Quantum phase transition and Resurgence: Lessons from 3d $\mathcal{N}=4$ SQED

We study a resurgence structure of a quantum field theory with a phase transition to uncover relations between resurgence and phase transitions. In particular, we focus on three-dimensional $\mathcal{N}=4$ supersymmetric quantum electrodynamics (SQED) with multiple hypermultiplets, where a second-order quantum phase transition has been recently proposed in the large-flavor limit. We provide interpretations of the phase transition from the viewpoints of Lefschetz thimbles and resurgence. For this purpose, we study the Lefschetz thimble structure and properties of the large-flavor expansion for the partition function obtained by the supersymmetric localization. We show that the second-order phase transition is understood as a phenomenon where a Stokes and anti-Stokes phenomenon occurs simultaneously. The order of the phase transition is determined by how saddles collide at the critical point. In addition, the phase transition accompanies an infinite number of Stokes phenomena due to the supersymmetry. These features are appropriately mapped to the Borel plane structures as the resurgence theory expects. Given the lessons from the SQED, we provide a more general discussion on the relationship between the resurgence and phase transitions. In particular, we show how the information on the phase transition is decoded from the Borel resummation technique.

Then we interpret the second-order phase transition as a simultaneous Stokes and anti-Stokes phenomena. Our results show that the resurgence theory works for describing the second-order phase transition of the SQED. Given the lessons from the SQED, we finally provide a more generic discussion on relations between the resurgence and phase transitions. In particular, we generally show that the orders of phase transitions are determined by how saddle points collide and scatter as varying a parameter through a critical point. We also show how the information of the phase transition is decoded from the Borel resummation technique. We believe that our results open up potential applications of resurgence to quantum field theories.
Let us finally comment on previous works closely related to this paper 4 . The work [147] studied thimble structures of simple fermionic systems such as zero-dimensional versions of the Gross-Neveu model and Nambu-Jona-Lasinio model, and one-dimensional gauge theory coupled to a massive fermion with a Chern-Simons term. In particular, it was found in the zero-dimensional Gross-Neveu model that there was a jump in the number of contributing thimbles at the second-order chiral phase transition point in the massless case. It was also demonstrated that there was an interesting link between anti-Stokes lines and Lee-Yang zeros. There are also interesting works on the two-dimensional pure U (N ) Yang-Mills theory on lattice [81,148,149], which is technically reduced to a unitary matrix model called the Gross-Witten-Wadia model [150,151] 5 . It was found that there occurred a condensation of complex saddle points at the third order phase transition point in the large-N limit. Historically, physicists have studied simple models to draw lessons and to uncover general laws. The field of resurgence and its relation to QFT phase transitions is not an exception either. Now it is a good time to broaden the reach of resurgence toward more realistic QFTs. The SQED studied in this paper should be a nice first step along this direction since it is more realistic and nevertheless its partition function is expressed in a simple manner thanks to the supersymmetry. This paper is organized as follows. In Sec. 2, we review the work by Russo and Tierz [143]. In Sec. 3, we provide interpretations of the phase transition from the viewpoints of Lefschetz thimble. In Sec. 4, we discuss relations between the phase transition and resurgence structures. In Sec. 5, given the lessons from the SQED example, we give a more generic point of view on relations between the resurgence and phase transitions. Sec. 6 is devoted to conclusion and discussions. In App. A, we explain details on calculations of 1/N f flavor expansion, where 2N f is the number of the hypermultiplets in the SQED. In App. B, we present thimble structures for larger arg(N f ) while the main text focuses around arg(N f ) = 0. In App. C, we make some comments on the Padé-Uniformized approximation based on comparisons with the standard Padé approximation in some simple examples. In App. D and App. E, we study resurgence structures of the 1/N f expansion from the viewpoint of a difference equation for finite values of the FI parameter η and the rescaled parameter λ = η/N f respectively. In App. F, we point out a possible relation between the Borel singularities and complex supersymmetric solutions found in [115].
2 Quantum phase transition in the 3d N = 4 SQED In this section, we review the arguments of [143] to find the quantum phase transition in the 3d N = 4 SQED with a large number of hypermultiplets. Let us consider a 3d N = 4 SUSY U (1) gauge theory coupled to 2N f hypermultiplets with charge 1. We turn on a Fayet-Illiopoulos (FI) term and a real mass associated with a U (1) subgroup of the SU (2N f ) flavor symmetry 6 .
Applying the SUSY localization [154], the path integral is dominated by saddle points and one can exactly compute the S 3 partition function of this theory as [155][156][157] where η is the FI parameter and m is the real mass. The integral variable σ is the Coulomb branch parameter and the factor in the denominator is the one-loop determinant of the hypermultiplets 7 . Now we are interested in the 't Hooft-like limit: For this purpose, it is convenient to write the partition function as where S(σ) is the "action" defined by S(σ) = N f −iλσ + ln(cosh σ + cosh m) . (2.4) In the large N f limit, the integral is dominated by saddle points satisfying S (σ) = N f −iλ + sinh σ cosh σ + cosh m = 0. (2.5) This equation is solved by where ∆(λ, m) = 1 − λ 2 sinh 2 m. (2.7) The action at the n-th saddle point is Note that the saddles and the action values are complex in general. Also, note that the action can be written as S(σ ± n ) = S(σ ± 0 ) + 2πnN f λ. (2.9) This implies that the most dominant saddle point for real λ is either σ = σ + 0 or σ = σ − 0 , if it contributes to the integral.
The authors in [143] have observed that dominant saddles change at λ = λ c with λ c ≡ 1 sinh m . (2.10) For the subcritical region λ < λ c , only a single saddle σ + 0 dominates the integral. For the supercritical region λ ≥ λ c , two saddles σ + 0 and σ − 0 contribute to the integral with equal weights. [143] numerically checked that these saddle approximations agree with the exact analytic expression of (2.1) at the large N f limit. The second derivative of the "free energy" jumps at λ = λ c as (2. 11) This implies that the system exhibits a second-order phase transition at λ = λ c . 7 The integral (2.1) can be done exactly as Z =  (cosh m) with the associated Legendre polynomial P m (x) [143] but this form does not seem particularly useful for our purpose.

Lefschetz thimble structures
In general, saddles with smaller Re S give larger weights. However, such saddles do not necessarily contribute to the path integral. This is because its original integration contour may not be deformed to the Lefschetz thimbles (steepest descent paths) associated with such saddles. Also note that the "free energy" of a partition function on a general manifold is not necessarily real since it is not interpreted as a thermodynamic one. This means that contributing saddles cannot be determined only by requiring the free energy to be real. For these reasons, we should study Lefschetz thimble structures to describe quantum phase transitions.
In this section, we interpret the quantum phase transition in terms of Lefschetz thimbles of the integral (2.1) obtained by SUSY localization. This provides a more precise justification for the arguments in [143] reviewed in the previous section. For this purpose, we first extend the Coulomb branch parameter σ ∈ R to complex values z ∈ C, since saddle points and the associated Lefschetz thimbles are complex-valued in general. As we have seen in the last section, we have infinitely many saddle points σ ± n satisfying the saddle point equation. Contributing saddles are determined by looking at the Lefschetz thimbles (or the steepest descents) obtained by deforming the original contour without changing the value of the integral. This can depend on the original integral contour, the parameters (λ, m) and properties of the (dual) Lefschetz thimbles as explained below.
The Lefschetz thimbles J ± n associated with the saddle points σ ± n are defined as solutions of the differential equation called the flow equation, which indicate that integrals along Lefschetz thimbles are rapidly convergent and non-oscillating. We can express the original contour C R as a linear combination of the Lefschetz thimbles If k ± n is nonzero, it implies that σ ± n contributes to the integral while we have no contributions from the saddle points with k ± n = 0. It is known that each expansion coefficient k ± n is an integer since k ± n is identified with the intersection number between the original contour C R and the dual thimble (or the steepest ascent contour) K ± n associated with σ ± n , which is defined by In general, k ± n depends on (g, m) but its dependence is not continuous since k ± n is an integer. Typically k ± n is a constant or a step function, and the latter case leads us to a Stokes phenomenon.

Real positive N f
First, let us briefly see the Lefschetz thimble structures for real positive N f i.e. arg(N f ) = 0. We have numerically solved the flow equations and drawn the results at some representative values of (m, λ) in  We immediately see that the Lefschetz thimbles pass multiple saddle points both in the subcritical (λ < λ c ) and supercritical (λ ≥ λ c ) regions. Although we have explicitly shown the results only at the two values of (m, λ), we have checked that this feature remains to hold unless the parameters cross the phase boundary. The thimble structures imply that the decomposition in terms of the thimbles is not well-defined at arg(N f ) = 0 and the Stokes coefficient has a discrete change. In other words, the present case arg(N f ) = 0 is on Stokes lines.
The appearance of the Stokes lines here is natural because we have infinitely many saddle points with the same imaginary part of the action at arg(N f ) = 0 although this is not sufficient but necessary to have the Stokes lines. This can be explicitly seen as follows. In the subcritical region λ < λ c , one can easily show that all the saddle points are purely imaginary and their actions are real: In the supercritical region λ > λ c , the imaginary parts of the actions at the saddles are nonzero but they satisfy Im(S ± n ) = −Im(S ∓ n ) and Im(S ± n ) = Im(S ± 0 ) for ∀ n and λ > λ c . (3.7) The above structure essentially comes from the fact that the action at the saddles depends on n only via the term "2πnλN f " as seen from (2.9). This motivates us to take complex N f to go beyond the Stokes lines and understand the thimble structures more precisely.

Complex N f
Let us take N f to be complex while keeping λ real 8 : and study the Lefschetz thimble structures. In the main text we study the thimble structures only around θ = 0. See App. B for the non-small θ case. Let us first focus on the subcritical region λ < λ c presented in Fig. 2. Regardless of the sign of θ, the dual thimble K + 0 intersects once with the original integration contour C R . This means that the original integral contour C R can be deformed to the thimble J + 0 . Indeed, we can apply Cauchy's integral formula since the integrand decreases at infinity of the upper-half plane. Thus, we find a unique thimble decomposition In other words, there is no Stokes phenomenon in the subcritical region. The structures in the supercritical region λ > λ c are shown in Fig. 3. We observe qualitatively different behaviors compared with the subcritical region 9 . Firstly, multiple saddle points contribute to the integral. In particular, two saddle points σ ± 0 always contribute to it as their dual thimbles always intersect with the original integral contour C R . This justifies the arguments by [143] reviewed in Sec. 2. Note that this fact is a priori nontrivial since the saddles are complex. Secondly, the intersection numbers k ± n≥1 jump discontinuously as the phase θ is changed. The thimble structures depend not only on the sign of θ but also on the absolute value |θ|. Specifically, a common feature for each sign ± of σ ± n is that the saddle points σ ± n≥1 do not contribute for sign(θ) = ∓. The dependence on the absolute value |θ| is more intricate. At θ = +0.025 shown in Fig. 3 [right top], we see that σ + 1 contributes while σ + n with n ≥ 2 do not contribute. Similarly, at θ = +0.015 shown in Fig. 3 [right bottom], we see that σ + 1 and σ + 2 contribute but σ + n with n ≥ 3 do not contribute. As we further decrease |θ|, we have found the following structure (although we do not explicitly show the plots). For finite θ > 0 (θ < 0), we have contributions from the saddle points σ + Contrary to the subcritical region, the contributing saddles change discontinuously. As the phase approaches θ → ±0, saddle points σ ± n with larger n contribute to the path integral.
In summary, the above analysis suggests that the thimble structure in the supercritical region is where This indicates an infinite number of Stokes phenomena at θ = 0 in the supercritical region.

Phase transition and thimble structures
We clarify a relation between the phase transition and the above Lefschetz thimble analysis. Whether the saddles contribute to the integral was a priori nontrivial since all of them are complex for λ > 0. Our Lefschetz thimble analysis showed that only a single saddle σ + 0 contributes to the integral in the subcritical region λ < λ c , while multiple saddles σ ± n with n ≥ 0 contribute in the supercritical region λ ≥ λ c . Among them, only two saddles σ ± 0 survive the large-flavor limit. Thus, the dominant saddles jump at the critical point λ = λ c from σ + 0 to σ ± 0 , which causes the phase transition. These provide a more precise interpretation of the phase transition in terms of the Lefschetz thimble analysis. All of these behaviors come from (anti-)Stokes phenomena and therefore these motivate us to study, in more detail, a relation between the phase transition and (anti-)Stokes phenomena. This will be summarized in Sec. 5.
Also, note that the periodicity of the action (2.4) along the imaginary axis causes an infinite number of Stokes phenomena at arg(N f ) = 0. Such property is typical in sphere partition functions of N ≥ 4 supersymmetric gauge theories with FI terms and without diagonal Chern-Simons terms. These infinite number of Stokes phenomena are inevitably related to the phase transition as we will see more details in Sec. 5.
Finally, we shortly provide some preparations for the next section. We have collected the thimble structures for larger values of arg(N f ) as summarized in App. B. For non-small arg(N f ), we have encountered a subtlety essentially coming from the logarithmic branch cuts in the action (2.4): when a thimble crosses the branch cuts once, the action changes its value by ±2πiN f where the sign depends on a direction of the crossing. Note that this modifies the condition for having Stokes phenomena as the imaginary part of the action is changed. App. B demonstrates that the Stokes phenomena due to this effect indeed happen in our problem. For instance, in the subcritical region (λ, m) = (0.4, 1), we have numerically found that the Lefschetz thimble associated with the saddle σ + 0 crosses one of the branch cuts once and then passes the neighboring saddle σ + 1 around arg(N f ) −1.190. The appearance of this Stokes phenomenon cannot be understood without taking the effect of the branch cuts into account as follows. The existence of the branch cut implies that the condition for having Stokes phenomena between the saddles σ + n and σ + 0 is modified as which is solved by  [158][159][160] although some of the assumptions there are violated in the SQED due to the logarithmic branch cuts in the action. We will discuss corresponding Borel singularities in the next section.

Borel singularities and resurgence structure
In this section, we consider the 1/N f expansion of the partition function (2.1) and study its resurgence structure from the viewpoint of the Borel resummation method. We numerically compute the 1/N f expansion up to 50th order and then study the structures of the Borel singularities. We confirm that the locations of the Borel singularities are consistent with the Lefschetz thimble structure. The resurgence structure of trans-series with respect to η or λ, instead of 1/N f , is discussed in App. D and App. E.

Numerical study of Borel singularities
Let us focus on the 1/N f expansion around the saddle point σ = σ + 0 . It can be computed in the standard way and the expansion takes the form (see App. A for details) The coefficients are given by This implies that the coefficient a grows factorially and the formal 1/N f expansion is not convergent. Therefore we apply the Borel resummation technique to control the divergence. Let us write the perturbation series as and define its Borel transformation by Then, the Borel resummation of the function F (1/N f ) is given by where BF (t) is a simple analytic continuation of the series (4.6) and the integration contour C is chosen so that arg(N f t) = 0. The Borel resummation discontinuously changes if the integration contour crosses a singularity of BF (t). From the viewpoint of resurgence, the Borel singularities must correspond to Stokes phenomena summarized in App. B.

Padé approximation and its improvement
As seen from (4.6), the Borel transformation is defined in terms of an infinite number of the perturbative coefficients. In practice, we often encounter the situation where we know only the finite number of the coefficients and have to estimate (the analytic continuation of) the Borel transformation from the limited perturbative data in some way. One of the standard ways to do this is the so-called Borel-Padé approximation, where we replace the Borel transformation BF (t) in (4.7) by its Padé approximation. The Padé approximation with degrees (m, n) is defined by a rational function where P m (t) and Q n (t) are polynomials of degrees m and n, respectively. The explicit forms of the polynomials are determined such that the small-t expansion of P m,n (t) agrees with the Borel transformation up to a desired order L: While there are various possible choices of (m, n) given L, it is empirically known that the Padé approximation often has better accuracy when m and n are close. Therefore we here take m = n = L/2 with even L and do not pursue (m, n)-dependence. This particular case is called the diagonal Padé approximation. In practice, we will present results for L = 50 as a representative.
In general, the Padé approximation is good at approximating meromorphic functions since it has only pole-type singularities. For cases with branch cuts, the Padé approximation typically becomes worse as it is impossible to express branch cuts in terms of a rational function in the exact sense. It is known that when the Padé approximation works for the cases with branch cuts, there appear sets of dense poles around locations of the branch cuts 10 . It is also known that the Padé approximation generically gives better descriptions for singularities closer to the origin t = 0. In particular, the location of the closest singularity is expected to be predicted when (m, n) is larger because this information is closely related to the radius of the convergence of the small-t expansion. In other words, it is typically hard to detect Borel singularities far away from the origin when (m, n) is not-so-large.
There are various ways to improve the Padé approximation (4.8). Here we use one of the improvements called the Padé-Uniformized approximation [161], which can be used when we know information on the location of a branch cut in the Borel transformation BF (t). This is constructed as follows 11 . Suppose that the function BF (t) has a branch cut ending at t = s. We send the Borel t-plane to a u-plane by the uniformization map which can be inverted as Note that the singularity at t = s in the t-plane is mapped to infinity in the u-plane. Then we construct the standard Padé approximation in the u-plane, meaning that we construct a rational approximation P m,n (u) such that Finally, we come back to the t-plane and approximate the Borel transformation as Re t . The same symbols beyond the real axis are their counterparts on different Riemann sheets obtained by shifting the ones on the real axis by 2πiZ. The colored symbols among them denote the saddles at which Stokes phenomena are expected to occur from the Lefschetz thimble analysis.
The uniformization map sends the branch cut singularity to infinity, where the standard Padé approximation does not see. Instead, it sends a region away from the branch cut singularity to a region around the origin. Thus, we can avoid pole resources of the Padé approximation being wasted on the branch cut. Also, thanks to the logarithm of the uniformization map, the order of Borel singularities affect only the scale of u. For example, 1/(1 − t) 1/n is mapped just to e u/n and logarithmic singularities are mapped to regular points. Another good example is ln(1 − t), which is simply mapped to −u. For the above reasons, the Padé-Uniformized approximation is safer than the standard Padé approximation in application to our problem.

Subcritical region
Let us focus on the subcritical region λ < λ c . In the left panel of Fig. 4, we present the locations of poles of the standard Padé approximation for the Borel transformation BF (t), which is expected to approximate the Borel singularities. The red crosses indicate the poles of the Padé approximant while the other symbols denote the values of the saddle point actions subtracted by the one of σ + 0 and their counterparts on different Riemann sheets 12 . The colored symbols among them denote the saddles at which Stokes phenomena are expected to occur from the Lefschetz thimble analysis in the last section. In other words, a symbol associated with a saddle σ ± n is colored if there exists arg(N f ) such that 13 From the viewpoint of resurgence, we expect that the Borel singularities are located at these color symbols. 12 Recall the arguments at the end of Sec. 3.3. The thimble structures for non-small arg(N f ) studied in App. B imply that we have to take care of situations that thimbles cross the branch cuts and then the action is shifted by 2πiN f Z. 13 For example, as discussed in the end of Sec In the left panel of Fig. 4, we see that a bunch of poles appear around the point corresponding to the saddle σ − 0 and are stretched along the negative real axis. According to general expectation on the Padé approximation, this signals that the Borel transformation has a branch cut type singularity ending on the point corresponding to σ − 0 along the negative real axis. This is consistent with the expectation that we have Stokes phenomena with σ − 0 . We also see a good agreement between the location of the poles and the action values at N f t = S(σ + 1 ) ± 2πi − S(σ + 0 ) as expected from the thimble analysis. However, it seems that we do not have a similar agreement for the other saddles in particular when we go away from the origin. One reason is that the Padé approximation becomes worse outside the convergence radius. Another reason is that there is a branch cut on the Borel t-plane. The Padé approximant has limited pole resources to resemble the genuine Borel plane structure. In our case, there are only L/2 = 25 poles. If there is a branch cut, a lot of poles are consumed to resemble it. Indeed, a lot of poles are accumulated on the negative real axis. Thus, it seems more appropriate to use the Padé-Uniformized approximation using the input that we have a branch cut ending on the point . In the right panel of Fig. 4, we show the result of the Padé-Uniformized approximation where we have eliminated the expected branch cut by the uniformization map (4.10). We first see that there is no longer a bunch of dense poles, which appeared in the standard Padé approximant. This confirms that the uniformization map has successfully removed the branch cut. Because of this, we expect that the Padé-Uniformized approximation has a better description of other singularities. Indeed the result shows better agreements between the locations of poles and the expected Borel singularities around the origin. In particular, note that there is no Borel singularity on the positive real axis. This is consistent with the Lefschetz thimble structure around arg(N f ) = 0 as shown in Fig. 1. However, we see some poles around the real axis which do not coincide with action values. It seems that they are artifacts by the Padé approximation. For details, see App. C.

Supercritical region
The results for the supercritical region λ ≥ λ c are shown in Fig. 5. As in the subcritical case, the left panels are the result of the standard Padé approximation while the right panels denote the one of the Padé-Uniformized approximation. The upper and lower panels are essentially the same but we plot them in different scales for convenience. Note that the actions of the saddles σ + n and σ − n are different by purely imaginary values even on the same Riemann surface.
Let us first focus on the result of the standard Padé approximation shown in the left panels of Fig. 5. We easily see that there are a bunch of poles around the point corresponding to the saddle σ − 0 again but now they are stretched along the upper imaginary axis in contrast to the subcritical case. This again implies that the Borel transformation has a branch cut ending on the point corresponding to σ − 0 along the upper imaginary axis. While this agrees with the expectation from the resurgence, we do not see good agreements beyond that saddle. Therefore we again improve the Padé approximation assuming the information on the branch cut as in the subcritical region.
The right panels of Fig. 5 show the result of the Padé-Uniformized approximation. The expected branch cut has been eliminated by the uniformization map (4.10) and one can check that a bunch of dense poles are indeed absent in this case. We now see better agreements: there are poles around expected locations of the Borel singularities. This is consistent with the Lefschetz thimble structure around arg(N f ) = 0 as shown Fig. 2. However, we still have missing singularities away from the origin. For details, see App. C.

Analytical study of Borel singularities for large λ
In Sec. 4.1, we have numerically found the Borel singularities at 2πiZ in the supercritical region as demonstrated in the right panel of Fig. 5. We interpret that this class of singularities corresponds to the saddle σ + 0 on different Riemann sheets. Here we provide an analytical justification for that: We analytically prove Re t that the Borel transformation of 1/N f expansion around the saddle point σ ± n has singularities at 2πiZ in the large λ limit.
Let us consider the large λ limit λ λ c and λ 1. The saddle point σ ± n in this limit is expanded as The action values at these saddle points are We are interested in the perturbative coefficients in the leading order of the large-λ limit. Let us expand the action around the saddle points: we can approximate S (n) (σ n ± ) as (4.21) Therefore the action becomes Then the perturbative series in the large λ limit is generated by  Noting that this equation is rewritten as we can write the solution as where W (z) is the Lambert W function defined as a solution of the following equation we find It seems natural to identify . • W (−e −1 ) = −1.
• Small z expansion of W (z) has a radius of convergence e −1 .
Thus the Borel transformation in the large λ limit has the branch cut singularities at t = 2πiZ. (4.31) One might wonder why we now do not have singularities beyond the imaginary axis which appeared in the numerical study represented in Fig. 5. This is because of the large λ limit: the singularities beyond the the imaginary axis go to infinity as λ → ∞. It is most transparent in the formula (4.17) for the asymptotic behaviors for the action.
We can see that the above Borel singularities come from the branch cuts in the σ-plane as follows. The variable t of the Borel plane is related to the σ-plane by the map (4.26). Therefore the origin t = 0 is associated with a saddle δσ = 0 while the infinity is associated with the branch cut singularity δσ = 1/iλ. An interval [0, 2πi] is associated with a closed loop which starts from the saddle δσ = 0 and runs around the branch cut singularity δσ = 1/iλ back to the saddle. Since there is a logarithmic branch cut on the δσ-plane, we reach the next Riemann sheet once we move along the closed loop. Thus the Borel singularities at t = 2πiZ are associated with the saddles on the different Riemann sheets. Such a relation should hold even when λ is not large as long as we are in the supercritical region.
The above structures technically come from the fact that the action (2.4) has the periodic structure and the logarithm branch cuts. Physically this type of factor is originated from one-loop contributions of hypermultiplets in the localization formula of S 3 partition functions [155][156][157]. This indicates that the above structures hold not only for the SQED but also for more general supersymmetric gauge theories.

Lessons from 3d N = 4 SQED
In this section, given the lessons from the SQED obtained in the previous sections, we provide a more generic discussion on relations between the resurgence and phase transitions. In particular, we discuss how the order of phase transitions are described from the viewpoint of (anti-)Stokes phenomena.

Phase transitions as collisions of saddles
Let us consider a generic theory whose partition function is described by a one-dimensional integral of the form e −N F (λ) = dσ e −NS(λ;σ) (5.1) whereS(λ; σ) is the "action" and (N, λ) are some parameters specifying the theory. Suppose that the theory undergoes a phase transition at λ = λ c in the limit N → ∞, accompanying a collision and a scattering of n saddles at σ = σ c . We do not consider phase transitions simply by anti-Stokes phenomena which have been often discussed in the context of the Lefschetz thimble analysis. Here we show that the order of phase transition is determined by the scattering angle of saddles. More specifically, we prove the following statement: if the n-saddles collide and scatter with a scattering angle βπ as we vary the parameter λ through the critical point λ = λ c (as illustrated in Fig. 6), then we have the phase transition of the order (n + 1)β , where x is the smallest integer larger than or equal to x. Before moving onto the proof, let us recall some basics on phase transition. We have an p-th order phase transition at λ = λ c when the p-th derivative of the "free energy" F (λ) becomes singular at λ = λ c given its non-singular lower derivatives, that is 2) Figure 6: An illustration of collision and scattering of saddles for n = 2. As the parameter λ is varied through the critical point λ = λ c , two saddles σ 0 , σ 1 collide at σ = σ c and scattered with an angle βπ. with The order p is related to a behavior of the free energy around the critical point as follows. Suppose that the free energy is expanded around the critical point as where A, B, C are complex constants and γ > 0. Note that the free energy is not necessarily real since it is not necessarily interpreted as the thermodynamic one for QFT on a generic manifold 14 . If γ / ∈ Z, the phase transition is of the order p = γ . Similarly, if γ ∈ Z and A = B, then the order of the phase transition is p = γ while there is no phase transition for A = B. In what follows, we consider only the case with A = B and the exponent γ independent of the sign of δλ. This is a consequence of a saddle collision as we will see soon.
Now we provide the proof. As mentioned above, we are interested in the situation that the n-saddles collide and scatter at σ = σ c in varying the parameter λ thorough the critical point λ c . This means that the saddle point equation at λ = λ c has the root with the degeneracy n. Therefore the action around σ = σ c is expanded as 15S (λ; σ = σ c ) = a 0 (λ) + a 1 (λ)δσ + · · · + a n+1 (λ)δσ n+1 + · · · . (5.5) The coefficient a i (λ) is constrained by the condition that the saddles collide at λ = λ c as Without loss of generality, one can shift the action by an appropriate constant to make a 0 (λ) independent of λ: Furthermore, using the condition that the saddle point equation has the root with the degeneracy n, we find a i (λ c ) = 0 (i = 1, . . . , n), where s m is a some constant and β = min α 1 n , α 2 n − 1 , . . . , α n 1 . (5.14) Around the critical point δλ = 0, each saddle acquires a phase (−1) β . This implies that the n saddles collide and scatter with an angle βπ. Then the action at the saddle σ m takes the valuẽ with a constant T m . At the phase transition point, there is a jump of contributions saddle points in various ways. For example, in the case where the contributing saddles jump as σ 0 → σ 1 , the free energy changes as In the case where contributing saddles jump as σ 0 → σ 0 , . . . , σ n−1 , the free energy changes as F c 0 + T 0 (δλ) (n+1)β for δλ < 0 c 0 + (T 0 + · · · + T n−1 )(δλ) (n+1)β for δλ > 0 . (5.17) In any case, the phase transition is of the order (n + 1)β and this completes the proof. Our argument also shows a connection between the order of the phase transition and the anti-Stokes line. The formula (5.15) for the action shows that the anti-Stokes line is given by Re (δλ) (n+1)β = 0. Thus one can also read off the order of the phase transition by looking at the anti-Stokes line.

Thimbles and Borel singularities around critical points
In this subsection, we demonstrate the discussion in the last subsection using the integral representation of the Airy function whose "action" is given bỹ We refer to this example as the Airy-type model. As we will see soon, this example corresponds to n = 2, α = 1/2, and has common features with the SQED (2.1) in the context of the argument in this section.

Lefschetz thimbles
The Airy-type model corresponds to The saddle points in this example are simply given by 20) which indicates that the two saddles collide at σ = 0 for λ = 0. Therefore, in the notation of the last subsection, we have and n = 2, α = 1 2 . (5.22) The action at the saddle σ ± is given byS We can find the (dual) thimbles by solving In Fig. 7, we show how the thimbles change as increasing the phase arg(δλ). As arg(δλ) is increased from a negative value, we encounter a Stokes phenomenon at arg(δλ) = −π, an anti-Stokes phenomenon at arg(δλ) = −2π/3, a Stokes phenomenon at arg(δλ) = −π/3 and an anti-Stokes phenomenon at arg(δλ) = 0. In particular, we observe the jump of the contributing saddles at arg(δλ) = −π/3 (as well as arg(δλ) = +π/3): we have a contribution only from σ = σ + for arg(δλ) < −π/3 while we have contributions from the two saddles σ = σ + and σ = σ − at arg(δλ) = −π/3 + 0. This is a manifestation of the Stokes phenomenon. The free energy also jumps as which implies the second-order phase transition. Next, let us increase δλ from −1 to 1, keeping Im δλ = 0. As δλ goes from −1 to 0, the two saddles (in the left top panel of Fig. 7) approach the origin along the imaginary axis. At δλ = 0, they collide and change their directions. As δλ goes from 0 to +1, the two saddles (in the bottom right panel of Fig. 7) depart the origin along the real axis. In other words, the two saddles collide with an angle π/2 at the phase transition. Also, we remark that, during the phase transition, we cross the anti-Stokes line arg(δλ) = −2π/3 and the Stokes line arg(δλ) = −π/3. Thus, the second-order phase transition is understood as a phenomenon in which an anti-Stokes and a Stokes phenomenon occur simultaneously. To summarize, the second-order phase transition in the Airy-type model is interpreted as follows i. Contributing saddles jump as σ + → σ + , σ − .
ii. The two saddles collide and scatter with a scattering angle π/2. iii. A Stokes phenomenon and an anti-Stokes phenomenon occur simultaneously.
Note that the coefficient grows factorially. The analytic continuation of its Borel transformation is This function has a Borel singularity (branch cut singularity) at This Borel singularity corresponds to the "non-trivial saddle" σ − = −λ 1/2 , and it collides with the origin corresponding to the "trivial saddle" at the critical point λ = 0. The scattering angle is −3π/2 ∼ π/2. After the collision (λ > 0), the Borel singularity is on the imaginary axis. This means that there occurs an anti-Stokes phenomenon: ReS + = ReS − = 0. Thus, the collision of saddles is appropriately encoded in the perturbative series as expected by the resurgence theory.

Second-order phase transition in the SQED revisited
In this section, we revisit the second-order phase transition in the SQED based on the previous subsections to clarify more the relationship between the phase transition and resurgence.

Lefschetz thimble analysis
From the Lefschetz thimble analysis in Sec. 3, we have seen that the SQED around the second-order phase transition point has the following properties i. Contributing saddle points jump as σ + 0 → σ + 0 , σ − 0 . ii. The two saddles collide and scatter with a scattering angle π/2.
iii. An infinite number of Stokes phenomena associated with saddles σ ± n>0 occur. The first two points are common with the Airy-type model in the last subsection. This is because the "action" of the SQED (2.4) has a similar expansion to one of the Airy-type models around the critical point. Thus, the second-order phase transition in the SQED is interpreted in a similar way as the Airy-type model. The third point is particular for the SQED. The difference essentially comes from the fact that the SQED has infinite number of saddles periodically distributed along the imaginary axis. Once thimbles run along the imaginary axis after a phase transition, they inevitably path through the periodic saddles. Such behavior of thimbles causes an infinite number of Stokes phenomena. Technically the appearance of the periodic saddles is due to the cosh factors originated from the one-loop determinant of the hypermultiplets in the SUSY localization of the S 3 partition function. Therefore we expect that the above features appear also in other SUSY gauge theories on S 3 .

Borel resummation
In the language of the Borel resummation, the second-order phase transition has the following features I. In the supercritical region, the two Borel singularities line up along the imaginary axis on the Borel plane.
II. The two Borel singularities collide and scatter with a scattering angle π/2 as we cross the critical point.
III. The 1/N f -expansion becomes Borel non-summable along the positive real axis in the supercritical region.
(I), (II) and (III) here correspond to (i), (ii) and (iii) of the Lefschetz thimble analysis, respectively, as expected from the resurgence theory. The first point means that the saddle points associated with the Borel singularities along the imaginary axis have the same real part of the actions and therefore contribute to the integral with the equal weights in the supercritical region. The second point is a counterpart of the collision of the two saddles from the viewpoint of the Borel resummation. The relation between Borel singularities and saddle points implies that the collision of the two saddles in the σ-plane leads to one of the two Borel singularities in the t-plane. Thus we can also decode the order of the phase transition purely from how the Borel singularities collide. The third point means that the thimbles cross the multiple saddle points for arg(N f ) = 0 as shown in Fig. 1. In the case of the SQED, the Borel non-summability detects the infinite number of periodic saddles which come from the contribution from the hypermultiplets. Finally, let us see the Stokes graph. The second-order phase transition is interpreted in terms of the Stokes graph as follows.
• A Stokes phenomenon and an anti-Stokes phenomenon associated with saddles σ ± 0 occur simultaneously. These points are analogous to the Airy-type model discussed in Sec. 5.2. The only difference is that the infinite number of Stokes phenomena associated with σ ± n>0 occur simultaneously.

Conclusions and discussion
We have studied the resurgence structure of a quantum field theory with a phase transition to uncover relations between resurgence and phase transitions. In particular we have focused on the three dimensional N = 4 SQED, which undergoes the second-order quantum phase transition in the large-flavor limit [143]. We have approached the problem from the viewpoints of the Lefschetz thimbles and Borel resummation. In the Lefschetz thimble approach, we have specifically studied the thimble structures of the integral representation of the partition function obtained by the supersymmetric localization [155][156][157]. We have first justified the assumption in [143] that all the dominant complex saddles contribute to the integral by the Lefschetz thimble analysis. Then we have found that there are a collision of the two saddles and a jump of the contributing saddle points as we cross the critical value of the parameter λ = η/N f . While this is the Stokes phenomenon, we have seen that an anti-Stokes phenomenon also occurs at the same time. Thus we interpret the second-order phase transition as the simultaneous Stokes and anti-Stokes phenomena. Our result also shows that the phase transition accompanies an infinite number of Stokes phenomena associated with the other saddles. This behavior technically comes from the fact that the action (2.4) has the periodic structure which is physically originated from one-loop contributions of hypermultiplets in the localization formula of S 3 partition functions. This indicates that the above structures hold not only for the SQED but also for more general supersymmetric gauge theories. In the Borel resummation approach, we have seen that the thimble structures are appropriately mapped to the Borel plane structures of the large-flavor expansion as expected from the resurgence theory. We have found the Borel singularities, two of which correspond to the two saddles. The two Borel singularities line up vertically along the imaginary axis after the phase transition. It is a sign that the two saddles contribute to the integral with equal weights. At the phase transition, the two Borel singularities collide as the two saddles. The scattering angle of the Borel singularities at the collision is related to the order of the phase transition. We have also seen that the large-flavor expansion becomes Borel non-summable along the positive real axis in the supercritical region, due to an infinite number of the Borel singularities. This reflects the infinite number of Stokes phenomena.
Given the lessons from the SQED, we have provided more generic discussion on relations between the resurgence and phase transitions. We have considered the one-dimensional integral of the form (5.1) and shown that if the n-saddles collide and scatter with a scattering angle βπ as we vary the parameter λ through the critical point λ = λ c , then we have the phase transition of the order (n + 1)β . Our argument has also shown that we have anti-Stokes phenomena at the critical point where the anti-Stokes line is given by Re (δλ) (n+1)β = 0. This implies that one can read off the order of the phase transition also by looking at the anti-Stokes line. We have also argued that the above behaviors are naturally translated into the language of the Borel plane. This means that the order of phase transitions can be determined also by tracking how Borel singularities move as varying the parameter. This implies that we can read off information on phase structures purely in terms of perturbative expansions. The above results apply to more general theories as long as they reduce to the form (5.1).
Finally, we have revisited the second-order quantum phase transition in the SQED from the above viewpoints. In the case of the SQED, the two saddles σ + 0 and σ − 0 collide and scatter with π/2 as we cross the critical point λ = λ c . Therefore we have (n, β) = (2, 1/2) in the formula (n + 1)β for the order of the phase transition and this agrees with the fact that the second-order phase transition occurs. From the viewpoint of Stokes graphs, the second-order phase transition is essentially described as the standard Airy-type graph. This clarifies that the second-order phase transition is understood as a phenomenon where the Stokes and anti-Stokes phenomena occur at the same time. This is a clear contrast to the common understanding that a first-order phase transition is associated with an anti-Stokes phenomenon. All of the above results support that resurgence works for describing the second-order phase transition in the SQED.
We have obtained a good news which may be useful to develop studies of resurgence on the technical side. Originally, the correspondence between saddles and Borel singularities in one dimensional integrals was shown in [158][159][160]. It is not guaranteed that we can naively apply the correspondence to the SQED because some of their assumptions are violated due to the logarithmic branch cuts in the action. Nevertheless our results suggest that the correspondence still holds even in the SQED. This seems to imply that one can extend the correspondence beyond the class of integrals studied in [158][159][160]. It would be interesting to pursue this direction.
We believe that our results give a good step to understand connections between phase transitions and resurgence. Yet there are still various questions and tasks which should be addressed as next steps. First, it is important to understand the physical meaning of the second order phase transition in the SQED. For instance, we have not understood yet whether there is a change of symmetries around the critical point, whether the critical point describes some conformal field theory and so on. Second, we have not identified interpretations of the saddle points in the SQED in the language of the original path integral. It seems that they are closely related to the complex supersymmetric solutions found in [115] as discussed in App. F. We need further studies to clarify the relations more precisely. Third, it would be interesting to study relations between Lee-Yang zeros and the Stokes graph. While the authors in [147] found that Lee-Yang zeros are on anti-Stokes curve in the zero dimensional Gross-Neveu-like model, the SQED studied in this paper does not seem to have such a property. This may suggest that the situation in the SQED is different from the zero dimensional Gross-Neveu-like model. Fourth, it would be illuminating to study resurgence structures with respect to other parameters in the SQED such as the FI parameter η. There may be interesting relations to the resurgence structure of the large flavor expansion as in the two-dimensional pure U (N ) Yang-Mills theory on lattice, where there were found interesting connections among expansions by 1/N , Yang-Mills coupling and 't Hooft coupling [81,148,149]. Finally it is technically important to improve the Padé-Uniformized approximation for cases with multiple branch cuts in Borel planes. It seems that an improvement of the approximation is hindered by the non-trivial topology of Riemann sheets due to the branch cuts. Such a problem often arises in the context of resurgence and therefore further studies are desired.
While this paper has focused on connections between resurgence and phase transitions, more generally, it would be very interesting to explore relations between resurgence and phases themselves rather than their transitions. It is known that information on phases in quantum field theories are partially captured by 't Hooft anomalies, including phases beyond the Ginzburg-Landau or Nambu paradigm. While 't Hooft anomalies are typically easy to calculate and give quite robust information on phases, it relies on existence of symmetries 16 . In contrast, analysis of resurgence does not require symmetries and gives detailed information while it is technically much more complicated. Therefore they play complementary roles. In this paper we have discussed that some features of phase transitions are captured by qualitative behaviors of the objects appearing in the analysis of resurgence. It would be great if one can find similar connections for 't Hooft anomalies 17 . It might open a door to a shining world of non-perturbative physics.

Acknowledgment
The authors would like to thank Okuto Morikawa, Naohisa Sueishi, Hiromasa Takaura and Yuya Tanizaki for valuable discussions. Preliminary results of this work have been presented in the KEK workshop "Thermal Quantum Field Theory and its Application" (Aug. 2020), the YITP workshop YITP-W-20-08 "Progress in Particle Physics 2020" (

A Details on large flavor expansion
In this appendix, we compute the coefficients of the 1/N f expansion of the partition function around a general saddle point σ * . First, to make N f -dependence transparent, we introduce Then we expandS(σ) as and regard the last term as a perturbation. Next we rewrite the contribution from the saddle σ * to the integral (2.1) as where J * is the Lefschetz thimble associated with σ * and To proceed, we also introduce Then, exchanging the integration and summation, the formal 1/N f expansion is computed as (−1) n Γ 1 2 + + n c 2 −n (n) Γ(1/2)Γ(n + 1) .
B Lefschetz thimble structures for larger arg(N f ) In this appendix, we study the thimble structures for larger values of θ = arg(N f ) than the one in Sec. 3.2 to understand the Stokes phenomena more precisely. It appears that the larger arg(N f ) region is not directly related to the phase transition itself since originally the parameters were real. However, this is essential to understand the Borel plane structures as discussed in Sec. 4. As mentioned in the last of Sec. 3.3, for non-small arg(N f ), we have to take the effects of the branch cuts into account. Namely, when thimbles cross the branch cuts, the action is shifted by 2πiN f Z, and this effect modifies the condition for having Stokes phenomena. We will see soon that the Stokes phenomena due to this effect indeed occurs in this problem.
In Fig. 8, we summarize the Lefschetz thimble structures for (λ, m) = (0.4, 1) with various −π < arg(N f ) < 0 as a representative of the subcritical region λ < λ c . Reflecting these figures along the vertical axis corresponds to flipping the sign of arg(N f ). Thus, these figures practically cover the full region −π ≤ arg(N f ) < +π. In the figures on the left side, Stokes phenomena occur. For example, at arg(N f ) = −1.190, the Lefschetz thimble associated with a saddle σ + 0 (blue line) passes also through other saddles σ + n>0 . This is nothing but a Stokes phenomenon. One can easily check that the codition for having Stokes phenomena: is satisfied by arg(N f ) = −1.190. In Fig. 9, we summarize the thimble structures for the supercritical region λ > λ c with various −π < arg(N f ) < +π (specifically (λ, m) = (1.    Im σ Figure 9: Illustrations of the Lefschetz thimble structures for the supercritical region λ > λ c (in these figures, m = 1, for which λ = 0.4). Larger phases are given −π ≤ θ ≤ 0 so that we can observe Stokes phenomena (left ones) and thimble structures between them (right ones).

C Comments on the Padé-Uniformized approximation
In the main text, we have seen that the Padé approximation becomes worse due to branch cuts. The branch cut singularities were associated with a saddle σ − 0 both in the subcritical and supercritical regions. To make matters worse, there are other signs of branch cuts on the Borel t-plane. Fig. 10 is an example. We set m = 10 (for which λ c = 9.0 × 10 −5 ) so that singularities gather around the origin and the Padé approximation works better. We can see signs of branch cuts associated not only with a saddle σ − 0 but also with four other saddles σ ± ±1 . We claim that even the Padé-Uniformized approximation is obstructed by these branch cuts. In this appendix, we discuss this point by studying some simple examples of applications of the Padé-Uniformized approximation.

C.1 Single branch cut
Let us start with the simplest case. Consider, for example, a function which has a single branch cut and a uniformization map In Fig. 11, we compare the standard Padé approximation with the Padé-Uniformized approximation for this example. We see that the Padé-Uniformized approximation works well in this case.

C.2 Multiple branch cuts
Next let us consider the following example with two branch cuts The result is shown in Fig. 12. We see that the approximation becomes worse. In the Padé-Uniformized approximation (right panel), we can see that there is a pair of singularities above and below the negative Re t However, since the approximation becomes worse away from the origin u = 0, found singularities (particularly with |n| > 0) are not sent back exactly to t = −1. As a result, the Padé-Uniformized approximation returns multiple singularities around t = −1. The pair of singularities in the right panel of Fig. 12 corresponds to n = ±1. Other pairs of singularities which correspond to larger |n| are missing simply because they are too far away from the origin u = 0. Such artifacts cause trouble since they are indistinguishable from other genuine singularities. Let us consider, for example, the following function . (C.8) The result of the Padé(-Uniformized) approximation is shown in Fig. 13. The two poles t = ±i are sent to Im t Figure 12: Comparison of the Padé and the Padé-Uniformized approximation (25,25) in the case with the two branch cuts. One of the branch cuts is removed by the uniformization map.
while the branch cut singularities t = ±1 are sent to u = − ln 2 + 2πin. (C.10) These two singularities u = − ln 2 + 2πin are further away from the poles u = − ln(1 ∓ i). Then, the approximation for the two singularities is disturbed by the poles. As a result, the Padé-Uniformized approximation returns the poles which originate in t = ±i, but with much worse artifacts which originate in t = ±1. Indeed, in the right panel of Fig. 13, the pair of artifacts is indistinguishable from the genuine poles. We claim that some of singularities found in Sec. 4.1 are these types of artifacts.

C.3 On elimination of multiple branch cuts
One possible way to avoid such multiple branch cuts is to consider a map u = ψ(t) = − ln(1 − t) + ln(1 + t), (C.11) which sends the singularities at t = ±1 to u = ±∞. However, this map causes other branch cuts on the u-plane. As a result, the Padé-Uniformized approximation returns a lot of artifacts along the imaginary axis as shown in Fig. 14. It seems that the problem resides in the non-trivial topology of the Borel t-plane due to multiple branch cuts. For the above reasons, the multiple branch cuts worsen the Padé-Uniformized approximation. Some artifacts are indistinguishable from other genuine singularities. Also, multiple branch cuts are not eliminated simultaneously at least by naive maps. Further improvements of the Padé-Uniformized approximation are left for future works.

D Transseries for a finite η
In this appendix we derive the transseries for the finite η from the viewpoint of difference equation. See [2,170] for technical details. We consider the formal transseries satisfying the difference equation (x = N f ).
Im t Figure 13: Comparison of the Padé and the Padé-Uniformized approximation (25,25) in the case of two branch cuts and two poles. One of the branch cuts is removed by the uniformization map. where We introduce P (x) = Z(x − 1), and eq.(D.1) can be written as a vectorial expression, given by where Z(x) = (Z(x), P (x)) , and the 2-by-2 matrix M (x) is defined as The transseries structure generally depends on the form of a difference equation and is uniquely determined from the change of asymptotic series by acting the shift operator. See Appendix D.1 in detail. In order to obtain the transseries based on the difference equation, one needs to diagonalize Λ and A. By using an invertible matrix U to diagonalize Λ, one finds that Re t where the transseries in terms of x is completely determined but it is not for other parameters such as m and η. Hence, σ s = σ(m, η), c s,n = c s,n (m, η) in general. Since c ±,0 is relevant only to the normalization, one can take c ±,0 = 1 without loss of generality. σ ± can be determined from the partition function, and they are given by σ + (m, η) = 0, σ − (m, η) = 2π(1 + cosh m).

(D.15)
Notice that c ±,n>0 are recursively determined from the difference equation. Notice that when g(x + 1) = g(x), the action of T to g(x) gives the identity map. As one can see easily, for example, the below type of transseress is closed under action of the shift operator T : .

(E.2)
If we ignore the extra dependence of Z(x, η) on x through η = λ/x, we find that the vectorial recursion relation for Z(x) = (Z(x), P (x)) is modified to

F A possible relation between the Borel singularities and complex SUSY solutions
In this appendix, we point out that a path integral interpretation of the Borel singularities appearing the main text may be complex supersymmetric solutions (CSS) found in [115]. It was proposed that Borel transformation of large Chern-Simons level expansion includes the following factor where n B (n F ) is the number of bosonic (fermionic) solutions. In the SQED studied in this paper, there are two types of CSS with n B − n F = 0, more precisely n B − n F = N f . Actions of the solutions are while the action at the saddle points σ ± n in the localization formula is Comparing this with (F.2), we find that these actions agree when λ is large λ 1. Thus it seems plausible that the Borel singularities correspond to the CSS in the original path integral at least for large λ. It would be interesting to extend the analysis in this appendix to finite λ.