Structure of Lefschetz thimbles in simple fermionic systems

Structure of Lefschetz thimbles in simple fermionic systems

Takuya Kanazawa Department of Physics, The University of Tokyo, Tokyo 113-0033, JapanTheoretical Research Division, Nishina Center, RIKEN, Wako, Saitama 351-0198, Japan    and Yuya Tanizaki Department of Physics, The University of Tokyo, Tokyo 113-0033, JapanTheoretical Research Division, Nishina Center, RIKEN, Wako, Saitama 351-0198, Japan

The Picard-Lefschetz theory offers a promising tool to solve the sign problem in QCD and other field theories with complex path-integral weight. In this paper the Lefschetz-thimble approach is examined in simple fermionic models which share some features with QCD. In zero-dimensional versions of the Gross-Neveu model and the Nambu-Jona-Lasinio model, we study the structure of Lefschetz thimbles and its variation across the chiral phase transition. We map out a phase diagram in the complex four-fermion coupling plane using a thimble decomposition of the path integral, and demonstrate an interesting link between anti-Stokes lines and Lee-Yang zeros. In the case of nonzero mass, it is shown that the approach to the chiral limit is singular because of intricate cancellation between competing thimbles, which implies the necessity to sum up multiple thimbles related by symmetry. We also consider a Chern-Simons theory with fermions in -dimension and show how Lefschetz thimbles solve the complex phase problem caused by a topological term. These prototypical examples would aid future application of this framework to bona fide QCD.


RIKEN-QHP-174 iTHES Research Group and Quantum Hadron Physics Laboratory, RIKEN, Wako, Saitama 351-0198, Japan

1 Introduction

Path integrals with complex weight appear in many branches of physics. Examples include the Minkowski path integral, QCD with chemical potential, Chern-Simons gauge theory, Yang-Mills theory in the theta vacuum and chiral gauge theories. Interest in quantum theories with complex actions was also stimulated by the advent of symmetry Bender:1998ke (); Bender:2002vv (). Despite the overwhelming significance of these theories, only partial progress has been made towards their first-principle understanding partly due to the incapability of numerical simulations based on Monte Carlo sampling to deal with complex weights, which impede a probabilistic interpretation.

A promising approach for handling complex actions is to complexify the field space. In a one-dimensional integral, the method of steepest descent (or stationary phase method) is well known, in which one deforms an integration contour into a more general path on the complex plane so that it passes through a critical point (saddle) of the integrand. This allows for an asymptotic evaluation of exponential integrals. The generalization of this method to higher dimensions is provided by the Morse theory (or Picard-Lefschetz theory) Pham1983 (); Nicolaescu2011 (), where contours of steepest descent are generalized to higher-dimensional curved manifolds called Lefschetz thimbles. Recently, a direct application of the Picard-Lefschetz theory to infinite-dimensional path integral in quantum field theory (QFT) was made by Witten Witten:2010cx (); Witten:2010zr (). He showed that Chern-Simons path integral can be defined nonperturbatively for complex gauge fields by taking Lefschetz thimbles as integration cycles. This work has provoked subsequent developments of numerical approaches to complex path integrals on the basis of Picard-Lefschetz theory Cristoforetti:2012su (); Cristoforetti:2013wha (); Mukherjee:2013aga (); Aarts:2013fpa (); Fujii:2013sra (); Cristoforetti:2014gsa (); Mukherjee:2014hsa (); Aarts:2014nxa (). They are reminiscent of the complex Langevin method Parisi:1984cs (); Klauder:1983zm (); Klauder:1983sp (), which is also based on the idea of field complexification, but there seem to be fundamental differences Aarts:2013fpa (); Aarts:2014nxa ().

The Picard-Lefschetz theory also provides a useful framework for obtaining a visual understanding of a subtle interplay between perturbative and nonperturbative saddles in asymptotic series Berry1991 (). See David:1992za (); Felder:2004uy (); Eynard:2008yb (); Marino:2008ya (); Marino:2009dp (); Pasquetti:2009jg (); Chan:2010rw (); Chan:2012ud (); Marino:2012zq (); Schiappa:2013opa () for related works in matrix models with applications to quantum gravity, Dijkgraaf-Vafa theory, ABJM theory and non-critical string theory. More recently, the relevance of Lefschetz thimbles is discussed in the context of semiclassical expansion in asymptotically free QFTs Dunne:2012ae (); Basar:2013eka (); Cherman:2014ofa (); Cherman:2014xia (); Dorigoni:2014hea (). For an inexhaustive list of references on complex path integral and Lefschetz thimbles, see Guralnik:2007rx (); Alexanian:2008kd (); Denbleyker:2010sv (); Nagao:2011za (); Harlow:2011ny (); Ferrante:2013hg (); Nishimura:2014rxa (); Tanizaki:2014xba (); Cherman:2014sba (); Gorsky:2014lia (); Sexty:2014dxa (); Tanizaki:2014tua ().

So far many of the works on Lefschetz-thimble approach to QFT seem to have been centered around bosonic theories Witten:2010cx (); Cristoforetti:2012su (); Cristoforetti:2013wha (); Mukherjee:2013aga (); Ferrante:2013hg (); Aarts:2013fpa (); Fujii:2013sra (); Cristoforetti:2014gsa (); Cherman:2014xia (). On the other hand, in view of possible future applications of the Picard-Lefschetz theory to QCD and QCD-like theories, it would be important to understand the behavior of Lefschetz thimbles for path integrals with fermionic degrees of freedom. In Mukherjee:2014hsa (); Aarts:2014nxa () Lefschetz thimbles in presence of a fermion determinant were studied numerically. These works were specifically focused on the Hubbard model Mukherjee:2014hsa () and on lattice fermions with chemical potential Aarts:2014nxa (), respectively. It would be certainly worthwhile to extend these analyses to a more general setting.

In this paper, we investigate the structure of Lefschetz thimbles in a variety of fermionic systems in zero and one dimension. We obtain complex critical points, determine associated Lefschetz thimbles and discuss their Stokes jumps, in a fermionic model with discrete chiral symmetry (with or without small mass term), in a fermionic model with continuous chiral symmetry (with or without small mass term), and in a Chern-Simons-like theory with fermions where a sign problem is caused by a topological term in the action. We expect that examples worked out here and lessons learned therefrom will be of value in future attempts to study the complex phase problem and spontaneous chiral symmetry breaking in QCD and QCD-like theories on the basis of Picard-Lefschetz theory.

This paper is organized as follows. In Section 2 a brief review of the Lefschetz-thimble approach to path integrals is given. We use simple toy integrals to illustrate that zeros and poles in the functional determinant in field theories do not obstruct the application of the Picard-Lefschetz framework. In Section 3 we study a zero-dimensional Gross-Neveu-like model with discrete chiral symmetry. We investigate chiral symmetry breaking, Stokes lines, monodromy and Lee-Yang zeros in the complex coupling space from a viewpoint of Lefschetz thimbles. In Section 4 a zero-dimensional Nambu-Jona-Lasinio-like model with continuous chiral symmetry is analyzed. We add a small mass term that breaks chiral symmetry explicitly and find that the approach to chiral limit is singular in terms of thimbles. In Section 5 a one-dimensional theory with a topological term is considered. Positions of infinitely many critical points are determined and the dependence of associated Lefschetz thimbles on the coefficient of the topological term is elucidated. We conclude in Section 6.

2 General remarks

In this section we give a short summary of the Picard-Lefschetz approach to complex integrals and discuss applicability of the method in presence of zeros and poles of the integrand, which is commonly caused by functional determinant in QFTs. We aim to provide a minimal background for later sections of this paper and to fix our notations and terminology. A more in-depth review on this topic may be found in Witten:2010cx (), to which we refer the interested reader for further details.

In physics we frequently confront the need to evaluate integrals of the form where is a parameter ( in quantum physics, in statistical mechanics, in matrix models and -vector models, and so on). The domain is a subset of for simplicity. When is real and is small, an asymptotic estimate of the integral is available by means of a saddle-point approximation. The saddle points of in dominate the integral, while those that lie outside of do not play a role. In addition, for real one can apply Monte Carlo sampling techniques by interpreting the measure as a probabilistic weight, which is useful especially when is large.

Things change substantially once becomes a complex function. Although the integrand at small is suppressed for large and enhanced for small , a naive saddle-point approximation based on saddles of fails because causes a rapid oscillation of the integrand which lessens contributions from the vicinity of these saddles. A viable asymptotic expansion should thus be done around a point where both and are stationary, but such point might not exist on . Moreover, the complex phase of the integrand makes a numerical evaluation of the integral quite challenging.

As is widely known, for the correct way of handling this analytically is to promote to a holomorphic function on , identify saddle points of on and deform the original integration contour so that it passes through the saddle in the direction of stationary phase: is constant on the deformed path. However, in the presence of multiple saddles it is far from trivial to see which one contributes to the integral and which does not; this is even more so when considering multidimensional integrals, i.e., .

A lucid way to organize asymptotic expansions for complex integrals is provided by the Picard-Lefschetz theory. For this to work we require that is a holomorphic function on and that all critical points of are non-degenerate.111The presence of degenerate saddles that stem from a symmetry of the action does not necessarily invalidate the Picard-Lefschetz theory, but requires a special treatment as we discuss in Section 4.2. The downward flow equation is then defined as


The essential property of (1) is that is constant but is monotonically decreasing along the downward flow. (This is a Morse flow with the Morse function.) Every critical point of is evidently a fixed point of the flow. For each critical point   () , we can define a Lefschetz thimble as the union of all flows that end at in the limit . This is a manifold of real dimension due to the fact that there are precisely directions around in which is increasing. (For , for instance, has one increasing direction and one decreasing direction. This generalizes to higher dimensions thanks to the holomorphy of .) Since is strictly decreasing along the flow, it must be that tends to in the limit .222An exception is when the flow meets another critical point. This is called the Stokes phenomenon Berry1989 () but does not occur for generic case and we momentarily ignore this. This implies that goes to zero at the ends of . In other words, gives an element of the relative homology for very large , where is the “good” regions in which the integrand decreases rapidly. It then follows, that for any linear combination of Lefschetz thimbles with integer coefficients , the integral


is convergent and well-defined. Importantly, the converse is also true: actually constitutes a basis of so that any cycle of real dimension without a boundary which is suitable as an integration cycle for can be decomposed into a sum of Lefschetz thimbles. Therefore, as long as the original integration cycle belongs to , one can always express an integral over it in the form


Now the integral on the RHS has a real positive weight (albeit on a curved manifold ) and is easily amenable to usual asymptotic analysis around saddles. At the same time the complex phase problem is resolved if we could perform efficient Monte Carlo sampling on the Lefschetz thimbles with .

Of course, this method would be useless if we do not know how to determine . Fortunately this is known at least for a finite-dimensional integral. Let us consider an upward flow equation


in which the sign on the RHS is flipped as compared to (1). As a result of this, along the upward flow, is conserved as in (1) while is monotonically increasing. For each critical point of , we define as the union of all upward flows which end in in the limit . Again this is a manifold of real dimension . The cycles may be seen as a dual of the Lefschetz thimbles in a homological sense and there is a natural intersection pairing between and Witten:2010cx (). Actually constitutes a basis of another relative homology for large , where is the “bad” regions. Of course, one cannot use as an integration cycle for  , because at the ends of (i.e., for ). Rather, the importance of lies in the fact that appearing in (3) can be computed as the number of (oriented) intersections between and .333In dimensions (), two general hypersurfaces with dimension intersect at isolated points. Consider two lines on a plane, for example. While the calculation of is tractable for low-dimensional toy models, it becomes a formidable task in the case of infinite-dimensional QFTs. In recent numerical Monte Carlo approaches to Lefschetz thimbles, only a single thimble associated with the perturbative vacuum was taken into account on the basis of universality Cristoforetti:2012su (); Cristoforetti:2013wha (); Fujii:2013sra (); Cristoforetti:2014gsa (); Mukherjee:2014hsa (). Although numerical results so far look quite promising, the validity of this approach still remains to be clarified.

In general applications of the Lefschetz thimble technique, we often face integrals that are not of a pure exponential form, but rather with another function . When is just an observable in field theories, it does not affect the saddle point analysis of the integral because grows extensively in the thermodynamic limit while is an quantity. In contrast, if grows in the thermodynamic limit this has to be taken into account in the saddle point analysis: examples of such are the quark determinant in QCD and the Vandermonde determinant in matrix integrals. In both cases has zeros and a simple rewriting looks subtle. Moreover may also have poles. In the remainder of this section, we discuss how the Lefschetz-thimble approach works in these cases.

Let us first consider the case when is holomorphic on . Then is defined except at zeros of .444Note that the set is typically of real dimension . The zeros of thus form a hypersurface in higher dimensions. Although is ambiguous up to integer multiples of , the gradient flow equations (1) and (4) are well-defined because the above ambiguity disappears by taking derivatives of , and the thimbles can be defined in the same way as before, provided that all the critical points of are non-degenerate. What is crucial here is that the “good” regions must be modified, by incorporating the vicinity of zeros of . This can be explained most easily on examples.

As a trivial case, consider . Of course this integral is identically zero, but this is irrelevant for the purpose of illustration. Casting this into the form with , we see that the critical points are located at . The Lefschetz thimbles , and the upward flow lines can then be defined.

Figure 1: Lefschetz thimbles (solid lines) and their duals (dashed lines) for simple one-dimensional integrals. Orange blobs are critical points of the action. Hatched areas are “good” regions where the integrand tends to zero. In both panels, the origin is a singularity of the flow.

They are schematically shown in Figure 1 (left panel). The three hatched areas are “good” regions, which includes the vicinity of . The Lefschetz thimbles run from one good region to another, thus providing a basis of the relative homology for very large in accordance with the general argument. By contrast, run from one “bad” region to another , thus providing a basis of for very large . Since intersect with the real axis, the integral receives contributions from both and . This is also intuitively obvious, for the original contour is a union of and . As a whole, the general framework of Lefschetz-thimble approach does not seem to be obstructed by the presence of zeros of the integrand.

There is one side remark here. We noted above that is conserved along a flow. While this is generally true, it does not imply that is constant over an entire upward/downward flow cycle. For illustration, let us note that is comprised of two distinct flow lines: one is stretching from to and the other from to . It is easily seen that is along the former and along the latter, owing to the fact that sits right on the branch cut of logarithm. Thus we conclude that although is locally conserved along a flow, it can jump by a multiple of at a point where two flows meet. (See also Aarts:2014nxa ()).

Next, let us turn to the case when is a meromorphic function with poles. An example of this is given by a bosonic functional determinant in QFTs. It is useful to once again employ a simple example to illustrate the general applicability of Lefschetz thimbles. Consider an integral where the contour is slightly uplifted from the real axis to avoid the pole at . Now, writing this as with , we see that the critical points are located at . The Lefschetz thimbles , and the upward flow lines can then be defined with respect to the flow equations for . They are shown in Figure 1 (right panel). Compared to the previous example, the geometrical structure of and are exchanged. Interestingly, now, and end at the origin because the area around was turned into a “bad” region by a pole. One can easily confirm that and again constitute the bases of relative homology. (Note that and are independent cycles, for they cannot be continuously moved to each other across the singularity at .) Since intersects with while does not, the integral only receives contribution from and not from . In summary, the presence of a pole in the integrand does not undermine the applicability of the Picard-Lefschetz theory.

We note that our discussion so far is concerned about mathematical aspects of the formalism and does not immediately suggest feasibility of numerical algorithms for Lefschetz thimbles. Actually the fermion determinant in QCD at finite density has dense zeros in the gauge configuration space and this could lead to a practical difficulty for the Lefschetz-thimble approach Cristoforetti:2012su ().

3 Gross-Neveu-like model

As a toy model for discrete chiral symmetry breaking, let us consider a zero-dimensional fermionic model similar to the Gross-Neveu (GN) model Gross:1974jv (). The partition function reads


where and are -component Grassmann variables with colors, is a coupling constant, is a bare mass, and is a constant matrix that mimics the effect of nonzero-momentum modes in higher dimensions. (As an explicit basis we use , and in the following.) At , the action in (5) is invariant under a chiral transformation . With a Hubbard-Stratonovich transformation,




The derivation of (7) is not only valid for but also for complex as long as . Now, let us discuss the cases with and separately.

3.1 Massless case

3.1.1 Lefschetz thimbles

For , the minimum of the action takes place at for and at for . A continuous “chiral transition” occurs at .

In order to analyze this transition from the viewpoint of Lefschetz thimbles, we lift to a complex variable . The critical points of the action are obtained as


Let us denote . Since the three critical points coalesce at and it makes the Lefschetz thimbles ill-defined, we will hereafter assume .

It can be easily checked that for


where the equality holds if and only if . Thus the nontrivial critical points always have a lower action than for , regardless of .

An important fact is that at all these critical points. This means that we are right on the Stokes ray: there are flow lines that connect distinct critical points Berry1989 (); Witten:2010cx (). To avoid the Stokes ray and make Lefschetz thimbles well-defined, we endow with a phase factor with so that the degeneracy of among the critical points is lifted.555It is not appropriate to rotate the entire action as because it allows the -ambiguity in the imaginary part of to induce an ambiguity in the real part of , which renders the integral ill-defined. A more comprehensive analysis of Stokes phenomena for complex will be presented in Section 3.1.2.

Figure 2: Lefschetz thimbles (black lines) and upward flow lines (red dashed lines) on the complex -plane for the GN-like model in the chiral limit. The three orange blobs at and are the critical points of , while the two red blobs at are the points where diverges. The background color scale describes .

In Figure 2 we show downward flow lines (i.e., Lefschetz thimbles) and their duals on the complex plane for and with . (Since , the system is in a chirally symmetric phase.) The displayed flow lines were obtained by first plotting the contours fulfilling and , and then discarding components that are not connected to the critical points.

There are several remarks concerning Figure 2. First of all, there are three Lefschetz thimbles , and and three upward flow lines , and , passing through each critical point. Among the three ’s, only intersects with the original integration cycle , which means that is the only Lefschetz thimble contributing to .666This is consistent with the general fact that Lefschetz thimbles associated with critical points that lie on the initial integration cycle contribute with a unit coefficient Witten:2010cx (). This is so even though is lower than .

Secondly, as we can see, and jump as crosses zero. This is a phenomenon called the Stokes jump. Since does not jump at , the partition function itself is analytic at .

Finally, it is worth an attention that the Lefschetz thimbles indeed provide a homological basis of convergent integration cycles even in the presence of logarithmic singularities, in agreement with our argument in Section 2. Here what we call convergent integration cycles are contours that start from and end in “good” regions (bright regions in Figure 2) which are the regions where grows to . In the present case, there are four “good” regions: (i) the vicinity of , (ii) the vicinity of , (iii) , and (iv) . As seen from Figure 2, the three ’s form a basis in the space of cycles that start from and end in those four “good” regions. This is the premise of the Lefschetz-thimble approach to complex integrals.

Figure 3: Same as Figure 2 but with .

Next, we consider a chirally broken phase with . To avoid the Stokes ray, we again attach a phase factor to . The resulting Lefschetz thimbles are shown in Figure 3. Notably, has rotated almost 90 degrees in comparison to Figure 2, and now it connects the two singular points at .

One can observe in Figure 3 that all the three ’s (red lines) intersect with the real axis, implying that now receives contributions from all of , and . Indeed it is visually clear that the union of the three thimbles is homologically equivalent to . Since is lower than (recall 10), the nontrivial saddles will completely dominate the behavior of in the large- limit, and fermions acquire a dynamical mass that breaks discrete chiral symmetry. The second-order chiral transition in this model along the line thus occurs through a jump in the number of contributing thimbles at . By contrast, we will see in Section 3.1.2 that the chiral transition for is generically first order and exhibits qualitatively new features.

We shall analyze the Stokes jump in Figure 3 in some details. Let us fix the orientation of as the direction of increasing and of as the direction from to . Then the homological jump of thimbles as is dialed from to can be summarized as


The meaning of this is that in the left panel of Figure 3 is equal to in the right panel of Figure 3, and so on. This actually implies that the real cycle can be expressed in two different ways, according to how we approach the limit of :


There are two remarks on (11) and (12).

  • The fact that does not jump across the Stokes ray (as one can see in Figures 2 and 3) has an intuitive explanation. From the definition of a downward flow and (10), we have for . This implies that the flow along has no chance to touch , so is insensitive to the presence of and shows no Stokes jump at all. By the same token, one can explain why do not jump.

  • The fact that the coefficients of in (12) do not jump across can be deduced in a simple way. For , the asymptotic behavior of at is dominated by the saddles since . Recalling that is a holomorphic function of at any finite , it follows that the contributions from cannot jump discontinuously. By contrast, such an argument does not constrain the exponentially smaller contribution from , and indeed the coefficient of does jump in (12). In short, the jump of coefficients can only occur for thimbles associated with subleading saddle points.

Both arguments have been presented by Witten (Witten:2010cx, , Sect. 3) in the context of bosonic integrals, and here we have highlighted their usefulness in a fermionic model.

3.1.2 Stokes lines and monodromy

So far we have described the Stokes phenomenon in the massless GN-like model for . It has been shown that the thimbles can be made well-defined if is given a small complex phase with . In this subsection, we study the Stokes phenomenon and some technical issues for a generic coupling .777A complex four-fermion coupling appears in studies of the vacuum in QCD Boer:2008ct (); Boomsma:2009eh ().

The necessary condition for a Stokes jump to occur is that the imaginary part of the action is degenerate for multiple critical points. In the present model with , this condition reads


This can be solved by and by . The union of these sets is displayed in Figure 4 as a blue line.

Figure 4: Stokes lines for the GN-like model with and (blue lines). The global topology of Lefschetz thimbles and their duals changes across the Stokes lines: and jump across the horizontal line, while and jump across the round curve. In the shaded area, only contributes to . Outside the shaded area, all thimbles contribute.
(The points and are excluded because the action is singular at and the critical points merge at .)

When these lines are crossed, some of the flow lines jump discontinuously. Which line jumps and which does not can be deduced along the line of arguments at the end of the last subsection. From (10), along the axis , hence and jump across this line while the others do not. It can also be checked numerically that is lower than along the kidney-shaped contour in Figure 4. Therefore, this time, and jump across this contour. We verified these expectations numerically.

In the shaded area of Figure 4, is the only thimble contributing to . The structure of thimbles looks like Figure 2. Outside the shaded area, and all contribute to and their structure resembles Figure 3. The boundary of the shaded area is where a jump occurs in the number of contributing thimbles ().

It has to be emphasized that the boundary curve in Figure 4 has nothing to do with the chiral phase transition. Roughly speaking, inside the curve , whereas outside the curve , at . In either case is entirely dominated by since on the boundary curve; a phase transition does not occur. It is known that the emergence of such subdominant exponentials across a Stokes line occurs smoothly Berry1989 (); Berry1989b (), i.e., is analytic around the Stokes lines. The correct identification of the chiral transition line will be made in Section 3.1.3.

So far we have tacitly assumed that originally defined for by the integral (7) can be extended to complex , but this requires some care. While (7) is convergent for , it apparently diverges for . Nevertheless, one can still expand (5) in Taylor series of and evaluate the partition function in a polynomial of order in , which is of course analytic over the entire complex -plane. The right procedure to fill this gap and define the integral (7) analytically for entire is as follows888Pedagogical reviews of this procedure for the Airy integral can be found in Marino:2009dp (); Witten:2010cx (); Marino:2012zq (). : as varies on the complex plane, the “good” regions on the complex -plane also rotates simultaneously. In order for the integral to converge, the integration contour (initially ) must have ends in those good regions, hence the contour should be rotated hand-in-hand with the variation of .

To be more explicit, let us consider a phase rotation of by on the complex plane, starting from some to avoid complications due to Stokes lines. Initially (with ), two of the good regions are located at and , where the weight goes to zero. When is dialed by and approaches the positive imaginary axis, these good regions are rotated by . When is rotated by , the good regions are rotated by : they are now specified by and . As returns to the positive real axis, the good regions return to their initial position.

What is interesting here is that the two good regions are permutated by this rotation. In other words, they are rotated by when is rotated by , and so are the Lefschetz thimbles: they return to themselves only after rotation of . Therefore the monodromy of Lefschetz thimbles around is of order 2. Related to this, note that although the contour itself is homologically equivalent to after a -rotation of , its orientation gets reversed. Thus the integral over the contour flips sign when is rotated by . At the same time, however, in front of (7) changes sign (i.e., when ) so that the integral (7) returns to its initial value. Thus the orientation of a thimble must be traced correctly to ensure the single-valuedness of the partition function for complex .

3.1.3 Anti-Stokes lines and Lee-Yang zeros

Now that the partition function is well defined for complex coupling , we can ask where is the boundary between a chirally broken phase and a chirally symmetric phase in the complex -plane. Considering that has three critical points , we would get a nonzero condensate in the large- limit if the following two conditions are both met:

  1. contribute to , and

  2. .

From the last subsection we know that the first condition is met for outside the shaded region in Figure 4. The second condition is necessary for the symmetry-breaking saddles to dominate the partition function at large . In general, a line on which exchange of dominance occurs between distinct saddles is called an anti-Stokes line, which should not be confused with the Stokes lines. In the present case the anti-Stokes line is specified by


and is shown in Figure 5 for , together with the Stokes lines from Figure 4.999The anti-Stokes line actually extends into the interior of the Stokes curve, but this part is not shown in Figure 5 because and do not contribute to there.

Figure 5: Anti-Stokes line for the GN-like model with and (green curve), overlaid with Lee-Yang zeros for (red bullets) and the Stokes line in Figure 4 (blue curve).

Chiral symmetry is broken at large for outside the green anti-Stokes curve, and is restored for inside the curve.101010It should be noted that a condensate for complex is a complex quantity and does not admit a physical interpretation as a dynamical mass of fermions. Since multiple saddles exchange dominance, the phase transition along the anti-Stokes curve is generally first order, with the only exception at where the transition is continuous. This point is quite special, as the Stokes curve and the anti-Stokes curve intersect there.

Next, we would like to explore a connection between the anti-Stokes line and zeros of the partition function. Since the seminal work by Lee and Yang YL1952 (); LY1952 (), it has been widely recognized that zeros of a finite-volume partition function in a complex parameter space, called Lee-Yang zeros, provides rich information on the phase transition in the thermodynamic limit. See Bena2005 () for a review and Fodor:2001pe (); Fodor:2004nz (); Ejiri:2005ts (); Stephanov:2006dn (); Denbleyker:2010sv () for applications to Yang-Mills theory and QCD. Connections between Stokes phenomenon and Lee-Yang zeros were investigated in Itzykson:1983gb (); Pisani1993 (); Guralnik:2007rx ().

In the present model, it is straightforward to evaluate (5) or (7) in the chiral limit to obtain a polynomial representation of the partition function:


We have numerically computed zeros of on the complex -plane. The result for and is presented in Figure 5. Clearly all the zeros are distributed in the vicinity of the anti-Stokes curve. For larger , zeros are observed to align on the anti-Stokes curve more and more densely. It is expected that they will form a continuous cut in the large- limit, marking a boundary between a chirally symmetric phase and a chirally broken phase. The overall picture is quite consistent with the Lee-Yang picture of a phase transition.

An important feature of the anti-Stokes curve in Figure 5 is that it has a kink at . It can be shown from (14) that the curve pinches the real axis at angle . According to a general theory of Lee-Yang zeros Bena2005 (), a kink occurs when the transition for this point is of higher order, and the angle implies that it is a second-order phase transition with a mean-field critical exponent. This is exactly what happens in this model at .

This completes our analysis of the GN-like model in the chiral limit.

3.2 Massive case

When the fermion mass is nonzero, the chiral symmetry is explicitly broken and the “condensate” is nonzero for any . Although a sharp phase transition is absent at , one can still find an interesting behavior of Lefschetz thimbles in the following. Without loss of generality, we assume .

To apply the Picard-Lefschetz theory we complexify to . The critical points (i.e., saddles) of are obtained as solutions to


This equation always has three roots, one of which is real and the other two are either both real or a complex-conjugate pair, depending on , and . With a bit of algebra, we find that the jump in the number of real roots () occurs when


under the condition that and .

Figure 6: Left: Phase diagram of the GN-like model. (The part can be obtained by reflection about the horizontal axis.) The red horizontal line represents a first-order phase transition line. The blue dashed line is a limit of metastability (not a phase transition), i.e., below this line (in the blue region), there are two local minima for the action. Above this line, the action has a single minimum. Concerning the points A and B, see Figures 8 and 8.
Right: The behavior of saddle points of when is increased at fixed . As we move out of the blue region in the left panel upward, two of the three saddles on the real axis merge and then migrate into the complex plane.

In Figure 6 (left panel) we show the phase structure of the model, together with the typical shape of in each region. The domain having three (one) real saddles are painted blue (white), respectively. Across the blue dashed line given by (17), the number of saddles on jumps. This is not a phase transition, but corresponds to the disappearance (or emergence) of a metastable state. Figure 6 (right panel) schematically shows the motion of critical points on the complex -plane when the line of metastability is crossed from below.

Now we are in a position to reveal the behavior of Lefschetz thimbles. We take points A and B in Figure 6 (left panel) as representatives of white and blue regions, respectively. Figure 8 shows the Lefschetz thimbles with at the point A. Interestingly, at no Stokes phenomenon occurs for and one can safely take without a complex factor. For this , the chirally condensate vanishes at . We observe that, just as we saw in Figure 2, there is only one thimble (the real axis, ) which contributes to the partition function. The saddle associated with this thimble gives rise to a condensate . The other two thimbles extend to the direction together, although they went in the opposite directions in Figure 2. Note that Figure 8 will be horizontally reversed for .

Figure 7: Same as Figure 2 but with and , corresponding to the point A in Figure 6 (left panel).
Figure 8: Same as Figure 2 but with and , corresponding to the point B in Figure 6 (left panel).
Figure 7: Same as Figure 2 but with and , corresponding to the point A in Figure 6 (left panel).

Figure 8 displays Lefschetz thimbles at the point B in Figure 6 (left panel). This time all the three critical points lie on the real axis and have , so is right on the Stokes ray.111111More generally, a Stokes phenomenon occurs everywhere in the shaded region of Figure 6 (left panel), since all critical points are real there. As before we rotated the phase of slightly to make the thimbles well-defined. Figure 8 shows that now the three thimbles all contribute to the integral. The overall structure of the thimbles is the same as in Figure 3 at . Among the three critical points, the right-most one has the lowest and hence governs the partition function and condensate at .

Between and there is a jump in the number of contributing thimbles. This occurs when one traverses the blue dashed boundary in Figure 6 (left panel). This is not a phase transition, since the critical point that gives the dominant contribution always sits on the positive real axis and moves smoothly with . Rather, there appears a new subleading contribution to the partition function, which is exponentially smaller than the leading one at .

It is worth an emphasis that the three Lefschetz thimbles in Figures 8 and 8 indeed form a homological basis of cycles connecting “good” regions, in accordance with the general argument in Section 2.

Finally we consider a special limit , in which the two singular points at merge into a single singularity at . The situation is simpler than for , because there are only two critical points at . The corresponding thimbles are shown in Figure 9. (Note that this is quite analogous to the example considered in Section 2.) The Lefschetz thimbles for are given by


They meet at the singular point and together constitute the integration cycle . In the limit , move to , hence chiral symmetry is spontaneously broken in the large- limit for any small , with . Such a non-analytic dependence on cannot occur at any finite order of expansion in and is a hallmark of nonperturbative physics.

Figure 9: Lefschetz thimbles and their duals for the GN-like model at .

This completes our analysis of the Lefschetz thimbles in the zero-dimensional GN model.

4 Nambu-Jona-Lasinio-like model

4.1 Model setup

Next, we consider two zero-dimensional toy models having continuous chiral symmetries, in analogy to the Nambu-Jona-Lasinio(NJL) model Nambu:1961tp (); Nambu:1961fr (). The first model is defined by the partition function


The definitions of symbols and variables are the same as in (5). In the chiral limit the action is invariant under a chiral rotation and . By introducing auxiliary fields and to bilinearize the action, we obtain




With this action enjoys an symmetry that rotates and . If we set by rotation, then the present model reduces to the GN-like model in Section 3. This will become important in the analysis of Lefschetz thimbles later.

The second model we consider is defined by the partition function


where and are two-component Grassmann variables with two flavors and colors, are the Pauli matrices, and the summation over flavor indices is implicitly assumed. At , this model has an exact chiral symmetry, which is broken explicitly to by nonzero .

Proceeding as before,


Under the chiral symmetry, rotates as a vector. One can rotate any such vector to by means of an unbroken rotation. The resulting integral over and is essentially equivalent to the former model (21) and does not entail a new feature. For this reason we focus on the first model in the following.

4.2 Massless case

For simplicity we begin with the chiral limit where the chiral symmetry is exact. The task is to identify the Lefschetz thimbles for and to figure out how to decompose the original integration cycle of (21) into a sum of Lefschetz thimbles in . Upon a complexification of variables, the action becomes


whose domain is . The set of singularities of logarithm forms a surface of real dimension 2 in rather than a set of isolated points. It is equal to the -orbit of the singular points of the massless GN-like model, where is a complexification of the group.

The downward flow equation reads


An important property of this flow is that it is symmetric under an rotation of although it is not under a general rotation. This will play a pivotal role in the construction of Lefschetz thimbles later.

The critical points may be obtained by solving (27) with . To avoid accidental degeneracy of critical points, we assume . The set of critical points then consists of two components:


is equal to the -orbit of the critical points in the massless GN-like model. It crosses the real plane if but has no crossing otherwise.

We now discuss how to determine the Lefschetz thimbles and their duals associated with and . Since we are considering a two-dimensional integral, the thimbles should be cycles of real dimension 2. As for we can apply the standard procedure since it has a Morse index 2, i.e., increases in two directions and decreases in the other two directions around . Then the Lefschetz thimble and its dual associated with can be defined as a union of downward and upward flows, respectively, flowing into . Since the flow preserves symmetry, may be simply obtained by rotating a Lefschetz thimble in the GN-like model by action:


and the same goes for . So much for .

The prescription for is a bit different. A general framework to handle a continuous manifold of critical points was developed in (Witten:2010cx, , Sect. 3) and we shall outline how to apply this framework to the present fermionic model with symmetry. First of all, at a point of , the Hessian matrix of has one positive eigenvalue, one negative eigenvalue and two null eigenvalues. Therefore, the set of points that can be reached by a downward/upward flow from any given point in is of real dimension 1.121212Note that no flow exists on , since all points on have the same value of . Therefore if we pick up a one-dimensional subset out of , the set of points that can be reached by a downward flow from that subset of forms a two-dimensional cycle, which gives an element of for very large , with . This cycle could be employed as the Lefschetz thimble for , say, .

The next question is how to choose a one-dimensional subset in in the first place. It is known that if a critical orbit is “semistable”, then it has a middle-dimensional homology of rank Witten:2010cx (). In the current setup, the condition of semistability for is that it should include a point where vanishes. Since this condition is trivially met ( at, say, ), is semistable and consequently its one-dimensional homology is of rank . This implies that the choice of a cycle in is essentially unique up to homologically equivalent ones. Adopting a canonical choice suggested in Witten:2010cx (), we shall take the set of points in where . It is given by . As this cycle is the -orbit of , we shall call it a compact orbit. It is pictorially shown as a blue circle in Figure 10.

Figure 10: A schematic illustration of relations between the -orbit , the Lefschetz thimble and its dual . The blue circle (red vertical line) represents a compact orbit (non-compact orbit), respectively. The slant lines are upward(red)/downward(blue) flow lines.

The Lefschetz thimble can now be defined as a union of downward flow lines emanating from the compact orbit. Recalling that the flow respects symmetry, can be simply obtained as an -revolution of the flow line emanating from , which is nothing but considered in Section 3. We thus conclude that


Since is conserved along a flow,131313 corresponds to the angular momentum if the Morse flow (27) is viewed as the Hamiltonian flow with Hamiltonian . vanishes everywhere on .

The dual cycle can also be constructed on the basis of Witten:2010cx (). Recall that we have used a rotation by to construct a compact orbit. Now, we shall use the complementary part of to build a non-compact orbit of 141414This could be replaced with any other point on the compact orbit.: it is given by and is depicted in Figure 10 as a red vertical line. (Generally is nonzero and varies along the non-compact orbit.) Then can be defined as a union of upward flow lines emanating from the non-compact orbit; see Figure 10 for an illustration of this. By construction it is ensured that and crosses exactly once, and thus intersection numbers in (3) are well defined. This completes our determination of and in the massless NJL-like model. It is intriguing that the number of Lefschetz thimbles, , is fewer than in the GN-like model, which can be attributed to the existence of chiral symmetry in the present model.

Behaviors of the thimbles, (29) and (30), for varying can be learned from Section 3 with no additional calculation. Here is a summary:

  • and jump on the Stokes lines in Figure 4. No modification of the figure is necessary. jumps across the circular curve and jumps across the horizontal line, respectively.

  • For inside the shaded area of Figure 4,