16 November 2000

Physics Letters B 493 (2000) 389–394www.elsevier.nl/locate/npe

Phase diagram of four-dimensional dynamical triangulationswith a boundary

Simeon Warnera,∗, Simon Catterallb

a T-8, Los Alamos National Laboratory, Los Alamos, NM 87545, USAb Department of Physics, Syracuse University, Syracuse, NY 13210, USA

Received 30 August 2000; accepted 29 September 2000Editor: H. Georgi

Abstract

We report on simulations of DT simplicial gravity for manifolds with the topology of the 4-disk. We find evidence for fourphases in a two-dimensional parameter space. In two of these the boundary plays no dynamical role and the geometries areequivalent to those observed earlier for the sphereS4. In another phase the boundary is maximal and the quantum geometrydegenerates to a one-dimensional branched polymer. In contrast we provide evidence that the fourth phase is effectivelythree-dimensional. We finddiscontinuousphase transitions at all the phase boundaries. 2000 Published by Elsevier Science B.V.

1. Introduction

Dynamical triangulation (DT) models arise fromsimplicial discretizations of continuous Riemannianmanifolds. A manifold is approximated by glueing to-gether a set of equilateral simplices with fixed edgelengths. This glueing ensures that each face is sharedby exactly two distinct simplices — the resultant sim-plicial lattice is often called a triangulation. In thecontext of Euclidean quantum gravity it is natural toconsider a weighted sum of all possible triangulationsas a candidate for a regularized path integral overmetrics. Physically distinct metrics correspond to in-equivalent simplicial triangulations. This prescriptionhas been shown to be very successful in two dimen-sions in which analytic methods have been comple-mented by simulation studies (see, for example, [2]).

* Corresponding author.E-mail address:simeon@lanl.gov (S. Warner).

In dimensions above two there are no known ana-lytic techniques for handling the sum over triangula-tions and we must rely on numerical simulation. Allof the latter studies have focused on elucidating thephase structure ofcompactmanifolds, principally thesphereS4. In this paper we investigate the phase struc-ture of four manifolds with the topology of a 4-disk— that is a four sphereS4 equipped with a singleboundary with topologyS3. As we will demonstratethe dimension of the parameter space of this modelis larger than the corresponding compact models. Thesimplest compact models exhibit onlydiscontinuousphase transitions precluding a continuum limit. Onemotivation for the current work was to see whetherthe richer parameter space of the non-compact modelscontains any continuous phase transitions. A naturallattice actionSb can be derived from the continuum ac-tion by straightforward techniques [9]. It contains boththe usual Regge curvature piece familiar from com-pact triangulations together with a boundary term. Theboundary term arises from discretization of the extrin-

0370-2693/00/$ – see front matter 2000 Published by Elsevier Science B.V.PII: S0370-2693(00)01154-0

390 S. Warner, S. Catterall / Physics Letters B 493 (2000) 389–394

sic curvature of the boundary embedded in the bulk.In four dimensions the curvature is localized on trian-gles. If TM denotes the set of triangles in the bulk ofthe 4-triangulation (excluding the boundary) andT∂Mthose in the boundary the action can be written

SEH = κ2

( ∑h∈TM

(2π − αnh)+∑h∈T∂M

(π − αnh))

(1)+ κ4N4.

The quantityα = arccos(1/4), N4 is the 4-volumeandnh is the number of simplices sharing the triangle(hinge) h. The curvature part of the action can berewritten in terms of the number of verticesN0, theboundary volumeNb3 and the number of boundaryverticesNb0 . With this in mind we shall consider thegeneral simplicial action

(2)Sb =−κ0N0+ κ4N4+ κbNb3 + κ0bN

b0 .

This form of the action contains both cosmologicalconstant terms and curvature terms for the bulk andboundary. Notice that the term involvingκ0

b is absentin three dimensions since it is related to the curvatureof a boundary two-sphere which is a topologicalinvariant. In this work we have always setκ0

b = 0and κ4 is used to tune the volume of the 4-disk.We are thus left with a two-dimensional phase spaceparameterized byκ0 andκb conjugate to the numberof vertices and the number of boundary tetrahedra.

The partition function for the system is then

(3)Z =∑T

e−Sb ,

where the sum is over triangulations,T .

2. Simulation

Our simulation algorithm is an extension of the al-gorithm for compact manifolds in arbitrary dimensiondescribed by Catterall [5] and is described in [11]. Tosimulate a triangulation with a boundary we actuallysimulate a compact triangulation with the topology ofthe sphere but consider onemarked vertexto lie out-side the bulk triangulation. This vertex may never beremoved during the course of the simulation and thesurface of the ball created by its neighbour simplicesconstitutes the triangulated boundary of the 4-disk.

The form of the action (Eq. (2)) can then be derived bywriting down an expression for the integrated curva-ture of the full sphere expressed as contributions fromthe 4-disk and the simplices around the marked vertex.The latter contributions can be evaluated explicitly interms of bulk and boundary simplex numbers and yieldthis simple form for the action. To perform a simula-tion we need a set of local moves which are ergodic onthe space of triangulations of the 4-disk. Fortunatelywe have such a set of moves — the usual moves on thesphere.

In four dimensions there are just 5 types of move:vertex insertion, vertex deletion, exchange of a linkwith a tetrahedron (two moves: link to tetrahedronand tetrahedron to link), and exchange of one trianglefor another triangle. Where these moves take placeon sections of the triangulation involving the markedvertex we take care to count changes in the numbers ofsimplices inside and outside the boundary so that wecan calculate the change in the action, but otherwisethe moves are the same as for the bulk.

The code is described in more detail in [11] where itwas used for the simulation of three-dimensional dy-namical triangulations with a boundary. The code waswritten for arbitrary dimension and earlier checkedagainst other workers’ results in two dimensions [1].

We have used the Metropolis Monte Carlo [10]scheme with usual update rule:

(4)p(accept move)=min{e−1Sb,1

}and in this way we explore the space of triangulationswith the actionSb.

Measurements. Here we define some of the mea-surements used to characterize the configurations ob-tained from our simulations. During the simulationswe store configurations at some interval which is suf-ficient to ensure that the configurations are indepen-dent. We check this by estimating the auto-correlationtime (τ ) of each measurement when calculating expec-tation values. Uncorrelated data would giveτ = 0.5,we quoteτ when it significantly exceeds 0.5.

We use two geodesic measures,davg anddbdy. Theaverage geodesic distance between simplices,davg,is measured by counting the smallest number ofsteps between adjacent simplices required to get fromone randomly selected simplex to another randomlyselected simplex and taking the mean over a sampleof such measurements. The mean geodesic distance

S. Warner, S. Catterall / Physics Letters B 493 (2000) 389–394 391

to the boundary,dbdy, is measured by counting thesmallest number of steps between adjacent simplicesrequired to get from one randomly selected simplexto the boundary and taking the mean over a sampleof such measurements. The last step from a boundarysimplex to the boundary is counted as 0.5.

To discuss the singular vertex structure observedin some phases we will consider acoordinationmea-sure,q , defined as the number of simplices sharing agiven vertex. Similarly we define〈qmax〉 as the expec-tation value of the coordination of the most highly co-ordinated vertex;〈qnextMax〉 as the expectation value ofthe coordination of the next most highly coordinatedvertex; 〈qavg〉 as the expectation value of the meancoordination of all vertices in the triangulation; and〈qbdy〉 as the expectation value of the coordination ofthe boundary by which we mean the average numberof unique 4-simplices being shared by a typical bound-ary vertex.

3. Phase diagram

We performed a set of simulations in four dimen-sions with action of Eq. (2). In all runsκ4 was used totune the nominal system volume,N4, for each givenκ0andκb. To map the phase diagram we usedN4= 1000;while to characterize the phases we usedN4 = 8000.In order to check the orders of transitions we have alsoused finite size scaling using additional simulations atN4= 2000 andN4= 4000.

Series of runs varying eitherκ0 or κb were madeand the vertex susceptibility used to search for phasetransitions. We define the vertex susceptibility,χ , to benormalized with respect to the number of 4-simplices:

(5)χ = 1

N4

(⟨N2

0

⟩− ⟨N0⟩2).

The points shown in Fig. 1 are taken from the positionsof peaks in the vertex and boundary susceptibilities.

In Fig. 2 there are four phases which we charac-terize as: crumpled, minimal boundary (CMB); bran-ched-polymer, minimal boundary (BPMB); boundarydominated (BD); and intermediate boundary (IB).

In CMB and BPMB phases the boundary is simply5 tetrahedra (3-simplices) connected to form a hyper-tetrahedral hole. The system is essentially a sphere

Fig. 1. Boundary size (〈Nb3 〉) and number of vertices (〈N0〉) for4-dimensional dynamical triangulation with a boundary. Nominalsimulation volume,N4= 1000.

Fig. 2. Phase diagram for 4-dimensional dynamical triangulationwith a boundary. Nominal simulation volume,N4 = 1000. Errorbars are from estimation of the positions of the susceptibility peaks.Where error bars cannot be seen they are smaller than the symbols;the lines are guides to the eye.

with one marked 4-simplex — the hyper-tetrahedralhole.

4. CMB phase

Here we show typical data forN4 = 8000 atκ0 =−4 andκb = 4 using 137 samples (a sample corre-sponds to 100000 attempted updates). The boundarysize is just 5 — a minimal hole and we see one ‘singu-lar’ vertex with〈qmax〉 = 2642(6) (the next most coor-dinated vertex has〈qnextMax〉 = 531(5), and〈qavg〉 =250.7(5)).

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The presence of singular vertices is similar to thebehaviour seen in compact triangulations in 4 dimen-sions [6,7]. In the compact case, there are two singularvertices which are equally coordinated. In simulationswith a boundary we find one singular vertex and theminimal-boundary appears to assume the role of theother singular vertex (the number of simplices sharedby boundary vertices is 1871(82)).

We find〈dbdy〉 = 9.18(4) (some correlation,τ = 1.7)and 〈davg〉 = 10.66(1). The approximate equality ofthese two distances tells us that in this phase the pres-ence of the boundary plays no crucial role (except forchanging the number of singular vertices). Further-more, we have observed that the typical manifold isvery crumpled, having only a logarithmic growth ofits mean size. Such a behavior may be characterizedby a large effective dimension and is reminiscent ofthe usual crumpled phase seen in simulations of thesphere.

5. BPMB phase

Again we characterize the phase using data forN4 = 8000 (116 samples have been acquired) withcouplingsκ0 = 5, κb = 6. We find〈dbdy〉 = 43.3(10)and 〈davg〉 = 45.6(5). Again, the distance to theboundary is comparable to the distance to any ran-domly selected simplex indicating that the former hasno distinguished role in the triangulation. Further-more, large mean-geodesics are consistent with pathsconstrained to follow long branches. In this phase thevertex coordinations show no sign of singular struc-ture: 〈qmax〉 = 387(7) and the next most highly coor-dinated vertex is about 50 lower,〈qavg〉 = 20.90(1).Thus, the physics of this phase is again rather likethe corresponding situation on the sphere, the typicalgeometries are one-dimensional polymers.

6. BD phase

In this phase the boundary is essentially maximal,the maximal value being obtained from an arbitrarilybranched chain of simplices such thatNb3 maximal=3× N4 + 2. This is easy to understand in terms ofa single chain of simplices — the 2 end 4-simpliceshave 4 boundary 3-simplices, and the(N4−2)middle

4-simplices each have 3 boundary 3-simplices. If onethen considers adding a branch, one boundary simplexis lost at the branch point but one boundary simplexis gained at the end simplex, so there is no net changein Nb3 .

To support this we list data from a run atN4= 8000and couplingsκ0 =−4, κb =−10, in which we havecollected 167 samples. We find that the boundary sizefluctuates only very slightly about a mean ofNb3 ≈24002= 3∗8000+2 and is very branched, possessing〈Nend〉 = 2534(2) end points. The maximum vertexcoordination〈qmax〉 = 103(2) is close to that expectedfor a flat lattice. Another indicator that the geome-try corresponds to narrow tubes is seen when we ex-amine the geodesic distances:〈dbdy〉 = 0.5038(2) and〈davg〉 = 125.8(2) The mean boundary distance tellsus that every simplex is a boundary simplex. Thusthe quantum geometry is effectively one-dimensional.We find 〈qavg〉 = 4.9975000(1) which we take as fur-ther indication that there are no sections of bulk inthis phase. The argument for this is as follows: in aminimal-width chain with no branches we have twoends which have a few vertices shared by less than 5-simplices — the coordination of vertices in the bulk ofthe chain. At each end there is 1 vertex with coordi-nation 1, 1 with coordination 2, 1 with coordination 3,and 1 with coordination 4. This adds up to a coordi-nation deficit of 10 for the 4 vertices at an end, 20 forboth ends. In a minimal-width chain consisting ofN4simplices we expectN0 = N4 + 4 and so withN4 =8000 we expect, and see,N0 = 8004. This allows usto calculate the expected〈qavg〉 = (N0× 5− 20)/N0,puttingN0= 8004 we get〈qavg〉 = 4.997501. Addingbranches to the chain results in more highly coordi-nated vertices at the branch point that exactly cancelthe deficit from the additional end.

7. IB phase

Here we show data forN4 = 8000 (86 samples)at couplingsκ0 = −4, κb = 0. We find that, onceagain, the boundary size scales linearly with the vol-ume (see Fig. 3) yieldingNb3 ≈ N4× 1.033. We alsofind one large singular vertex, with〈qmax〉 = 5721(9)(〈qnextMax〉 = 233(2), 〈qavg〉 = 133(1)). The measure-ments of geodesic distances show that the boundaryplays a preferred role in the triangulation〈dbdy〉 =

S. Warner, S. Catterall / Physics Letters B 493 (2000) 389–394 393

Fig. 3. Plot showing scaling of boundary size with simulationvolume for the intermediate boundary (IB) phase,κ0=−4, κb = 0.The error bars are much smaller than the symbols.

0.834(2) while 〈davg〉 = 11.36(1). These numbersare incompatible with an extremal branched polymershape. However, these measurements by themselveswould not be inconsistent with a ‘fat-branch’ model.If we add to this the presence of a single singular ver-tex and the observation that about 40% (3495 (9) outof 8000) of the simplices have one boundary face wetentatively conclude that the typical geometry in thisphase isthree-dimensional— the boundary of the sys-tem coinciding with the boundary of the 4-ball sur-rounding the singular vertex.

8. Phase transitions

The phase transition between the CMB and BPBMphases has been studied in the simulations of compactsystems in three and four dimensions and found tobe discontinuous (first-order) in both cases (3d [3],4d [4,8]). Our simulations have minimal boundary inboth phases and so we expect the same behaviour. Thiswas verified in three dimensions [11].

We have also investigated the transition betweenCMB and IB phases and find good evidence herealso for a discontinuous phase transition (Fig. 4,top; N4 = 1000, κ0 = −4, κb = 0.9). Similarly, thetime series close to the BPMB–BD phase boundary(Fig. 4, middle;N4 = 1000,κ0 = 4, κb = 1.64) alsoshows signs of bistability indicative of a discontinuoustransition.

At N4 = 1000, 2000 and 4000 we found no suchsignals near the BD–IB boundary. However, we foundlinear scaling of the height of the peak in the boundary-size susceptibility which indicates a first-order tran-sition. To confirm this we extended our simulationsto N4 = 8000 and found bistability in the time series(Fig. 4, bottom;N4= 8000,κ0=−4, κb =−1.862).

9. Concluding remarks

We have simulated four-dimensional simplicial gra-vity on manifolds with the topology of a 4-disk. Ouraction contains both bulk curvature terms and a bound-ary cosmological constant term. We have identifiedfour phases in the model within the range of couplings−6< κ0 < 6 and−4< κb < 8. The observed phasesinclude the crumpled and branched-polymer phasesseen in triangulations of compact manifolds, and theboundary dominated phase seen in three dimensions.The latter consists of a maximally branching tree. Wehave also identified a fourth phase which resembles asingular 4-ball in which essentially all simplices sharea common bulk vertex and the physics is dominated bythe three-dimensional boundary.

All of the boundaries between these phases appearto be associated with discontinuous phase transitions.This is the same situation as for four-dimensionalsimplicial gravity on manifolds withS4 topology andmeans that we cannot take a continuum limit in thevicinity of any of the observed phase transitions.Notice that all four phase boundaries appear to meet(within our errors) at one unique point. The simplestexplanation for this feature is to assume that our phasediagram contains only two independent transition lineswhich would then generically intersect at a singlepoint. Glancing at the nature of the phases one wouldassociate one line with a boundary dividing latticeswith singular vertices from those without. The secondline would then correspond to a dividing line betweengeometries with extended boundary and those withminimal boundary.

Notice also, that in all the phases we have identified,the number of vertices on the boundary is stronglycorrelated with the boundary volume. This need notbe the case and it would be interesting to vary thecouplingκ0

b away from zero to see whether a phase

394 S. Warner, S. Catterall / Physics Letters B 493 (2000) 389–394

Fig. 4. Simulation time series showing bistability in the boundary size (Nb3 ) at the CMB–IB phase boundary (upper;N4 = 1000,κ0 = −4,κb = 0.9); the BPMB–BD phase boundary (middle;N4 = 1000,κ0 = 4, κb = 1.64); and the BD–IB phase boundary (lower;N4 = 8000,κ0=−4, κb =−1.862). We take this as indication of the discontinuous nature of these transitions.

could be found which exhibited a classical scaling ofboundary size with 4-volume.

Acknowledgements

Simon Catterall was supported in part by DOE grantDE-FG02-85ER40237. Simeon Warner was supportedin part by the DOE under contract W-7405-ENG-36.

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