Finite temperature phase transition of a single scalar field on a fuzzy sphere
aa r X i v : . [ h e p - t h ] J un IMSc/2007/06/7
Finite temperature phase transition of a single scalar field on a fuzzysphere
C. R. Das, ∗ S. Digal, † and T. R. Govindarajan ‡ The Institute of Mathematical Sciences, C.I.T. Campus, Taramani, Chennai 600 113, India
We study finite temperature phase transition of neutral scalar field on a fuzzy sphere using MonteCarlo simulations. We work with the zero mode in the temporal directions, while the effects of thehigher modes are taken care by the temperature dependence of r . In the numerical calculationswe use “pseudo-heatbath” method which reduces the auto-correlation considerably. Our resultsagree with the conventional calculations. We report some new results which show the presence ofmeta-stable states and also suggest that for suitable choice of parameters the symmetry breakingtransition is of first order. PACS numbers: 12.60.Rc; 12.10.-g; 14.80.Hv; 11.25.Wx; 11.10.Hi
I. INTRODUCTION
QFT’s on non-commutative spaces have been studied from various perspectives recently [1, 2,3, 4, 5, 6, 7, 8]. Most frequently studied NC space is the well known Groenwald Moyal space [9] R dθ and various issues like, renormalisation, causality, solitons, statistics have been analysed in theliterature [10, 11, 12, 13]. The conventional quantisation of fields on these spaces have led to aninteresting behavior known as IR/UV mixing. The phase structure of fields on such a space revealsa new phase known as strip phase [14]. Alternative quantisation which preserves a twisted Poincaresymmetry in these theories avoids such a difficulty [15, 16].On the other hand the fields on fuzzy spaces like fuzzy spheres, fuzzy CP n etc are explicitly finiteand do not have the IR/UV mixing [17, 18]. But there is an anomaly in the finite case which revealsitself as generating the IR/UV mixing. There is lot of confusion about taking the limit of continuumin these models and it has been pointed out various possibilities do exist [17, 18, 19, 20].The QFT on fuzzy sphere is a matrix model and easily amenable to simulations and numericalstudies [21, 22, 23]. We study a real scalar field on the fuzzy sphere using Monte Carlo simulations.The earlier studies involved metropolis algorithm to ensure the randomness of fluctuations butthe autocorrelations are reduced using over-relaxations [23]. But we will use another techniqueextensively used in the study of Higgs model - known as pseudo-heatbath [24]. Using this algorithmwe have been able to reproduce previous results from different studies. Apart from this we areable to characterise the order of the transitions between the different phases. In particular we findthe transition between order ↔ non-uniform transition is of first order which is mostly due to thepresence of many meta-stable states at low temperatures in the model. Also we find new results forthe structure of the phase diagram as well as for scaling of the location of the triple points in thecontinuum limit.The paper is organised as follows: the sec(2) introduces the QFT on fuzzy spheres; sec(3) discussesthe pseudo-heatbath technique and brings out its salient features. Sec(4) reproduces the known ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] results and discusses our results on the nature of phases and the transitions. In sec(5) we presentour conclusions.
II. THE MODEL
We use the following action for the massive, neutral, scalar field on the fuzzy sphere of radius R [19, 21, 23], S (Φ) = 4 πN Tr (cid:2) Φ [ L i , [ L i , Φ]] + R (cid:0) r Φ + λ Φ (cid:1)(cid:3) . (1)Here Φ ∈ Mat N is a N × N hermitian matrix. The first term in the action is (kinetic) coming fromthe variation of the Φ field on the fuzzy sphere. The quartic term represents the self interaction ofthe Φ field. For the thermal behavior of the Φ field one needs to study the system in 2 + 1 d . Howeverone can consider the above action as the dimensionally reduced version of a 2 + 1 dimensional actionwith the effects of temperature going into the temperature dependence of r . Finite temperaturebehavior of the Φ field with fluctuations included is then studied at different r . In the mean fieldapproximation the expectation value of Φ field which minimises the action is given by, h Φ i = ± φ , φ = r − r λ (2) φ ≥ r and zero for r ≥
0. The system is in ordered phase for r < r ≥
0. So at the mean-field level there are only two phases. h Φ i decreasescontinuously to zero in the limit r →
0. At r = 0 the system undergoes a second order phasetransition with mean field critical exponents, β = 1 / , α = 0 etc.. In the disorder phase h Φ i = 0 sothe Z symmetry of the model is restored. Even though the above form of h Φ i minimises the actionthere are additional local minima or meta-stable states. For these states, form of h Φ i is non-identityin general. Some of these states will have same potential as the ground state. In the mean-fieldapproach these states do not play any role in the phase transition. However they become importantwhen fluctuations beyond mean-field are considered.Hence the next step in the calculations is to consider effect of thermal fluctuations beyond meanfield. It is important then to ask if the results of the mean field analysis survive. One expects thatthe fluctuations will destroy a non-zero h Φ i even for r less than the mean-field critical value which iszero. Further more the fluctuations may lead to new phases and different types of phase transitions.These are some of the issues of intense numerical investigations recently [19, 21, 23]. So far theresults show the appearance of a new phase called non-uniform ordered phase. These studies aremostly done using a standard Monte Carlo simulations with metropolis algorithm.In the present work we study the fluctuations in the above model using a different numericaltechnique known as “pseudo-heatbath” method. Like the previous studies we also observe the non-uniform phase. However our results seem to indicate that there are phase structure within thenon-uniform phase. These phases can be probed using different operators/order parameters. As aconsequence there will be multiple triple points in the λ − r plane. In the following we describe the“pseudo-heatbath” method. Subsequently we present and discuss our results next section. III. NUMERICAL TECHNIQUE
Effects of the fluctuations beyond mean field are computed from the partition function, which inthe path integral approach is given by,
Z ∝ Z D Φ e − S (Φ) . (3)The standard numerical methods adopted for this integration are Monte Carlo simulations. In theMonte Carlo algorithms, one generates an “almost” random sequence of Φ matrices by successivelyupdating elements of Φ taking into account the measure and the exponential in the integral above.This sequence of Φ is then used as an ensemble for calculating averages of various observables. Forexample, thermal average of Φ is given by, h Φ i = 1 N m N m X m =1 Φ m , (4)here, Φ m is the m th element of the ensemble. Usually there are different ways to generate theensemble. Previous studies of this model have considered the metropolis algorithm [21, 23]. In themetropolis updating usually there is a substantial correlation between Φ’s in the sequence. For agood ensemble the auto correlation between the configurations in the sequence must be really small.Though this auto correlation can be reduced by using some over relaxation programme [23]. Theauto-correlation is greatly reduced, however, when “heatbath/pseudo-heatbath” type of algorithmsare used [24]. This method is very much common in the non-perturbative study of Φ theories inconventional lattice simulations. It gives better sampling and is efficient at least for smaller λ values.This is why we make use of “pseudo-heatbath” technique. In the following we explain the algorithmin greater detail.In the “pseudo-heatbath” algorithm, given a Φ we update the elements of this matrix one at atime. Advantages of updating matrix elements were demonstrated in Ref. [25]. Keeping in mindthat Φ is hermitian we update Φ ij and Φ ji simultaneously. We update Φ ij using the probabilitydistribution, P (Φ ij ) = e − S (Φ ij ) where S (Φ ij ) = α (Φ ij − A ) + λB (Φ ij − C ) , (5) A , B , C may depend on the elements of Φ (other than Φ ij ). α is a parameter chosen so as tomaximise the efficiency of updating. In the first step a random number is generated using thedistribution, e − α (Φ ij − A ) . (6)In the second step the newly generated random number is accepted or rejected using the secondterm of S (Φ ij ). In our calculations we get for some choice of the parameters, in particular small λ acceptance rate up to 95%. Over relaxation can also be easily incorporated into this algorithm. Inthe over relaxation process we flip the element φ = Φ ij in the following way, φ ′ = A − φ (7)then accept it with the probability exp( − δS ). δS is the change in action due to flipping. For small λ this amounts to changing Φ by large amount with only a small change in the total action. Evenwithout using the over relaxation method we get very small auto correlation. In the following wepresent and discuss our numerical results. IV. NUMERICAL RESULTS AND DISCUSSION
To study the phase diagram and transitions we make measurements of various observables suchas
T r (Φ) , T r (Φ ) , T r ( S ) (8)at various values rR for different choices of ( N, λR ). In order to check our algorithm we consideredsome of the parameters used in previous calculations [19, 21, 23], and found that our results matchreasonably well with previous results. The results also agreed with mean-field away from the tran-sition point. In Fig. 1(a) we show Monte Carlo history of T r (Φ) for N = 2, R = 1 . λ = 0 . r = − . T r ( Φ ) Measurement N = 2 R = 1.0 λ = 0.636620 r = -1.530502-3-2-1 0 1 2 3 0 4000 8000 12000 16000 20000 (a) Monte Carlo history of T r (Φ). -3-2-1 0 1 2 3 0 5000 10000 15000 20000 t r( Φ ) MeasurementSample Monte Carlo history of the trace, with overrelaxation and ergodic updates N =2 R =1 λ =0.636620 r =-1.530502 (b) Monte Carlo history of T r (Φ) from [23].
FIG. 1: Comparison between pseudo-heatbath method and Metropolis method
Order ↔ non-uniform transition Having reproduced some of the previous results we considered various values of the parameters(
N, λR ) to study the phase diagram. For large values of λR we found multiple transitions [19].For low temperatures, or r ≪
0, the average value of Φ is essentially identity matrix indicatingthe ordered phase. For larger values of r the average of T r (Φ) vanishes while the average of someelements Φ ij is non-zero. Such a form of Φ average indicates a non-uniform phase which breaksspatial rotation spontaneously. Since the average of T r (Φ) is non-zero in the order phase and zero inthe non-uniform phase one can use it as an order parameter for the order ↔ non-uniform transition.As for the order of the non-uniform ↔ order transition we find it is strong first order for larger λR . This can be seen from the hysteresis effects of T r (Φ). In Fig. 2(a) we show the hysteresis loopof
T r (Φ) for the set of parameters N = 25 and λ = 0 .
8. The value of r corresponding to the middleof the hysteresis loop is take to be the critical(transition) value r = r for this transition. By doingsimulations for different values of λ we find that the strength of this order ↔ non-uniform transitionvaries. For smaller λ the transition becomes weaker. For example, for λ = 0 . 〈 | T r ( Φ ) | 〉 r 0 5 10 15 20 25 30-2.2 -2 -1.8 -1.6 -1.4 -1.2 -1 (a) Hysteresis loop of T r (Φ) for N = 25, R = 10 . λ = 0 . 〈 | T r ( Φ ) | 〉 Measurement-25-20-15-10-5 0 5 10 15 20 25 0 2000 4000 6000 8000 10000 (b) Monte Carlo history of
T r (Φ) for N = 25, R = 10 . λ = 0 . r = − . FIG. 2: First-order Phase Transition
The history of measurement of
T r (Φ) and its distribution is shown in Fig. 2b for r close to thecorresponding critical value r . One clearly sees three degenerate ground states here. Out of thesetwo are connected by Z symmetry and the 3rd has T r (Φ) peaked around zero and represents anon-uniform ordered phase. This distribution indicates that the order and non-uniform phases docoexist suggesting first order nature of the transition between these two phases. Even though thetransition is first order, it is weaker compared to the previous example.
Non-uniform ↔ disorder transition In the non-uniform phase
T r (Φ) keeps fluctuating around zero. One can clearly see measuredvalues of
T r (Φ) form bands symmetrically situated around zero as shown in figure Fig. 3(a). Theband structure are not seen in the Monte Carlo history of
T r (Φ ) and S (Φ). This band structure of T r (Φ) we saw mostly in the case when there were many meta-stable states before transition in theordered phase, i.e for larger N . In Fig. 3(b) we show the histogram of T r (Φ) which clearly showsa peak close to zero. This implies the state with lowest
T r (Φ) is the ground state of the systemand other bands are meta-stable states. The meta-stable bands tend towards zero as we increase r further. At the same time some bands disappear and/or others merge with the middle one.In the basis we choose to work with even though T r (Φ) fluctuates around zero both Φ and Φ NN fluctuate around non-zero Z symmetric values. So the symmetry of Φ is not restored yet. Whenwe increase r the non-zero values around which the first and last diagonal of Φ fluctuate approachsmoothly to zero. Beyond certain value of r = r all elements of Φ fluctuate around zero restoringthe Z symmetry. In Fig. 4 we show the distribution of these elements of Φ both below and above r . Given this behavior of Φ and Φ NN one can consider any of them as the order parameter forthe non-uniform ↔ disorder transition. This also implies that the higher spherical harmonics arebecoming important for this transition in our basis [19]. When r is approached from below thepeaks of the distribution of Φ , Φ NN smoothly approach zero indicating only one ground state atany particular value of r around the transition point. So this transition is a continuous transition. Phase diagram and triple points
When we analysed the data for fluctuations of
T r (Φ ) we found these peaked at a certain value T r ( Φ ) Measurement-6-4-2 0 2 4 6 8 0 20000 40000 60000 80000 100000 (a) Monte Carlo history of
T r (Φ) at N = 16, R = 15 . λ = 0 . r = − . H ( T r ( Φ )) Tr ( Φ ) 0 1000 2000 3000 4000 5000 6000 7000 -6 -4 -2 0 2 4 6 (b) Histogram of T r (Φ) at
T r (Φ) at N = 16, R = 15 . λ = 0 . r = − . FIG. 3: Monte Carlo history and histogram for r < r < r H ( Φ ) Φ r = -0.05r = -0.04625 0 5000 10000 15000 20000 25000 30000 35000 40000 45000-1.5 -1 -0.5 0 0.5 1 1.5 (a) Distribution of Φ for N = 12, R = 10 . λ = 0 . H ( Φ NN ) Φ NN r = -0.05r = -0.04625 0 5000 10000 15000 20000 25000 30000 35000 40000 45000-1.5 -1 -0.5 0 0.5 1 1.5 (b) Distribution of Φ NN for N = 12, R = 10 . λ = 0 . FIG. 4: Non-uniform ↔ disorder transition of r within range r < r < r (Fig. 5b). This suggests finer structure or phases in the non-uniformphase. The peak in T r (Φ ) then corresponds to the transition between these phases.We did not see any dramatic change in the variables such as T r (Φ) or the diagonal elements ofΦ. The finiteness of the peak implies that the transition is a continuous transition. For larger λ thetransition point was far from both r and r . We anticipate that the non-uniform phase has morestructure than what we see from the behavior of T r (Φ ). This finer structure could be exploredby appropriate operators such as multi-trace operators. Note that this fine structure becomes moreprominent for larger N , which implies that it will survive in the continuum non-commutative limit.As r is increased from some large negative value the system explores all these phases for large λ .For smaller λ some of these phases will not appear when r is varied. This leads to presence of triplepoints in the λ − r plane. For some small λ there is transition directly between order ↔ disorder χ r 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1.4 -1.3 -1.2 -1.1 -1 -0.9 -0.8 -0.7 (a) Susceptibility of T r (Φ) for N = 25, R = 10 . λ = 0 . χ r 0.16 0.2 0.24 0.28 0.32 0.36 0.4-1.4 -1.3 -1.2 -1.1 -1 -0.9 -0.8 -0.7 (b) Susceptibility of T r (Φ ) for N = 25, R = 10 . λ = 0 . FIG. 5: Transition between non-uniform phases phases, i.e | r − r | vanishes. This leads the triple point which has the lowest value of λR . In Fig. 6awe show the phase diagram for N = 25 in the λR vs rR plane. This is the triple point studiedin previous works. In these studies the triple point was obtained by using numerical results for theorder-non-uniform transition and the analytic results which takes into account only the potentialterm [19]. In our case both transitions lines are from our simulations.Conventional lattice regularization of the model does not show any evidence of non-uniform or-dered phase. So it is imperative to study what happens to the non-uniform ordered phase in thecontinuum limit. If the non-uniform phase survives this limit then only it can be physically relevant.We have studied the N dependence of the triple point. Since it’s not practical to do simulations forvery large N , one must study scaling to find out the limiting position of the triple point for larger N values. In the Fig. 6b we show value of Y tri = λR , X tri = rR corresponding to the triple pointfor different N . The values of N considered for our simulations are N = 4 , , , ,
25. When N was increased the triple point moved away from the origin. Our results suggest that the triplepoint scales with ( N µ , N ν ) with µ ≃ . ≃ ν . We did not observe any universal scaling of the phaseboundaries in the phase diagram for different N .For smaller λR there is only one transition, the order ↔ disorder transition. For larger N the distribution of the observables such as T r (Φ) , Φ , Φ NN , close to the critical point, show aplateau around zero with highly non-gaussian features. This can be seen in Fig. 7 where haveshown the histogram of T r (Φ) , Φ around the critical temperature. The parameters consideredhere are λ = 0 . , R = 10 , N = 12. We take the plateau structure around zero as an indicationof a transition which is stronger than second order transition. However it does not rule out thepossibility of second order phase transition for smaller values of λ and N . Meta-stable states
We also studied cases with very large values of λR . When r is large negative we find differentaverage values for T r (Φ) for different initial choices of Φ. These different values correspond to localand global minima of the effective action. The barrier between these states inhibits the transitionamongst them. The number of these states which we observed grow with N . This can be moreor less seen from the analysis of the action itself as fluctuations are not much important for small r . The state with highest T r (Φ) found to satisfy h Φ i ∝
1l and has the lowest value for the action, λ R rR Disorder Non-uniformOrder 0 5 10 15 20 25 30 35 40 45 0 10 20 30 40 50 60 70 80 (a) Phase diagram for N = 25. T r i pp l e po i n t c oo r d i na t e s NX tri Y tri (b) Scaling of the triple point with N . FIG. 6: Phase diagram and scaling of the triple point H ( T r ( Φ )) Tr ( Φ )r = -0.05r = -0.0475r ≈ r c = -0.04875 0 5000 10000 15000 20000 25000 30000 35000 40000-15 -10 -5 0 5 10 15 (a) Distribution of T r (Φ) H ( Φ ) Φ r = -0.05r = -0.0475r ≈ r c = -0.04875 0 5000 10000 15000 20000 25000 30000 35000 40000-1.5 -1 -0.5 0 0.5 1 1.5 (b) Distribution of Φ FIG. 7: hence is the ground state. So we conclude that the state with largest
T r (Φ) is the global minimumof the system. Other states, which are basically the non-uniform phases, are meta-stable. We thinkthat the meta-stability increases with decrease in the average of
T r (Φ). In Fig. 8(a) we show a briefMonte Carlo history of Φ, after thermalization, for different initial Φ’s. For higher r the bands persistbut move slowly towards zero. After certain value of critical r most of these states are observed onlyfor sometime in the Monte Carlo history and then the values of T r (Φ) jumped to zero as seen inFig. 8(b).
V. CONCLUSIONS
We have developed a “pseudo-heatbath” algorithm to study the finite temperature phase tran-sitions of Φ theory on a fuzzy sphere. The results from Monte Carlo simulations clearly show T r ( Φ ) Measurement-5 0 5 10 15 20 25 0 1000 2000 3000 4000 5000 (a) Monte Carlo history of
T r (Φ) at N = 16, R = 15 . λ = 0 . r = − . T r ( Φ ) Measurement-4-2 0 2 4 6 8 10 12 14 0 2000 4000 6000 8000 10000 (b) Monte Carlo history of
T r (Φ) at N = 16, R = 15 . λ = 0 . r = − . FIG. 8: Monte Carlo history for r < r and for r ∼ r finite temperature transitions. For some range of λR , in particular, for large values one clearlysees stable non-uniform phases for some intermediate temperature, intermediate values of r . Thevarious phases are characterised by different properties of Φ. In the ordered phase this behaves likea identity matrix. All non-uniform phases have zero T r (Φ). Their existence is confirmed by thepeak in the fluctuation of
T r (Φ ). T r (Φ) serves as an order parameter for the order ↔ non-uniformtransition while Φ and Φ NN describe the non-uniform ↔ disorder transition.The order-non-uniform transition is found to be first order. This transition was found to be strongfirst order for larger values of λR . We conjecture that the first order nature of the transition hasto do with the presence of meta-stable states discussed above. In fact the state with T r (Φ) = 0 ismeta-stable for small temperatures when Φ ∝
1l is the absolute ground state. Fluctuations can onlystabilise if it is a stable configuration at higher temperature, so there is always a barrier with theordered phase, leading to first order transition. For smaller values of λR the barrier between thestable and meta-stable phases is not high so thermal fluctuations make Φ hop between the differentstates. In this case we rather study the distribution of T r (Φ) to infer the transition value of r for theorder ↔ non-uniform transition. From the distribution we find that ground state is discontinuouslychanging. Moreover the distribution of T r (Φ) is very non-gaussian, rather flat near zero, suggestingthat there are degenerate states characterised by zero and non-zero values
T r (Φ). So from ourresults the transition of ordered phase to other phases is always first order for the parameter spacewe have explored. All other transition in model appeared to be continuous transitions.The results from previous studies have shown that by doing simple scaling phase boundaries fordifferent N coincide [19]. This expected scaling with N does not occur up to the value of N we havestudied, though it is the largest so far. We studied the behavior of the triple point for large N and itscales approximately linearly with N . However our results seem to agree with the previous studiesin that the non-uniform phase survives the continuum limit. In our analysis we considered primarily T r (Φ),
T r (Φ ), Φ etc.. However analysis of the full matrix may result in better understanding ofthe phase structure, such as variants of the non-uniform ordered phases.0 VI. ACKNOWLEDGEMENT
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