Exploring the new phase transition of CDT

This work focuses on the newly discovered bifurcation phase transition of CDT quantum gravity. We define various order parameters and investigate which is most suitable to study this transition in numerical simulations. By analyzing the behaviour of the order parameters we present evidence that the transition separating the bifurcation phase and the physical phase of CDT is likely a second or higher-order transition, a result that may have important implications for the continuum limit of CDT.

C o n ten ts 1 I

In tro d u ctio n
Assum ing only key aspects of q uantum mechanics and general relativity, and including few additional ingredients, causal dynam ical triangulations (CD Ts) define a particularly simple approach to quantum gravity. T he sim plicity of construction and plenitude of results has m ade C D T a serious contender for a n o nperturbative theory of quantum gravity. There now exists strong evidence th a t C D T has a classical lim it th a t closely resembles general relativity on large distance scales [1], while on short distance scales it has produced some exciting hints about the n atu re of spacetim e a t th e P lanck scale, including evidence th a t th e num ber of spacetim e dim ensions m ay dynam ically reduce [2], a result th a t has also been reported in num erous other approaches to quantum gravity [3][4][5][6]. CD T gives an approxim ate description of continuous spacetim e via th e connectivity of an ensemble of locally flat n-dim ensional simplices. In order to reproduce general relativ ity in th e classical lim it it seems th e introduction of a causality condition is a necessary requirem ent [7], such th a t th e lattice can be foliated into spacelike hypersurfaces of fixed topology. By only including geom etries in th e p a th integral m easure th a t adm it such a foliation, th e unphysical features observed in dynam ical triangulations w ithout a causal ity condition (see [8][9][10] and more recently [11,12]) ap p ear to be suppressed, yielding a semi-classical geom etry th a t closely resembles general relativity. CD T discretises th e continuous p a th integral into a p artitio n function [13] Z e = E C -e~SEH, CD T defines two types of 4-dim ensional triangulations, th e (4,1) and (3,2) simplices (see ref. [7] for m ore details). T he num ber of (4,1) simplices in eq. ( 1.2) is given by N 4,i, the num ber of (3,2) simplices is denoted by N 3,2 and th e num ber of vertices in a triangulation is given by N 0. E quation (1.2) is a function of three bare coupling constants: ko, A and k4. ko is inversely proportional to N ew ton's constant, A defines an asym m etry param eter quantifying th e ratio of th e length of space-like and tim e-like links on th e lattice and k4 is related to th e cosmological constant, and is typically tuned in num erical sim ulations to a (pseudo-)critical value. Fixing k4 in this way allows one to take an infinite-volume limit, leaving a two-dim ensional param eter space spanned by th e bare couplings ko and A.
T he param eter space of CD T has now been m apped out in some detail, as shown in figure 1, and consists of four distinct phases. Phases A and B are generally regarded as lattice artifacts containing unphysical geom etric properties [7]. P hase C, however, closely resembles 4-dim ensional de S itter space on large distance scales [1]. T he possibility of taking a continuum lim it w ithin phase C seemed a real possibility following the discovery of a second-order phase tran sitio n dividing phases B and C. However, the discovery of a fourth so-called bifurcation phase (D) existing betw een phases B and C makes it difficult or impossible to approach this second-order tran sitio n from w ithin th e physically interest ing phase C. This m otivates th e need to investigate th e location and order of the (C-D) bifurcation phase transition, since if th e tran sitio n was second-order it would re-establish th e possibility of taking a continuum lim it in CDT. O v e r v ie w In order to locate and study th e critical behaviour of th e tran sitio n dividing th e bifurcation and de S itter phases we seek an order param eter (O P) th a t is approxim ately zero in one phase and non-zero in th e other. Hence, by taking th e n th -o rd er derivative of an appropri ately defined order param eter one should in principle be able to determ ine th e order of the transition . For exam ple, in the infinite volume lim it a first order tran sitio n is characterised by a discontinuity in th e first order derivative a t th e tran sitio n point, whereas a continu ous function should be observed for higher-order transitions. In num erical sim ulations one respectively. We have analysed all of th e above OPs, finding sim ilar qualitative behaviour.
In th e following sections we will focus on a particu lar com bination, nam ely In order to analyse th e bifurcation tran sitio n we perform ed a series of m easurem ents of this O P for a range of bare coupling constants th a t begin in phase D and end in phase C. We study a particu lar p a th w ithin th e phase diagram for which we fix k0 = 2.2 and vary A. Therefore O P o given by eq. (2.2) , which is conjugate to A in th e bare CD T action ( 1.2) , seems to be a particularly good choice. T he sam e order param eter was also used in refs. [14,15] to analyse th e form er B -D 1 tran sitio n in a sim ilar way.
T he second group of order param eters focuses on microscopic geom etric properties of th e underlying CD T triangulations. It was shown in ref. [16] th a t th e distribution of volume in th e bifurcation phase is m arkedly different th a n in phase C, w ith spatial 1As we now know th a t phase D exists, th e former "B-C" tran sitio n now becomes th e B-D transition.

JHEP02(2016)144
volume concentrated in clusters connected by vertices of very large coordination num ber (the num ber of 4-simplices sharing a given vertex). This change of th e geom etric stru ctu re can be exploited to signal th e phase transition. Inside th e bifurcation phase b o th the average scalar curvature R (t) = 2 n -C (where C = 6 arccos(1/3) -2n > 0) and the m axim al coordination num ber of a vertex O (v (t)) differ significantly betw een spatial slices of odd and even tim e t, whereas there is no such difference in phase C. One can quantify this difference by defining the order param eters [16] and where th e (integer) tim e to is chosen to be th e closest to th e centre of volume of a triangulatio n .2 A detailed analysis of all three order param eters is presented in section 2.3.   such a procedure is presented in figure 3 where we plot th e error in th e m easurem ent of the susceptibility Xo p 2 a t th e point closest to the phase transition. T he error is estim ated by a jackknife procedure for each block size and is plotted as a function of th e num ber of blocks. T he error typically increases w ith the num ber of blocks, eventually stabilising around a co n stan t, as shown for the sm aller volume ensemble presented in figure 3 (left). In some cases th e largest error is observed for a small num ber of blocks, which appears to be the case for th e larger volume ensemble close to th e phase tran sitio n point (see figure 3 (right)). As already discussed, for this em pirical d a ta we observe only two m etastable transitions in th e order param eter over th e entire sim ulation period, this likely m eans th e jackknife procedure is overestim ating th e error. We adopt a cautious a ttitu d e and take th e highest value as our error estim ate.

.3 R e s u lt s
We now present th e results of our order param eter studies. We focus on three order param eters defined in section 2. Figure 4 shows In figure 5 we plot th e susceptibility %OP of each order p aram eter defined in eq. (2.1) . A clear signal of th e phase tran sitio n is observed only for th e O P 2, where one can see a peak of susceptibility at th e (pseudo-)critical points A crit(80k) = 0.30 ± 0.01 and A crit(160k) = 0.35±0.01. Interestingly, if one plots the ratio %OP/(O P ) one can also observe th e tran sitio n peaks using O P 1 (see figure 6) .
T he above results indicate th a t for th e bifurcation tran sitio n th e details of th e geom etry play an im p o rtan t role, and therefore order param eters based solely on global properties of th e triangulation do not capture these details. T he central difference betw een phase C and phase D is related to th e form ation of periodic clusters of volume around singular vertices, which form a kind of tu b e stru c tu re (see ref. [16] for details). Such a stru ctu re does not exist in phase C, but is a generic property of phase D. Therefore, in order to observe the phase tran sitio n it is im portant to analyse th e microscopic simplicial geometry. Even order param eters such as O P 1 only cap tu re general features of th e geom etry (i.e. th e difference

D iscu ssio n and o u tlo o k
S ta rtin g from a point in th e param eter space w ith good semi-classical features, th e hope is th a t one can establish a continuum lim it by approaching a second order transition, thereby defining a sm ooth interpolation betw een the low and high energy regimes of CD T. The infinite correlation length associated w ith such a tran sitio n should allow one to shrink the lattice spacing to zero while keeping observables fixed in physical units. Such a continuous tran sitio n has been shown to exist in th e C D T param eter space [14,15] and was originally thought to divide th e semi-classical phase C from phase B. However, recent results [16,17] show ) and th a t the frequency of such jum ps decreases w ith increasing volume. This result m ay suggest th a t th e tran sitio n is first order. This is illustrated in figure 7 where we plot a histogram of the O P 2 (norm alised by th e lattice volume) m easured for two different volumes N 4,i = 80,000 (blue) and N 4,4 = 160,000 (red), respectively. By fitting a double G aussian function to th e m easured d a ta we observe two clearly separated peaks.4 T he peak separation is slightly sm aller for a larger to ta l volume. A sim ilar situation was previously observed a t th e 'old' B-C (now called th e B-D) phase tran sitio n (which is very likely second order) [15], where the peak separation reduced w ith increasing volume. M easuring th e behaviour of th e order p aram eter for a num ber of different lattice vol um es will enable us to calculate critical exponents and to analyse th e order of th e phase tran sitio n in detail. This work is still in progress, however prelim inary results are prom is ing. In figure 8 we plot th e position of (pseudo-)critical points A crit as a function of lattice volume N 4,1. Using this em pirical d a ta we fit th e function A crlt(N4,i) = A critM -a ■ N 4,i-1/v (3.1) 4As we are able to establish th e phase tran sitio n point only w ith finite precision th e height of th e two peaks is different. T he peaks would be th e same height a t th e (pseudo-)critical point. F igu re 7 . A histogram of the O P 2/N 4ji order param eter m easured at the phase transition point for two different lattice volumes N4ji = 80, 000 (blue) and N4ji = 160, 000 (red). We fit the histogram d ata to a double Gaussian function (solid line). The position of the two peaks is marked by dashed lines. The peak separation appears to shrink slightly with increasing lattice volume.
F igu re 8 . Prelim inary results of C-D phase transition dependence on lattice volume. The (pseudo-)critical points A crlt were estim ated for fixed k0 = 2.2 and various total volumes N 4ji by looking at peaks in susceptibility x OP2 as described in section 2. The solid red line corresponds to a fit of eq. (3.1) to the m easured d ata (v = 2.6), while the dashed blue line uses the same fit but with a critical exponent of v = 1. and estim ate th e critical exponent v = 2.6 ± 0.6 (solid red line in figure 8) . This value of v suggests a continuous transition. For com parison we also m ade a fit using a fixed value of v = 1 th a t would correspond to a first order tran sitio n (dashed blue line in figure 8), which cannot be com pletely excluded b u t appears much less likely. We are currently collecting d a ta a t th e C -D tran sitio n for additional lattice volumes as well as increasing statistics of previous m easurem ents. U nfortunately, this process is com putationally very tim e consum ing and a com prehensive study of the bifurcation tran sitio n order will be presented in a separate article. O p e n A c c e s s . This article is distrib u ted under the term s of th e C reative Commons A ttrib u tio n License (CC-BY 4.0) , which perm its any use, d istribution and reproduction in any m edium , provided the original author(s) and source are credited.