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GROUND STATE OF 2D QUANTUM

arXiv:hep-th/9212114v1 18 Dec 1992PUPT-1354TAUP-2013-92hep-th/9212114December 1992GROUND STATE OF 2D QUANTUMGRAVITY AND SPECTRAL DENSITYOF RANDOM MATRICESMarek Karliner ∗Alexander Migdal †Boris Rusakov ∗∗School of Physics and AstronomyRaymond and Beverly Sackler Faculty of Exact SciencesTel-Aviv University, 69978 Tel-Aviv, Israel† Physics Department andProgram in Applied and Computational MathematicsFine Hall, Princeton UniversityPrinceton, NJ 08544-1000AbstractWe compute the exact spectral density of random matrices in the ground stateof the quantum hamiltonian corresponding to the matrix model whose double scalinglimit describes pure gravity in 2D. We show that the non-perturbative effects are verylarge and in certain cases dominate the semi-classical WKB contribution studied inthe earlier literature.

The physical observables in this model are the loop averageswith respect to the spectral density. We compute their exact ground-state expectationvalues and show that they differ significantly from the values obtained in the WKBapproximation.

Unlike the alternative regularizations of the nonperturbative 2D quan-tum gravity, based on analytic continuation of the Painlev´e transcendent, our solutionshows no pathologies.1

The discovery [1] of the double scaling limit of the matrix models [2] of 2D Gravity andits relation [3] to the KdV hierarchy opened the way to the study of the nonperturbativephenomena in Gravity and String theories. These phenomena are supposed to be describedby the so called string equationx =Xtl(2l + 1)Rl [C](1)where x is the scaling combination of cosmological constant and parameter 1/N of the genusexpansion, tl are the mass parameters of the theory and Rl [C] = Cl + derivatives are theGelfand-Dikii differential functionals corresponding to the specific heat C(x) as the potentialof the Schr¨odinger operator ( see e.g.

[3] for details).The string equation is in general a differential equation for C(x) of the order correspond-ing to the highest l in the sum. As observed in [4], for odd l there is the natural boundaryconditionC(x) →x1/lx →±∞(2)which uniquely determines the solution.For the even l case situation is more complicated.

The above boundary condition wouldlead to a complex solution, corresponding analytic continuation of the divergent matrixintegral [5]. The attempt to define the even l case as the limit tl+1 →0 of the odd case [6]does not work, since the limit does not exist.One way to interpret this paradox is to claim the even case and in particular the l = 2 caseof pure 2D Gravity inconsistent.

That would be an exciting possibility : the nonperturbativecreation of matter in Gravity. However, the matter here is not quite physical, as it does notcorrespond to the unitary multiplets of conformal field theory.

The partition functions ofthe l > 2 models with higher genus are not positive [7].So, in fact, this interpretation implies that no 2D Gravity could exist at nonperturba-tive level. This is too serious a statement to make without investigating the alternativeinterpretations.One such interpretation, suggested by Marinari and Parisi [8], along the lines of the gen-eral method by Greensite and Halpern [9], seems particularly appealing.

They do not addany ad hoc terms to stabilize the equation, but rather modify the basic definition of thematrix model to make it meaningful for arbitrary potential. In Ref.

[8] the spectral densityof the resulting quantum hamiltonian was studied in the semi-classical WKB approximation.It turns out to be possible to go beyond the WKB approximation and to obtain the exactspectral density of this hamiltonian. The necessary framework was set up in Ref.

[10], inwhich the set of equations generalizing the string equation was derived and studied numer-ically. Quantum effects, invisible in the WKB approximation, turned out to be extremelyimportant.The prescription of Refs.

[8],[9] has attracted a considerable amount of attention, [11]-[21]. In particular, there have been some interesting attempts to study the properties ofthe model numerically at finite N. The problem encountered in these studies is the sameas with all numerical studies of the matrix models, namely the double scaling limit requiresvery large N, because of the N 1/5 dependence of physical observables.2

The goal of the present paper is to apply the analytic and numerical methods of thenote [10] to derive the exact spectral density of random matrices in the ground state of2D Gravity. As a result, we are able to demonstrate that the non-perturbative effects arevery large and in certain cases completely dominate the semi-classical WKB contribution.The physical observables in this model are the loop averages with respect to the spectraldensity.

We compute their exact ground-state expectation values and show that they differsignificantly from the values obtained in the WKB approximation.We begin by discussing the method of Greensite and Halpern for the simplest example ofa one dimensional integral. Consider a positive action S(x) depending on a single variablex:Z =Zdx exp [−S(x)](3)By definition, the average of an operator Q(x) in the action S is given by⟨Q⟩Z ≡Zdx Q(x) exp[−S(x)]Zdx exp [−S(x)](4)If we defineψ0 ≡exp [−S(x)/2] /√Z(5)the expectation value (4) can be written⟨Q⟩Z = ⟨ψ0| Q |ψ0⟩(6)ψ0 is always positive and therefore it is always possible [22],[9],[8] to construct a quantum-mechanical hamiltonian H = −d2/dx2+V (x), such that ψ0 is its ground-state wave-functionwith the ground-state energy equal to zero:Hψ0 = 0−ψ′′0(x) + V (x)ψ0(x) = 0(7)V (x) = ψ′′0(x)/ψ0(x)It is instructive to rewrite this hamiltonian in a manifestly positive form, as a product of anoperator and its hermitean conjugate,H = −ddx + S′(x)2!

ddx + S′(x)2! (8)ψ0 is annihilated by the second factor and hence Hψ0 = 0.

When the action S is unboundedfrom below, the average (4) is ill-defined. Formally, (5) is still annihilated by (8), but ψ0 isnow not normalizable and therefore cannot be the ground state of H. On the other hand,H is a positive hamiltonian by construction and therefore a normalizable ground state Ψ0must exist.

Clearly the true ground state Ψ0 must be different from ψ0 and must have aneigenvalue e0 > 0. Consider the action S(x) = x2 −g2x4.

The average (4) is ill defined, but3

the power-series expansion in g2 exists and can be computed explicitly. Each order in theexpansion in powers of g2 involves only moments of x with a gaussian measure,exp[−S(x)] = exp(−x2)∞Xk=0(g2x4)nn!⟨Q⟩lZ =Zdx Q(x) exp(−x2)lXn=0(g2x4)n/n!Zdx exp(−x2)lXn=0(g2x4)n/n!

(9)Any sensible definition of average with the action S(x) ought to reproduce the perturbationexpansion (9) and it should reduce to (4) for a positive action. The quantum-mechanicalexpectation value in the true ground-state Ψ0 of H satisfies both constraints, we thereforetake it as the definition of the average with the action S:⟨Q⟩Z = ⟨Ψ0| Q |Ψ0⟩(10)This prescription can be applied to matrix models,eF = Z =Zdφ exp [−βtr Uk(φ)](11)where φ is a hermitean N × N matrix and the critical potentials are given byUk(φ) =Z 10dtt1 −1 −t(1 −t)φ2k(12)At large φ the φ2k term dominates, with the coefficient (−1)k+1 R 10 dt tk−1(1 −t)k. Thereforethe k-even critical potentials are unbounded from below and it is necessary to provide aprescription for defining the average (11) beyond perturbation expansion.Following thework of Greensite and Halpern [9] and Marinari and Parisi [8], we adopt the prescription(10).

After all, the only existing justification of the matrix models as theories of gravityis perturbative, in the sense of the genus expansion. The quantum mechanical definitionis a priori as good as the statistical one, but has the advantage of being guaranteed tomake sense for all models.

The extra bonus of this prescription is the relation to the onedimensional supersymmetric string theory [8], with dynamically broken supersymmetry inthe even l case. For practical purposes the supersymmetry seems to be useless so far, butone may hope to relate it to the physical supersymmetric string theories.The simplest case of an unbounded potential in a matrix model is the cubic potentialU(φ) = φ22 −gφ33(13)In order to proceed, it is useful to change variables from the hermitean N × N matrices totheir eigenvalues, {λ1, .

. .

, λN}:eF = Z =Z YidλiYi

The factorYi

The integration measure is then simplyZ Yidλi. It follows thatthe effective wave function corresponding to (14) isΨeff=Yi

The fermionicproperty is generic – it is due to the Jacobian of the transformation from the matrices φ totheir eigenvalues. On the other hand, the fact that the effective fermions are free is only truewhen the original potential U(φ) is cubic in φ.The effective potential depends on the free variable λ and on three parameters of theoriginal matrix-model action: N, β and g, Veff= Veff(N, β, g; λ).

We are interested in thecritical properties of the theory in the double scaling limit,x = β −NN1/5 ;N →∞;x ∼N0(17)with the critical point at x = 0. For small positive x the effective potential Veff(N, β, g; λ)has a double-well shape.

In the double scaling limit (17) the depth of the left well scales like∼N2, while the depth of the right well scales like ∼x3/2 ∼N0. Despite this, it is the right,tiny well, which is responsible for all the interesting critical phenomena in the double scalinglimit.

All the cubic potentials of the type (13) are in the same universality class. At thecritical point the depth of the second well goes to zero, and the potential has an inflectionpoint, instead of a double minimum.Since we are interested in having the critical point at x = 0, it is necessary to chose avalue of g such that for x = 0, i.e.

for β = N, the extremum is an inflection point:∂Veff∂λλ=λ0= 0 ;∂2Veff∂λ2λ=λ0= 0(18)leading tog =1q12√3(19)λ0 =√3 + 14√3(20)We are interested in the scaling properties of the Fermi energy, or ground state, of thehamiltonian (16). The scaling properties are determined by the tiny right well, but the bulkof energy levels is in the left, bigger well.

The Fermi energy, eF can be written aseF = e0F + esF(x)(21)5

where the bulk part, e0F , is determined by the left well and esF(x) is the x-dependent scalingpart that we are interested in. The depth of the left well ∼N2, while the depth of the rightwell ∼N0.

In terms of magnitude, eF is completely dominated by e0F. The hamiltonian(16) can only be solved by some approximation procedure, analytical or numerical.

It ismandatory to first isolate and “magnify” the scaling part, otherwise any approximate resultfor eF will be dominated by e0F and the x dependence of the fine-structure will be lost.To isolate the scaling part, we expand the effective potential in x and in λ −λ0 aroundthe inflection point and obtain the scaling potential v(y):Veff(N, β, g; λ) = Veff(N=β, g; λ=λ0) + β4/5α2/5v(y)(22)v(y) = y33 −ǫ y(23)where x is given by (17) andy = (λ −λ0)α1/5β2/5,ǫ = xα−3/5,α = 2 +√344√3(24)The critical properties of the theory are thus determined by the scaling hamiltonianh = −∂2∂y2 + v(y)(25)The cubic potential v(y) (not to be confused with the original cubic potential (13) ) is for-mally unbounded from below. This is not a problem, however, since v(y) is only meaningfulin the scaling region and the full potential Veff(λ) is bounded.

The strategy for computingthe fine structure of the Fermi energy is then as follows. An exact numerical solution for thespectral density of the scaling hamiltonian (25) can be obtained through the powerful meth-ods of Gelfand and Dikii [24].

The exact solution can then be compared with the spectraldensity obtained from the WKB approximation. The two differ in the scaling region only,since WKB is completely adequate in the large left well.

Their difference converges fast,yielding the contribution of the scaling region.In order to obtain the density of states, we first write down the general equations of theFermi-gas theory in one dimension. It is convenient to scale out β from the Hamiltonian,and introduce the resolvent,G(e, y) = ⟨y| (h −e)−1 |y⟩(26)The particle density ρ(e, y) is related to the imaginary part of the resolvent,ρ(e, y) = 1π Im G(e + i0, y)(27)and the spectral density ν(e) is given by the integral of ρν(e) =Z ∞−∞dy ρ(e, y).

(28)6

The normalization of ν(e) is fixed by the fact that (15) and (16) describe N non-interactingfermions. We rescale the spectral density by N and the equation for the Fermi energy eF istherefore1 =Z eF−∞de ν(e).

(29)The WKB particle density is given byρW KB(e, y) =12πqe −v(y)(30)with the corresponding WKB spectral density νW KBνW KB(e) =Z y1−∞dy12πqe −v(y)(31)where y1 is the first root of e −v(y). νW KB(e) can be expressed in terms of elliptic integrals.When there are 3 real roots y1 < y2 < y3 of v(y) −e = 0, from eqs.

(3.131.1) and (8.112.1)of Ref. [25] we obtainνW KB(e) =1π√y3 −y1K(p);p =sy3 −y2y3 −y1(32)When there is one real root y1 and two complex-conjugate roots y∗2 = y3, one can useeq.

(8.126.1) of Ref. [25] to transform (32) into the formνW KB(e) =1π√η K(sin(φ/2));y2 −y1 = ηeiφ(33)The “Fermi energy” corresponding to the WKB solution, eW KBFis defined by1 =Z eW KBF−∞de νW KB(e).

(34)Outside the scaling region the exact spectral density is equal to the WKB density,ν(e) = νW KB(e)fore < elow,(35)where elow is some large negative value of e. The WKB solution for the Fermi energy eW KBFfor ǫ > 0 was found in [8]. It exactly coincides with the bottom of the second well, which inour normalization iseW KBF= −23ǫ3/2.

(36)Combining (29), (34), (35) and (36), we obtain the final implicit equation for the Fermienergy eF as determined by the scaling region:Z eW KBFelowde νW KB(e) =Z eFelowde ν(e). (37)7

The exact solution for spectral density can be obtained from the Gelfand-Dikii equation [24]for the resolvent G(e, y)−2G∂2G/∂y2 + (∂G/∂y)2 + 4(v −e)G2 = 1(38)Under usual circumstances, when the parameters are not fine-tuned to magnify the scalingregion, the G-D equation is rather useless, as the direct solution of the Schr¨odinger equationis simpler. However, in our case it is just what we need.

Differentiating (38) with respectto y, dividing by G and taking the imaginary part, we obtain an ordinary third-order lineardifferential equation for the continuous particle density ρ(e, y) in the double scaling limit,ρ′′′ = 2v′ρ + 4(v −e)ρ′(39)At large |y|, the general asymptotic form of the solution for ρ, correct up to terms ∼O(1/|v|),can be written in terms of three integration constants c±1 , c±2 and c±3 :ρ →c±1 exp2Z y √vdy+ c±2 exp−2Z y √vdy+ c±3q|v|;y →±∞(40)To chose the proper boundary conditions, physical intuition about the system must be used.At y →∞the potential v(y) grows large, so only the decaying exponential is left, whilec+1 = c+3 = 0. This leaves one free parameter, c+2 , the overall normalization of ρ, whichcan be determined as follows.

We start from the asymptotic solution (40) at some largepositive y = y0, with c+1 = c+3 = 0 and some arbitrary initial value of c+2 . We then solvethe differential equation (39) numerically, evolving down to large negative values of y, wherev(y) < 0 and √v is complex.

The solution there is of the form (40), with c−1 , c−2 , c−3 ̸= 0, i.e.it contains two oscillating exponentials, plus a powerlike term. The oscillations representa pure quantum effect, invisible in WKB expansion.

If we average ρq(e −v) over theseoscillations, the c−1 and c−2 terms disappear and we should obtain12π, according to theWKB solution. This means that the solution ought to be multiplied by a constant such thatc−3 = 1/2π.

This is the missing normalization condition for the density.In practice averaging over the oscillations is rather tricky, as it involves delicate cancel-lations between the positive and the negative contributions. There is, however, a better wayof extracting c−3 from ρ.

If ρ is given by (40), then, up to terms ∼O(1/|v|),c−3 = √e −v ρ −ρ′′4(e −v)! (41)The prescription (41) has the advantage that it is local in y and only requires the knowledgeof ρ′′, which is readily available in any code used for solving differential equations.

Thereliability of (41) can easily be tested, by verifying that the result for c−3 is independent of y.We employed the differential equation solving routine ODE, described in detail in Ref. [26].The routine is very stable and extremely easy to use.

It is based on a variable-step, variable-order Adams method (explicit linear multistep method). The variable-step feature is essen-tial, because of the rapid crossover from smooth to oscillatory behavior.8

Once ρ(e, y) is known, ν(e) is in principle given by (28). In practice, it is more efficientto solve for the integral of ρ(e, y):ξ(e, y) = −Z ∞ydy ρ(e, y) ;ρ(e, y) = ∂ξ(e, y)/∂y(42)which satisfies the 4-th order differential equationξ′′′′ = 2v′ξ′ + 4(v −e)ξ′′(43)yielding both ρ(e, y) and ν(e) at the same time,ν(e) = −limy →−∞ξ(e, y)(44)The boundary conditions for the derivatives of ξ are those for ρ and its derivatives.Inaddition, ξ(y0, e) = 0, where y0 is large and positive.While solving eq.

(43) for the various values of e, we store the values of ρ(e, y) on a two-dimensional grid in the [e, y] plane. These are to be used later in obtaining the distributionof the eigenvalues of a random matrix (see eq.

(45) below).The tail of the integral in (42) converges rather slowly, since ρ(e, y) ∼|y|−3/2 for largenegative y. In that region, however, the integral of ρ(e, y) is very well approximated by theintegral of ρW KB(e, y), eq.

(30). The corresponding WKB tail τ(y) can expressed in termsof elliptic integrals, which we calculated numerically using standard methods.The solution of the G-D equation for ρ at ǫ = 1 and e = −23, which corresponds to thebottom of the second well is presented in Fig.

1. The quantum effects are huge!Once ν(e) is known, we obtain the Fermi energy eF(ǫ) from eq.

(37). The result is shownin Fig.

2.We can now obtain the distribution Ωǫ(y) of the eigenvalues of a random matrix in thedouble scaling limit. This is done by integrating ρ(e, y) with respect to e, up to eF(ǫ):Ωǫ(y) =Z eF (ǫ)−∞de ρ(e, y)(45)In practice the integrand is obtained by interpolating ρ(e, y) from the previously stored two-dimensional grid in the [e, y] plane.

The resulting Ωǫ(y) is shown in Fig. 3. and comparedwith the corresponding WKB density,ΩW KBǫ(y) =Z eW KBF(ǫ)−∞de ρW KB(e, y) = 1πqeW KBF−v(y)(46)The contribution of the non-perturbative effects in physical observables can most easilybe seen by computing loop averages for positive l:Wǫ(l) = 1ZǫZ ∞−∞dη Ωǫ(η)elη(47)whereZǫ =Z ∞−∞dη Ωǫ(η)(48)9

The integral in (47) converges fast on both ends. For large positive y, one is in the classicallyforbidden region, where Ωǫ decreases faster than exponentially (see Fig.

3), and for largenegative y the exponential factor in the integrand ensures exponential convergence. Thusin practice it turns out to be sufficient to take a finite upper limit of the integration aty = yright, i.e.

the value of y at which Ωǫ(y) becomes sufficiently small.Comparison of Wǫ(l) with its WKB analogue, defined by eqs. (47),(48) with ΩW KBǫ(y)instead of Ωǫ(y), shows that the non-perturbative part of spectral density takes over theWKB part for large l. The results for Wǫ(l) for 0 < l < 6 are shown in Fig.

4.Note, that the exact loop average, unlike the WKB one, reaches the minimum, and thengrows! This is a direct consequence of the fact that the scaling eigenvalue density Ωǫ(y) isdifferent from zero in the classically forbidden region y > 0.In physical terms, this striking phenomenon is explained as follows.The larger theboundary l of our two dimensional space, the more area it could encircle, and hence, themore handles could be attached to it.

The number of surfaces with unlimited genus growsas exponential of the fifth power of the area [7], and a typical area grows as the square ofthe length of its boundary. So, we might expect a very fast growth of the loop average dueto nonperturbative contributions.Another comment: in the literature, doubts were raised [27] , whether the Marinari-Parisiprescription (in the WKB approximation) could preserve the positivity of the loop average.We did not observe such pathologies.

Moreover, we believe that with correct nonperturbativedefinition of the loop average the positivity is guaranteed.The point is that the spectrum of the random matrix in the Marinari-Parisi model isinfinite, it covers the whole real axis. Our spectral integral for the loop average is manifestlypositive definite.However, in the WKB approximation, studied in [27], there are several disconnectedregions of the classically allowed motion with gaps in between.

Our prescription is to addall these contributions with positive sign. Being rewritten as a contour integral, this wouldproduce a set of loops encircling each allowed region anticklockwise, or a single contour goingin the imaginary direction to the right of all the regions.For positive l one can close the contour in the left half plane, yielding the original spectralintegral.

For negative l one could close the contour in the right half plane, yielding zero, asthere are no singularities of the integrand.AcknowledgementsThis research was supported in part by grant No. 90-00342 from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel, and by the Basic ResearchFoundation administered by the Israel Academy of Sciences and Humanities.One of us(A.M.) would like to thank the Institute for Advanced Study at the Tel-Aviv University forhospitality.

This work was partially supported by the National Science Foundation undercontract PHYS-90-21984.10

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Figure Captions• Figure 1. The particle density ρ(e, y), solution of eq.

(39), for ǫ = 1 and e = −23.Dash-dotted line denotes the WKB solution.• Figure 2. The Fermi energy, eF(ǫ), obtained from eq.

(37). The diamonds denote theactual values computed, the continuous curve is plotted to guide the eye.

Dash-dottedline denotes the Fermi energy in the WKB approximation, eW KBF(ǫ).• Figure 3. The scaling eigenvalue density Ωǫ(y).

The WKB approximation is plottedas a dash-dotted line.a) Ωǫ(y) and ΩW KBǫ(y), ǫ = −1, 0, 1.b) ∆Ωǫ(y) = ΩW KBǫ(y) −Ωǫ(y), ǫ = −1, 0, 1.• Figure 4. Loop average Wǫ(l), eq.

(47), for ǫ = −1, 0, 1. The WKB approximation isplotted as a dash-dotted line.13

y

y

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