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[EQUATION] Here [MATH] lives in the subspace [MATH] of [MATH] (This subspace is [MATH] for [MATH] .) The notation [MATH] means nonnegative linear span. |
The [MATH] -Cambrian fan [MATH] is the collection consisting of the cones [MATH] and their faces, for all [MATH] -sortable elements [MATH] The following theorem is the concatenation of 28 , Corollary 5.15] with Theorem 2.5 Versions of 28 , Corollary 5.15] appear as 27 , Theorem 9.1] 33 , Theorem 1.10] (see 33 , Remark ... |
Theorem 4.1 If [MATH] is acyclic with associated Coxeter element [MATH] , then [MATH] is a simplicial fan, and in particular a subfan of [MATH] |
4.2. Cambrian scattering diagrams Continuing the notation from above, we now use the sortable/Cambrian machinery to directly construct a (transposed) scattering diagram equivalent to [MATH] |
The walls of the scattering diagram will be the codimension- [MATH] faces of the [MATH] -Cambrian fan. Each such face is the intersection of two maximal faces [MATH] and [MATH] , and thus is contained in [MATH] for some root with [MATH] and [MATH] Given a [MATH] -sortable element [MATH] , by 27 , Proposition 5.2] , the... |
[EQUATION] The notation emphasizes that the information in [MATH] is equivalent to the information in the Cartan matrix [MATH] and Coxeter element [MATH] This is a scattering diagram in the transposed sense of Section 2.2 |
Example 4.2 Figure shows [MATH] for all cases with [MATH] , up to Proposition 2.4 and the symmetry of swapping the indices [MATH] and [MATH] For the associated root systems, see Figure |
Our goal is to prove the following theorem. Theorem 4.3 If [MATH] is acyclic of finite type with associated Cartan matrix [MATH] and Coxeter element [MATH] , then [MATH] is equivalent to [MATH] |
Since [MATH] is a finite collection of walls, it is a scattering diagram. Since none of the maximal cones of [MATH] intersect any hyperplane [MATH] in their interior, and since [MATH] is a finite, complete fan, each [MATH] is the union of finitely many walls of [MATH] , each with function [MATH] Up to equivalence, thes... |
First, we need some more background on the form [MATH] in the acyclic case. In this case, we write [MATH] for [MATH] , where [MATH] is the Coxeter element associated to [MATH] The point is that we can fix the Cartan matrix [MATH] while changing [MATH] as in the following lemma, which is 27 , Lemma 3.8] |
Lemma 4.4 If [MATH] is initial in [MATH] , then [MATH] for [MATH] and [MATH] We rewrite Lemma 4.4 in terms of the dual action on [MATH] as follows. |
Lemma 4.5 If [MATH] is initial in [MATH] , then for any [MATH] , the action of [MATH] sends [MATH] to [MATH] We now prove part of Theorem 4.3 |
Proposition 4.6 Every wall of [MATH] not contained in some [MATH] is outgoing. This proposition is closely related to the notion of [MATH] -alignment in 21 , Theorem 4.1] —see also 27 , Theorem 4.3] —but here, we give a simple recursive proof. |
Proof. Each wall is [MATH] with [MATH] for [MATH] and thus [MATH] , but there is another cone [MATH] sharing that wall as a facet and having [MATH] Thus it is enough to show that for every [MATH] -sortable [MATH] and every positive root [MATH] , the vector [MATH] is not in [MATH] We argue by induction on [MATH] and on ... |
Suppose [MATH] is [MATH] -sortable and [MATH] is a positive root in [MATH] Let [MATH] be initial in [MATH] First consider the case where [MATH] If [MATH] is not a positive root, then [MATH] , because [MATH] are the only roots whose sign changes under the action of [MATH] This is ruled out by hypothesis, so [MATH] is po... |
If [MATH] , then by induction on [MATH] , the vector [MATH] is not in [MATH] If on the other hand [MATH] for some [MATH] , then [MATH] because otherwise [MATH] We have [MATH] , where [MATH] is the exchange matrix defined by [MATH] and [MATH] The Cartan matrix entry [MATH] is nonzero, because otherwise [MATH] Since [MAT... |
In either case, [MATH] by ( 4.3 ). But [MATH] is equal to [MATH] by Lemma 4.5 , so [MATH] is not in [MATH] It remains to check the case where [MATH] In this case, [MATH] , so we need only consider positive roots [MATH] By induction on [MATH] , the vector [MATH] is not in [MATH] Therefore, there exists [MATH] such that ... |
The proof of the last piece of Theorem 4.3 uses a result 27 , Theorem 9.8] on the local structure of the [MATH] -Cambrian fan [MATH] to reduce to the case where [MATH] We now explain the case of 27 , Theorem 9.8] that we need. |
Suppose [MATH] is a face of [MATH] of codimension [MATH] If [MATH] is in the interior of [MATH] and [MATH] is in the relative interior of [MATH] , then as [MATH] approaches [MATH] , the vector [MATH] remains in the interior of some maximal cone [MATH] of [MATH] We call [MATH] the [MATH] -sortable element above [MATH] T... |
The cone [MATH] is contained in [MATH] , and since [MATH] and [MATH] , the intersection [MATH] is a codimension- [MATH] face of [MATH] Taking [MATH] in the relative interior of [MATH] and [MATH] as before, as [MATH] approaches [MATH] , the vector [MATH] remains in the interior of some cone [MATH] for [MATH] We call [MA... |
Given any cone [MATH] and any [MATH] , the linearization [MATH] of [MATH] at [MATH] is the set of vectors [MATH] such that [MATH] is in [MATH] for sufficiently small [MATH] For any cone [MATH] in [MATH] , we fix [MATH] in the relative interior of [MATH] and define the star of [MATH] in [MATH] to be the collection [MATH... |
[EQUATION] Here [MATH] is a Coxeter element of the parabolic subgroup [MATH] and [MATH] is the [MATH] -Cambrian fan constructed in [MATH] , which we have identified with the subspace of [MATH] spanned by [MATH] The fan [MATH] is the collection of cones [MATH] such that [MATH] is a cone in [MATH] |
Proposition 4.7 [MATH] is consistent. Proof. For the purpose of defining path-ordered products, generic paths in [MATH] are indistinguishable from sequences [MATH] of maximal cones in [MATH] with [MATH] and [MATH] adjacent for [MATH] We write [MATH] for the path-ordered product [MATH] such that [MATH] is a generic path... |
We continue the notation that was used above to define the [MATH] -sortable element above [MATH] , the element below [MATH] , etc. Since every wall crossed by [MATH] contains [MATH] , the path-ordered product [MATH] acts as the identity on [MATH] when [MATH] is in the linear span of [MATH] The span of [MATH] is complem... |
We write [MATH] for the sequence of wall functions encountered in evaluating [MATH] Let [MATH] be the sequence of wall functions encountered along a cycle about the origin in [MATH] , where [MATH] is the restriction of [MATH] to the rows and columns of [MATH] and [MATH] Then ( 4.5 ) implies that [MATH] and (when we cho... |
to see that [MATH] is consistent. In particular, the path-ordered product for the cycle in [MATH] fixes both [MATH] and [MATH] The path-ordered product on the corresponding cycle [MATH] in [MATH] differs only by replacing the functions [MATH] for [MATH] with the corresponding functions [MATH] for [MATH] The effect is t... |
This completes the proof of Theorem 4.3 Remark 4.8 When [MATH] is acyclic and of finite type, the [MATH] -Cambrian fan coincides with [MATH] (This was conjectured and partially proved in 26 , Section 10] and proved first in |
and then as 28 , Corollary 5.16] .) By Theorem 2.5 (which follows from results of ), in finite type, the transposed scattering diagram coincides with [MATH] In particular, the [MATH] -Cambrian fan and the transposed scattering diagram coincide, and Theorem 4.3 follows immediately by Theorem 2.8 Here we have chosen to m... |
However, using the [MATH] -vector fan may be useful in not-necessarily-acyclic finite type. In that setting as well, if one constructs [MATH] , then one obtains the transposed cluster scattering diagram easily by putting functions on the codimension- [MATH] faces according to Theorem 2.8 For example, the main result of |
is a construction of the [MATH] -vector fan for the oriented cycle (a non-acyclic exchange matrix of finite type D) using a root system of affine type A. In that case, putting the function [MATH] on each codimension- [MATH] face of the fan normal to the root [MATH] yields a scattering diagram. |
Alternately, still in not-necessarily-acyclic finite type, if one constructs what one thinks is the [MATH] -vector fan, then one can prove that it is indeed the [MATH] -vector fan by putting functions on the codimension- [MATH] faces according to Theorem 2.8 , then showing consistency, checking that each [MATH] is cove... |
Remark 4.9 In light of Theorem 4.3 , we can compute cluster variables (or more generally cluster monomials) in acyclic finite type as explained in Theorems 2.5 and 2.9 Thus the cluster monomial with [MATH] -vector [MATH] (encoded as usual as an element of the weight lattice) is [MATH] , as defined in Section 2.3 Altern... |
We compute [MATH] using a sequence of maximal cones of [MATH] , as in the proof of Theorem 4.3 Choose some [MATH] -sortable element [MATH] such that [MATH] We want a sequence [MATH] such that [MATH] and [MATH] are adjacent for each [MATH] and such that [MATH] , or equivalently [MATH] is the identity. Ideally, we would ... |
4.3. Shards Our construction of [MATH] made each codimension- [MATH] face of the [MATH] -Cambrian fan into a wall. Typically, there is more than one codimension- [MATH] face orthogonal to each root, so one expects that one could combine some of the codimension- [MATH] faces into a smaller number of walls. In this secti... |
The walls in this scattering diagrams will be shards —or, more specifically, the shards not removed by the [MATH] -Cambrian congruence [MATH] Shards play a role in the combinatorics, geometry, and lattice theory of the weak order on a finite Coxeter group, and play a key role in papers including |
The [MATH] -Cambrian congruence is the key player in a lattice-theoretic approach to sortable elements and Cambrian fans. Here, we will not go into details about shards and [MATH] -Cambrian congruences. Rather, we simply quote results that make the connection. |
First, by definition—see for example 23 , Section 3] —shards are codimension- [MATH] closed convex cones defined by hyperplanes [MATH] for [MATH] (In |
, the shards were relatively open, but in later references, closures were taken as part of the definition.) Second, again by definition, for each hyperplane [MATH] , the shards contained in [MATH] exactly cover [MATH] with non-intersecting relative interiors. Third, in each hyperplane [MATH] , the union of all codimens... |
Combining this information on shards and Cambrian fans with Theorem 4.3 , we have the following corollary, which shows in particular that [MATH] is equivalent to a scattering diagram with exactly one wall orthogonal to each positive root. |
Corollary 4.10 If [MATH] is acyclic of finite type with associated Cartan matrix [MATH] and Coxeter element [MATH] , then [MATH] is equivalent to [MATH] |
Corollary 4.10 makes possible explicit constructions or computations of scattering diagrams in finite acyclic type. Explicit inequalities defining shards can be derived using 18 , Lemma 3.7] Details in type A are found in |
Each hyperplane [MATH] (for [MATH] ) typically has many shards, and exactly one of them is a wall in the scattering diagram [MATH] If the decomposition of [MATH] into shards is known, we can pick out the shard [MATH] using the bilinear form [MATH] Call a shard (or a wall) [MATH] |
gregarious if the vector [MATH] is in the relative interior of [MATH] Since the relative interiors of shards in [MATH] are disjoint, there can be at most one gregarious shard in [MATH] |
Remark 4.11 The term “gregarious” alludes to the notion of an outgoing wall in a scattering diagram. Recall that a wall [MATH] is said to be outgoing if the vector [MATH] is not in [MATH] Thus in most cases, a wall containing the opposite vector [MATH] is particularly outgoing. A gregarious shard [MATH] can fail to be ... |
Proposition 4.12 If [MATH] is acyclic of finite type, then for each positive root [MATH] , the shard [MATH] is the unique gregarious shard in [MATH] |
The proof of Proposition 4.12 requires additional background on shards that we will not give here. The structure of the proof is an induction on length and rank similar to the proof of Proposition 4.6 However, since we are dealing with shards instead of walls of the [MATH] -Cambrian fan, we need 22 , Observation 4.7] ,... |
We rephrase Proposition 4.12 as a statement about scattering diagrams. Corollary 4.13 If [MATH] is acyclic of finite type, then [MATH] can be constructed entirely of gregarious walls, with exactly one wall in each hyperplane [MATH] , where [MATH] runs over all positive roots. |
For trivial reasons, the property that every wall is gregarious also holds in rank [MATH] Acknowledgments Thanks to Gregg Musiker for helpful suggestions and to Tom Sutherland for helpful answers to questions. Thanks to Dylan Rupel and Salvatore Stella for correcting errors in an earlier version. Thanks to the anonymou... |
# Source: arxiv 1806.05301 # Title: Localization of topological charge density near $T_c$ in quenched QCD with Wilson flow # Sections: all # Downloaded: 2026-03-03T05:17:16.487700+00:00 |
Localization of topological charge density near [MATH] in quenched QCD with Wilson flow Abstract We smear quenched lattice QCD ensembles with lattice volume [MATH] by using Wilson flow. Six ensembles at temperature near the critical temperature [MATH] corresponding to the critical inverse coupling [MATH] are used to in... |
[MATH] , the masses are [MATH] and [MATH] respectively, they are consistent with results from conventional methods. topological structure, localization of topological charge density, pseudoscalar glueball mass, flow, HS calorons |
Introduction Topological properties of the QCD vacuum are believed to play an important role in QCD. For example, the topological susceptibility has the famous Witten-Veneziano relation, which can explain the U(1) anomaly and the large mass of the [MATH] meson WittenVeneziano1 WittenVeneziano2 WittenVeneziano3 . The to... |
A usual way to study the topological structure is investigating the localization of topological charge density, such as BPST instantons-like localized topological lumps at zero temperature. Instanton is a semi-classical solution of the QCD Lagrangian in Euclidean space instantons1 . Isolated instantons are zero modes o... |
Since the topological structure is connected with chiral symmetry breaking and confinement, we are interested in the behavior of topological structures when the temperature is near the critical temperature [MATH] . The temperature in lattice QCD is given by: |
[EQUATION] in which [MATH] is the lattice spacing in the temporal direction, and [MATH] is the temporal lattice size. Therefore we can change [MATH] or [MATH] to vary the temperature [MATH] If we change [MATH] , because [MATH] cannot be too large the temperature will be changed coarsely. Thus we cannot get different en... |
wilsonflow1 , where [MATH] is the flow time. Recent works cooling_and_flow1 cooling_and_flow2 cooling_and_flow3 show that the gradient flow is consistent with standard cooling, therefore like using cooling we can also use the gradient flow to study topological structures. Then we can compare the topological structure o... |
In our work, we used the Harrington-Shepard (HS) caloron solutions HScaloron1 to filter the localized topological lumps, which is the generalized form of BPST instantons at finite temperature with periodic boundary condition at the temporal direction. We also used the inverse participation ratio (IPR) IPR1 |
to investigate the topological localization. The IPR is defined by: [EQUATION] in which [MATH] is the topological charge density. In this work we use the gluonic definition for [MATH] |
[EQUATION] in which [MATH] is the Levi-Civita symbol, [MATH] is the trace running over the color space, and the field tensor [MATH] is defined by: |
[EQUATION] in which [MATH] is the average of the four plaquettes on the [MATH] plane. When all topological charges focus on one lattice site [MATH] , IPR would decrease if the topological charge density becomes more delocalized. Finally it will equal to [MATH] when the topological charge density distributes uniformly. |
The topological charge density correlator(TCDC) of quenched QCD can be used to extract pseudoscalar glueball masses at zero temperature with Wilson flow gbmass_fit1 . In our work, we extracted the pseudoscalar glueball mass from TCDC at finite temperature with Wilson flow. The results are compared with those from Ref. ... |
II Locating the HS caloron-like topological lumps II.1 Find the critical inverse coupling [MATH] First, we need to find the critical temperature [MATH] . In other words we need to determine the critical inverse coupling [MATH] . We use pure gauge ensembles that have lattice size [MATH] in our work. We use the susceptib... |
[MATH] [MATH] is defined as [EQUATION] in which [MATH] is the [MATH] rotated Polyakov loop: [EQUATION] where [MATH] is the usual Polyakov loop of each configuration. |
In Table the 6 ensembles we used to find [MATH] are listed. The lattice size is [MATH] . We expect that the finite volume effects are negligible. The lattice spacing [MATH] is found by using Wilson_Scale |
[EQUATION] where [MATH] is set to be [MATH] from Ref. Sommer_R0 . Obviously Table shows that [MATH] is near [MATH] . The critical inverse coupling [MATH] is obtained by interpolating to the location where [MATH] is maximum. We use a B-spline interpolation and obtain [MATH] which is compatible with [MATH] in Ref. critic... |
II.2 HS caloron-like topological lumps In this paper we use the HS caloron solutions to filter the localized topological charge density lumps. The localized topological lumps are defined by sites that have maximum absolute value of |
[MATH] in a [MATH] hypercube centered at site [MATH] . The center [MATH] is also mentioned as peak. After applying the HS caloron filters in the following, we can get calorons-like topological lumps. |
In SU(2) gauge theory at temperature [MATH] , HS caloron solution of gauge field [MATH] has the exact form as HScaloron1 [EQUATION] |
where [MATH] is the center of a HS caloron, [MATH] is the size of a HS caloron. It satisfies the (anti-)self-dual condition [MATH] |
[MATH] is the ’t Hooft symbol: [EQUATION] When the temperature [MATH] , it approaches the BPST instanton solution [MATH] instantons1 . Similar things happen when we constrain our study at the region [MATH] . Therefore when we use the center and its 8 closest neighbour sites on the lattice to filter the topological lump... |
[EQUATION] where [MATH] represents the color rotations embedding the SU(2) BPST instantons into SU(3). The topological charge density near the center of an isolated instanton approximates |
[EQUATION] where the ” [MATH] ” sign is for instanton, ” [MATH] ” for anti-instanton. Then at the center [EQUATION] Therefore we can get the relation |
[EQUATION] In this paper we use the peak and the 8 closest neighbour sites on the lattice to fit Eq. ( 13 to get the size [MATH] |
Like in Ref. Find_instanton , we also use 3 filter conditions to find HS caloron-like topological lumps: [EQUATION] which comes from Eq. ( 12 ). |
[EQUATION] where the normalized action density [MATH] , the normalization factor [MATH] comes from the action of a single HS caloron [MATH] with |
[MATH] To avoid double countings of two peaks of a single but distorted HS caloron, we filter peak [MATH] by [EQUATION] The topological lump centering at [MATH] will be filtered. |
III Localization of topological charge density We use the HS calorons filter conditions and IPR to investigate the localization of topological charge density. Ensembles in Table would be used every ten configurations, which means that every ensemble includes 200 configurations and each configuration is separated by 100... |
[MATH] . But we have used parameters varied in the regions [MATH] . These results are consistent with the discussion in the following. We choose [MATH] |
since the results are stable around them. The gradient flow we used is of Wilson action, which means that we use Wilson flow to smear the gauge fields. The effective smearing radius [MATH] runs from 0.3fm to 0.9fm. |
In Fig. , we present the topological charges [MATH] of ten configurations versus Wilson flow in every ensemble, the topological charges [MATH] of the original configurations have also been presented. Obviously when [MATH] runs from 0.3fm to 0.9fm, the topological charges |
[MATH] approach to integers. At the same time the topological charges [MATH] don’t drop down to the value zero. Therefore the long-ranged topological structures should be preserved during the Wilson flow. |
III.1 Investigating the HS caloron-like topological lumps In Fig. , we show the three quantities of HS caloron-like topological lumps versus [MATH] : the average density [MATH] , the average size |
[MATH] and [MATH] , which is the average absolute value of topological charge density on the peak. The three quantities with different effective smearing radius are marked with different colors or shapes. |
With the increase of the effective smearing radius [MATH] , the average density [MATH] decreases monotonically, the average size [MATH] grows monotonically. Unlike |
[MATH] and [MATH] [MATH] of the ensembles at higher temperatures decreases at first, then becomes to increase instead as [MATH] increases. |
The phenomena that [MATH] decreases monotonically and [MATH] grows monotonically can be expected. Since with the increase of [MATH] , more and more small topological lumps would be smoothed out. |
When [MATH] is large, we find that the three quantities of HS caloron-like topological lumps are consistent at [MATH] and [MATH] . It indicates that the localization of topological charge density is stable. When [MATH] , we find that the three quantities change significantly as the the temperature increases. It means t... |
Since when [MATH] is small, the short-ranged fluctuations may not be suppressed enough, we needn’t pay much attention to the behaviors of the three quantities of the HS caloron-like topological lumps at small [MATH] |
The decrease of the average density [MATH] when [MATH] means that the topological excitation is suppressed. It may explain why the topological susceptibility starts to drop down near [MATH] |
Xiong-guangyi001 Noting that [MATH] , the average volume occupied by one HS caloron-like topological lump, is always close to [MATH] , the average volume of the HS caloron-like topological lumps. It means that the HS caloron-like topological lumps are not sparse but dense. |
Since the chiral condensate [MATH] Instantons1996 , the decrease of [MATH] and the increase of [MATH] as the temperature increases at [MATH] indicate that the absolute value of chiral condensate will drop down as the temperature rises. It is consistent with the fact that the chiral symmetry will restore at high tempera... |
IPR has also been used to study the localization of [MATH] , and conclusions from both methods are consistent. III.2 Average IPR versus [MATH] with Wilson flow |
In Fig. we show the average inverse participation ratio [MATH] versus [MATH] with Wilson flow. Theoretically, when a certain structure is embedded in a finite [MATH] space discretized by lattice spacing [MATH] , the IPR of the structure obeys [MATH] as [MATH] |
IPR1 , where [MATH] denotes the dimension of the structure. But the dependence of IPR on the volume of the finite [MATH] space is small IPR1 . However, when we use gradient flow to smear the configurations in a space discretized with different lattice spacings, the average IPR of [MATH] with same [MATH] would be almost... |
among different temperatures can’t result from the lattice discretization with different lattice spacings. The manifest differences can only result from the different localizations of topological charge density at different temperatures. |
In Fig. we find that when [MATH] is large, [MATH] increases as [MATH] increases when [MATH] . It is just the same transition point that we found in Sect. III.1 . Obviously, this behaviour of [MATH] should come from the fact that the topological localization was enhanced by the increase of temperature. The ensembles at ... |
[MATH] compatible for all used [MATH] . It means that the localization of [MATH] hasn’t changed yet when [MATH] , just like the behaviours of the three quantities of HS caloron-like topological lumps in Fig. |
By using the two different methods, we get the conclusion that the localization of topological charge density near [MATH] doesn’t change when [MATH] , and starts to change significantly when [MATH] |
IV Extracting the pseudoscalar glueball mass from the TCDC at high temperature The topological charge density correlator (TCDC) is defined by |
[EQUATION] In the negative tail region of the TCDC, it can be approximated by the pseudoscalar propagator Cqq_mass [EQUATION] where [MATH] is the modified Bessel function, it has the asymptotic form as |
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