Lefschetz Thimbles and Quantum Phases in Zero-Dimensional Bosonic Models

In this paper, by analyzing the underlying Lefschetz thimble structure, we study quantum phases in zero-dimensional scalar field theories with complex actions. Using first principles, we derive the Lefschetz thimble equations of these models, and discuss the issues when we apply the same calculations for more complicated systems. We also derive the conditional expressions involving relations among the parameters of the model, that would help us predict quantum phase transitions in these systems.

We encounter path integrals with complex actions in many branches of physics. The prominent examples are the Minkowski path integral, Yang-Mills theory in the theta vacuum, Chern-Simons gauge theories, chiral gauge theories, and QCD with chemical potential. There are also quantum theories with complex actions that exhibit PT symmetry [1][2][3]. It would be very useful to have a formalism that offers a promising tool to solve field theories containing complex path-integral weights.
A recently developing method to deal with quantum field theories with complex actions uses the complex analog of Morse theory from differential topology [4,5] 1 . There, the objects of primary interest, the so-called Lefschetz thimbles, are a set of sub-manifolds associated with a function that satisfy the Morse flow equation for the real part of the function. The central idea behind using this formalism is to recast the path integral in terms of a finite set of non-oscillatory integrals. Recent work on complex path integrals and connections to Lefschetz thimbles, including applications to quantum tunneling and scattering amplitudes can be seen in Refs. [15][16][17][18][19][20][21][22][23][24][25][26]. In Refs. [27][28][29][30][31][32] the Lefschetz-thimble approach has been employed to study bosonic quantum field theories, and in Refs. [33][34][35][36][37][38] models including fermions were studied. The relevance of Lefschetz thimbles in the context of semi-classical expansion in asymptotically free quantum field theories is discussed in Refs. [39][40][41][42][43].
In this paper we explore the quantum phases in zerodimensional scalar field theories with complex actions, containing quartic interactions and a source term. The Lefschetz thimble equations are derived, using first principles, in these models for various values of the coupling parameters. The set of conditions on the coupling parameters of the model can lead to discontinuities in the curve equations for thimbles. This in turn make the solutions simultaneously being either the thimble or the 1 There exists another compelling method to deal with models containing complex actions. It is based on complex Langevin dynamics. See Refs. [6][7][8][9][10][11][12][13][14] for recent developments in using complex Langevin dynamics in quantum field theories with complex actions.
anti-thimble depending on the region in the complex plane under consideration. We refer to this interesting behavior as the 'piecewise behavior' of the solutions, since they appear themselves as piecewise thimbles/antithimbles/ghosts. Whether a thimble solution shows piecewise behavior or not depends on the set of coupling parameters of the model. We also derive the conditional expressions involving relations among the parameters of the model, that in turn, would help us predict quantum phase transitions in these systems. We also see that the underlying thimble structure undergoes a drastic change while the system is going through such a phase transition. The paper is organized as follows. In Sec. II we provide a primer on Lefschetz thimbles by introducing the gradient flow equations of the given action. In Sec. III we introduce the model of our interest, a zero-dimensional bosonic model with complex action containing quartic interactions and a source term. The thimble equations for this model are derived next in Sec. IV. We discuss analytic expressions for the thimble and anti-thimble equations, and the ghost solutions. The model exhibits an interesting piecewise behavior for the thimbles and antithimbles when the parameters of the action take certain values. The partition function and observables of the model are also discussed. In Sec. V we discuss the phase transition boundaries starting with the Airy integral as a simple example. We discuss the boundaries of phase transitions for various combinations of the values of the coupling parameters. This includes the interesting case when the complex action exhibits PT symmetry. We provide a few example of phase transition boundaries in Sec. VI. We see that the structure of the thimbles undergoes a drastic change when the system goes through a quantum phase transition. In Sec. VII we provide a summary of the main results. In Sec. VIII we conclude and indicate possible future directions.

II. A PRIMER ON LEFSCHETZ THIMBLES
Intuitively, we can relate the Lefschetz thimbles to the original integration cycle of the quantum field theory in the following way. Let us denote the original integration cycle as M R . We 'complexify' this manifold to M C , that is, we take a complex manifold M C that contains the original manifold M R as a submanifold, with the requirement that the complex conjugate of an element of M R is the element itself. One can think of M R = R n and M C = C n for ease of understanding. An example more useful to gauge theories is M R = SU(n) and M C = SL(n, C), which is obtained by letting the field over which the Lie algebra of SU(n) is defined to be C instead of R.
Post complexification, we identify the Morse function [44]. The Morse function in a loose sense determines these thimbles. A natural function to consider is the action 2 . Given a Morse function, we identify its critical points -points in M C where the Morse function is locally extremized. The next step, visually, can be thought of as continuously deforming M R , the deformation being controlled by the Morse function through the Morse flow equations where g ij is the metric on M C and z i are a set of local coordinates around the critical points of S. It can be checked immediately that the imaginary part of the action S is constant along the solution to the above equations.
As the final result of this construction, we obtain a pair of sub-manifolds, called the thimble and anti-thimble, associated with each critical point. The thimble is the 'stable' solution. That is, the action goes to infinity sufficiently rapidly along a thimble, so as to keep the integral involving exp(−S) to be convergent. The anti-thimble is the 'unstable' solution. An example familiar in physics is the method of steepest descent, and thus the Lefschetz thimbles formalism can be thought of as the generalization of the steepest descent method. A rigorous treatment of this construction can be found in Refs. [44][45][46].
An integral involving the action on the sub-manifold M R can now be written as a linear combination of integrals over the Lefschetz thimbles. In this language, the expression for the partition function associated with a system with action S is given as the weighted sum of contributions from the critical points φ i of the action where the integral denotes integration over the Lefschetz thimble J i , which is associated with the i-th critical point φ i of the action. The weight (also known as the intersection number) n i is an integer that decides the contribution of a particular critical point to the partition function. Assuming that the critical points do not share a common gradient flow, given in Eq. (1), n i is given by the number of times the anti-thimble intersects the original integration cycle M [47]. That is, An advantage of using Lefschetz thimbles is that on these thimbles, as discussed before, the imaginary part of the action remains constant. This is certainly a desirable property since, in the (Euclidean) path integral formalism of quantum field theories, the constant imaginary part of the action, Im(S), in the integral, Eq. (2), can be pulled out as a phase factor, and the remaining integral becomes a non-oscillatory integral 3 .
In zero spacetime dimensions the formalism simplifies greatly. For the majority of the situations considered in this work, the original integration cycle is the real line, R. In this case, we end up dealing with curves in the plane of allowed degrees of freedom for the fields (i.e., C) that satisfy the gradient flow equation where t is a parameter and the overline represents complex conjugation. The thimble J i associated with the critical point φ i of the action is defined as the solution to Eq. (4) that satisfies By definition, the thimbles always end inside regions of stability 4 , while anti-thimbles end inside regions of instability.

III. QUARTIC MODEL WITH A SOURCE TERM
We are interested in the action, which is complex, with quartic interaction and source terms, in zero spacetime dimensions where h, α and β are complex coefficients. For convenience, we also express α = a + ib and β = c + id.
The motivation for considering this particular action is two-fold. First, the above action acts as an excellent toy model for understanding systems with complex actions, in the path integral formalism [48,49], and how Lefschetz thimbles help mitigate the sign problem, while also being not too trivial and allowing us to showcase a lot of rich dynamics that accompany the Lefschetz thimble analysis. Second, for the method employed in our calculations, quartic interactions are the highest, exactly solvable terms due to the Abel-Ruffini theorem in algebra [50][51][52] that states that there are no closed-form expressions for solutions to general polynomial equations of degree five or higher. Further, the inclusion of a source term ensures that we exhaust all physically possible situations for a system with quartic interactions.
The regions of stability (sometimes referred to as the Stokes wedges [41,53]) are determined as follows. Since the integral in Eq. (2) involves the expression exp(−S), the integral is convergent in regions where, as φ approaches infinity, Re(S[φ]) ≥ 0. Since the highest order in our action is four, we get four wedges on the complex plane where the integral is convergent. This is shown schematically in Fig. 1.
One way to find the (anti-)thimble associated with a critical point is to solve the gradient flow equation, Eq. (4), for (anti-)thimbles. This, however, quickly becomes very complicated, even for simple forms of actions, due to the coupling between the real and imaginary parts of the differential equation. Instead, we exploit a very crucial property of (anti-)thimbles: the imaginary part of the action remains constant along these (anti-)thimbles. Therefore, to solve for the thimbles, we look for solutions to We restrict our calculations to cases where h (the parameter controlling the linear term in the action) is small compared to α and β. We further restrict h to be either real or purely imaginary. This allows us to approximate the three critical points 5 of the action as The critical point φ 0 is close to the origin (that is, φ = 0) for small h while the position of φ ± depends on the choice of the parameters. Let us denote the imaginary part of the action at a given critical point by ρ i . That is, For the particular action we are considering, they take the following forms 3 There is a possibility that the integral can pick up an oscillatory nature due to the Jacobian that transforms the integration measure. This, however, is much milder compared to the original integral and is referred to as the mild sign problem [27]. 4 Regions of stability are defined as regions in the complex plane where the integral in Eq. (2) remains convergent. 5 In our discussion here, the critical points are the points in the φ plane where the action gets extremized, as defined in Sec. I. They are not the points in the parameter space corresponding to phase transitions. (2) is convergent. In general, the position and shape of these wedges are controlled by the parameters h, α and β in the action.
We note that the convergence of the partition function integral given in Eq. (2) requires the real part c of β to be positive when the original integration cycle is R. However, when c is negative, which is the case when the action possesses PT symmetry (which we will see later), the standard procedure is to take an integration cycle about the angles 5π/4 and 7π/4 in the complex plane (that is, in the third and the fourth quadrant, respectively) [54,55]. This choice ensures that the partition function integral remains convergent.
Parametrizing the field as φ = x + iy, this amounts to choosing our integration cycle in the cases where c is negative, as

A. Thimble Equations
As discussed in Sec. III, we solve for the (anti-)thimble by equating the imaginary part of the action at a generic value of φ to the imaginary part of the action at one of its critical points. The equation for the (anti-)thimble corresponding to φ 0 when h = 0 and d = 0 was derived by Aarts in Ref. [31]. We recreate those results here as a primer, and for completeness.
Substituting ρ 0 into the constraint given in Eq. (7), having set h = 0 and d = 0, we obtain the constraint Solving for y as a function of x, we obtain the thimble and anti-thimble, J 0 and K 0 , respectively, associated with the critical point φ 0 Here θ ∈ {− π 3 , 0, π 3 }. In Fig. 2 we show the three curves corresponding to the three values of the parameter θ. The thimble corresponds to θ = − π 3 and the antithimble corresponds to θ = π 3 . The curves for θ = 0 are paths of constant Im S that are neither thimbles nor anti-thimbles. We shall refer to these curves as the ghost solutions or ghosts.
Similarly, when solving for φ ± , we obtain Eq. (13), but with Eq. (13d) now changed to the following form In this case, θ = 0 corresponds to the thimbles for both φ + and φ − . The curve has two branches, one for x < 0 and the other for x > 0. The anti-thimble associated with φ + has θ = − π 3 . The anti-thimble associated with φ − has θ = π 3 . In Fig. 2 we show the thimbles, antithimbles and ghosts for all the critical points, φ 0 and φ ± , of the action for the parameters a = 1, b = 1, c = 1, d = 0, and h = 0.
So far we have restricted the model to the case where h = 0 and d = 0. Let us now do away with the restriction on d while still maintaining the constraint h = 0. The thimble equation given in Eq. (12) is now modified as Rearranging the above equation in the form we obtain the equation of curves, y k (x), k = 1, 2, 3, 4, for (anti-)thimbles as a function of x Although the solutions to the thimble equation given in Eq. (15) exist in the form of Eq. (16), there are a few caveats we would like to stress on. There are too many conditions 6 to keep track of due to the requirement that x, y ∈ R. These conditions could potentially lead to discontinuities in the curve equations, y k (x), k = 1, 2, 3, 4, for the thimbles in Eq. (16). Further, the requirement of keeping track of these conditions manifests itself as the four solutions simultaneously being either the thimble or the anti-thimble depending on the region in the complex plane under consideration. We will refer to this as the 'piecewise behavior' of the solutions since they appear themselves as piecewise thimbles/anti-thimbles/ghosts. Whether a solution shows piecewise behavior or not depends on the set of parameters {a, b, c, d}.
Let us consider the examples illustrated in Figs. 3 and 4. We see that the solution y 1 for φ 0 gives the thimble for x < 0 and a ghost for x > 0, y 2 gives the anti-thimble for x < 0 and a ghost for x > 0, y 3 gives the anti-thimble for x > 0 and a ghost for x < 0, and y 4 for gives the anti-thimble for x > 0 and a ghost for x < 0. Similarly, for φ ± , the solutions y 1 and y 2 give the thimble for both x < 0 and x > 0, and y 3 and y 4 give the thimble for both x < 0 and x > 0. However, they still exhibit the piecewise behavior. There definitely are parameter sets {a, b, c, d} for which the piecewise behavior might not be exhibited. One such case is when a = 1, b = 1, c = 1, and d = 1; six of the eight solutions do not exhibit this behavior. (We show this in Figs. 5 and 6.) These cases, however, seem to be exceptions rather than the norm.
From the thimble/anti-thimble/ghost solutions given in Eq. (16), we see that obtaining the curves for the case h = 0 is straightforward. If h is real, then C changes from (ax + cx 3 ) to (h + ax + cx 3 ) while E remains the same, except for the change in ρ i . If h is purely imaginary, then C remains unchanged and E gains an additional hx term apart from the change to ρ i . This situation also suffers from the issues discussed earlier for the case where h was taken to zero while d was non-zero.

B. Partition Function and Observables
Let us consider the action given in Eq. (5) for the case h = 0 (the so-called quartic model) We can construct an n-point function the following way Consider the following integral associated with the above action along the original integration cycle R Since the integrand is of odd parity under the exchange x → −x when n is odd, the above integral is non-zero only for even values of n. The partition function is recovered when n = 0, and the observables for the system are related to the above integral as The exact result of the integral is known in terms of modified Bessel functions for the cases n = 0 and n = 2, (a) Thimbles, anti-thimbles and ghosts for all the critical points, φ 0 and φ ± , of the action.
Here K is the modified Bessel function of the second kind.
In the case where Re(α) < 0, we replace K in Z with I, the modified Bessel function of the first kind.
Integrating Eq. (19) by parts, rearranging, and dividing by Z, we obtain a recursion relation for observables of the theory Thus, since the closed-form expressions for the partition function and the observable φ 2 are known, all observables of the theory are known and can be written in terms of the two using Eq. (23). The relation could potentially be used to determine the partition function of the action with sources. The partition function is given (a) Thimbles, anti-thimbles, and ghosts for all the critical points, φ 0 and φ ± , of the action.    (a) Thimbles, anti-thimbles, and ghosts for all the critical points, φ 0 and φ ± , of the action.   by Taylor expanding the first exponential, we get and from the recurrence relation derived above, Z sources can be written solely in terms of I 0 and I 2 . However, this is not valid, as will be shown in Sec. VI, due to a very subtle change in the behavior of the critical points away from the origin (that is, the critical points φ + and φ − ).

V. COMPUTING THE INTERSECTION NUMBERS
The intersection number n i , defined in Eqs. (2) and (3), being an integer, greatly controls the behavior of the partition function and observables of the model. As the parameters of the action are changed, the intersection number corresponding to a critical point could potentially change, which results in an abrupt change in the value of the partition function, and as a result, an abrupt change in the values of the observables of the system. The most dramatic among these is the case when the intersection number takes the value zero, and this in turn results in the corresponding critical point not contributing to the dynamics of the system. This change in the intersection number is referred by the name Stokes phenomena 7 [4,60] and this phenomenon points at quantum phase transitions in the system.
Despite having a few issues as discussed in Sec. IV, the power of using Eq. (7) to solve for (anti-)thimbles is the fact that it captures the information about these intersection numbers. Using this, we can look for the values of the parameters around which the intersection number changes, and thus allowing us to predict the boundaries of phase transitions.
One of our main results will be the analytic expressions for the combined intersection number of thimbles and 7 An alternative, and equivalent, definition used frequently in the literature in the context of integration by the method of steepest descent is the change in the asymptotic formula for the same analytic function when the parameters of the function are changed [58,59].
anti-thimbles of the zero-dimensional scalar field theory, with quartic interactions and a source term.
To arrive at these expressions, for the cases where c is positive, we use the fact that the original integration cycle R corresponds to y = 0. Substituting this in Eq. (7), we obtain a polynomial equation in x of degree four or lower. Looking at the number of real solutions to the polynomial equation (remember, x ∈ R) gives us the information about the number of times the thimbles and anti-thimbles 8 intersect the original integration cycle. When we look at the action with PT symmetry, we substitute Eq. (11) in Eq. (7) and repeat the analysis.

A. A Simple Demonstration using Airy Integral
Before we present our results for the action Eq. (5), we demonstrate quantum phase transitions in the Airy integral as a primer.
Consider the following integral 8 The number of solutions could potentially also contain information about the number of times a ghost solution intersects the original integration cycle. However, we have not come across a situation where a ghost solution intersects the real line. This is explained by the observation that a ghost solution always has one end inside the region of stability and the other end inside the region of instability. This, along with the fact that these curves do not intersect either the thimble or the anti-thimble of the same critical point tells us that the ghosts are always away from the real line.
where we restrict λ to take real values. The integral in Eq. (26) is equivalent to taking our action (after continuation to the complex plane) as There are two critical points of this action, namely At these critical points, the action takes the values Using our previous notation for φ as φ = x + iy, the imaginary part of the action is To look for phase transition boundaries, we look for the number of real solutions to the equation which is equivalent to putting y = 0 in Eq. (7). Thus we look for real solutions to the equation For cubic equations, the number of solutions depends only on the sign of the discriminant, which for the above equation is When the discriminant is negative, the number of real solutions to the cubic equation is one, and when the discriminant is positive, the number of solutions is three. Thus we expect a phase transition at λ = 0. In fact, this phase transition coincides with the change in the asymptotic expansion of Ai(λ). There is a very subtle detail that must be noted. For real λ, it is not possible to find the thimble for φ − and the anti-thimble for φ + without deforming λ into the complex plane as λ + i for small . This occurs because when λ is real, the two critical points are always connected by a Stokes ray. More specific to our method, this problem arises because when λ is real, Eq. (31) has the RHS = 0. This leads to four curves being described by a polynomial equation of degree three, which cannot occur unless the critical points are connected by a flow. This is referred to as being connected by a Stokes ray [4]. The deformation λ → λ+i moves the critical points away from the Stokes ray, allowing us to find the thimbles and anti-thimbles.
We can further complicate this case and take λ in Eq. (26) to be complex. This shows a similar phase transition structure, where the phase boundary is |arg(x)| = 2π/3, and was shown explicitly using the Lefschetz thimbles formalism in Ref. [47].
We now move on to our action with quartic interactions. Due to the differences in algebraic calculations and physical interpretations, we divide our results into multiple cases, and provide the detailed calculations that led to the results separately in Appendix A.
B. In the Absence of the Source Term

Real Coupling
We begin with the case where the parameters satisfy h = 0, α, β ∈ C, Re(β) ≥ 0, and d = 0. The thimble equation, Eq. (7), gives a quadratic in x, from which the intersection numbers can trivially be found based on the conditions given below This is the easiest of the cases that have been considered. In further analyses, the possibility of d = 0, where the equations reduce to a quadratic instead of the original quartic in x, is not considered since repeating the calculation by requiring d = 0 is straightforward.

Complex Coupling
Upon relaxing the condition on d while maintaining h = 0, α, β ∈ C, and Re(β) ≥ 0, the polynomial obtained from Eq. (7) is a bi-quadratic in x.
Let us define the variables ∆, Π and Σ, which are related to the discriminant, product of roots, and sum of roots, respectively as outlined in Appendix A, as Then the intersection number for the critical point φ 0 is determined using the conditions in Table I, and the intersection number for the critical points φ ± is determined using the conditions in Table II. There are two comments to be made about these results. First, in both the Tables I and II (and later), we have extensively used the fact that (anti-)thimbles pass through the corresponding critical points. Further, in the situations discussed in this section, the (anti-)thimbles are not connected by the same flow equation, except for points in the parameter space at which the intersection  number changes. Thus we have also used the fact that a (anti-)thimble of a particular critical point does not pass through any other critical point. Second, if a condition given in these tables does not provide any condition for a specific relation between the parameters (for instance, Π and ∆ in Table I), it is to be understood that the value of that particular relation does not affect the intersection number. II: Constraints on the intersection number for the critical points φ ± when h = 0, α, β ∈ C, Re(β) ≥ 0, and d = 0.
As an illustration, let us determine the boundary at which the Stokes phenomena occurs for the choice of constants a = 1, c = 0, and d = 1.5, as derived by Fukushima and Tanizaki in Ref. [60] 9 . Conditions for Π given in Table II imply that all three thimbles to contribute when b ∈ (−∞, −1) ∪ (1, ∞). Conditions for Σ in Tables I  and II further require b < 0, which implies that when b ∈ (−∞, −1), all three thimbles contribute, and that Stokes phenomena is observed around b = −1.

Real Source Parameter
We now relax the condition on h to h ∈ R. The obtained equation, like the previous case, is a bi-quadratic but with a change to the part independent of x.
Again let us introduce the variables ∆, Π and Σ as Here ρ i is the imaginary part of the action, as defined in Eq. (10). The intersection number for each critical point φ i is now determined by the conditions given in Table  III.
For the situation where ∆ > 0, Π ≥ 0, Σ = 0, the intersection number depends on the critical point under question. For φ 0 , the intersection number will be equal to one, while for φ ± , the intersection number is zero.

Imaginary Source Parameter
Let us consider the case when the source parameter is purely imaginary. Defining we obtain the conditions on the intersection number. They are provided in Table IV. IV: Constraints on the intersection number for φ 0 , φ ± when h ∈ C, Re(h) = 0, α, β ∈ C, Re(β) ≥ 0, and d = 0.

D. In the Presence of PT Symmetry
We now specialize to actions that possess so-called PTsymmetry, where P is the parity symmetry and T is the time reversal invariance. In zero dimensions, any r eal function of ix is symmetric under PT transformation [54]. That is, our action should be of the form 10 S = n −a n (ix) n , with n denoting integers and a n representing real numbers. Comparing Eq. (44) with Eq. (5), we see that Eq. (44) corresponds to the case with h ∈ C, Re(h) = 0, and α = a, β = c ∈ R, such that c < 0. This is equivalent to replacing h → ih and c → −c in Eqs. (5), (8) and (10), and maintaining h ∈ R, c > 0. This leads to The values of ρ i depend on the value of a.
We first explore the case where a is positive. In this situation we obtain a set of quadratic equations. Solving for each sector in Eq. (11), and combining the results, 10 We have only considered polynomials in ix but any function with real powers of ix is PT -symmetric.
we obtain the conditions in Tables V and VI, where we have defined Combining the intersection numbers for each sector is highly non-trivial, and more information on how they were combined can be found in Appendix A.  The situation when a ≤ 0 is far more delicate than the previous situations we have considered. In the region for the these values of the parameters, all the three critical points lie on the imaginary axis (x = 0). Further, one of the solutions to the thimble equation, Eq. (7), is x = 0. Since c < 0, this solution lies outside the regions of stability, and is an anti-thimble as illustrated in Fig. 7. The main assumption in deriving Eq. (3) was that the critical points do no share a common gradient flow. This assumption is violated when a ≤ 0, resulting in the possibility In both the figures, the green solid curves represent the thimbles, red dashed curves represent the anti-thimbles, and the grey solid curves represent the ghosts. The shaded regions represent the regions where Re(S) ≥ 0. The anti-thimble x = 0 has been offset to x = 0.01 for better visibility. We see that there is a drastic change in the underlying thimble structure as the system passes through a phase transition.

Condition Intersection number
of critical points sharing a common (anti-)thimble, and the (anti-)thimbles of two different critical points intersecting with each other. Thus, the intersection number cannot be determined using the method employed in our calculations.

VI. QUANTUM PHASE TRANSITION AND CHANGE IN THIMBLE STRUCTURE
In this section, we demonstrate the usefulness of the results in Sec. V with the help of a few examples. We choose to fix c, d, and h, and vary either a or b in order to maximize the number of conditions that need to be checked.
First, consider the situation where β = 1 and h = 0. Equation (34) tells us that the intersection number depends only on the relative sign of a and c, and that b has no effect on the intersection number. Thus, choosing b = 1 and plotting the partition function as a function of a, we clearly observe a discontinuity (or kink) at a = 0 as demonstrated in Fig. 8, and thus for the given choice of parameters, the system undergoes a phase transition at a = 0. Looking at the corresponding change to the structure of the thimbles, shown in Fig. 9, the discontinuity in Z is due to the change in the intersection number of φ ± from zero for a > 0 and one for each critical point when a < 0. The blue curve represents the real part, the red curve represents the imaginary part, and the green curve represents the absolute value. Clearly, there is a discontinuity/kink at a = 0.
We now choose a = 1, c = 1, d = 1, and h = 0. The Based on the conditions given in Table I, we expect a sudden change in the value of the partition function when b = 0. From Table II, we expect that this should happen when b = 0, −1 − √ 2. Note that although it seems like we can expect a phase transition around b = 1 and b = −1 + √ 2, in the vicinity of these points, the intersection number does not change. On plotting the partition function for these parameters, we observe a discontinuity at b = −1 − √ 2. (See Fig. 10.) The explanation for why we do not obtain a discontinuity is that at b = 0, the change in the number of solutions is reflected in J i , R instead of K i , R . This explains why we have mentioned everywhere that the intersection number is less than or equal to a certain integer.
We can further complicate the situation and try to see where we observe discontinuities as we vary both a and b simultaneously. The expressions given in Eq. (35) in terms of a and b for c = 1, d = 1, and h = 0 are A naive expectation would thus straightforwardly be that when a = b or when b 2 − a 2 + 2ab = 0 (correspond- ing to the case ∆ = 0 and Π = 0), the partition function will have a discontinuity. (See Fig. 11.) Plotting the partition function as a function of a and b, we observe that this expectation is valid in certain cases, and in certain cases there is no discontinuity.
Let us now turn on the source term. We maintain a = 1, c = 1, and d = 1. Choosing h = 0.01, Eq. (10) becomes where We do not expect a phase transition with respect to φ 0 since here ρ 0 = 0, and we have already fixed our choice of a and c. Corresponding to the critical points φ + and φ − , we have, from Eq. (36) As is evident, when the source term is real, the equivalence between the critical points φ + and φ − gets lifted while φ 0 remains untouched. Solving the equations using a symbol interpreter, we get the points where the phase transitions could be expected as The partition function for this action, from Eq. (25), is given as This has a discontinuity in the vicinity of b = −2.4. However, the point at which the partition function is discontinuous does not match exactly with either b = −2.38621 or b = −2.44278. In fact, it matches exactly with our previous example where the boundary was at b = −1 − √ 2. We believe the issue is with the expansion of Z sources and not the method used to find the points of phase transitions, due to the fact that the perturbative expansion with respect to h in Eq. (25) depends on the partition function and observables of the action without sources. These are not sensitive to the lifting of equivalence between φ + and φ − .

VII. SUMMARY OF RESULTS
We have presented a lot of small results in the previous sections. Let us summarize them and put them into perspective.
For an action in zero dimensions with scalar fields, where the information about the background manifold is irrelevant 11 and the complexified degrees of freedom of the fields are in C, the thimbles can be found analytically by exploiting the most crucial property of these curves -the imaginary part of the action remains constant on them. However, solving for the thimbles using this method has its own problems as illustrated in Sec. IV where for more general situations, it is difficult to clearly distinguish between the solutions as they can either be thimbles or anti-thimbles, based on the region in the complex plane under question. We called this the 'piecewise behavior' of the solutions. The method is also extremely restricted to a small set of toy problems since it is only valid for polynomial actions of order less than five in zero dimensions. It would be interesting to comment about the piecewise behavior of thimbles when dealing with models in higher dimensions.
Despite these issues, there are advantages of employing the Lefschetz thimbles method since it provides a lot of ancillary information about the system. Since the weights in Eq. (2) are in general integers, changes in the weights correspond to discontinuities in the partition function, indicating the existence of different phases. We 11 Rather, non-existent.
used the simple method of solving Eq. (7), massaged in a way to access the information on the weights as outlined in Sec. V and Appendix A, to find conditional expressions involving relations between the parameters of the system that characterize the different phases of the system. A few examples showcasing the effectiveness of this method was presented in Sec. VI.
Although from the results, it is evident that phase transitions occur in the system, comments on the thermodynamical nature of these transitions cannot be made for the model we have chosen since thermodynamic quantities such as the free energy cannot be consistently defined in zero dimensions. However, there are two main observations about the behaviour of the phases. First, the boundaries of phase transitions are completely determined by the parameters h, α, and β. Thus, any symmetry involving the field φ remains a symmetry post phase transition. Second, these phase transition boundaries correspond to distinct changes in the topological structure of the thimbles and anti-thimbles. (We show this feature in Fig. 9.) These observations are a clear indication of the existence of "quantum phases" and "quantum phase transitions" [15,20,33,40,41]. Further, regions within the phase boundaries are akin to wall chambers and a phase transition corresponds to wall crossing.
We also, in passing, mentioned how looking at the action with sources as a perturbative expansion in terms of the action without sources fails, which was not obvious in Eq. (25) but emerged during our demonstration in Sec. VI. This further validates the power of using Lefschetz thimbles when compared to simple, perturbative analysis of systems.

VIII. CONCLUSIONS AND FUTURE DIRECTIONS
Although the Lefschetz thimbles formalism is a great mathematical tool to deal with the sign problem in quantum field theories with complex actions, performing actual calculations might involve many difficulties. The flow equation, Eq. (1), even in its zero-dimensional version, Eq. (4), is in general very difficult to solve since it involves complex variables. In this paper, we instead exploited the properties of these thimbles to analytically demonstrate how the Lefschetz thimbles formalism can be used to predict phase transitions for scalar theories in zero dimensions, and connected it to quantum phase transitions. An immediate extension would be to explore the same problem for supersymmetric theories in zero dimensions. The logarithmic term in the effective action, once the fermionic degrees of freedom have been integrated out, could lead to a few issues due to a pesky arctan term in Eq. (10). We still believe that the system would be solvable, post a few clever deformations that mitigate the problem of dealing with the arctan.
The definition of the thimble and anti-thimble also makes it difficult to solve numerically since any large number as the choice for the parameter t in Eq. (1), that is not infinity, will give a constant solution. An alternative has been suggested to deal with the issue by introducing a computational parameter that makes the flow equation easier to handle computationally [61]. Further, there are computational tools in place to heuristically find these thimbles using Monte Carlo methods [29,35,62,63], which seem to have some success in finding these thimbles without having to solve the flow equation. However, these solutions are numerical in nature, and in general it is very difficult to find these thimbles analytically.
In zero dimensions, except for showing that the partition function and observables develop discontinuities, comments on the thermodynamical nature of phase transitions cannot be made. Thus a more non-trivial and highly elucidatory extension would be to study phase transitions in higher dimensional systems, where the information of the background manifold becomes important and thermodynamic quantities can be defined consistently. There has been some success in effecting these calculations numerically using hybrid Monte Carlo simulations for the one-dimensional Thirring model [62], and there are numerous demonstrations of connections between Lee-Yang zeroes and Stokes phenomena in the context of chiral phase transitions [15,33,64,65]. However, a completely analytic and general demonstration of phase structures of higher dimensional systems, their relation to the structure of thimbles/anti-thimbles, and a relation with the thermodynamics of the system if any, is desired. It is to be noted that we have chosen to omit overall factors (such as that of 2 in Σ) since what is relevant is only the sign of these quantities. When the discriminant is positive, we have two distinct real solutions for w. In this case, when the product of roots is positive, either both solutions are positive (giving a combined intersection number of 4) or both solutions are negative (intersection number is zero). This is checked using Σ. When the discriminant is zero, we only get one real root for w. Again, Σ helps in determining whether the root is positive or negative. When the discriminant is negative (which for this particular case is never possible), there are no real roots of w and the intersection number is zero. These end up giving the conditions mentioned in Tables I and II. When the source term is non-zero, and the parameter h is real, the analysis remains exactly the same. The only change is the change to ρ i . If the source term is purely imaginary, Eq. (A4) now becomes Since the equation is now a purely quartic equation (in the sense that it is not reducible to a bi-quadratic), we have a complicated set of conditions for n i . We refer the reader to the conditions in Ref. [56] to arrive at the results in Table IV.
When the action possesses PT -symmetry, we can obtain its critical points and the imaginary part of the action by substituting h → ih and c → −c in Eqs. (8) and (10), which gives us when a > 0, i = 0, −h a |c| when a > 0, i = +, +h a |c| when a > 0, i = −.

(A16)
The imaginary part of the action, given in Eq. (A1), upon making these substitutions becomes Im S(x, y) = hx + axy − c x 3 y − xy 3 . (A17) As outlined in Sec. III, the standard procedure for dealing with this action is to take an integration cycle about the angles 5π/4 and 7π/4 (in the third and the fourth quadrant, respectively). We have chosen it to be (See Eq. (11)) y(x) = x for x ≤ 0, −x for x > 0.
Substituting the above integration cycle in Eq. (A17) and equating it to the imaginary part of the action at a critical point, we obtain hx + ax 2 + ρ i = 0 for x ≤ 0, (A19) hx − ax 2 + ρ i = 0 for x > 0.
For the case where a > 0, we split the intersection number into two parts. The number of times the thimble and anti-thimble intersect the part of the integration cycle where x < 0 is called n L i , and the number of times the thimble and anti-thimble intersect the part of the integration cycle where x > 0 is called n R i . The total intersection number thus is n i = n L i + n R i . For x < 0, since the associated (anti-)thimble equation is a quadratic in x, we define the discriminant, product of roots, and sum of roots as We obtain similar expressions for x > 0 Here we look for negative real solutions for the Eq. (A19) and positive real solutions for the Eq. (A20). Standard analysis of quadratic equations gives us the conditional expressions in Tables VIII and VIII. Combining these conditions is slightly non-trivial since there are cases where two conditional expressions cannot be satisfied simultaneously. (For example, Σ L > 0 and Σ R > 0 is not simultaneously possible.) Having taken care of such situations, we arrive at the results in Table  VI.