Schwinger model on an interval: analytic results and DMRG

Quantum electrodynamics in $1+1$ dimensions (Schwinger model) on an interval admits lattice discretization with a finite-dimensional Hilbert space, and is often used as a testbed for quantum and tensor network simulations. In this work we clarify the precise mapping between the boundary conditions in the continuum and lattice theories. In particular we show that the conventional Gauss law constraint commonly used in simulations induces a strong boundary effect on the charge density, reflecting the appearance of fractionalized charges. Further, we obtain by bosonization a number of exact analytic results for local observables in the massless Schwinger model. We compare these analytic results with the simulation results obtained by the density matrix renormalization group (DMRG) method and find excellent agreements.

Quantum electrodynamics in 1 + 1 dimensions (Schwinger model) on an interval admits lattice discretization with a finite-dimensional Hilbert space, and is often used as a testbed for quantum and tensor network simulations. In this work we clarify the precise mapping between the boundary conditions in the continuum and lattice theories. In particular we show that the conventional Gauss law constraint commonly used in simulations induces a strong boundary effect on the charge density, reflecting the appearance of fractionalized charges. Further, we obtain by bosonization a number of exact analytic results for local observables in the massless Schwinger model. We compare these analytic results with the simulation results obtained by the density matrix renormalization group (DMRG) method and find excellent agreements.

I. INTRODUCTION
Quantum electrodynamics in 1 + 1 dimensions, also known as the Schwinger model [1], is one of the simplest non-trivial gauge theories. Since its introduction in the 60's it has been widely studied. These days it is often used as a toy model to benchmark numerical techniques for quantum gauge theories, such as tensor network and quantum simulations. See, for example, .
With the recent rapid development of quantum devices, quantum simulation of gauge theory is becoming more feasible. For this purpose, as in classical simulation, we need to discretize the gauge theory and put it on a finite lattice. In the noisy intermediate-scale quantum (NISQ) era [26], the number fo available qubits and the physical volume of the space on which the gauge theory is simulated will be limited. For this reason, simple (1 + 1)dimensional gauge theories such as the Schwinger model are natural targets of quantum simulation. Putting these theories on a spatial interval rather than a circle has an advantage, because the Gauss law constraint allows us to remove gauge fields completely on an interval, while on a circle there remains an infinite-dimensional Hilbert space. The spatial interval for the continuum model corresponds to the open boundary condition of the lattice model. It is thus desirable to know the precise correspondence between the theories in the continuum and on the lattice. To compare the continuum and lattice formulations, it also helps to have analytic results that take into account the strong effects of the boundaries and the finite volume. Rather surprisingly, the study of such effects in the literature is limited. 1 With these motivations, in this paper we study the Schwinger model on a finite interval and clarify the precise mapping between the continuum (original and bosonized) and lattice models. In particular, we show that the commonly used Gauss law constraint [31] in the lattice formulation induces fractionalized charges on the boundaries, and demonstrate that for an alternative constraint [32] the boundary charges are also modified. 2 Along the way we establish the precise correspondence between the boundary conditions in different formulations. We also derive a number of analytic expressions for physical observables in the ground state in the massless case. This is possible because bosonization maps the massless Schwinger model to a free scalar theory [34,35]. Some of these analytic results were used in [36] to compare with the results of digital quantum simulation of the lattice Schwinger model on a classical simulator.
The paper is organized as follows. In Section II we review the continuum Schwinger model in the original formulation. In Section II B we study the Schwinger model on an interval using bosonization and derive some analytic results. Section III contains our study of the Kogut-Susskind lattice formulation of the Schwinger model on a finite lattice with the open boundary condition. We review two equivalent formulations, one based on the staggered fermion and another based on spin variables. We compute by the density matrix renormalization group (DMRG) [37,38] some physical observables in the ground state and find agreement with the analytic results from Section II B, using the original and modified Gauss law constraints. We conclude the paper with discussion in Section IV. In Appendix A we calculate the energy in the presence of probe charges using the method of images. In Appendix B we show that the general lattice QCD in the Kogut-Susskind formulation [39] enjoys an exact one-form symmetry for the part of the center of the gauge group under which the matter fermions are neutral. (3), corresponding to probe charges +q at x = 0 and −q at x = 0 + .

II. CONTINUUM SCHWINGER MODEL ON AN INTERVAL
In this section we study the continuum Schwinger model on an interval. We first review the original fermionic formulation of the model. Then we review the bosonized version and derive a number of new analytic results for local observables.

A. Review of the fermionic formulation
We use notations x 0 = t, x 1 = x for spacetime coordinates and use the Minkowski metric η µν = diag(1, −1) to raise and lower indices. The dynamical fields in the Schwinger model are the gauge field A µ (µ = 0, 1) and the Dirac fermion ψ = (ψ u , ψ d ) T which is a two-component spinor. Let g be the gauge coupling and m the fermion mass. The model is defined by the action We use the notations andψ = ψ † γ 0 . We allow the theta angle to be positiondependent and denote it by Θ(x). Consider, for example, See FIG. 1. The discrete changes in the theta angle Θ(x) correspond to the presence of probe charges. Indeed we can rewrite the relevant part of the action as where we explicitly see the point-like sources for the gauge field.
with s = ±1. 4 We work in the temporal gauge A 0 = 0, where the Gauss law constraint δS/δA 0 = 0 should be imposed on physical states. Varying A 0 , we find the Gauss law in the bulk. Composite operators such asψγ 0 ψ should be defined by some normal ordering [41]. Throughout this paper we make this implicit and omit normal ordering symbols for the fermion. We will specify the b.c.'s on F 01 at x = 0 and x = L in Section II B where we study the continuum model in the bosonized formulation. The boundary terms in (1), which we do not write explicitly, should be chosen so that they are compatible with the b.c.'s. The canonical momentum conjugate to A 1 (= −A 1 ) is The density H of the Hamiltonian H = Let us denote the expectation value of the operator O in the ground state by O . Local observables of the continuum Schwinger model on an interval include the energy density H , the charge density ψ γ 0 ψ , the chiral condensate ψ ψ , and the electric field F 01 .

B. Bosonized Schwinger model
In this subsection we study the Schwinger model in the bosonized formulation. There is some overlap with the appendix of [36] that uses the same convention, and we refer the reader to that paper for details omitted here.
The bosonized Lagrangian density is (cf. [42]) We choose an appropriate boundary condition on the gauge field so that the solution to the Gauss law constraint is The Hamiltonian density is given as where Π φ is the canonical momentum conjugate to φ and µ ≡ g/ √ π. We write : • : ∞ for the normal ordering (see below) with respect to the creation-annihilation operators defined in the infinite volume and used the relation where γ 0.58 is the Euler constant. The particular numerical coefficient e γ /(2π 3/2 ) is correct for this choice of normal ordering. 5 We study the bosonized model with m = 0 and the Dirichlet boundary conditions We set k n := πn/L. Let us define Let us consider the Fourier expansions Θ n sin (k n x) .
5 See [43] for a general discussion of normal ordering.

The Hamiltonian becomes
ω n a † n a n + where ω n = µ 2 + k 2 n and a n = √ Lω We have [a n , a † n ] = δ nn . The ground state |0 satisfies a n |0 = 0 and has a divergent energy due to the terms proportional to ω n , which are independent of Θ.
The energy density 0|H(x)|0 is also UV divergent. Let x denote the largest integer smaller than or equal to x. With a cut-off k n ≤ Λ, 6 the regularized energy density is which is quadratically divergent. On a full infinite line without probe charges, the corresponding regularized energy density is, with ω(k) := k 2 + µ 2 , 7 We define the renormalized energy density as An expression for the chiral condensate was found in [36]: 6 For plots throughout the paper, we use Mathematica to evaluate regularized sums numerically by setting [LΛ/π] to 10 4 . 7 Explicitly, E line For the electric field we have Below, we consider special and limiting cases. a. Two probe charges on an interval. For probe charges on an interval, one can evaluate the sums above. As an example, let us consider which represent a pair of charges ±q placed at x = (L ∓ )/2, i.e., Θ pair = Θ q,θ0=0 | 0=(L− )/2 . We impose the boundary conditions φ = 0 at x = 0, L corresponding to w 0 = w 1 = 0. The non-zero Fourier coefficients are for j ∈ Z >0 . The total energy E pair , defined as the energy computed from (16) by removing terms proportional to ω n , was obtained in [36]: 9 The energy density E(x) in (20) computed for (26) is plotted in FIG. 2(a). 10 The chiral condensate ψ ψ(x) in (21) corresponding to (26) is plotted in FIG. 2(b). For the cosine in (21), the residue method gives its argument explicitly as By several manipulations, one may rewrite (23) as . (22) where η 0 (x) := (x − L− 2 )/L, η 1 (x) := (x − L+ 2 )/L, and {η} := η − η . The charge density ψ γ 0 ψ(x) that corresponds to (26) is plotted in FIG. 3(a). The summation in (24) can be performed explicitly to give The electric field that corresponds to (26) is plotted in FIG. 3(b). Performing the summation in (25), we obtain the explicit expression Behaviors near a boundary. We now consider the massless Schwinger model on a half-line [0, ∞) with Θ(x) = 0 and the boundary condition φ(x = 0) = 0. Let us begin with the energy density. The regularized energy density on a half-line is obtained from (18) by sending L to infinity: The modified Bessel function K 0 (z) has the asymptotics Thus the energy density diverges logarithmically near the boundary and decays exponentially away from it. From we get for the chiral condensate which is the result obtained in [30]. The condensate diverges as ψ ψ(x) = O(x −1/2 ) near the boundary and decays exponentially away from it. The charge density and the electric field simply vanish an half-line for Θ(x) = 0 and the boundary condition φ(x = 0) = 0. c. Behaviors near a probe charge. Let us consider the system with a single probe charge q at x = x 0 on an infinite line, which is represented by the positiondependent theta angle  (31) and (36), respectively. (b) Chiral condensate ψ ψ(x) given by (21) for the same set-up. The local behaviors near each boundary and each pole are given by (34) and (37), respectively.  (24) or equivalently by (29) for the same set-up as for FIG. (2). The local behaviors near the probes are given by (38). (b) Electric field F01 given by (25) or equivalently by (30) for the same set-up.
By taking an appropriate limit of (18) or by repeating the steps leading to (18), we obtain the energy density (renormalized by subtracting the value without a probe) In a similar manner one can obtain expressions for other local observables. We obtain, as in Appendix D of [36], We also have which previously appeared in (4.12) of [27]. Integrating this, one obtains d. Behaviors near a boundary charge. We now consider the massless Schwinger model on a half-line [0, ∞) with Θ(x) = 0 and the boundary condition (or equivalently Θ(x) = √ πq and φ(0) = 0). This can be obtained by setting w 0 = q/2 and taking the limit L → ∞. We find that the energy density is the sum of (31) and (36) while the charge density and the electric field are respectively given by (38) and (39)  Let us turn to the Kogut-Susskind lattice formulation of the Schwinger model [39,44] with a position-dependent theta angle. We wil and follow the conventions of [36].
We consider a one-dimensional spatial lattice with N sites, labeled by integers n = 0, 1, . . . , N − 1. 11 The two components ψ u (x) and ψ d (x) of the Dirac fermion ψ = (ψ u , ψ d ) T are replaced by the staggered fermion χ n at the even and odd sites with x = na respectively, according to the correspondence On the n-th link, which connects the sites n and n+1, we introduce the link variables U n and L n satisfying U † n = U −1 n , L † n = L n according to the correspondence These operators satisfy canonical (anti-)commutation relations, among which the non-trivial ones are We also introduce the lattice version ϑ n of the positiondependent theta angle on the n-th link: The Hamiltonian of the lattice theory is which is the direct counterpart of (9). There is a relation between N and the fermion boundary conditions. As in (42) we identify ψ u (ψ d ) with χ even (χ odd ). Since n in χ n runs from 0 to N − 1, we effectively have χ −1 = χ N = 0. Thus we have ψ d = 0 on the left and ψ u = 0 (ψ d = 0) on the right for N even (odd), namely there is a correspondence, leading to to the NS (R) boundary condition. 12 11 We will see that the behavior of the model depends strongly on whether N is even or odd. 12 We also checked that the DMRG computation of the spectra of the XY model, which is equivalent to the free fermion model via the Jordan-Wigner transformation for open b.c.'s, reproduces the expected spectra of the continuum Dirac fermion obeying the NS (R) boundary conditions for large N even (odd).
As in the continuum theory, the physical Hilbert space is obtained by the Gauss law constraint. The standard choice [31] of the Gauss law constraint is 13 We impose the boundary condition L −1 = 0 and fix the gauge U n = 1 to eliminate (L n , U n ). The term (−1) n /2 in (47) represents a site-dependent background charge.
In the bulk, the spatially averaged background charge density vanishes in the continuum limit, but we will see that there remains a non-trivial localized charge on a boundary and induces a background electric field. We convert the fermions into spin variables by the Jordan-Wigner transformation [45] where X n , Y n , Z n respectively denote the Pauli matrices σ x , σ y , σ z associated with the n-th site. Besides the theta angle, g and m as in Section II, the lattice introduces the lattice spacing a as an extra parameter. The length L of the spatial interval is given by L = (N − 1)a. The Gauss law constraint reads We solve this with the boundary condition The Hamiltonian in terms of the spin variables is Note the structural similarity between (11) and (51). We have the following correspondence for the local observables of the continuum theory and the spin model. 13 Here, the presence of external charges is accounted for, as in (3), by the position-dependent theta angle ϑn in (46). Cf. (B4).
The quantities on the left and right hand sides are for a continuous theory and a lattice model respectively, requiring renormalization (normal ordering) in the former. We will often consider the particular form of the position-dependent theta angle corresponding to probe charges ±q located at the sites n =ˆ 0 n = (ˆ 0 +ˆ ):

B. Spin lattice versus bosonized continuum models
Let us compare the Gauss law constraints (49) and (10) in the spin and bosonized formulations, respectively. The correspondence in (43) suggests the correspondence The operator φ n rotates the X j Y j planes for j ≤ n. The comparison of (11) and (51) suggests the correspondence Taking the commutators of the both sides of (58) and (59) gives another correspondence where the expression on the right arises from [φ m , h n ] = (i/a)δ mn π n . We note the commutation relation [φ m , π n ] = −πiδ mn h n .
This reduces to the canonical commutation relation between φ(x) and Π φ (x) in the continuum limit because the density of the kinetic term diverges as −1/(πa 2 ) 14 so that we can replace h n by −1/(πa) in (61). The lattice Schwinger model described in Section III A should correspond, in the continuum limit, to specific values of w 0 and w 1 in (13). In the appendix of [36], it was argued that 14 The divergence can be computed by the free fermion because it is not affected by g or m, which only appears as ga or ma.
The charge Q is conserved and can be treated as a cnumber within a fixed charge sector. In fact, if the value of ν in (4) is ν 0 (ν 1 ) at x = 0 (x = L), we have 15 The relations (62) and (63) are non-trivially consistent with the correspondence between with parity of N and the fermion boundary conditions found in Section III A. We will also explicitly confirm the identification (62) by comparing the charge densities computed by DMRG and by bosonization. We note that the eigenvalue of Q is even (odd) if N is even (odd). Thus the winding number w 1 − w 0 is an integer (a half-integer) if N is even (odd).

C. Comparison of DMRG and analytic results
Here we compare the DMRG results based on the spin formulation in Section III and the analytic results based on bosonization in Section II B. For our implementation of DMRG, we used the ITensor library [47]. See [48] for a related study.
For the chiral condensate ∝ (−1) n Z n , we plot the DMRG results including the extrapolated values and the analytical results in FIG. 4(b) for the case with no probe charges. After extrapolation, the DMRG and analytical results match well.
For the charge density, we plot the DMRG and analytic results in FIG. 4(a). We see that they agree very well. This gives strong evidence for the identification (62). Near the right boundary, the charge density profile is identical to that near a probe of charge −1/2 for N even and +1/2 for N odd. We note that the parameters w 0 and w 1 parametrizing the Dirichlet boundary conditions for φ in 13 are related to the boundary charges q L and q R on the left and right boundaries as generalizing (40). Therefore, the charges on the boundaries are half-integral, signifying charge fractionalization. Indeed, for N both even and odd, the charge density near left boundary has a spatial profile identical to the charge density near a probe charge +1/2.

D. DMRG with a modified Gauss law constraint
Above, we used the standard Gauss law constraint (47), or equivalently (49), for DMRG. The constraint (47) is chosen [31] so that it is satisfied by the ground state |GS 0 in the "strong coupling limit" (ga → +∞ with m/g 2 a fixed) [44] with vanishing L n . In terms of the fermion occupation numbers χ † n χ n , |GS 0 corresponds to |010101 . . . .
In this subsection we consider a modified version of the Gauss law constraint [32] Compared with the standard choice (47), we dropped the term −(−1) n /2, which affects the boundary value of the scalar φ, as argued in the appendix of [36].
If the periodic b.c. is chosen, as explained in Appendix B.4 of [33], this modification of the Gauss law constraint is equivalent, via a shift of L n by (−1) n /4, 17 to a shift of the mass parameter such that the theory with a vanishing shifted mass enjoys a discrete chiral symmetry and a faster convergence to the continuum limit. While one has to allow L n to take non-integer values to satisfy the modified Gauss law (66), one can require the shifted version to take integer values. Solving the modified constraint with the boundary condition L −1 = 0 and fixing the gauge, we obtain the modified Hamiltonian A direct calculation shows that Comparing with (51) we see that, within the fixed charge Q sector, the modification (66) of the Gauss law is equivalent to a shift of the mass m → m − (g 2 a/8) [33] and a shift of the theta angle ϑ n → ϑ n − (π/2). The latter shift would be further modified if we chose a boundary condition other than L −1 = 0. FIG. 5 displays the profiles of the chiral condensate ψ ψ(x) and the charge density ψ γ 0 ψ(x) computed by analytic formulas and DMRG for m = 0. Contrary to FIG. 4(b), extrapolation is unnecessary because the modification of the Gauss law, which is partially equivalent to the mass shift of [33], makes the convergence to the continuum limit much faster.

IV. SUMMARY AND DISCUSSION
In this work, we studied three formulations of the Schwinger model: the original fermionic formulation, the bosonized formulation, and the Hamiltonian lattice formulation. We computed analytically physical observables in the ground state using the bosonized formulation and found excellent agreements with the DMRG computations in the lattice formulation. We clarified the correspondence between boundary conditions in different formulations. We studied a non-standard Gauss law constraint (66) in the lattice formulation, and showed that it is equivalent to the mass shift of [33] and a shift of the theta angle. In accordance with [33], we found that the 17 The corresponding manipulation for the open b.c. is (68). modification of the Gauss law makes the convergence to the continuum limit faster.
As for future directions, it would be interesting to rederive our analytic results in the path integral formalism, along the line of [28]. It would also be worthwhile to establish the faster convergence more firmly by computing the precise difference between the lattice and continuum Hamiltonians. This should be possible by classifying the potential counterterms to the local observables along the line of [49,50] that deals with the Euclidean path integral. Finally, one should be able to perform DMRG in a similar manner to compute local observables in nonabelian lattice gauge theories in 1 + 1 dimensions.
In this appendix we compute the ground state energy of the massless Schwinger model with probe charges, using the effective potential obtained in [42].
By integrating out the matter field and restricting to a static gauge field, the effective Lagrangian, on an infinite spatial line, is found to be where we introduced the density ρ(x) of external charges. For two charges q 1 and q 2 separated by distance , , the solution to the Euler-Lagrange equation gives the two-body potential [27] V q1,q2 ( ) = − π 2 To compute the energy on an interval [0, L], we extend the domain of the charge density ρ(x) from to (−∞, ∞) as an even periodic function of period 2L, ρ(−x) = ρ(x), ρ(x+2L) = ρ(x). We solve (A2) for A 0 using the Green's can be evaluated by summing the two-point potential between 1) the probe charges in the interval [0, L], and 2) the probe charges in the interval and image charges.
As an example let us consider the charge distribu- We extend ρ pair (x) to an even periodic function, which is depicted in FIG. 6. The energy (A4) reproduces (28).
As another example, let us modify the set-up in the previous paragraph by adding charges q L and q R to the left and right boundaries, respectively. We are interested in the q-dependent part of the energy. To compute it, we sum the two-point potentials between 1) the probe charges in the interval [0, L], 2) the probe charges in the interval and their image charges, and 3) the probe charges in the interval and the boundary charges including their images. The two-point potential between boundary charges is q-independent and we drop it. Compared with the previous paragraph, the new contribution is from 3):  (13), the ground state energy computed from (16)) turns out to be This is consistent with (A5) by the relations in (65), i.e., w 0 = q L /2 and w 1 = −q R /2. For w 0 = −w 1 = 1/4, ∆E pair = 0. This result appeared and was used in [36]. We checked that the DMRG computation of the ground state energy with large N agrees well (as a function of ) with E pair + ∆E pair for (q L , q R ) = (1/2, −1/2) if N is even, and for (q L , q R ) = (1/2, 1/2) if N is odd.
Appendix B: Exact one-form symmetries in lattice QCDs in 1 + 1 dimensions In this appendix, we show that the general lattice QCD in the Kogut-Susskind formulation [39] enjoys an exact one-form C 0 symmetry, where C 0 consists of the elements of the center of the gauge group G under which the matter fermions are invariant. The presence of such a center 1-form symmetry well-known in the continuum limit [51] and is also known for the charge-q lattice Schwinger model [33]. As the discussion for the general lattice QCD is rather abstract, we begin with the charge-q Schwinger model, which can be understood more intuitively.
The charge-q Schwinger model, i.e., the U (1) gauge theory with a single Dirac fermion of charge q, has attracted attention in recent years. See, e.g., [52,53]. As the defining action we take Unlike in Section II, we define probe charges using couplings separate from the theta angle. We also take q p to be integers so that the corresponding Wilson lines are genuine line operators rather than boundaries of topological surface operators. The bosonized Lagrangian is The Gauss law constraint reads The theory possesses a Z q one-form symmetry, whose generator can be expressed in the bosonized form [52,53] V q := exp 2πi qg This is piecewise constant as a topological operator should be, and labels the distinct decomposed sectors of the theory called "universes" [54][55][56][57][58].
The corresponding Hamiltonian of the lattice theory in the presence of probe charges is Again, we work in a formulation slightly different from Section III and [52] and implement the effects of probe charges by adding the corresponding terms in the Gauss law constraint In terms of spin variables we have The lattice generator of the Z q one-form symmetry, corresponding to (B4), is [33] where we used the correspondences (57) and (58) and the fact that Z i + (−1) i vanishes mod 2. As in the continuum case, the Gauss law (B5) implies that V q acting on a physical state is almost constant as a function of the position but gets multiplied by a phase as one crosses probe charges q p (temporal Wilson lines) at n = n p . This means, by the Wick rotation and the exchange of space and time, that V q obeys the expected commutation relations with the Wilson lines. 18 We now turn to an arbitrary Hamiltonian lattice gauge theory with a general gauge group G and a fermion in representation ρ [39]. For G we only require that it is compact: it can be non-abelian, discrete, a product, a quotient, or something more complicated. The Lie algebra g of G decomposes into simple Lie algebras The maximal torus T of G has the Lie algebra t = b t b , where g and g b are the Cartan subalgebras of g and g b , respectively. The center C of G has the Lie algebra c = i c i , where c i R is the Lie algebra of the "U (1) factor" labeled by i. In general ρ is reducible: where ρ f is an irreducible representation of G. Again we consider a one-dimensional lattice with sites labeled by n = 0, 1, . . . N − 1. The Hilbert space of the theory is the tensor product of the local Hilbert spaces associated with sites n ∈ {0, . . . , N − 1} and links n ∈ {0, . . . , N − 2}. On each site n, we have a fermion Fock space, possibly tensored with the representation space for a probe charge. The fermionic Fock space is generated by the fermion χ n = (χ f n ) f in representation ρ = f ρ f and its hermitian conjugate. In addition, if we place a probe charge in representation R p on site n p , we tensor the Fock space with the representation space V p of R p . On each link n we have the space of square-integrable functions on G. The total Hilbert space is thus of the form The Hamiltonian takes the form 19 where w = 1/2a, J α = g 2 b a/2, g b is the coupling constant for g b , ϑ i is the theta angle for c i , and m f is the mass for the fermion labeled by f . The trace tr is taken in 19 We do not include a kinetic term for the discrete part of the gauge group. Thus if the whole gauge group is discrete and there is no matter, the gauge theory is topological.
a faithful irreducible representation of G, in which g n is represented by a unitary matrix U n . (The g b part of) the "left' canonical momentum L n = ⊕ b L b n conjugate to g n ∈ G can be expanded as L n = L αn T α and obeys (in our sign convention) the canonical commutation relation [g m , L αn ] = δ mn T α g m , where T α = κ αβ T β , the matrix κ αβ is the inverse of κ αβ = tr(T α T β ), which is a Killing form of g. Let us define R n := g −1 n L n g n . Then one can show that R αn defined by R n = R αn T α satisfies the commutation relation [g m , R αn ] = δ mn g m T α . (B12) The group G of gauge transformations is the product of copies of G, each associated with a site n. The gauge transformation h n ∈ G on site n acts as χ n → ρ(h n )χ n , g n → h n g n h −1 n+1 , L n → h n L n h −1 n , R n → h n+1 R n h −1 n+1 , and leaves the Hamiltonian invariant and the canonical commutation relations invariant. For the continuous part of the gauge group, the Gauss law constraints are where T α p are the generators in the representations R p for probe charges. It is possible to consider, as in (66), the modified version of the Gauss law constraint where the term containing (−1) n is dropped.
If the gauge group G contains as a factor the cyclic group Z d = {e (2πi/d)j | j = 0, 1, . . . , d − 1}, on each link there exist operators Z n and X n such that Z m X n = exp[(2πi/d)δ mn ]X n Z m , Z d n = X d n = 1. The Gauss law constraint takes the group form where D is a diagonal matrix of Z d charges (integers modulo d) for the fermion, and q s are the Z d charges of the probes at n = n s . For a general gauge group (including non-abelian discrete groups such as the dihedral group D 4 ) instead of imposing the Gauss law constraint in terms of operators, we can simply project the total Hilbert space H total onto the physical Hilbert space H phys , which is the subspace of H total invariant under the group G of gauge transformations [59].
To study one-form symmetries, let C 0 consist of the elements of the center C (of the gauge group G) under which the fermion χ n is invariant. Since C 0 is abelian and compact, it is of the form C 0 = U (1) M × Γ, where Γ is a product of cyclic groups. On each site n and for c ∈ C 0 , let us consider the operator Gauge n (c) implementing the gauge transformation corresponding to c −1 . 20 It is of the form Gauge n (c) = Left n (c −1 )Right n−1 (c)R p (c −1 ) δnn p , where Left n (h) (resp. Right n (h)) is the operator corresponding to the left (resp. right) action of h on the copy of G on link n. The appearance of Left n (c −1 ) and Right n−1 (c) can be understood from (B13). The operator R p (c −1 ) δnn p represents the action of c −1 on the representation space V p for the probe p. Because c is in the center, in fact we have Left n (c −1 ) = Right n (c −1 ) =: V n (c). On the physical Hilbert space H phys , which is invariant under gauge transformations, we have the equality V n (c)V n−1 (c) −1 R p (c −1 ) δnn p = 1 or equivalently V n (c) = V n−1 (c)R p (c) δnn p . (B15) Since c is in the center and R p is an irreducible representation, R p (c) = exp iα Rp (c) is in fact a c-number corresponding to the charge under C 0 . Equation (B15) establishes that V n (c) is the generator of the one-form symmetry for C 0 . It is constant between probe charges, and obeys the expected commutation relation between Wilson line operators W R = Tr R P exp i A : which we rewrote as an operator relation via a Wick rotation and a rotation in the Euclidean spacetime.