A modified c = 1 matrix model with new critical behavior

1 December 1994

PHYSICS LETTERS B ELSEVIER

Physics Letters B 340 (1994) 35-42

A m o d i f i e d c = 1 matrix m o d e l w i t h n e w critical b e h a v i o r Steven S. Gubser, Igor R. Klebanov Joseph Henry Laboratortes, Prmceton University, Princeton, NJ 08544, USA Recewed 18 July 1994 Editor. H. Georgl

Abstract

By introducing a f dt g (Trq~2(t))2 term into the action of the c = 1 matrix model of two-dimensional quantum gravity, we find a new critical behavior for random surfaces The planar limit of the path integral generates multiple spherical "bubbles" which touch one another at single points. At a special value of g, the sum over connected surfaces behaves as A210gA, where A is the cosmological constant (the sum over surfaces of area A goes as A -3). For comparison, in the conventional c = 1 model the sum over planar surfaces behaves as A2/log A.

1. Introduction

In recent years a considerable effort has been devoted to understanding the statistical mechanics of random surfaces. This problem is relevant to Polyakov string theory in non-critical dimensions. Thanks to the matrix model techniques, we now have a thorough understandmg of random surfaces embedded in one dimension [ 1-3] or less [4]. Little is known, however, about the physically more interesting higher dimensional embeddings. With this in mind, it is important to continue formulating and solving new matrix models. One interesting modification of the conventional one-matrix model was solved in Ref. [5]. A new type of critical behavior arises when a term of the form g ( T r qb2)2 is added to the action, leading to an integral over the N × N hermitian matrix • of the form log

f

0370-2693/94/$07 00 © 1994 Elsevier Science B V All rights reserved

SSD10370-2693(94)01 192-3

ex, E (1) The large N limit of this quantity has an interesting geometrical interpretation. Feynman graphs of the perturbation theory in A generate the usual connected closed random surfaces, while the g ( T r qb2)2 term can glue a pair of such surfaces together at a point. This point can be resolved into a tiny neck (a wormhole), so that the network of such touching surfaces can be assigned an overall genus. In the leading large N limit one picks out the surfaces of genus zero, which look like trees of spherical bubbles such that any two bubbles touch at most once, and a bubble is not allowed to touch itself. The authors of Ref. [ 5 ] found a critical line in the (A, g) plane where the free energy Eq. (1) becomes singular. There exists a critical value gt such that, for g < g , the singularity in the function of A is characterized by Tst~= - 1/2. In this phase the touching of random surfaces is irrelevant and one finds the

S.S. Gubser, I R Klebanov / Physics Letters B 340 (1994) 35-42

36

conventional c = 0 behavior. For g > gt, on the other hand, y,t~= 1/2, and one finds branched polymer behavior, which is dominated by the touching. Most interestingly, for g =g,, the authors of Ref. [5] found a new type of critical behavior with 7s~ = 1/3. The interpretation of this is not completely clear, but it has been suggested that at this point one has an effective theory for random surfaces embedded in more than one dimension [6]. In fact, after some fine tuning, theories with ys~ = 1/n can be formulated [5-7]. In view of these results, a continued study of the g(Tr 42) 2 term is warranted. In this paper we investigate its effect on random surfaces embedded in one dimension. The relevant model is the matrix quantum mechanics defined by the path integral _2"= f ~ 4 ( t ) T/2

×exp[-N

f

dt(Tr(142+142-¼A44)

--T/2

\q - g ( T r 4 2) 2/1 N ]d

(2)

where 4 is an N × N hermitian matrix. T will be taken to oo in the end, but we retain it as a finite quantity for the time being. In the large N limit we find a critical line where the free energy is singular. As in the model Eq. ( 1 ), there exists a critical value gt which separates the conventional matrix model behavior (in this case c = 1) from the branched polymer phase. For g = gt we find a new critical behavior: the sum over connected surfaces of genus zero behaves as A21og A, where A is the cosmological constant. This should be compared with the usual c = 1 behavior, A2/log A. Our modified critical behavior is much simpler after we carry out the inverse Laplace transform: the sum over surfaces of fixed area A behaves as A - 3. The corresponding formula for c = 1 is more complicated due to logarithmic corrections. The organization of this paper is as follows. In Section 2, we show how the modified c = 1 model can be reduced to a fermionic system. In Section 3, we find conditions on the ground state of the fermionic hamiltonian which allow us to find the critical points of the model and evaluate Ys~.We end with a brief discussion in Section 4.

2. R e d u c t i o n to f e r m i o n s

The sum over connected surfaces is given, up to a factor, by the free energy 1

5 r = ~ log . ~ .

(3)

In the N ~ oo limit, the leading term of 5 r (which is the only term we will consider in this paper) is O(N2). This dominant term generates touching spherical bubbles embedded in one dimension. To be more explicit, the single trace terms in Eq. (2) generate planar, quartic Feynman graphs embedded in one dimension: these are the individual bubbles, which look just like ordinary c = 1 surfaces. The double trace term, g(Tr 42) 2, gives rise to a pair of "touching" propagators. When these propagators are incorporated into two different bubbles, we interpret the bubbles as touching. Arrangements where a pair of bubbles touch in more than one place, or where a bubble reaches around and touches itself, are suppressed by a factor 1 / N 2 and thus correspond to O( 1) corrections to 5 r. The path integral in Eq. (2) can be written as a transition amplitude . ~ = (f[ e x p ( - N H T ) l i ) , 1

02

1

2N2042 + T r ( ~ 4

H= -

2

1

-zA4

g ( T r 4 2 ) 2.

4

) (4)

N In the T--+oo limit, ~r, is - N times the ground state energy of this hamiltonian. The ground state must be a SU(N)-invariant function of 4, which is to say a symmetric function of the eigenvalues x, of 4: dependence on "angular" degrees of freedom can only raise the energy. As in the c = 1 model, the key step is to pass to a fermionie system, in which the ground state is an antisymmetric function of the x,. If A(x,) is the Vandermonde determinant of the eigenvalues x, of 4, then 1

H= - -

zl(x,)

Hfa(x,)

where He is the fermionic hamiltonian:

(5)

S.S Gubser, I.R Klebanov/ Physlcs Letters B 340 (1994) 35-42

)

2N z Ox 2 + ~x, - ~Ax a,

Hf = t=l

g (Tr qb2) 2. N

(6)

The (Tr ~ 2 ) 2 term introduces interactions among the fermions. These interactions can be taken care of by a self-consistent field approach, analogous to Hartree-Fock calculations in multi-electron atoms. For the purpose of finding the ground state energy of Hr to leading order in N, it is permissible to make the replacement (Tr qb 2) 2--~ 2(Tr q~ 2)Tr q~ 2 - (Tr • 2) 2 .

(7)

An intuitive way to justify this replacement is to consider Tr 4 2 as varying slightly around its expectation value: Tr q~ = (Tr q~) + 6Tr cl~. Expanding (Tr q~)2 to first order in 6 T r q~ yields Eq. (7). Applying Eq. (7) to Eq. (6) turns Hf into a¢ 3nstant plus a sum of N single-particle hamiltonians:

H f - - , g N (1TrCI)2> 2

_~_

5(1°

,=1

--

2

1

N p(x) = - ~/2[er - U(x) ] .

2NEOx2 + ~x, - ~Ax,

Given A and g, the values of eF and c are determined by a normalization condition and a self-consistency condition:

if

dxp(x)=l,

,f

~1

.

(11)

=gc2+eF - - ~1 I d x { 2 [ e F - U ( x ) ] } 3/2.

(12)

As a check on the self-consistent field method, one may easily show that

41fdxx~o(x)

a<;Tr4>

4

Og

1 02 2N 20x 2 + l ( 1 - - 4 g c ) x 2 - - 1 A X 4 ,

C= ( 1 T r y 2 )

dxxZp(x)=c.

If we let E be 1/N times the ground state energy of Hf, then J ---=E N2

(8)

We write the single-particle hamfltonian as

(10)

77"

OA

4

-2g(1Trcl)2)x2).

h(x) =

techniques. We regard the N fermions with single-particle hamiltonian h(x) as a Fermi gas in the potential U(x) = ½( 1 - 4gc)x 2 - ~Ax 4. If the Fermi energy is e~, then the particle density in x is

OF. 1

37

2

(9)

Since our modified matrix model reduces to free fermions, its solution proceeds essentially along the lines of the usual c = 1 solution. However, the necessity of imposing a self-consistency condition adds an extra ingredient which, as we will show, can modify the nature of the critical behavior.

These results agree with what one finds by explicit differentiation of Eq. (2) with respect to A and g. At this point it is convenient to rescale x and h(x) as follows: y=

X,

h(y)

=

A ( 1 - 4 g c ) 2 h(x)

1 02 2~ 20y 2

+ V(y)

,

( 1 - 4 g c ) 3/2 N

3. Leading order solution

3=

To obtain the ground state energy of Hf to leading order in N as N--* o,, it suffices to use semi-classical

V(y) = ½ y 2 _ ¼ y 4 .

,X

(14)

S.S. Gubser, LR Klebanov / Physlcs Letters B 340 (1994) 35-42

38

We also define a rescaled Fermi energy A PF = (1 --4gc) 2 eF. The highest possible value for PF is Pc = 1/4: at this energy, the fermions completely fill the local well of

V(y). Upon rescaling, Eq. ( 11 ) becomes yc

A

( 1 - 4 g c ) 3/z

= 1 f 7r

dy~12[pF_V(y)]

-yc

---l ( p F ) , yc

A2c

( 1 -- 4gc) 5/2

_ 1 f -a"

dyy2x/2[pv_V(y)l

--y¢

- J(PF),

(15)

V/1

where Yc = - v/1 - 4 / X F . c Can be eliminated from these equations to yield

I2(A~I/3[1-(A)2/3]

(16)

g= TW Eqs. (15) and (16) represent necessary conditions on the ground state, but as we shall see, they are not quite

30

sufficient, as there are sometimes two possible values of PF for a given A and g. It will not be hard to tell which represents the true ground state. Eq. (16) determines a family of curves in A-g space that depends on the single parameter PF- Fig. 1 shows twenty of these curves. The outer envelope of all the curves is the locus of critical points where the perturbation series that generates the random surfaces begins to diverge. To see this, suppose one varies A and g from 0 toward criticality. As long as the point (A, g) is inside the outer envelope, there is a ground state of Hp and E is finite. But as soon as (A, g) crosses the envelope, there is no solution to Eq. (15), so E and hence ~ r are undefined. There are two regions of the critical curve. For g g,, the critical curve is tangent to a curve given by Eq. (16) with PF < Pc. Because of the different critical behaviors we shall find in these two regions, we shall call the region g < g, the c = 1 region and the region g > gt the branched polymer region. While the critical behaviors in these two regions have been found in other models, right at g = g, we find something new. In all the calculations we present, we continue to use the quartic potential, A Tr qb4, that we started out with in Eq. (2). As a check on universality, however, we have performed the same calculations with a cubic potential, and we found the same regions of the critical curve and the same three types of critical behavior.

3.1. Critical behavior in the c = 1 region

24£

~A

3rr

Fig. 1. The family of curves representing all possible solutions (A, g) to the self-consistent field equations Each curve has constant Fermi energy p , . The critical curve is the outer envelope of all these curves The transition point from the c = 1 region to the branched polymer regton is (A,, g,) = (( 1/31r) (~)3/2, ( 1 / 12~') (3~ ) a/2), where curves with different/~ start crossing one another When two curves do cross, the one with the l o w e r / ~ (or, eqmvalently, more negative slope) represents the physical ground state. Thus, for each curve, only the part to the right of the contact with the envelope is physical.

Let us pick a point (Ac, go) on the c = 1 part of the critical curve. To determine the critical behavior of E as A ~ Ac with g fixed at go, it suffices to calculate OE/ 0A, which from Eq. (13) can be shown to be

0,~.

13 '

'I

yc

K = K ( p F ) - --

"rr

dyy4~12[pF_V(y)] .

(17)

To analyze the beha~'for of I, J, and K for PF near Pc, let us set PF = Pc -- P and write

S.S. Gubser, I.R. Klebanov / Physics Letters B 340 (1994) 35-42

yc

(1 -y4>dr

_yc AYf

-Y2)

=- fi 7r

39

I

+-\/2’ dy( 1 +y2) IT

I


-1

as Z.L+O.

(20)

Z(cL,-p)=Zc+~Z,

Keeping terms of order CLand larger, we find

J(pc-p)=Jc+a,

w= Sz+3zCJ.L,

K(pc-/.L)=K,+SK.

(18)

,. Y..

1

dY i _-yr421Pu, - V(Y) 1

a/L=-,

=5

log p+O(l)

g:

,

(19)

(22)

Using this along with Eq. (2 1) one can derive

SA A,=2 To calculate W and SK, let us first define r,=Gz?F, so that = +(Y:

-Y*>


-y2)

Now, aJ --_ acL

=z(l-22).

$3(3;+15$

which means

PF -v(Y)

(21)

Now we perform a perturbation of A away from A, with g fixed at g,. If one defines z= (h/Z)“3 and zC= ( A,/ZC)“3, Bq. ( 16) becomes

A standard result from c = 1 calculations is aZ

SK= SZ+4Z,p.

az aP

.

5_622SJ +45 1-z: 1-32: 1-32: z,

P*

(23)

In this and all the rest of our calculations in this section, we retain terms only up to O(p) . Before continuing with the calculation of the critical behavior of E, let us note how Elq. (23) can be used to determine the point (A,, gt) where the Z+ = Z.L= curve ceases to coincide with the critical line. Eq. (23) actually applies to any point (A =Ig,) on the /+ = pCcurve, except when 1 - 32: = 0. If we set SA= 0, then Eq. (23) becomes (24)

(1 -Y’)dY \/(YZ -Y2)(z

+--ti’ TT

I

dy = 31,

-1

as Z.L-+O, aK -ap

aZ z

-yZ)

which is a condition on A, that determines where the Z+ = pC- Z.Lcurve intersects the ,FL~ = Z.L~ curve. Now, (A,, g,) is the point where the Z+ = pC curve just starts to meet curves with lower pr. Hence A, can be determined by letting Z.L -+ 0 in Eq. (24). Since the dominant term on the right hand side of Bq. (24) is the one involving SZIZ,, we must have zz +5/6, whence A,=(1/3~)($)3’2andg,=(1/12~)($)3’2. Now let us see how E behaves near criticality. Retaining terms up to 0( Z.L)as usual,

40

S S. Gubser, LR Klebanov /Phystcs Letters B 340 (1994) 35--42

OE

1 K

OA

4Z 2 13

E = (analytic in A) at- ~

1 K~[

-

4Z2 g

(26

1+

---6

l_--~Z2 ] ~ -

t - ( ~ - - 3 0 ~ 3 / z ]" l - 3 z c ]

+ (higher order terms) . (25,

Eqs. (23) and (25) lead to the existence of two different types of critical behavior. First, let us take A~> At (g¢
1 K~ 4z 2 13 ( X

22-15zcESA 701-3z~ ) 1 + 3 5 - 6 Z c 2 A~ + 3 5 - 6 z ~ z ~

(26)

As in the case of ordinary c = 1, we define the cosmological constant A by A = A c - A . To leading order in A,

/z =

7r 1 1 - 3 z ~ 2 A V/2 z 3 5 - 6z~z log A '

(27)

which is the behavior typical of ordinary e = 1. The leading nonanalytic behavior of E is E = (analytic in A) 97r3

1 (1-3Z~ 2 A 2

32V~ Z~5 ~ , ]

log A

+ (higher order terms) ,

(28)

which again is typxcal of c = 1. Our conclusion is that, for g < g,, the touching of random surfaces is irrelevant in the sense that it does not destroy the c = 1 behavior. If we fix g = g, and let A ~ At, then we get a different critical behavior: Eq. (23) becomes 6A/At = - 5/z, so that the leading order relation between A and/x is just A /z= 5)%

"r/'3z~2 log A

l-zZ~'~81

(29)

The fact that A and/x are simply proportional to each other is the crucial feature of the new critical behavior. In contrast to Eq. (28), the universal part of E is now

(30)

Similar calculations are possible for perturbations of g away from criticality with A held fixed. If one sets g =go-/'and A = Ao then in the c = 1 region one finds that the singularity of E is ~ - F 2 / l o g / ' . For A = At, E ~ F 2 log F instead. 3.2. The branched polymer region It is clear from Fig. 1 that the region above the curve given by Eq. (16) with ~F =/x~, but below the critical curve, is covered not once but twice by the family of curves in Eq. (16). Given (A, g) in this region, there are thus two values of/z~ and c that satisfy the selfconsistent field equations Eq. (15). The true ground state of H corresponds to the solution with the lower energy E. It is an unsurprising but slightly non-trivial fact that the solution with lower E is the one with lower /-tF. At any given point in Fig. 1 where two curves cross each other, the more down-sloping curve is the one with lower/XF, and thus represents the ground state. Let us fix g and consider how A varies with/ZF. The singularity of the perturbation theory in A occurs at the maximum of A (/XF). For any g, A(0) = 0. The behavior of A with increasing/ZF, however, depends on the value ofg. For g < gt, A is an increasing function of P-F and the maximum occurs at the end-point/~F =/z~. The slope diverges logarithmically as /XF approaches/Zc, which leads to the c = 1 critical behavior. For g = gt, A. is an increasing function of/Zv with a slope that is finite everywhere. The finiteness of the slope at/Zv =/zc leads to the modified critical behavior. For g > g,, A(/XF) increases to a quadratic maximum at a value of/xv less than /Zc. This maximum corresponds to A reaching the critical c u r v e : A(/,/,m) = A c. If we increase/zv past/Zm, A(/xF) decreases; this region represents only unphysical solutions to the self-consistent field equations. In other words, if for a given value of A there are two possible values of/xF, then we pick the smaller one because it is analytically connected to the perturbative region near A = 0. A graph of A (/zF) with g = ~2gt is shown in Fig. 2.

S.S. Gubser, LR Klebanov / Physics Letters B 340 (1994) 35-42

41

4. Discussion

\

By performing the inverse Laplace transform of Eq. (30) we find that, for g =gt, the sum over connected genus zero surfaces of area A is 36

~-(a) = ~

(3] 1/2 [~,] ~3 A

-3

(34)

This is precisely the KPZ scaling [ 8 ] with Ystr= 0. For the conventional c = 1 matrix model we instead have

[2] 1 Fig. 2. A graph of A versus p~ for g fixed at ~g,. Only the points with / ~ < P,mrepresent physical solutions to the self-consistent field equations

The main point is that we can perturb A away from its critical value Ac by lowering ~,LF: if we write A = A ~ - A and/ZF = /Zm-- /Z, then A = a/z 2 + (higher order terms),

(31 )

where a > 0 is some constant. From Eq. (17) and Eq. (31) it is simple to show that OE

1

0A

A~2/3 _..c 1

"

~ a d/xF (32)

~r(A) ~ A3(IogA) 2 .

(35)

Thus, the critical behavior we have found is actually simpler than the conventional c = 1 behavior. Is it possible that we have penetrated the c = 1 barrier, as suggested in Ref. [6] ? Although we do not have a definite answer, it seems likely to us that the new critical point can be described by two-dimensional string theory. The scaling violations for c = 1 have been attributed [9] to the unusual dependence of the tachyon potential on the Liouville field, T(~b) ~ q~e6. It appears that, if the tachyon potential has the ordinary Liouville form, T(~b) ~ e 4', then the simpler scaling of Eq. (34) follows. Clearly, more work is needed to settle the stringy interpretation of our new critical theory. We believe that there are more interesting calculations that can be performed in the modified c = 1 matrix model. Such calculations should shed light on the critical behavior for higher genus surfaces and, ultimately, on the double-scaling limit.

K / I 7/3 always has positive derivative, so the universal

behavior of E is

Note added

E = (analytic in A) - ¢xA 3/2 + (higher order terms) ,

(33)

where a is a positive number. Predictably, the same power law occurs in the universal part of E when we perturb g = g c - F with A fixed at A¢: the leading nonanalytic behavior of E is ~ -/-,3/2. It is remarkable that as soon as gc exceeds g,, the Fermi level/~F starts falling. Since the universal behavior of the c = 1 model is driven by the Fermi energy approaching criticality, we regard the dropping of the Fermi energy as the reason for the transition to the branched polymer phase.

After this paper was completed, we learned of an interesting paper by Sugino and Tsuchiya [ 10] m which results similar to ours were obtained by means of collective field theory.

Acknowledgement We thank David Gross for interesting discussions. This work was supported in part by DOE grant DEFG02-91ER40671, the NSF Presidential Young Inves-

42

S.S. Gubser, LR. Klebanov /Physics Letters B 340 (1994) 35-42

tigator A w a r d P H Y - 9 1 5 7 4 8 2 , J a m e s S. M c D o n n e l l Foundation grant No. 91-48, and an A.P. Sloan Foundation R e s e a r c h Fellowship.

References [ 1] D.J. Gross and N. Miljkovlc, Phys. Lett. B 238 (1990) 217, E. Brezin, V. Kazakov and AI. B. Zamolodchikov, Nuel. Phys B 338 (1990) 673; P. Ginsparg and J. Zmn-Justin, Phys. Lett. B 240 (1990) 333; G. Pansi, Phys. Lett. B 238 (1990) 209. [2] D J. Gross and I.R. Klebanov, Nucl. Phys. B 344 (1990) 475

[3] I.R. Klehanov, String Theory in Two Dimensions, in: String Theory and Quantum Gravity '91, eds. J. Harvey et al. (World ScienUfie, River Edge, NJ, 1992) [4] D.J. Gross and A.A. Migdal, Phys Rev. Lett. 64 (1990) 717; M. Douglas and S. Sbenker, Nucl Phys. B 335 (1990) 635; E. Brezin and V. Kazakov, Phys. Lctt. B 236 (1990) 144. [5 ] S.R. Das, A. Dhar, A M. Sengupta and S R. Wadia, Mod. Phys Lett A5 (1990) 1041. [6] L. Alvarez-Gaum6, J.L. Barbon and C Cmkovle, Nucl. Phys. B 394 (1993) 383. [ 7 ] G. Korcbemsky, Mod. Phys. Lctt. A 7 (1992) 3081; Phys. Lett. B 296 (1992) 323. [8] V.G. Knizhnik, A.M. Polyakov and A.B. Zamolodehikov, Mod. Phys. Lett A3 (1988) 819. [9] J Polehinski, Nucl. Phys. B 346 (1990) 253. [ 10] F. Sugino and O Tsuehiya, Cntical behavior in c = 1 matrix model with branebang interactions, UT-Komaba 94-4, hep-th9403089 (1994).