# Extracting topological information from momentum space propagators

## Abstract

A new topological invariant quantity, sensitive to the analytic structure of both fermionic and bosonic propagators, is proposed. The gauge invariance of our construct is guaranteed for at least small gauge transformations. A generalization compatible with the presence of complex poles is introduced and applied to the classification of propagators typically emerging from non-perturbative considerations. We present partial evidence that the topological number can be used to detect chiral symmetry breaking or deconfinement.

## 1 Introduction

Topology has been used since a long time in the study of condensed matter physics and, nowadays, it has become the main theoretical tool for the description and classification of topological insulators (see [1] for a comprehensive review). This is because it is possible to construct topological invariant quantities, i.e. , quantities that are invariant under “smooth deformations” (homeomorphisms). In this manner the space of parameter-dependent results is naturally divided into disjoint sectors, invariant with respect to homeomorphisms, that are characterized by different values for the topological invariants. Also in a particle physics, or more general, (quantum) field theory context, the importance of topology cannot be underestimated; see [2] for the typical illustrative examples.

From the physical point of view, different sectors correspond to different physical phases. Similarly, momentum space topology (MST) has been applied to the classification of the ground state of relativistic quantum field theories (see [3], where MST is first applied to lattice fermions) into universality classes, however, without the same resonance as in condensed matter [4, 5, 6, 7, 8, 9]. The MST framework might be a very powerful tool and has also been applied to the investigation of emergent gravity, [10, 11]. Indeed relativistic quantum fields share some topological properties with topological materials [4]. The Standard Model (SM) below the electroweak scale, for example, belongs to the same universality class as of the superfluid \(^{3}\)He and of the three-dimensional topological insulators obeying time-reversal symmetry [5]. However, up to now, topological invariants constructed in MST have not been applied to bosonic systems. In a sense, this could be interpreted as one of the major drawbacks of MST, which only applies to “half of the world”, for reasons that will become clear in the next sections. On the contrary, it would be desirable to apply MST to bosonic fields, especially considering that the analytic structure in momentum space of bosonic propagators might encode extremely valuable physical information.

Inspired by the power of MST in the classification of the vacuum of quantum field theories, we propose a new topologically invariant object that is not only sensitive to the full analytic structure of fermionic propagators, but that can also be applied to bosonic field propagators. Particularly, we will consider here the case of certain classes of fits to lattice fermion propagators [12, 13, 14, 15] and gluon propagators [16, 17, 18, 19], although the proposed tool can be applied to any relativistic fermionic or bosonic propagator. Hence, the present results give rise to a considerable enhancement of the power, as well as applicability, of MST. The paper is organized as follows: in Sect. 2 we recall the definition and properties of the momentum space topological invariant. In Sect. 3 we extract, departing from the standard definition, a specific representation for the MST invariant that is very useful for a generalization to the bosonic sector. In Sect. 4, we generalize this definition to make it compatible with the potential presence of complex poles and we apply this latest definition to several concrete propagators in Sect. 5. Finally, in Sect. 6, our conclusions can be found.

## 2 The momentum space topological invariant

*K*stands for the matrix representation of a vacuum symmetry;

*G*is the two-point Green function of the fermion field; the integral is on a three/dimensional hypersurface \(\Sigma \), in our case defined by imposing \(p_{0}=0\). From now on, we assume the following notation: \(p^{2}=p_{i}p^{i}\) and Open image in new window, with \(i=1,2\) and 3 (the spatial indices); finally,

*N*is a normalization factor. Notice that the foregoing implies that \([K,G]=0\). A rather similar, albeit not exactly the same, topological invariant was also studied in the lattice context of [22].

At first glance, the topological invariant defined above can be constructed only for fermion propagators, since the \(\gamma \)-matrices are indispensable to get a non-trivial result. A direct computation reveals that if one tries to make sense of the above expression for bosonic propagators with internal indices, the result vanishes identically while for fermions the \(\gamma \)-matrices save the day by generating the completely antisymmetric tensor.

Our main goal now for the following sections is to propose a new, and more general, topological invariant that can also be applied to bosonic fields.

*G*, can be written as

*G*must fulfill so that \(\mathcal {N}_{3}\) is invariant under small (continuous) deformations of

*G*. In Sect. 3, we shall derive a more specific condition for a particular, but general enough, class of fermionic propagators. These have been used to fit lattice data for the quark propagator in QCD; see [12, 13, 14, 15].

A priori, one might question the physical relevance of the topological invariants discussed here as they are explicitly based on gauge variant input, viz. propagators that will depend on a chosen gauge. In practice, this will boil down to a possible dependence on a gauge parameter. Indeed, changing the gauge parameter can always be achieved via consecutive application of small (infinitesimal) gauge variations (or BRST variation, if you wish), which is just a special class of continuous deformations leaving the topological number untouched. We will have nothing to say about large gauge transformations and invariance of \({\mathcal {N}}_3\) w.r.t. those.

## 3 Topological invariant for fermionic case: old and new results

*SU*(

*N*) gauge theory, with most general parameterization (assuming Lorentz invariance of course):where

*AB*are internal Dirac fermion indices.

^{1}It can be shown that the boundary condition (2) is recovered if [8]

Assuming the above-mentioned asymptotic behavior of \(Z(p^{2})\) and \(\mathcal {M}(p^{2})\), one can verify directly that propagators of the kind (3) fulfil condition (4).

*K*will be considered as being the one that accounts for the CT-symmetry, representable by \(\gamma ^{5}\gamma ^{0}\) (the four-dimensional Euclidean Dirac matrices). A nice overview of CPT symmetry can be found in [30]. Notice that \(\gamma _{5}\gamma _{0}\) does commute with the general Dirac fermion propagator given at Eq. (3). Then, since we already showed that \({\mathcal {N}}_3\) is a topological invariant, we can set \(Z(p^2)=1\), as this \(Z(p^2)\) for \(p^2>0\) is a smooth deformation of 1. One can again appreciate this from non-perturbative functional or lattice computations; see e.g. [12, 14, 24, 25, 26, 27, 28, 29]. Therefore, after some algebraic manipulations, we can reduce

*phase boundaries*[31]. One concrete example of such phase boundary is shown in Fig. 1, where the two-dimensional surface is characterized by the equation \(M^{3}+m^{2}\mu =0\) on the three-dimensional space of parameters.

^{2}On such surface the topological invariant is \(\mathcal {N}_{3}=0\). Notice that this is also the topological invariant’s value in the chirally invariant case of a simple massless Dirac quark propagator. As such, the topological number \(\mathcal {N}_3\) can discriminate between chirally symmetric and chirally broken phases.

More general, if the parameters take (critical) values such that the denominator of the integrand in Eq. (6) has singular behavior for positive values of \(p^{2}\), there will be a transition \(\Delta \mathcal {N}_{3}\ne 0\) of the topological invariant for a small variation of the critical parameters. As \(\mathcal {N}_{3}\) is an integer number, this transition must come with another integer value for the topological number.

## 4 A generalized topological invariant

^{3}Such an extension is a topological invariant, applicable either to fermionic or to bosonic systems.

Despite the similarity, it is not clear to us if both objects encode the same physical information. Furthermore, notice that the integrand of Eq. (10) has at least two branch points, 0 and \(\infty \), but depending on the analytical structure of \({\mathcal {M}}(z)\), the number of non-removable singularities can increase. Likewise, it seems that also the quantity defined in [37] is susceptible to the appearance of new branch points, regarding the analytical expression of the propagator.

*f*(

*z*), indeed,

*f*(

*z*) given at Eq. (16), and as such it represents the difference between the number of zeros and poles of

*f*(

*z*) as the latter function is per assumption meromorphic within the contour \(\Gamma \) and has no poles or zeros on the contour \(\Gamma \).

### 4.1 Invariance of \(\mathcal {N}_{\Gamma }\) under small deformations of the parameters

Now, in order to derive the boundary condition for \({\mathcal {N}}_{\Gamma }\) to be invariant under smooth variations of \({\mathcal {M}}\) (with respect to the parameters of \({\mathcal {M}}\)), we will rely on two procedures. In this Subsection, we use general complex analysis results, while the second argument, *cf*. Appendix A, makes direct use of expressing Eq. (10) through the function \(\theta (z)\) defined in Eq. (16).

^{4}

*z*; we did not write it explicitly to avoid notational clutter. In order to have something proportional to \(\delta \mathcal {M}\), we need to integrate by parts the term with \(\partial _{z}\delta \mathcal {M}\). This is achieved via

### Lemma 4.1

Let *F*(*z*) be a complex function with a pole at \(z=a\) with finite multiplicity *m* and analytic in the punctured disc \(D'(a,r)=\{z\in \mathbb {C},\;0<|z-a|<r\}\).

Then \(F'(z)\) has a pole with multiplicity \((m+1)\) and zero residue at \(z=a\).

### Proof

*m*, we may consider the Laurent series of

*F*(

*z*) in \(D'(a,r)\) (see for instance [38])

*F*(

*z*) at \(z=a\). Therefore,

## 5 Applications

In this Section we will apply our complex topological object (10) on both fermionic and gluonic sectors as a matter of exercise, but, being the form of the propagator for Dirac fermions at finite temperature more complicated because of the absence of Lorentz invariance, the Dirac fermions will be treated at zero temperature while the gluonic case, only, will be treated at finite temperature.

### 5.1 A Dirac quark propagator that fits lattice data

For general values of the parameters \((M^{3},m^{2},\mu )\), we find \({\mathcal {N}}_{\Gamma } = -2\) and \({\mathcal {N}}_{\Gamma } = +2\). More precisely, \({\mathcal {N}}_{\Gamma }=-2\) was associated to a negative mass at zero momentum, and also with a massless quark at zero momentum evolving to a strictly negative mass. It is possible to find combinations of the parameters so that the mass function is divergent at some point. In these cases, if the mass starts from negative values (considering \(p^{2}\) from zero to positive values), \({\mathcal {N}}_{\Gamma }\) also acquires the value \(-2\). On the other hand, we verified that \({\mathcal {N}}_{\Gamma }=+2\) is associated to configurations of the parameters such that the dynamical mass is positive at zero momentum, and also if it is a massless quark at zero momentum with an increasing positive mass. For example, for the specific values of \((M^{3},m^{2},\mu )\) that makes the quark propagator fit to lattice data, [12, 13], our topological invariant acquires the positive value \(+2\).

### 5.2 A gluon propagator that fits lattice data

In order to explicitly show that our generalized topological object (10) can in fact be applied to the topological analysis of bosonic relativistic quantum fields within the MST framework, we perform here the topological classification of the space defined by the two-point Green function of gluons, whose analytic expression is one that has been used in the literature to fit lattice data at zero and finite temperature, according to [13, 19, 39, 40]. Such an analytic expression can be derived by means of the refined Gribov–Zwanziger framework; see e.g. [41, 42].

As mentioned before, since our proposed \({\mathcal {N}}_{\Gamma }\) is constructed in the complex plane and explicitly depends on the expression of the (analytic continuation of the) mass function of the propagator \({\mathcal {M}}(p^{2})\), it is quite possible that non-removable singularities appear, others than 0 and \(\infty \), as in the case of the present example. Given the expression of the complex mass function \({\mathcal {M}}(z)\), at Eq. (21), one can check that the integrand of Eq. (10) has also the new branch points \(-\frac{1}{d}\) and \(\frac{-b^2 d - d r2^2}{-1 + 2 d r^2}\), which are real and, therefore, lie outside the region bounded by \(\Gamma \).

In this example we restrict ourselves to the finite temperature results of [39]. The values of (*r*, *b*, *d*, *c*) are taken from their Table II. Namely, the authors of [39] assessed the following values of the temperature (in unities of the critical temperature): \(T/T_{c} = 0.65\); \(T/T_{c} = 0.74\); \(T/T_{c} = 0.86\); \(T/T_{c} = 0.99\); \(T/T_{c} = 1.20\); \(T/T_{c} = 1.48\); \(T/T_{c} = 1.98\); and \(T/T_{c} = 2.97\). For all the temperature values with the exception of the highest three, they set \(b=0.0\). Notice that for \(T/T_{c} = 1.20\) (clearly above the critical temperature) the parameter *b* is also set to zero.

For the values reported in [39], our topological invariant acquires the following values: \({\mathcal {N}}_{\Gamma } = 0\) whenever \(b=0\), since the propagator then displays only a (negative) real (double) pole. Such configuration appears for temperatures \(T/T_{c} = 0.65\); \(T/T_{c} = 0.74\); \(T/T_{c} = 0.86\); \(T/T_{c} = 0.99\); \(T/T_{c} = 1.20\). On the other hand, we find \({\mathcal {N}}_{\Gamma } = +2\) for the three highest temperatures \(T/T_{c} = 1.48\); \(T/T_{c} = 1.98\); and \(T/T_{c} = 2.97\) (where \(b \ne 0\)). Therefore, these results suggest a phase transition at a critical temperature somewhere between \(T/T_{c} = 1.20\) and \(T/T_{c} = 1.48\), different from the deconfinement phase transition found by the authors of [39].

Interestingly, the authors of [37] found \(N_W = 0\) for gauge propagators displaying only real poles; and \(N_W = -2\) for gauge propagators with a set of complex conjugate poles. Here, again, we point out that our topological invariant assumes the same absolute values as the one of [37], despite that they do not necessarily represent the same topological quantity. Therefore, following a completely different approach from the one in [37] (although both are aimed to the momentum space topological analysis of the gauge field two-point Green’s function), we could find a phase transition between different regimes of the gauge propagator, in agreement with [37].

## 6 Conclusions

We developed a topological classification of relativistic quantum systems within the momentum space topology framework, starting with the usual topological invariant \({\mathcal {N}}_{3}\) as applied to Dirac fermions. This topological invariant can experience integer jumps when one crosses different open regions, which are limited by *phase boundaries*. This change can be related to the non-analyticities appearing in the integrand of (6).

Afterwards, we proposed a new topological invariant, \({\mathcal {N}}_{\Gamma }\), that is sensitive to the existence of complex poles of the propagator of the relativistic quantum field, be it fermion or boson. This topological invariant was constructed by performing the analytic continuation \(p^{2} \rightarrow z\) to the complex plane of the usual \({\mathcal {N}}_{3}\). As a result, \({\mathcal {N}}_{\Gamma }\) depends on the closed contour \(\Gamma \) and on the analytic expression of \({\mathcal {M}}(z)\). In order to ensure the invariance of \({\mathcal {N}}_{\Gamma }\) with respect to smooth variations of \({\mathcal {M}}(z)\), we found that \({\mathcal {M}}(z)\) and \(\delta {\mathcal {M}}(z)\) must be constant in the limit \(|z| \rightarrow \infty \). We applied this topological number \({\mathcal {N}}_{\Gamma }\) to a model Dirac quark propagator whose analytic expression can fit quite well corresponding lattice data. In addition, we also applied it to a gluon propagator form that can fit zero and finite temperature lattice data. We made use of the rational function fits according to [39], and we could find a phase transition at a temperature within the range \(T/T_{c} = 1.20\) and \(T/T_{c} = 1.48\), which is slightly above the thermodynamic critical (deconfinement) temperature found by the authors.

While eq. (1) has the clear physical meaning of being related to a non-dissipative conductivity (Hall, spin Hall, etc, *cf*. [43] and [44]) at this stage the physical interpretation of our invariant is less obvious. What is clear is that it is sensitive to the presence of complex conjugate poles thus, as shown in refs. [37] and [45], it “measures” the possible occurrence of a violation of the “reflection positivity condition” (the Euclidean counterpart to the positive definiteness of the norm in the Hilbert space of the corresponding Wightman quantum field theory, see the Osterwalder–Schrader axioms [46, 47]) in terms of the Schwinger function. Violation of the reflection positivity condition is regarded as a signal for gluon confinement (see Section VI of [37] and Section V and VI of [45], as well as [48]).

For these reasons we believe that our topological invariant proposed in Eq. (10) may be interpreted as an indication of whether the quantum system is in a confined or deconfined phase. However, for the moment, this is only an *intuition* without a rigorous analytic proof, which deserves further investigation.

For future work, it would be interesting to use improved fits to more recent finite temperature gluon data, to see if the phase transition coincides with the deconfinement transition. That such is possible can be inferred from the preliminary fitting report of [40], where in contrast to the here used numbers of [39], there are complex poles below (up to \(T=0\)) and around \(T_c\). More takes on the analytic structure of gluon and/or quark propagators can be found in *e*.*g*. [45, 53, 54, 55, 56, 57, 58, 59, 60]. Although these rational function fits never capture the renormalization group-controlled logarithmic tails, we explained how our construct is insensitive to these structures anyhow.

In addition, it would also be rather interesting to follow the temperature-evolution, if any, of the quark topological number based on the rather recent lattice output of [32], to see if one can find a topological signal of the deconfinement and/or chiral transition in the quark sector. We hope to come back to these issues in the foreseeable future.

## Footnotes

- 1.
Usually, fermions belong to the fundamental representation of the internal

*SU*(*N*) group. - 2.
One can check that this condition translates into the Green function having a zero mass pole.

- 3.
Evidently, this is a big assumption. For example, a Monte Carlo-based lattice computation will never give direct access to the full analytic structure in the complex momentum squared plane. But using dedicated inversion schemes can shed light on this anyhow; see e.g. [35], in some cases also (numerical) solutions over the complex momentum squared plane of the Dyson–Schwinger equations [35, 36] can be obtained.

- 4.
The contour \(\Gamma \) itself depends on the integrand, in the sense that \(\Gamma \) is chosen in such way that it does not encircle or pierce any cuts or has poles on it. As before, this means that the renormalization group related logarithms affecting the integrand are analytic within \(\Gamma \) and as such can be considered as smooth deformations. Stated otherwise, these logarithmic terms can again be neglected being irrelevant for determining the topological number \({\mathcal {N}}_\Gamma \).

## Notes

### Acknowledgements

P.P. , D.D. and L.R. acknowledge the warm hospitality of the Centro de Estudios Científicos (CECs) in Valdivia, Chile, during different stages of this work. P. P. is supported by the project High Field Initiative (CZ.02.1.01/0.0/0.0/15_003/0000449) from the European Regional Development Fund. I. F. J. is supported by Fondecyt Grant No. 3170278. This work was partially supported by the research Grant Number 2017W4HA7S “NAT-NET: Neutrino and Astroparticle Theory Network” under the program PRIN 2017 funded by the Italian Ministero dell’ Istruzione, dell’ Università e della Ricerca (MIUR) and also by the Initiative “TAsP: Theoretical Astroparticle Physics” funded by the Italian Istituto Nazionale di Fisica Nucleare (INFN). This work has been funded by the Fondecyt Grant 1160137. The Centro de Estudios Científicos (CECs) is funded by the Chilean Government through the Centers of Excellence Base Financing Program of Conicyt.

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