Physical properties of the massive Schwinger model from the nonperturbative functional renormalization group

We investigate the massive Schwinger model in $d=1+1$ dimensions using bosonization and the nonperturbative functional renormalization group. In agreement with previous studies we find that the phase transition, driven by a change of the ratio $m/e$ between the mass and the charge of the fermions, belongs to the two-dimensional Ising universality class. The temperature and vacuum angle dependence of various physical quantities (chiral density, electric field, entropy density) are also determined and agree with results obtained from density matrix renormalization group studies. Screening of fractional charges and deconfinement occur only at infinite temperature. Our results exemplify the possibility to obtain virtually all physical properties of an interacting system from the functional renormalization group.

However the FRG is not restricted to the study of universal properties. When the microscopic theory is known or with the help of additional input (e.g. the knowledge of some physical quantities of the microscopic theory), one can compute the expectation value of observables even when the system is far away from a critical point. The FRG has been used in many models of quantum and statistical field theory ranging from statistical physics and condensed matter to high-energy physics and quantum gravity [10]. Besides the interest in models where perturbative approaches or numerical methods are difficult for various reasons, there is an ongoing effort to characterize and quantify the efficiency of the FRG by considering well-known models of field theory.
In this paper, to exemplify the predictive power of the FRG formalism, we determine both universal and nonuniversal properties of the Schwinger model, at zero and finite temperature. Our analysis is based on a previous study of the sine-Gordon model [25] and goes beyond previous FRG applications to the Schwinger model [26,27]. The results compare well with numerical studies except for the finite-temperature phase diagram where, contrary to the expectation, we find a region with spontaneous symmetry breaking (SSB) at T > 0. This observation is likely an artifact of the FRG calculation using a truncated derivative expansion, as we will discuss.
After briefly reviewing the Schwinger model in Sec. II, we start our discussion in Sec. III by making the identification of the parameters between the fermionic and the bosonic theory precise. We then briefly review the FRG using a derivative expansion (DE) at finite temperature in Sec. IV. In Sec. V we investigate the phase transition of the Schwinger model and determine the critical ratio (m/e) c and the critical exponents. We discuss the observation of a nonconvex effective potential in the SSB-phase in Sec. VI and then, still at zero temperature, determine the θ dependence of nonuniversal observables in Sec. VII. This calculation is extended to finite temperature in Sec. VIII where we also consider the issue of SSB at finite temperature. Finally we conclude the paper in Sec. IX.

II. THE SCHWINGER MODEL
The Schwinger model [28] describes quantum electrodynamics (QED) in d = 1 + 1 space-time dimensions. It was originally introduced to show that a gauge field, for which an explicit mass term is forbidden by gauge invariance, can acquire a mass dynamically through the chiral anomaly. With both massless and massive fermions, it demonstrates confinement [29,30] and thus serves as an important toy model for real confining theories such as quantum chromodynamics (QCD) in 3+1 dimensions.
The microscopic action of the Schwinger model in Euclidean space is given by [31] S[ψ, ψ, A] = x −ψγ µ (∂ µ + iA µ )ψ − mψψ where A µ is the usual U (1) gauge field and F µν = ∂ µ A ν −∂ ν A µ the corresponding field strength. Moreover, ψ and its conjugateψ are two-component Dirac spinor Grassmann fields describing the charged fermions. The Dirac matrices and the real antisymmetric tensor µν are taken in Euclidean space such that 1 2 {γ µ , γ ν } = g µν = diag(1, 1) and 10 = − 01 = 10 . The two free parameters of this theory are the fermion mass m and the electric charge e (the latter has dimension of mass in d = 1 + 1). Contrary to 3+1-dimensional QED, the Schwinger model permits a topological term linear in the field strength, that gives rise to an additional free parameter, the vacuum angle θ. As the model is a super-renormalizable theory (positive mass dimension of the coupling constant e), it can be defined without an explicit ultraviolet (UV) cutoff (Λ → ∞). At low energies, the Schwinger model can be mapped onto the massive sine-Gordon model [32] defined by the action with β = 2 √ π, the Schwinger mass M = e/ √ π and the parameter u, linearly related to the fermion mass m. The bosonic action in Eq. (2) is also written in Euclidean space. Contrary to the fermionic action, it is only welldefined with a UV regularization and the relation between m and u depends on the implementation of this cutoff. The exact relation between these two parameters is not well documented for the path integral formalism in the literature, as the equivalence is most commonly proven in the operator formalism [29,30] or in mass perturbation theory [33] which both, at least implicitly, rely on a normal ordering choice [34]. This freedom is of course absent in the path integral formalism and we shall discuss the exact relation in Sec. III. The equivalence between the theories in Eqns. (1) and (2) has also been proven at finite temperature [35].
From the form of the bosonic action, Coleman first conjectured a continuous phase transition between a symmetric phase and a phase in which the charge conjugation symmetry φ → −φ at θ = π is spontaneously broken [32]. In the SSB phase "half-asymptotic" fermions can exist as widely separated kink-antikink pairs, as long as they are ordered in the right way. In the symmetric phase, no kink solutions exist. Fermions can only exist in bound meson states, so there is confinement in the weak sense. The byname weak refers to the fact that the fermionantifermion pair in the bound state can still be separated at finite energy cost, through the creation of a new pair of particles from the vacuum. The newly created particles form new bound states with the original particles to screen their electric field. Fractionally charged external particles cannot be screened by pair creation [36,37] at nonzero fermion mass and therefore experience confinement in the strong sense, i.e. they can only be separated for infinite energy cost. In the massless model, arbitrary charges can be screened though.
Numerically the existence of a phase transition when θ = π was shown using various approaches [38][39][40][41]. In recent years density matrix renormalization group (DMRG) techniques have become quite popular and provide the most accurate result for the critical value (m/e) c = 0.3335(2) [40]. Furthermore, estimates for the critical exponents were calculated [40,42], showing that the phase transition belongs to the universality class of the two-dimensional Ising model.
The DMRG method also allowed the precise calculation of infrared (IR) observables, both at zero [41] and finite temperature [43][44][45]. By the findings of Ref. [45] we expect that at finite temperature there is no phase transition and the Schwinger model has a unique ground state, see Fig. 1.

III. MAPPING OF PARAMETERS
To deduce the parameters of the low-energy theory (2) from the microscopic model (1) one has to compare renormalized parameters or expectation values in the IR obtained in the two approaches. The Schwinger model offers two limits which we know how to solve exactly and can be used to relate u, M and Λ to m and e. One is the weak coupling limit e/m → 0 + or, in the bosonic formulation, M Λ/u → 0 + where the Schwinger model is equivalent to the sine-Gordon model at K = β 2 /8π = 1/2 (origin of Fig. 1). In the strong coupling limit m/e → 0 + or u/M Λ → 0 + (infinitely far into the radial direction of Fig. 1), i.e. for massless fermions, the bosonic action is that of a free field. In the following two subsections, we show that both limits yield the same relation between the fermionic and bosonic parameters.

A. Weak coupling limit
In the sine-Gordon model the soliton mass, i.e. the mass of the fermion, is known exactly [46], where b is a nonuniversal scale factor that depends on the implementation of the UV cutoff [25]. For the special case K = 1/2, which corresponds to the Schwinger model, this reduces to We assume a hard momentum cutoff at with the Euler-Mascheroni constant γ 0.5772. A discussion of more general cutoff functions can be found in the supplemental material of Ref. [25]. The relation between m and u is then given by Λm.

B. Strong coupling limit
Even though in the strong coupling limit there is neither a mass term in the fermionic model nor a cosine term in the bosonic model, we can still compare the relation between u and m through the expectation value of the chiral density. In the massless Schwinger model the expectation value for the chiral density at θ = 0 is given by [47,48] In the bosonic theory, this expectation value is obtained from the expectation value of the exponential operator where the average on the right-hand side is taken with the (Gaussian) action of the field φ. The propagator must be calculated in the presence of the same cutoff that was used in the calculation in the weak coupling limit, i.e. a hard cutoff, We deduce that which agrees with the result obtained from the weak coupling limit in Eq. (6).
In the fermionic theory, there is no UV cutoff, so it is evident that one of the parameters e and m sets the overall mass scale of the theory, while physics and especially the critical point of the phase transition may only depend on the ratio m/e. In the bosonic theory, this translates into the fact that, provided Λ e, m, the expectation values and the critical point only depend on the ratio and not, as one might naively expect, also on the dimensionless ratio u/M 2 [49].

IV. FUNCTIONAL RENORMALIZATION GROUP APPROACH
We apply the FRG approach to the bosonic form of the Schwinger model (2) in Euclidean space where the imaginary-time dimension has a finite length given by the inverse temperature, with x = (r, τ ). Wilson's idea of the renormalization group (RG) is realized by including a scale dependent IR regulator ∆S k [φ] in the partition function The flowing effective action which is defined as the modified Legendre transform of ln Z k [J], then smoothly interpolates between the microscopic action Γ kin [φ] = S[φ] for k in Λ and the macroscopic quantum effective action Γ k=0 [φ] = Γ[φ]. To ensure this property ∆S k [φ] must behave as a masslike term of order k for small momenta p and Matsubara frequencies ω n , i.e. for y = (p 2 + ω 2 n /c 2 k )/k 2 1, and become negligible for high energy modes y k 2 . Here c k is a flowing velocity that encodes the difference in scaling between momenta and frequency and will be defined below.
We realize this requirement by choosing with q = (p, ω n = 2πT n) and the cutoff function We further define the combined momentum integral and Matsubara sum At T = 0 the frequency sum turns into an integral. The prefactor Z k appearing in the cutoff function is defined below. With the free parameter α we can probe the dependence of the final results on the precise form of the regulator.
The idea behind the FRG approach is to integrate Wetterich's exact flow equation [5][6][7], from the microscopic initial conditions Γ kin to the effective action Γ k=0 . On the right hand side of equation (19), Γ k is the second-order functional derivative of Γ k . This functional partial differential equation can usually not be solved exactly so we employ the derivative expansion which is a commonly used, systematic approximation scheme [8,11,12,50]. At second order it relies on the following ansatz for the flowing effective action which is defined by three k-dependent functions of φ: Z k (φ), X k (φ) and U k (φ). At finite temperature the O(2) Euclidean symmetry is broken, such that we have to admit different renormalization functions for the spatial and temporal derivatives: Z k (φ) = X k (φ). The initial conditions for these functions are fixed by the microscopic action (2), One can split the effective potential into the mass term and a periodic contribution: (19) only depends on the second derivative of Γ k , which is periodic in the field, the periodicity of Z k (φ), X k (φ) and V k (φ) is preserved during the RG flow and the mass term does not renormalize: As in the study of the sine-Gordon model [25], it is convenient to use the dimensionless functions where and · · · φ denotes the average over [−π/β, π/β]. We also define One can also introduce dimensionless spacetime coordinatesr where c k = X k /Z k is a running velocity. The latter is however not the actual, physical, velocity which is defined from the minimum of the potential: U k (φ 0,k ) = 0. In practice, the difference between c k and c k,phys is small. Note that we do not rescale the field. This agrees with the fact that φ is an angular variable in the sine-Gordon and massive Schwinger models and has therefore a vanishing scaling dimension (as evident from the fact that φ appears inside a cosine). These dimensionless functions however do not allow us to find a fixed point associated with a transition that belongs to the universality class of the two-dimensional Ising model. For example, it is well known that the field has scaling dimension 1/4 at this transition, which seems to be incompatible with the angular character of φ in the massive Schwinger model. Thus, to obtain a fixed-point solution with nonvanishing scaling dimension of the field, one has to allow for a rescaling of the field, at least for fields near the origin, which is the important field range to understand a second-order transition. At the critical point (anticipating parts of the discussion in Sec. V), we indeed find that the minimum of the potential vanishes as φ 0,k ∼ 1/ Z k (0) ∼ k η/2 with η = lim k→0 η k , so that the rescaled variableφ 0,k = Z k (0)φ 0,k reaches a fixedpoint valueφ * 0 . This clearly shows that Z k (φ 0,k ) has the usual meaning of a field renormalization factor near the transition and η can be identified with the anomalous dimension. In practice, we define the rescaled fieldφ by and redefine Z k and X k as The change of variable (28) preserves the periodicity of the flow equations whereas φ ∼φ/ √ Z k for φ → 0. The redefinition (29) is necessary to identify η = lim k→0 η k with the anomalous dimension. From a practical point of view, the rescaled field allows us to zoom in on the smallfield region even using an equally spaced grid inφ, which is necessary to study the vicinity of the critical point where φ 0,k becomes extremely small. The change in the definition of Z k effectively only changes the regulator's amplitude, i.e. α, to which the RG flow is insensitive far from the critical point. Near the critical point α needs to be optimized anyway (see Sec. V and Appendix A) and the change in the definition of Z k only affects the optimal value of α. We emphasize that the nonlinear transformation (28) is only employed when looking at the phase transition. All expectation values in Secs. VII and VIII are obtained using a regular grid in φ.
The flow equations for the functions U k , Z k and X k can be obtained from Eq. (19). They differ from those of the sine-Gordon model [25] only in the initial conditions. For the determination of the function X k a discrete derivative, using the smallest Matsubara frequency, is used. The numerical results confirm that this choice performs better than a continuous derivative. We display concrete expressions for the flow equations in Appendix B.

V. PHASE TRANSITION
We study the continuous phase transition at zero temperature, i.e. on the T = 0 plane of Fig. 1. By choosing θ = π, we further restrict ourselves to the axis of the phase diagram where only the parameter u/M Λ needs to be fine-tuned to arrive at the critical point (red dot of Fig. 1). This point is defined by an infinite correlation length, i.e., lim k→0 U k (0)/Z k (0) = 0. Using Eq. (12) we find the critical ratio which is in good agreement with the most accurate result found in the literature (e/m) c = 0.3335(2) [40]. We also determine the critical exponents η and ν. The anomalous dimension η has been defined in the previous section. The exponent ν characterizes the divergence of the correlation length and can be obtained from the flow ofŨ k (φ) near the critical point and forφ ∼φ 0 : at long RG time |t| (t = ln(k/Λ)). The exponents were optimized using the principle of minimal sensitivity (PMS) on the parameter α. More details can be found in Appendix A.

VI. CONVEXITY IN THE ORDERED PHASE
In the following, we want to determine nonuniversal expectation values in the IR, where the effective action should have converged to a convex functional. However as can be seen in Fig. 2, the effective potential does not become convex in the ordered phase. This is caused by η k tending towards 2 (from below) on the concave part of the potential. As a consequence Z k (φ) is unbounded in this region, while on the convex part η k reaches zero and thus Z k (φ) a finite value, leading to nonanalyticity inZ k near the potential minima.
More importantly, unless η k converges to 2 extremely slowly, i.e. lim k→0 (2 − η k ) ln k → −∞, which is not realized in practice, the regulator function R k will approach a finite value in the IR, R k = Z k k 2 yr(y) ∼ k 0 yr(y) for fixed y and k → 0. This violates our initial requirement, R k=0 (q) = 0, that ensured the convergence of Γ k to the quantum effective action of the Schwinger model, and allows the effective potential to remain concave. On the other hand, it is not possible to choose a point of renormalization Z k = Z k (φ r ) with φ r on the convex part of the potential. Since Z k (φ) is unbounded, the regulator cannot ensure the positivity of the propagator and a pole is hit at finite RG time. Note that, except in a few cases [52][53][54], the DE does not in general guarantee the convexity of the effective action Γ k=0 and nonconvex potentials have been obtained in other models [55]. So far it is unclear whether a regulator can be chosen such that both these problems can be avoided.
However, the concavity is only conceptually problematic. In practice, the mass M effectively stops the flow of physical observables at a finite scale where the lack of convexity and the pathological small-k behavior of R k are irrelevant. Many IR observables can therefore be accurately predicted as shown below.

VII. DEPENDENCE OF OBSERVABLES ON VACUUM ANGLE θ
In this section, we present our results for various observables at T = 0 as a function of the vacuum angle θ. Our discussion thus extends to the entire bottom plane of Fig. 1. We compare with the DMRG results of Ref. [41].
The string tension σ Q , which characterizes confinement, is defined as the change of energy per unit length  when two probe charges ±Q are introduced in the system. In the limit where the two charges are infinitely separated, they generate an additional background electric field that can be taken into account by a redefinition of the vacuum angle: θ → θ + δ where δ = 2πQ/e. The string tension can thus be obtained by comparing two ground-state energies at different θ. Since the energy per unit length is nothing but the effective potential in the FRG formalism, one obtains where the notation emphasizes that the k = 0 effective potential U θ depends on the vacuum angle. Here, φ 0,θ is the value of the field at the minimum of U θ . The string tension is a periodic function of θ. In the following we restrict ourselves to θ = 0 and consider (with the relabeling δ → θ) As the order parameter, the electric field E = − e 2π φ 0,θ is also an interesting observable.
The chiral density, which serves as an order parameter for chiral symmetry breaking, is of particular interest in the massless Schwinger model since it is nonvanishing due to the chiral anomaly. In the massive model, the chiral symmetry is broken explicitly. Nevertheless, the anomaly still leaves an imprint on the chiral condensate. It can be obtained by taking a derivative of the partition function with respect to u or, equivalently, the fermion mass. In terms of the effective potential this yields This derivative can be obtained either by taking the finite difference after integrating the flow equations or, as we discuss in Appendix C, by integrating the flow equation for ∂ m U k but the results are less accurate. Finally, we are interested in the particle spectrum. This was discussed originally by Coleman [32], who counted the number of stable particles in all phases. Unfortunately only the first excitation is available in the derivative expansion approximation. Its mass is given by the pole of the propagator, These four observables are shown as a function of θ in Fig. 3 where they are compared with the DMRG results of Buyens et al. [41]. The string tension and the electric field show an excellent agreement. From this, we can conclude that the effective potential has very well converged on its convex part, even though part of it remains concave.
The mass gap is also very well reproduced in the symmetric phase. In the broken phase, there is a significant difference, especially for θ near its critical value θ c = π. Since the convex part of the effective potential seems to have converged to its infrared value, it seems that the problem must come from Z k (φ). Indeed we see in Fig. 2 that in the SSB phase Z k (φ) approaches a nonanalytic function. While it remains regular for any finite k, its exact structure exhibits pronounced peaks, which are hard to resolve numerically. This is less of an issue in the sine-Gordon model; the minimum being always at φ = 0, it is maximally far away from the difficult region. The error on the mass, which is about 10% in the sine-Gordon model for K = 1 2 [25], gets amplified to more than 50% at θ = π by the mass term of the Schwinger model, which pushes the minima of U k (φ) towards the peaks of Z k (φ).
Numeric results for the string tension, electric field, and chiral density using matrix product states were also reported in Ref. [56]. They are compatible with ours but concern only the symmetric phase.
Our results are robust against a change of the regulator amplitude α. Note that α must be chosen greater than 2 to ensure that the propagator remains positive definite [54].
As we expect, the electric field, as the order parameter of the phase transition, is discontinuous in the ordered phase for θ = π. The string tension exhibits a cusp at this point. A larger fermion mass allows for a larger electric field and also stronger confinement. The larger string tension is easy to interpret as heavier fermions imply a higher threshold for pair production, responsible for a partial screening of fractional charges. The same is true for the electric field, as the quark-antiquark pairs try to screen the background electric field. Both quantities lose their θ dependency in the limit m → 0 as is expected from the massless Schwinger model. This also holds for the mass gap M 1 , which approaches the Schwinger mass M in this limit. Near the phase transition, the theory becomes almost massless at θ = π. For the chiral density, the opposite is true. In the zero-mass limit, it is strongly dependent on the vacuum angle due to the chiral anomaly. At finite mass, the chiral symmetry is explicitly broken, but in the weak coupling limit m/e 1, the theory becomes a free fermionic theory, where the gauge field and therefore also the vacuum angle become unimportant.

VIII. TEMPERATURE DEPENDENCE OF OBSERVABLES
Finally we extend the discussion to finite temperatures, i.e. the vertical dimension of Fig. 1.
In Fig. 4 we benchmark our finite temperature implementation by comparing our result for the chiral density at m = 0 with the exact analytic expression by Sachs and Wipf [48], Our result agrees both in the zero-temperature limit, where it approaches the value mentioned in Eq. (7), and in the high-temperature limit, where the chiral density decays exponentially, At a value of about T /e ∼ 6 machine precision is reached, so the finite difference method used to compute the derivative in (34) breaks down at this point. When using the calculation based on the flow equation (see Appendix C), the lack of numerical precision manifests itself as a constant offset of the function ∂ m U θ (φ) which can be easily removed and the calculation therefore continued beyond this limit. We now consider nonzero values of m where the string tension and the electric field are nonvanishing. Additionally, we will also calculate the entropy density, which is related to the string tension by using the finite difference method with ∆T = 10 −7 Λ to compute the temperature derivative. We first consider the limit θ → 0. To allow for nontrivial electric field, we consider a small, but nonvanishing, value α = θ/2π = 0.05. Our results in this limit are shown in Figs. 5 and 6. They compare rather well with the finite-temperature DMRG study by Buyens et al. [45].
In the low-temperature regime, the agreement is better for higher values of m/e. However, we believe that the FRG results are more reliable than the DMRG ones for all values of m/e in this limit; our small, finite temperature results converge to our zero-temperature result, which are in accurate agreement also with Buyens' more recent zero-temperature study [41], as we have shown in Fig. 3. In the high-temperature limit, our results agree very well with Ref. [45] for the smaller values of m/e, but for higher values the amplitude is no longer accurately reproduced. Nevertheless the correct exponential decay is exhibited for all values of m/e (Fig. 6). As compared to Ref. [45] we can continue our results to higher temperatures, but eventually fail at the threshold of numerical precision as discussed before for the case m = 0. The difference with the results of Ref. [45] is more pronounced in the crossover region between the low-and high-temperature regimes, especially for the chiral con- densate. For the three higher values of m/e, our results for the peaks are systematically higher, even though the zero temperature limits agree. Now turning to the case θ = π we observe a major qualitative difference with the DMRG [45] result: For values m/e > (m/e) c we find that thermal fluctuations are no longer able to restore the symmetry of the effective potential below a critical temperature T c (m/e). The dependence of the critical temperature on the ratio m/e is shown in Fig. 7 along with the nonvanishing order parameter. This is unexpected since at the scale k = 2πT the model becomes effectively one dimensional as the nonzero Matsubara frequencies become heavily suppressed. Although, as discussed in Ref. [45], neither the Mermin-Wagner theorem nor the Peierls argument apply to the Schwinger model, a one dimensional field theory at T > 0 corresponds to a quantum-mechanical one-particle problem, where SSB is usually assumed to be absent. Also, in Ref. [45] numerical evidence was found for the restoration of symmetry.
At finite temperature, the flow is eventually dominated by the vanishing Matsubara frequency ω n=0 = 0 (quantum-classical crossover). One can then approximate the Matsubara sum by T ωn f (ω n ) T f (0). The flow equations become one-dimensional and the prefac- and is suppressed when η k > 1 (T = T /c k k is the dimensionless temperature). This will occur whenever in the quantum part of the flow the system is attracted by the T = 0 fixed point of the ordered phase characterized by lim k→0 η k = 2. In that case the T = 0 fixed point remains attractive and the symmetry is not restored. This phenomenon is thus strongly related to the problem of convexity discussed in Sec. VI. As noted in Ref. [57], a similar difficulty occurs in the FRG approach to the onedimensional sine-Gordon model. The instantons, which are responsible for the restoration of the symmetry in the one-dimensional sine-Gordon model [58], are therefore not properly taken into account in the derivative expansion although they are accurately described in the two-dimensional case [25]. Interestingly, recent studies have shown using symmetry arguments that some specific quantum mechanical systems, like a particle on a ring with magnetic flux going through the ring, can indeed have a degenerate ground state [59,60]. The authors were able to map the free Schwinger model on a torus with no coupled fermions, corresponding to the limit m/e → ∞, to this particular quantum-mechanical model. Using Lorentz symmetry, this result is then also applicable to finite temperature via a Wick rotation, showing that the free Schwinger model has a degenerate ground state at T > 0. This was shown to hold also for the gauge field coupled to a matter particle [60]. However a requirement of their discussion is that the matter particle must have a charge greater than the fundamental charge, e.g. q = 2e. Since the model discussed in our present work has the fundamental charge, these arguments are not applicable. We are therefore convinced that the lack of symmetry restoration in the Schwinger model at finite temperature is an artifact of our approach.

IX. CONCLUSION
The nonperturbative functional renormalization group approach has proven to be an efficient method to study the sine-Gordon model. In particular, it gives an accurate estimate of the mass of the soliton and soliton-antisoliton bound state (breather) in the massive phase, which is exactly known thanks to the integrability of the model. It also allows us to quantify the amplitude of quantum fluctuations in agreement with the Lukyanov-Zamolodchikov conjecture [25]. In this paper, we have shown that the FRG approach is also a very powerful method, notably at zero temperature, to determine the physical properties of the massive sine-Gordon model, which is the bosonized version of the massive Schwinger model. More generally, the FRG is very useful to study quantum systems in d = 1 + 1 dimensions where solitonlike excitations may play a crucial role [61].
By computing the critical exponents, we confirm that the phase transition occurring in the massive Schwinger model for a value θ = π of the vacuum angle belongs to the two-dimensional Ising universality class, as had been shown in several previous studies [40,42]. Though the precision of the critical exponents is typical of a secondorder derivative expansion (DE), more accurate results could be obtained by pushing the expansion to higher orders [11,12]. On the other hand, we find that the phase transition occurs for a critical ratio (m/e) c = 0.318 (13) between the mass and the charge of the fermions, which is in agreement with the highest precision results available in the literature [40].
We have also computed the temperature and vacuum angle dependence of various physical quantities: string tension, electric field, chiral density, mass gap of the lowest-lying excitation and entropy density. At zero temperature, our results are in excellent agreement with DMRG lattice calculations [41]. The finite temperature results near θ = 0 are also in good agreement with the lattice calculations [45], though less accurate. At θ = π and for m/e > (m/e) c , we do not observe the restoration of the symmetry as expected for a quantum system in one space dimension at finite temperature. We ascribe this failure to the inability of our limited truncation to properly account for all the effects of instantons in the one-dimensional Schwinger and sine-Gordon models [57].
Whether or not topological excitations, in particular when they are responsible for symmetry restoration, can be captured by truncations like the DE is a very interesting issue in many models. There is no doubt that the DE, which yields very accurate estimates of the critical exponents [11,12], captures the topological excitations of the three-dimensional O(3) model in which the hedgehog singularities are known to be essential [62,63]. The situation is less clear in lower dimensions. Nevertheless, the kinks of the one-dimensional ϕ 4 theory [64] and the vortex excitations (responsible for the Berezinskii-Kosterlitz-Thouless transition) in the two-dimensional O(2) model [65] are, at least partially, captured. In the two-dimensional sine-Gordon model solitons and antisolitons, as well as their lowest-lying bound state (breather), are well described [25]. The possibility to accurately describe the topological excitations and their consequences for the one-dimensional sine-Gordon model, and therefore the finite-temperature Schwinger model (in d = 1 + 1), remains currently an open issue.

X. ACKNOWLEDGEMENTS
We thank Zohar Komargodski for his valuable insights on the issue of SSB at finite temperature, and Bertrand Delamotte for a useful discussion.
The work of S.

Appendix A: Regulator Dependence
The critical exponents were calculated using the principle of minimal sensitivity and three different regulators, which all have the same limit r(y) → α/y for y → 0 (Θ denotes the step function). The result is shown in Fig. 8. The first regulator, also defined in Eq. (17), shows the least sensitivity of the three regulators and was thus chosen with α = 2.