Clearly, we can see that the slope for q_p = 2 is similar to that for q_p = −1 in the large gℓ regime if we remove a few data points near the boundary of the lattice extent. The string tension obtained by the linear fit in terms of ℓ is summarized in Table 2. We can see a good agreement between q_p = −1 and q_p = 2 as expected from the -form symmetry, although it depends more or less on the fitting range.

Lastly, we study what causes the difference between the potentials V(ℓ)/g for q_p = −1 and q_p = 2. To see this, we investigate the local energy density of each site. Figure 6 shows the local energy density E(n)/g defined in Eq. (73) for q_p = −1 (black circles) and q_p = 2 (red squares) at each site in the case of ℓ/a = 10, 12, 14. The gray regime in each panel depicts the regime inside the insertion points of the probe charges or, equivalently, the Wilson loop. We can see that E(n)/g with q_p = −1 and q_p = 2 are almost consistent with each other inside and outside the Wilson loop, respectively. Thus, the discrepancy in the potential between q_p = −1 and q_p = 2 comes only from the boundary contributions of the Wilson loop. Moreover, Fig. 6 clearly shows that the energy density inside the Wilson loop is lower than that outside, as suggested by the negative string tension. For q_p = −1, the energy excess at the probe charges is not so large. For q_p = 2, on the other hand, the energy excess at the probe charges becomes much bigger. We see that the difference in the probe charges is localized near the insertion points, and the local energies for q_p = −1 and q_p = 2 give the same values in the bulk. Because of this difference in the localized energy at the probe charges, the q_p = 2 case has a larger offset for the linear potential compared with the q_p = −1 case. If we remove the boundary effects, then it confirms the selection rule as predicted by the 1-form symmetry on the lattice.

Let us discuss the structure of the site-dependent energy density in more detail. In the limit of the massless fermion, the mass gap is given by . Since the fermion mass makes the mass gap larger, the correlation length ξ can be estimated as (76) The lattice constant for N = 25 is set to ga = 0.40, and this suggests that only a few sites around the edges of the Wilson loop are affected by the boundary effect. This is roughly consistent with Fig. 6.