Comment on "The phase diagram of the multi-matrix model with ABAB-interaction from functional renormalization"
CComment on “The phase diagram of the multi-matrixmodel with ABAB-interaction from functionalrenormalization”
Carlos I. Perez-Sanchez
Faculty of Physics, University of Warsawul. Pasteura 5, 02-093 Warsaw, PolandEuropean Union
E-mail: [email protected]
Abstract:
Recently, [1, JHEP
131 (2020)] obtained (a similar, scaled version of) the( a, b )-phase diagram derived from the Kazakov–Zinn-Justin solution of the Hermitian two-matrix model with interactionsTr ! a p A ` B q ` b ABAB ) , starting from Functional Renormalization. We comment on something unexpected: thephase diagram of [1] is based on a β b -function that does not have the one-loop structureof the Wetterich-Morris Equation. This raises the question of how to reproduce the phasediagram from a set of β -functions that is, in its totality, consistent with Functional Renor-malization. A non-minimalist, yet simple truncation that could lead to the phase diagramis provided. Additionally, we identify the ensemble for which the result of op. cit. wouldbe entirely correct. a r X i v : . [ h e p - t h ] F e b ontents We prove the following: For a Hermitian two-matrix model including the “
ABAB -interaction” b p ABAB q “
AB AB the only one-loop, one-particle irreducible (1PI) diagram of order b that has connectedexternal leg structure—that is, such that it contributes to a connected-boundary correlationfunction—is (1.1)Its external leg structure is the cyclic word ABBA , represented by
AA BB . Thisimplies that the β b -function given in [1, Eq. 3.41], namely β b “ p η A ` η B ` q ˆ b ´ “ p ´ η A q ` p ´ η B q ‰ ˆ b loooooooooooooooooomoooooooooooooooooon ñ coeff. of b must be 0 , (1.2)does not have the one-loop structure of the Wetterich-Morris equation in Functional Renor-malization (this is relevant, since the phase diagram [1, Fig. 1] relies on β b ). For thequadratic term b to be present in this equation, the graph (1.1) would need to have anexternal ABAB -structure. In other words, for eq. (1.2) to hold, after “filling the loop” Here η A and η B are the anomalous dimensions for the matrices A and B , but this is irrelevant, sincethe coefficient of b should identically vanish. – 1 –n (1.1) and shrinking the disk to a point, the remaining graph should read, just like thevertex, AB AB
Different infrared regulators might lead to different coefficients (containing non-perturbativeinformation), but the one-loop structure in Functional Renormalization should be evident;this means that the coefficient of b in β b must vanish.Next, in Section 2, these statements are presented in detail in Claims 2.1, 2.2 and2.3 (at the risk of being redundant) and proven. In Section 3 we further propose a moregenerous truncation to obtain the phase diagram. We compute in the large- N limit withoutfurther notice. Feynman graphs turn out to be quite useful also for “non-perturbative [2] renormalization”.In the case of (multi-)matrix models, Feynman graphs are (edge-colored) ribbon graphs orfat graphs. The two-matrix model in question is Z “ C N ij H N ˆ H N e ´ N p Tr A ` Tr B q´ NS int p A,B q d A d B (2.1a)with C N a normalization constant and d A and d B both the Lebesgue measure on the space H N of Hermitian N ˆ N matrices. The representation of the interaction S int p A, B q “ ´ a p A ` B q ´ b p ABAB q (2.1b)in terms of ribbon vertices reads : ´ S int p A, B q “
AA AA ` BB BB ` AB AB . (2.2)Since the above graphical representation will be used only as a cross-check, we ignore thesymmetry factors (strictly, we should put a root on an edge of each interaction vertex) andalso absorb the couplings in the vertices. This representation, due to ’t Hooft [3] (see alsoits relation to discrete surfaces or maps studied by Brezin-Itzykson-Parisi-Zuber [4]), iswell-known to be of paramount importance both in physics and mathematics; applicationsare also worth mentioning [5, 6]. The theory around ribbon graphs can be formulatedin extremely precise way [7], but in this comment it suffices to notice that these verticeshave a cyclic orientation—typically denoted by a disc or a planar neighborhood. This way, We respect here the notation of [1], the exception being renaming their p α, β q to p a, b q as to avoid “ β β ”. In case of color-blindness, light (green) and dark (red) represent A and B , respectively. – 2 –ibbon graphs, unlike ordinary ones, “detect” non-cyclic reorderings of the edges (see e.g.[8, Fig. 1 and eq. (18)]). This means that AB AB ‰ AA BB , which faithfully represents the obvious: Tr p ABAB q ”{ Tr p ABBA q . (2.3)In 1999, the model (2.1) was exactly solved by Kazakov and (P.) Zinn-Justin, whopresented in [9, Fig. 4] a phase diagram of right-angled trapezoidal form for the couplings( a, b ), called there ( α, β ) as well as in [1]. A similar phase diagram with trapezoidal form (i.e. predicting « {
10 for both critical exact a ‹ “ b ‹ “ { π values) is the result presentedin [1], who addressed the model (2.1) using Functional Renormalization.The phase diagram [1, Fig. 1] obtained from p β a , β b q follows from a correct expressionfor β a but also from β b “ ` η A ` η B ` ˘ b ´ rp ´ η A q ` p ´ η B qs b , which is [1, Eq. 3.41].We prove here that the β b -function is incompatible with the well-known one-loop structure[2] of Wetterich-Morris Functional Renormalization Group Equation [11, 12]. While thebehavior β g „ g ` . . . is indeed commonly found for other quartic operators g O , this doesnot happen for the coupling b .Next, some terminology. A face of a ribbon graph is said to be unbroken if no (uncon-tracted) half-edges are incident to it; the face is otherwise broken. Thus, vacuum graphscan only have unbroken faces. The external leg structure of a ribbon graph is the cyclicword(s), in the random matrices, read off from the uncontracted half-edges going along theboundary-loop of a broken face, respecting their order. This process is known in the matrixfield theory literature [13] (illustrated in [14, Sec. 5]) and a generalization to multi-matrixmodels requires to additionally list the half-edges respecting the coloring .A concrete example is the following: from the broken face inside the red loop of thegraph (2.4)one gains A ; the same from the outer face. Thus, the external leg structure of (2.4) is thedisjoint union of A with A (the general idea is depicted in Fig. 1). In contrast, the face The statement follows from the form of the Wetterich-Morris equation and appears e.g. in [1, Sec. 3.1§1] “As the full propagator enters, non-perturbative physics is captured, despite the one-loop structure”but also in almost any introduction to Functional Renormalization, e.g. [10]. Additionally, once an orientation is fixed, the words at the two boundaries should be read in oppositecyclic order, but here we will not find graphs for which this would be needed. The propagators are shaded and they respect the color of the associated matrix. These are usually operators; for this concrete example Tr p A q ˆ Tr p A q . In general the disconnectedunion of the words leads to multiple traces. Here, however, we are interested only in the combinatorial classof these invariants. – 3 – Figure 1 : The gray part is an abstract representation of a multi-matrix model planar graph(after “vulcanizing” the unbroken faces and forgetting everything but boundaries) with twofaces broken by uncontracted half-edges, represented by white circles. These boundariesdetermine the external leg structure decorating the cylinder. The 4-leg boundary in theupper right corner yields the cyclic word
ABBA (or
AABB , or . . . ); the 6-legged one is
AABABA (or
AAABAB, . . . ). The external leg structure is the disjoint union of thesewords.inside the loop of the next graph G : “ (2.5)is unbroken. Gluing a disk to the loop of G and shrinking it one gets: Claim 2.1.
The external leg structure of the graph G given by (2.5) is ABBA , i.e.
AA BB
Claim 2.2.
The unique one-loop 1PI graph of order b having a connected external legstructure is the graph G defined by (2.5). Proof.
By assumption, the graph has two interaction
ABAB -vertices; let us name V and V the two copies. The 1PI-assumption constrains the p propagators implied in the graphto p ą
1, while the one-loop condition implies p ă
3. Since the graph in question is 1PI,the p “ V with V . If they connect two equalcolors, we get a disconnected external leg structure, i.e. either the graph (2.4) or its A Ø B (i.e. green Ø red) version. Therefore, by assumption, the two propagators connecting V with V must have different color. The only such graph having also a connected externalleg structure is G defined by (2.5). Claim 2.3.
For the particular operator ´ b Tr p ABAB q , the quadratic term b in β b isimpossible in Functional Renormalization. – 4 – roof. Working with the effective action imposes the 1PI condition from the onset. Then,Wetterich-Morris Equation imposes the one-loop structure on any (non-vertex, i.e. non-linear) term appearing in each β -function. For the β b -function, in particular, a secondcondition is the (cyclic) external leg structure being ABAB . These two conditions aremutually exclusive. Indeed, by Claim 2.2, the only such O p b q graph is G , which by Claim2.1 has an external leg structure ABBA . Remark . Some closely related, but not essential points:• The correlation functions of matrix [13] and tensor field theories are indexed by boundarygraphs [15]. The terminology makes sense graph theoretically but also geometrically inboth the matrix and tensor field [14] contexts. In the case of matrix models, boundarygraphs coincide with what we call here external leg structure . In matrix models, thereare as many β -functions as correlation functions, hence the importance of the external legstructure. The map defined by “taking the boundary” seems also to play a role in otherrenormalization theories, like Connes-Kreimer Hopf algebra approach [16]. In that theoryfor matrices (related construction appears in [17]) taking the boundary seems to be theresidue map in terms of which one can define the coproduct of the Hopf algebra (also true[18] for the Ben Geloun-Rivasseau tensor field theory [19]).• Other graphs might appear for real symmetric matrices, but the ribbons corresponding toHermitian matrices remain untwisted, which played a role in the uniqueness of the graphabove. It is not exaggerated to stress the reason for this rigidity, which is explained by thedifference in the propagators of matrix models in ensembles t M P M N p K q | M : “ M u fordifferent fields K , i.e. h M ij M kl i K “ N $’’’&’’’% ´ ij lk ` ij lk ¯ K “ R p symmetric real q ij lk K “ C p Hermitian q (2.6) We recall that in [1] the truncation is minimal. Thus, the running operators are also thosein the bare action in eqs. (2.1a)-(2.1b).In that truncation [1], if one computes correctly, β b „ b (the value of the vertex); thisis not wrong, but only says that such truncation threw away useful information. In ordernot to “waste” the b term, we propose to add the operator that captures it, so the neweffective action reads:Γ N r A, B s “ Z p A ` B q ´ ¯ a p A ` B q ´ ¯ b p ABAB q ´ ¯ c Tr p ABBA q , (3.1)where bar on quantities means unrenormalized and Z is the (now common) wave functionrenormalization. The ABBA -operator also respects the original Z -symmetries: p A, B q ÞÑ p
B, A q , p A, B q ÞÑ p´
A, B q , p A, B q ÞÑ p A, ´ B q . (3.2)– 5 –ow, b does appear, but in the β c -function. In fact β c „ b ` c ` ac . This relation wasobtained with the (“coordinate-free”) method presented in [20, Corollary 4.2]. Removingthe symmetry condition ( g AAAA “ a “ g BBBB ) we initially imposed on the couplings for A and B , and writing in full g w for the coupling of Tr t w p A, B qu , being w a cyclic wordin A, B , one has β p g ABBA q ´ g ABBA p η ` q „ g AAAA ˆ g ABBA ` g BBBB ˆ g ABBA ` p g ABAB q ` p g ABBA q , (3.3a)obtained without the use of graphs. The missing coefficients (hidden in „ and implying theanomalous dimension η “ ´ N B N Z ) are regulator-dependent and contain non-perturbativeinformation, but the essential point is that now each β -function has a transparent one-loopstructure; to wit, the RHS of (3.3a) corresponds (respecting the order in that sum ) with ` ` ` . (3.3b)It is also clear that the cyclic external leg structure of each of these terms is ABBA . Sinceit is easy to confuse the order of the letters, we stress that this is already an extended ver-sion of the
ABAB -model to exemplify the one-loop structure of another coupling constant.But expressions (3.3) also give the (by Claim 2.2 unique) β -function where the b actuallyhas to sit.Adding the ABBA operator modifies the flow (now β b ´ b p ` η q „ bc ) but in order toget the desired fixed points, higher-degree operators might still be required. As pointed outin the paragraph before [1, Sec. 3.3] when addressing higher-degree operators, Tr rp AB q s isindeed forbidden. However, there are degree-six operators that do preserve the symmetries(3.2), concretely Tr p ABABAA q or Tr p ABABBB q , and contribute to the β b -function (see[20, Thm. 7.2]), thus enriching the truncation. Remark . Some closing points one could learn from [1]:• Notice that if we could somehow make
ABAB indistinguishable from
ABBA , then the β b -function [1, Eq. 3.41] would be, in that case, correct; see (2.3). This happens for asub-ensemble of pairs of Hermitian matrices A, B such that AB is Hermitian (for then, A and B commute). Or β p g ABBA q ´ g ABBA p η A ` η B ` q „ g AAAA ˆ g ABBA ` g BBBB ˆ g ABBA ` p g ABAB q ` p g ABBA q incase that colors distract the reader. Of course, the rotation of reflection of these graphs plays no role, since only the cyclic order matters.We display some vertically, and others horizontally for sake of the reader’s comfort: this way, she or he canstart reading the word from the upper left corner anticlockwise and this order will coincide with the orderof the word in the corresponding coupling constant. – 6 – Notice that the graph (3.4)would yield the b term needed in [1, Eq. 3.41]. However, this graph is not possible, sincethe Ising operator σ Tr p AB q , denoted by the bicolored bead and responsible for “changingcolor”, is not in the truncation behind that equation; moreover, if added, it violates twosymmetries in (3.2), on top of b being screened by σ (that graph is a 1-loop containingfour operators: two Ising, two ABAB , alternated). However, it seems plausible that thecontamination of the running operators (i.e. considering operators that are not unitaryinvariant) might effectively lead to a color-change, as in (3.4).
We showed that the connection that [1] established between the Functional RenormalizationGroup (FRG) and a phase diagram—identified there (due to its resemblance) with [9,Fig. 4]— relies on a β -function that does not have the 1-loop structure of the FunctionalRenormalization Equation. In Section 3 above, we proposed to extend the minimalisttruncation of [1] in order to find a FRG-compatible set of β -functions. Accomplishingthis proposal would provide a sound bridge, in the intention of [1], between the FRGand Causal Dynamical Triangulations [21] through the ABAB -model [22, 23]. Finally, weprovided in Remark 3.1 the condition one would need to add in order for the β b -functiongiven by [1, Eq. 3.41] to be correct. Acknowledgements
The author was supported by the TEAM programme of the Foundation for Polish Sci-ence co-financed by the European Union under the European Regional Development Fund(POIR.04.04.00-00-5C55/17-00).
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