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[MATH] modulo 2 and [MATH] for every [MATH] . Therefore, [EQUATION] This implies that the Laurent series [MATH] satisfies the following quadratic equation over [MATH] |
[EQUATION] Similarly for [MATH] , one obtains from the relations [MATH] that [EQUATION] Thus, [MATH] satisfies the quadratic equation |
[EQUATION] In the same paper as where the [MATH] –adic Littlewood Conjecture was first stated , De Mathan and Teulié established that [MATH] –LC holds for quadratic irrational power series. This is therefore in particular the case for the series [MATH] and [MATH] above. Of course, this claim extends naturally to any fi... |
It should also be noted that much more is currently known about the occurrence of windows in the Number Walls of quadratic series — see for further details the paper by Kemarsky, Paulin and Shapira |
, which uses dynamics on Bruhat–Tits tree. The connections to the present work are explained in 25 , p.5] Appendix : Implementation and Code |
It is worthwhile to mention several matters related to the implementation of the algorithms described in Figures and . The complete Magma program is contained in the file dragon_wall_B.mag available at |
. The results have been confirmed by an independent implementation in Sage, for which the program is available at ). The Wall Builder |
The wall builder described in Figure is implemented as a Magma program procedure NumberWall ( [MATH] seq, mlo, mhi, nlo, nhi, [MATH] wal) where seq and wal hold respectively the sequence [MATH] and the wall entries [MATH] and mlo and mhi (resp. nlo and nhi ) the ranges of [MATH] (resp. of [MATH] ) in the output segment... |
The natural boundary of a wall segment with, say, [MATH] and [MATH] is trapezoidal, descending from a sequence segment of length [MATH] |
at [MATH] to length [MATH] at [MATH] (see also § 5.1 for details). To avoid complicated program logic where square window frames cross the boundary, two further effectively infinite sentinel windows are attached along left and right segment edges. This permits filling a rectangular boundary with lengths roughly |
[MATH] partially containing spurious entries which can ultimately be ignored. Finally, after pruning to lengths [MATH] , the rectangle contains only valid entries. |
Counting Deficiencies in a Wall A separate Magma program procedure WallDeficiencies ( [MATH] wal, mlo, mhi, nlo, nhi, [MATH] mulset) |
computes as a multiset mulset the number of windows of each deficiency [MATH] (where [MATH] is the side length of a window). Broken windows (where the pane crosses the boundary of the segment) are represented temporarily by [MATH] Not all cases with [MATH] are recorded; the purpose is to accurately detect all visible w... |
Finding Patterns in Sequences and Tilings The tile builder described in Figure is implemented as procedure SquareTiling ( [MATH] tab, mlo, mhi, nlo, nhi, tel, cid, |
[MATH] codes, [MATH] tetrads, [MATH] stab) . Here, with the notation of Theorem 5.1 tab holds the wall [MATH] codes returns the images [MATH] of the tiles [MATH] under the 13–coding [MATH] |
stab returns the tiling [MATH] restricted to the region [MATH] tetrads returns the images under the 2–substitution [MATH] of the 2353 tiles [MATH] followed by the rest of the 2–patterns in stab that are not obtained as such images; |
tel , which stands for tile edge length , is the value taken by [MATH] , where [MATH] is the parameter used to refer to the coding [MATH] as an [MATH] –coding (thus, tel [MATH] for the coding [MATH] in Theorem 5.1 ). In practice, the parameter tel is restricted to even values; |
cid is the distance between the centers of two overlapping encoded tiles [MATH] and [MATH] in the tiling [MATH] (from Figure for instance, cid =8 in the case under consideration). In practice, the parameter cid is also restricted to even values. |
Note that the value of the parameter [MATH] when referring to the coding [MATH] as being [MATH] –consistent is then [MATH] (hence, [MATH] in Theorem 5.1 ). It is the width of overlap between two adjacent encoded tiles. |
Adjusting some Parameters As the 2–dimensional substitution [MATH] is meant to generate the Paper–Folding sequence along the zeroth row of the Number Wall, and as the Paper–Folding sequence can be obtained as a one dimensional 2–substitution followed by a coding — see Example 4.6 —, it is natural to impose that [MATH] ... |
The translation by the vector [MATH] in the definition of the coding [MATH] in \tagform@ 5.2 is due to the fact that the algorithm described in Figure is actually run in |
with a slightly different definition of the coding [MATH] defined in \tagform@ 4.6 ; namely, with the coding [EQUATION] Here, [MATH] is a tile in [MATH] and |
[EQUATION] are integers ( [MATH] denotes the integer part of a real number [MATH] ). Note that when [MATH] and [MATH] as in Theorem 5.1 [MATH] and [MATH] . Definition \tagform@ 7.1 ensures that the center of the square with side length [MATH] determined by [MATH] coincides with a point nearest to the center of the squa... |
In the case of Theorem 5.1 , the coding \tagform@ 7.1 presents the advantage of providing a tiling [MATH] which coincides exactly with the Number Wall under consideration. The theory developed in Section 4.3 with the coding [MATH] can nevertheless be seen as more robust. Indeed, on the one hand, the choice of the top–l... |
Canonical Order in a Two Dimensional Array One matter of a purely algorithmic nature is the ordering of the tiles in the set [MATH] . In Pass 1 of the algorithm described in Figure , not much importance is given to this ordering. However, in practice, imposing a certain canonical label to the elements of [MATH] is of c... |
A general method of assigning a canonical index to a given tile in [MATH] is based on the “matrix Manhattan metric” and is applicable to symmetric segments in any dimension. This method is described here in dimension two in the context of Theorem 5.1 with the notation introduced therein. |
Imposing that the parameters tel and cid introduced in Section Finding Patterns in Sequences and Tilings should be even guarantees that the squares [MATH] which, from Proposition 5.4 , appear in the tiling [MATH] , are centered at integer points in the plane (this follows from the definition of the coding [MATH] in \ta... |
Consider the portion of the tiling lying in the region defined by the conditions [EQUATION] and let there be a square [MATH] obtained as the image under [MATH] of a tile in [MATH] . Note that from the injectivity of the coding [MATH] (this is guaranteed by the algorithm in Figure — see also Section 5.2 ), the underlyin... |
[EQUATION] where [EQUATION] with [EQUATION] and where the minimum in \tagform@ 7.3 is taken over all centers [MATH] of squares identical to [MATH] lying in the region \tagform@ 7.2 |
It is elementary to establish that [MATH] is independent of the region \tagform@ 7.2 provided this region should be sufficiently large along each axis. The values assigned in \tagform@ 7.3 allow reordering the tiles according to their “earliest” appearance in [MATH] , first by least Manhattan (or [MATH] ) distance from... |
Code Design Primary design targets for the present program were robustness and portability, rather than optimum deployment of computer time or space, leading to several similar instances of deliberately inefficient consumption of resources. One is the preliminary rectangular wall created and then pruned by the wall bui... |
[EQUATION] proves adequate for this purpose. Another non–optimal step is the naive machinery selecting coordinates [MATH] of centers of encoded tiles and converting those to Magma array addresses [MATH] (which start from 1). The ensuing time penalty remains negligible, but the improved transparency is significant for m... |
Similar considerations prompt limitations on the Number Wall and tiling models employed above, mostly motivated by maintaining symmetry, but easily relaxable at the cost of some increase in complexity. As mentioned in Section using Hankel determinants instead of Toeplitz determinants in the definition of a Number Wall ... |
One case where this minimalist philosophy has been abandoned concerns the canonical ordering of tiles as described in Section Canonical Order in a Two Dimensional Array . Assigning these required extra program to build a ‘mini–wall’ of tiles, as well as an extra stage to sort the output. But the enhanced tile builder n... |
Faustin Adiceam School of Mathematics The University of Manchester Alan Turing Building Manchester, M13 9PL United Kingdom faustin.adiceam@manchester.ac.uk |
Erez Nesharim Department of Mathematics University of York Heslington York YO10 5DD United Kingdom erez.nesharim@york.ac.uk Fred Lunnon Department of Computer Science Maynooth University Co. Kildare Ireland fred@cs.may.ie |
# Source: arxiv 1806.04493 # Title: Chiral symmetry restoration at finite temperature within the Hamiltonian approach to QCD in Coulomb gauge # Sections: all # Downloaded: 2026-03-03T05:17:18.735020+00:00 |
Chiral symmetry restoration at finite temperature within the Hamiltonian approach to QCD in Coulomb gauge Abstract The chiral phase transition of the quark sector of QCD is investigated within the Hamiltonian approach in Coulomb gauge. Finite temperature [MATH] is introduced by compactifying one spatial dimension, whic... |
[MATH] Introduction Understanding the phase diagram of quantum chromodynamics (QCD) is still one of the most challenging problems in particle physics Karsch ( 2002 ); Fukushima and Hatsuda ( 2011 . Lattice calculations can shed some light on its structure for vanishing baryon density but still suffer from the so-called... |
In ref. Reinhardt and Vastag ( 2016 , the dressed Polyakov loop, the order parameter for confinement, and the chiral quark condensate, the order parameter for the spontaneous breaking of chiral symmetry, have been evaluated within this approach for vanishing chemical potential (i.e. baryon density). Thereby, finite tem... |
In the present paper, we solve the quark part of these equations numerically. Thereby, we ignore the coupling of the quarks to the (transversal) spatial gluons. This corresponds to a confining quark model – the so-called Adler–Davis model Adler and Davis ( 1984 – which was considered in refs. Davis and Matheson ( 1984 ... |
The organization of the rest of this paper is as follows: In section II , we briefly review the essential ingredients of the novel approach to finite-temperature Hamiltonian quantum field theory developed in ref. Reinhardt ( 2016 and its application to QCD in Coulomb gauge given in ref. Reinhardt and Vastag ( 2016 . Th... |
II The quark sector of finite-temperature QCD Below, we briefly discuss the main ingredients of the Hamiltonian approach to the quark sector of QCD when finite temperatures are introduced by compactifying a spatial dimension, for which we choose w.l.o.g. the 3-axis. For a more detailed description and a discussion of f... |
Let [MATH] be the QCD Hamiltonian in Coulomb and Weyl gauge on the compactified spatial manifold [MATH] , where [MATH] denotes the inverse temperature. One can then show Reinhardt ( 2016 that the grand canonical partition function at finite temperature [MATH] and chemical potential [MATH] is given by |
[EQUATION] where [MATH] is the length of the uncompactified spatial dimensions and [MATH] is the smallest eigenvalue of the pseudo-Hamiltonian |
[EQUATION] Here, [MATH] denotes the usual Dirac matrices and [MATH] is the quark field which has to fulfill the anti-periodic boundary condition |
[EQUATION] on the compactified manifold, while for the bosonic fields [MATH] the periodic condition [EQUATION] holds ( [MATH] and [MATH] are color indices in the fundamental and adjoint, respectively, representation). Furthermore, we have introduced the short-hand notation |
[EQUATION] for the spatial integration. Let us stress that the novel finite-temperature Hamiltonian approach proposed in ref. Reinhardt ( 2016 and leading to eq. ( ) is equivalent to the familiar finite-temperature (imaginary-time) approach for any [MATH] (i.e. relativistic) invariant quantum field theory. It is, howev... |
[EQUATION] (with [MATH] being the Hamiltonian on [MATH] and [MATH] being the fermionic particle-number operator) over the whole Fock space. Rather the thermal quantities like the partition function ( ) are obtained from the vacuum state on [MATH] alone (see below). Thus, the novel approach avoids introducing additional... |
for more details. This fact will be exploited in the discussion following eq. ( 27 ) below. The QCD Hamiltonian [MATH] entering eq. ( ) is given by Christ and Lee ( 1980 |
[EQUATION] where [EQUATION] is the quark single-particle Dirac Hamiltonian with [MATH] being the strong coupling constant, [MATH] the bare quark mass, [MATH] the usual Dirac matrix and [MATH] the color generator in the fundamental representation. The second term in eq. ( ) is the gluonic Yang–Mills Hamiltonian |
[EQUATION] where [MATH] is the canonical momentum operator (which agrees with the color electric field), [EQUATION] is the color magnetic field and |
[EQUATION] denotes the Faddeev–Popov determinant with [EQUATION] being the covariant derivative in the adjoint representation. Finally, |
[EQUATION] is the so-called color Coulomb interaction which contains, besides the color density [EQUATION] of quarks and gluons, the non-Abelian Coulomb kernel |
[EQUATION] From eq. ( ) it follows that all thermodynamical quantities can be obtained from the ground state [MATH] of the pseudo-Hamiltonian [MATH] which fulfills the functional Schrödinger equation [MATH] |
Reinhardt ( 2016 . Solving the functional Schrödinger equation is, thus, the aim of the Hamiltonian approach. On the compactified manifold [MATH] , this has been first tackled in ref. Heffner and Reinhardt ( 2015 for the Yang–Mills sector and was recently extended to full QCD in ref. Reinhardt and Vastag ( 2016 . There... |
In the present paper, we will give the numerical solution of the finite-temperature variational equations of motion for the quark sector and calculate the chiral condensate from it. Since the numerical cost is substantially higher for solving the full coupled equations, we will thereby neglect the coupling between quar... |
Neglecting the coupling between quarks and transverse gluons, the fermionic part of the QCD Hamiltonian reduces to [EQUATION] where [MATH] [Eq. ( )] is the free Dirac Hamiltonian and [MATH] follows from the Coulomb term ( 13 ) after substituting [MATH] [Eq. ( 14 )]. Note that this implies the cancellation of the Faddee... |
[EQUATION] which plays the role of a confining quark potential, [MATH] at [MATH] , where [MATH] is the Coulomb string tension Epple et al. 2007 |
Neglecting the coupling between quarks and transversal gluons, the ansatz for the fermionic part of the vacuum wave functional from ref. Reinhardt and Vastag ( 2016 reduces to the BCS-type functional |
[EQUATION] where [MATH] is a scalar variational kernel, [MATH] denotes the positive/negative spectral projection of the quark field [MATH] and [MATH] is the bare vacuum of the Dirac sea, fulfilling [MATH] . This type of ansatz together with the Hamiltonian ( 16 ) corresponds to the confining quark model (Adler–Davis mo... |
[EQUATION] where [MATH] is the planar momentum and [EQUATION] are the fermionic Matsubara frequencies resulting from the Fourier transformation of the (compactified) spatial component [MATH] . Furthermore, we have introduced the short-hand notation [ [MATH] |
[EQUATION] In the following, we focus on the limit of vanishing chemical potential [MATH] ) and chiral quarks ( [MATH] ). From the variational principle [MATH] one finds then the following integral equation for the variational kernel [MATH] |
[EQUATION] where [MATH] is the value of the quadratic Casimir of the color group [MATH] Reinhardt and Vastag ( 2016 and [MATH] . For the numerical solution it is, however, more convenient to rewrite the scalar kernel [MATH] in terms of the effective quark mass function |
[EQUATION] which transforms the gap equation ( 22 ) to [EQUATION] Assuming the linearly rising form [MATH] for the non-Abelian Coulomb potential 17 ), its Fourier transform in the gap equation is given by |
[EQUATION] The Coulomb string tension [MATH] entering this expression sets the overall scale in the present model. Lattice and continuum calculations |
Nakagawa et al. 2006 ); Greensite and Szczepaniak ( 2015 ); Golterman et al. 2012 ); Burgio et al. 2015 2017 favour values [MATH] in terms of the Wilson string tension [MATH] , with the rather large uncertainties coming from the extrapolation of the lattice Coulomb potential in the deep infrared. With the standard valu... |
[MATH] in the range [MATH] In the present work, we will use a standard value of [MATH] corresponding to [MATH] but we should be aware that this stipulation easily has uncertainties of up to [MATH] |
For a numerical evaluation, eq. ( 24 ) is not directly useful, since the entire calculation is dominated by the pole of the Coulomb potential for the single frequency [MATH] which – in constrast to the [MATH] equation discussed below – is not lifted by the integration measure. We thus have to introduce a small mass par... |
[EQUATION] valid for a suitable test function [MATH] . If we use this equation to replace the Matsubara sum we find, after combining terms, shifting the loop momentum [MATH] and moving the Poisson sum outermost: |
[EQUATION] For any index [MATH] , the integral under the sum is bound by the [MATH] contribution, i.e. the [MATH] limit. The zero temperature equation is, however, known to be both ultraviolet and infrared finite Reinhardt and Vastag ( 2016 ); Campagnari et al. 2016 and the same must hence hold for each integral in the... |
To close this section, we note that the ansatz ( 18 ) leads to the following expression for the chiral quark condensate: [EQUATION] |
III Numerical Method In this section, we sketch the numerical techniques necessary to solve eq. ( 27 ). To fix our notation and discuss some numerical optimiziation, we briefly revisit the [MATH] case. |
III.1 The zero temperature case revisited The zero-temperature gap equation is simply the [MATH] contribution from eq. ( 27 ). To study it numerically, we measure all dimensionfull quantities in units of the mass scale |
[EQUATION] As explained earlier, this stipulation has rather large uncertainties from the lattice calculations of [MATH] so will have all absolute numbers quoted in the present work. In the discussion, we will also present results for the quark condensate and the critical temperature, when [MATH] is adjusted to match t... |
Next, we introduce spherical coordinates and exploit the rotational symmetry of the [MATH] system to eliminate the azimuthal angle. This gives |
[EQUATION] where [EQUATION] and we have indicated that the scalar mass function can only depend on [MATH] due to spherical symmetry. The prefactors in the mass scale eq. ( 29 ) were chosen such that all clutter is removed from the Coulomb potential, which now simply reads |
[EQUATION] It is easy to see that the momentum integral in eq. ( 30 ) is ultraviolet convergent as long as [MATH] is bounded at [MATH] In the infrared, the superficial [MATH] pole in the integrand disappears after integration over [MATH] , and the equation is infrared finite as well. However, solving eq. ( 30 ) by iter... |
quotient form by collecting all pieces that contain the mass as a function of the external momentum, [EQUATION] At first sight, this seems like a very bad way to rewrite the equation, since both the numerator and denominator are now infrared divergent If we regularize the divergence by a lower cutoff [MATH] to the mome... |
[EQUATION] Iterating eq. ( 33 ) therefore produces changes to the mass function which are of order [MATH] , i.e. the small infrared regulator also limits the speed at which the iteration progresses. This is what gives eq. ( 33 ) its inherent stability: any form of the gap equation in which the integrals are infrared fi... |
As often, stability does not automatically imply efficiency: since eq. ( 33 ) makes very small progress in each step, a large number of more than 7,000 iterations is still necessary to solve it. This can, however, be cured by using sequence accelerators for the iteration, which improve convergence speed without sacrify... |
Aitken ( 1926 and Anderson’s higher degree secant method Anderson ( 1965 . Both algorithms must be vectorized , i.e. their transformation must affect the entire solution at all momenta [MATH] uniformly, because the convergence speed would otherwise differ at different [MATH] and the solution [MATH] would become progres... |
The accelerators generally buffer a certain number of iteration elements, and predict an improved estimator for the next iteration based on its history. Anderson’s method, in particular, comes with a level [MATH] that describes the dimension of the sub-space in which the univariate secant method is applied. It requires... |
below a threhsold of [MATH] , even when using overrelaxation. If we combine it with Aitken’s method, the overall iteration count is reduced to about [MATH] at the same accuracy, while Anderson’s method requires only [MATH] |
iterations to reach a residual of [MATH] . The sequence accelerator thus gives a higher final accuracy and easily saves us a factor [MATH] of CPU time in the present case. We have plotted this situation again in the left panel of Fig. , where we show the iteration history, i.e. the distance of the intermediate result a... |
The resulting solution to the [MATH] equation is shown in the right panel of Fig. . As mentioned in the introduction, this mass function originates from the instantaneous part of the quark propagator in Coulomb gauge. It cannot be compared directly to the constituent mass function in Landau gauge, and attempts to match... |
[MATH] observed here could still be compatible with the standard findings in Landau gauge Campagnari and Reinhardt ( 2018 . The mass function computed here gives rise to a chiral condensate Reinhardt and Vastag ( 2016 |
[EQUATION] if the standard scale eq. ( 29 ) corresponding to [MATH] is used. III.2 The finite temperature case: Poisson resummation |
At non-zero temperatures, the presence of the heat bath singles out a rest frame and the original spatial [MATH] symmetry is broken to [MATH] . As explained earlier, we have put the heat bath in the spatial [MATH] -direction and compactified this dimension. The use of polar coordinates [MATH] |
is, however, discouraged since the mass function would then depend on two non-compact coordinates [MATH] and [MATH] , which complicates the UV and, in particular, the IR limit considerably. A better strategy is to keep the spherical coordinates [MATH] . The remaining axial |
[MATH] symmetry of rotations about the [MATH] -axis entails that we can place the external momentum into the [MATH] -plane and set the azimuthal angle |
[MATH] . Also, the mass function must be invariant under the reflection [MATH] , as a remainder of the original [MATH] symmetry and we can take |
[MATH] without loss of generality. The mass function is hence [EQUATION] For the loop integration, we also adopt spherical coordinates |
[MATH] . The angles only enters through their cosine via the scalar product [EQUATION] where we have introduced the cosines [EQUATION] |
and defined the useful abbreviation [EQUATION] In these coordinates, the Poisson re-summed gap equation ( 27 ) in quotient form becomes |
[EQUATION] with the shifted momentum [EQUATION] If we compare this to the [MATH] version in eq. ( 30 ), it is evident that the [MATH] term in the Poisson sums reproduces the [MATH] limit, if we assume that the mass function does not depend on [MATH] because of the restored |
[MATH] symmetry. Also, the shifted momentum eq. ( 42 ) agrees with the [MATH] limit in the equation below ( 30 ) when [MATH] , i.e. when the external momentum points into the direction of the heat bath. This direction of the external momentum therefore gives the closest analogue of the [MATH] mass function, and we will... |
Eq. ( 41 ) is not yet suited for numerical investigation. As we have explained in the previous section, a small regulator [MATH] for the infrared divergence of the quotient form is required and provides for a stable iteration, |
[EQUATION] At finite temperatures, however, the infrared divergence also leads to a poor behaviour of the Poisson series, whose terms typically decay very slowly when [MATH] . It is therefore convenient to subtract an analytic helper function in the integrands of eq. ( 41 ), which will render the [MATH] -integral IR fi... |
[EQUATION] where the integrands read [EQUATION] and the subtractions are compensated by the inhomogeneities [MATH] and [MATH] . An obvious choice for the subtractions is the [MATH] limit of the integrands, |
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