Grassmann Matrix Quantum Mechanics
GGrassmann Matrix Quantum Mechanics
Dionysios Anninos (cid:52) , Frederik Denef † ,(cid:93) and Ruben Monten (cid:93), † (cid:52) School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA. † Department of Physics, Columbia University, 538 West 120th Street, New York, New York 10027 (cid:93)
Institute for Theoretical Physics, KU Leuven, Belgium
Abstract
We explore quantum mechanical theories whose fundamental degrees of free-dom are rectangular matrices with Grassmann valued matrix elements. We studyparticular models where the low energy sector can be described in terms of abosonic Hermitian matrix quantum mechanics. We describe the classical curvedphase space that emerges in the low energy sector. The phase space lives on acompact K¨ahler manifold parameterized by a complex matrix, of the type dis-covered some time ago by Berezin. The emergence of a semiclassical bosonicmatrix quantum mechanics at low energies requires that the original Grassmannmatrices be in the long rectangular limit. We discuss possible holographic in-terpretations of such matrix models which, by construction, are endowed with afinite dimensional Hilbert space. a r X i v : . [ h e p - t h ] D ec Introduction
Models with matrix like degrees of freedom make numerous appearances throughoutphysics. Applications range from the study of the spectra of heavy atoms to modelsof emergent geometry [1, 2, 3, 4, 5, 6]. In this paper we will concern ourselves with aparticular class of quantum mechanical models whose degrees of freedom are purelyfermionic rectangular matrices ψ Ai , with A = 1 , ..., M and i = 1 , ..., N . The matricestransform in the ( M, N ) bifundamental representation of a U ( M ) × SU ( N ) symmetrygroup. In a Lagrangian description of the system, transition amplitudes can beexpressed as path integrals over Grassmann valued paths ψ Ai . Grassmann matricesnaturally appear as the supersymmetric partners of bosonic Hermitian matrices insupersymmetric matrix quantum mechanical theories such as the low energy worldlinedynamics of a stack of N D0-branes in type IIA string theory [3, 7] or the Marinari-Parisi matrix model [8]. Our interest is in quantum mechanical models consisting of only the Grassmann matrices.Ordinary integrals over Grassmann matrices were studied extensively in [9, 10, 11].There, it was shown how the problem of Grassmann matrix integrals at large N , M can be expressed as an eigenvalue problem for the composite N × N matrixΦ ij = (cid:80) A ¯ ψ iA ψ Aj , which is effectively bosonic. Unlike bosonic matrices, a Grass-mann valued matrix cannot be diagonalized and characterized in terms of eigen-values. Instead, the authors were able to analyze the model by diagonalizing Φ ij .Certain features of the Φ ij integral, such as a contribution to the potential of theform tr log Φ, were shown to be universal and specifically related to the Grassmannnature of the original problem. Along a similar vein, emergent bosonic matrices fromspin systems were considered in [12, 13]. The models of interest in our work can beviewed as multi-particle quantum mechanical models of fermions which can occupy afinite set of single particle states | A, i, α (cid:105) , labeled by the matrix indices. In particularthe Hilbert space is finite dimensional. Fermionic multi-particle models often ariseas lattice models in condensed matter physics, where there is typically an assump-tion about some sort of nearest-neighbour interaction between the fermions reflectingspatial locality. In contrast, the class of models of interest in our paper have no suchnotion of spatial locality. They are described by actions of the form: S = (cid:90) dt i (cid:88) A,α,i ¯ ψ αiA ∂ t ψ αAi − tr N × N V (cid:88) A,α,β ¯ ψ αiA σ αβ ψ βAj . (1.1)1he potential V ( x ) is an N × N matrix valued function. The index α is an spinor indexassociated to the d -dimensional rotation group, but we will focus on the particularcase of d = 3 and take the σ αβ to be the ordinary Pauli matrices. We will also demandthat the potential V ( x ) be SO (3) invariant. An example of such a model was studiedin [14]. The objects we wish to understand are path integrals over { ¯ ψ αiA ( t ) , ψ αAi ( t ) } rather than simple integrals. In particular, we study to what extent the Grassmannmatrix models at large N and M can be described in terms of a composite bosonicmatrix degree of freedom. We then describe several features of the emergent bosonicmatrix quantum mechanical systems. We focus on the case where V ( x ) is quartic inthe Grassmann matrices, but the techniques we develop can be used more generally.As mentioned, our models have a finite dimensional Hilbert space. In this sensethey differ from many of the quantum mechanical models studied in the context ofholography, such as the D0-brane quantum mechanics or N = 4 super Yang-Mills,where the systems have an infinite space of states, even at finite N . On the otherhand, several proposals have been made throughout the literature suggesting that theholographic dual of a de Sitter universe (or at least its static patch) is indeed a systemwith a finite dimensional Hilbert space [15, 16, 17, 18, 19, 20]. Our considerationsare particularly similar, in spirit, to those of [15, 16] where the basic building blocksare also taken to be a large collection of fermionic operators. Part of our motivationis to understand to what extent systems with a finite Hilbert space can give rise toa holographic description with a dual gravitational theory in an appropriate large N type limit. In order for this to be the case, bosonic variables (such as the Hermiteanmatrices) should emerge from the discrete variables, at least at low energies and inan appropriate large N limit. The models studied in this work serve as toy modelswhere this can be seen explicitly, and we can examine to what extent the bosoniceffective degrees of freedom adequately capture the physics and when this descriptionbreaks down.The first part of the paper provides a detailed study for the N = 1 case, in whichthe degrees of freedom are organized as vectors. We derive several results regardingthe physics of the effective composite degree of freedom ¯ ψ αA σ αβ ψ βA . We show towhat extent the theory is described by three bosonic degrees of freedom x = ( x, y, z )transforming as an SO (3) vector. The Euclidean path integral is expressed as a pathintegral over x and a low velocity expansion is developed at large M . We study the Part of the reason for choosing an SO (3) index is to mimic the examples of matrix quantummechanics that appear in holography, where the matrices are labeled by a similar rotational index.We discuss this further in the outlook. CP . Some of theresults in this section have appeared in several contexts (see for example [21, 22, 25]).However, certain aspects of our treatment are novel and furthermore our treatmentnaturally generalizes to the matrix case. This is studied in the second part of thepaper, where now the effective theory becomes that of three bosonic Hermitian N × N matrices Σ aij , with a ∈ { x, y, z } . The matrix Σ aij transforms in the adjoint of SU ( N )and is an SO (3) vector. The matrix analogue of the emergent classical phase space isidentified as a compact K¨ahler manifold, first introduced by Berezin [26]. The K¨ahlermetric is parameterized by a complex N × N matrix Z ij . We discuss how the Z ij and Z † ij relate to the description of the system in terms of the Σ aij as well as the originalGrassmann matrices. The volume of the K¨ahler metric computes the dimension ofthe Hilbert space captured by the (quantized) classical phase space. It is shown toprecisely match the dimension of the U ( M ) invariant Hilbert space of the originalGrassmann theory. We end with an outlook discussing speculative connections of ourmodels to holography. In this section we discuss a quantum mechanical model in which the degrees of free-dom are a vector ψ αA of complex Grassmann numbers, with A = 1 , . . . , M and α = 1 ,
2a spinor index of SU (2), the double cover of the rotational group SO (3). Our systemhas a 2 M complex-dimensional Hilbert space of states. The purpose of the section isto analyze a simplified version of the matrix model studied in the next section, whichhowever still retains some of the salient features.We focus on an action with quartic interactions of the specific form: S = (cid:90) dt i ¯ ψ αA ∂ t ψ αA + g (cid:16) ¯ ψ αA σ aαβ ψ βA (cid:17) (cid:16) ¯ ψ γB σ aγδ ψ δB (cid:17) , (2.1)where it is understood that the A and α indices are summed over and the σ aαβ = { σ xαβ , σ yαβ , σ zαβ } are the three Pauli matrices. The model has an SU (2) × U ( M ) globalsymmetry group. The ( ¯ ψ αA ) ψ αA transform in the (anti-)fundamental representationof U ( M ) and SU (2).Upon canonical quantization, the non-vanishing anti-commutation relations be-tween the fermionic operators are given by { ¯ ψ αA , ψ βB } = δ αβ δ AB . The SU (2) genera-3ors working on these operators are given by ˆJ a = ¯ ψ αA σ aαβ ψ βA /
2. The U ( M ) generatorsare given by: ˆ J n = ¯ ψ αA T nAB ψ αB + c ˆ I δ n , n = 0 , , . . . , M − . (2.2)The T nAB with n > SU ( M ) subgroup of U ( M ), and T AB = δ AB generates the U (1) subgroup of U ( M ). c is a normal ordering constantthat appears as a possible central extension of the U (1). As expected, [ ˆ J n , ˆ J a ] = 0.We take g > g so that g = 1. The Hamiltonian of the system is proportional to the normal ordered square of theangular momentum operator:ˆ H = − : ¯ ψ αA σ aαβ ψ βA ¯ ψ γB σ aγδ ψ δB := − ˆJ · ˆJ := − J · ˆ J + 3ˆ n , (2.3)where ˆ n ≡ ¯ ψ αA ψ αA , commutes with the ˆ J a . If we view the index A as a lattice site, thesystem above is describing two-body SU (2) spin-spin interactions of spin-1/2 fermionsbetween all M possible lattice sites, each with equal strength. From (2.3), it followsthat the the eigenstates | J, m ; n (cid:105) can be labeled by their total angular momentum J , their angular momentum m in the z -direction and their eigenvalue n with respectto the ˆ n operator. The energy of | J, m ; n (cid:105) is simply E = − J ( J + 1) + 3 n . For M >
1, the ground states | g (cid:105) are the ( M + 1) states in the maximally spinning spin- M/ J = 0 state with n = 2 M has maximal energy. We canconstruct the full Hilbert space by acting with the ¯ ψ αA operators on the particular J = 0 state | (cid:105) , defined to be the state annihilated by all the ψ αA . For instance theground state with maximal spin- z angular momentum is | M/ , M/ M (cid:105) = (cid:81) A ¯ ψ A | (cid:105) and has energy E g = − M ( M − A we have two states with vanishing angular momentum in the z -direction, and a spin-1 / H = (0 ⊕ / ⊕ ⊗ M . The degeneracies for a given angular momentum inthe z -direction can be obtained from the partition function: Z [ q ] = tr q (cid:80) A J zA = M (cid:88) k =0 (cid:18) Mk (cid:19) q M/ − k/ (2.4)From the above partition function, we can also obtain the degeneracies of the multi-4
10 15 J5.0 × × × × d J Fig. 1: Plot of d J vs. J for M = 70.plets with total spin J : d J = (cid:18) MM + 2 J (cid:19) − (cid:18) MM + 2( J + 1) (cid:19) . (2.5)Indeed, there is exactly one state with m = M/
2, which is part of the maximallyspinning (ground state) multiplet. There are 2 M states with m = ( M − /
2, eachof which is part of a spin-( M − / M (2 M − m = M/ −
1, one is already part of the maximally spinning multiplet,leaving (2 M − M −
1) spin-( M − / J z yields the formula above. As expected, (cid:80) J (2 J + 1) d J = 2 M and d M/ = 1. At large M , using the Stirling approximation, we find a large degeneracyof 2 M /M J = 0 states. Moreover, for small J/M , we can use the approximations: (cid:18) MM + 2 J (cid:19) ≈ (cid:18) MM (cid:19) e − J /M , (cid:18) MM + 2( J + 1) (cid:19) ≈ (cid:18) MM (cid:19) e − J +1) /M . (2.6)From these we can derive that d J peaks at J ≈ (cid:112) M/
8. We show a plot of thedegeneracies d J in figure .The d J are the exact degeneracies for the operator ˆ˜ H = (cid:16) ˆ H − n (cid:17) , with eigen-values ˜ E J = − J ( J + 1). At large M , the d J are also approximately the degeneraciesof ˆ H for several of its lowest lying states. For example, the energy difference betweenthe ground state with J = M/ J = ( M − / M to leading order. The ˆ n operator does not split the energies of the ( M + 1)-fold degenerate states in the ground state multiplet, but it does split the energiesof the 2 M distinct J = ( M − / M multiplets sep-5rated by an O (1) amount in energy. Since the energies of both the J = M/ J = ( M − / − M at large M , to leading order in M the d J are agood approximation of the degeneracies of ˆ H for the two lowest lying states. Moregenerally, considerations similar to those leading to (2.5) lead to the formula for thedegeneracies of distinct J -multiplets with a given n : d J,n = (cid:18) M n + J (cid:19)(cid:18) M n − J (cid:19) − (cid:18) M n + J + 1 (cid:19)(cid:18) M n − J − (cid:19) , (2.7)where n = 2 J, J + 2 , . . . , M − J . When J ∼ M/ J is large enough to cause overlaps betweentheir energy levels and those of multiplets with different J . For example, the J = 0states have energies ranging between E ∈ [0 , M ] which can easily be seen to overlapwith the energy levels of the J = 1 / U ( M ) symmetry, the spectrum wouldhave changed significantly. For instance, by selecting the normal ordering constant c = − M , the only gauge invariant states are the ( M + 1) maximally spinning groundstates. We would now like to recast the Euclidean path integral of the theory as a Euclideanpath integral of a bosonic (mesonic) variable and understand several features of themodel in terms of the bosonic degree of freedom. The Euclidean path integral com-putes features in the low energy sector the system. For instance, the generatingfunction of vacuum correlation functions is given by: Z [ ξ αA , ¯ ξ αA ] = (cid:90) D ¯ ψ αA D ψ αA e − S E [ ¯ ψ,ψ ] − (cid:82) dτ ¯ ξ αA ψ αA − (cid:82) dτ ¯ ψ αA ξ αA , (2.8)where the Euclidean action S E is obtained from − iS by a Wick rotation t = − iτ .Upon introducing an auxiliary three-vector x and integrating out the Grassmannvariables, this can be recast as: Z [ ξ αA , ¯ ξ αA ] = (cid:90) D x det ( − ∂ τ + σ · x ) M e − (cid:82) dτ r / e − (cid:82) dτ ξ αA ( − ∂ τ + σ · x ) − αβ ¯ ξ βA , (2.9) As a simple check, (cid:80) n d J,n = d J reproduces (2.5). Furthermore, (cid:80) J d J,n (2 J + 1) = (cid:0) Mn (cid:1) , where J = n/ , n/ − , . . . covers positive integer or half-integer values, depending on whether n is evenor odd. r = | x | . From the partition function we can read off the effective action forthe x degree of freedom: S eff = − M Tr log ( − ∂ τ + σ · x ) + (cid:90) dτ r . (2.10)As it stands, the above action is highly non-local in τ . We would like to understandunder what conditions this action can approximated by a small velocity expansion.Generally speaking there is no a priori reason for this to be the case in a quantumsystem, given that the spectrum is discrete and one cannot continuously change thekinetic energy. However, one may hope that it would be a valid approximation atlarge M . We will see that this is the case. It is useful to diagonalize the 2 × x · σ for each τ . Since the σ are traceless, we take some U ∈ SU (2) such that U † σ · x U = r σ z for each τ . The U matrix is parameterized by a unit vector n = (sin θ cos φ, sin θ sin φ, cos θ ). Explicitly: U = (cid:32) cos θ e − iφ sin θ e iφ sin θ − cos θ (cid:33) . (2.11)It then follows that:det ( − ∂ τ + σ · x ) M = e M Tr log ( − ∂ τ − U † ˙ U + r σ z ) . (2.12)Notice that we can transform the above functional determinant under the time repa-rameterization symmetry τ → f ( τ ) , r ( τ ) → ˙ f ( τ ) r ( f ( τ )) , U ( τ ) → U ( f ( τ )) , (2.13) e M Tr log ( − ∂ τ − U † ˙ U + r σ z ) → e M Tr log ˙ f e M Tr log ( − ∂ τ − U † ˙ U + rσ z ) . (2.14)The first factor on the right-hand side of (2.14) is independent of U and r and canbe absorbed into the overall normalization of the path integral. The above symmetrycan therefore be used to set r to a constant in performing a small velocity expansionof the functional determinant. It follows from this that no time derivatives will begenerated for r . In other words, if we view the symmetries (2.13) as (0 + 1)-dimensional diffeomorphisms of theworldline, r ( τ ) becomes the einbein which can always be gauge fixed to a constant.
7e expand (2.12) in powers of υ a σ a = i U † ˙ U by expanding the logarithm. Thezeroth order term is the effective potential governing r . Going to Fourier space, thecomputation becomes: V eff = − M (cid:90) dω π log (cid:0) ω + r (cid:1) + r − M r + r , (2.15)where we have regulated the ω -integral by differentiating once with respect to r andre-integrating it back while setting the constant of integration to zero. Note that theeffective potential is minimized at r = 2 M for which V ( min ) eff = − M . To leading orderin M this agrees with the exact ground state energy of the system E g = − M ( M + 2).The first order term in the velocity expansion is given by: S (1) kin = − M (cid:90) dω π ( − iω + rσ z ) − αβ iσ aαβ ˜ υ a (0) = i M (cid:90) dτ (1 − cos θ ) ˙ φ , (2.16)where ˜ υ a ( l ) is the Fourier transform of υ a at frequency l . The linear velocity piece S (1) kin is the phase picked up by a unit charge moving on the surface of a two-sphere,in the presence of a magnetic monopole of strength M/ S (2) kin = M (cid:90) dτ r (cid:0) ( υ x ) + ( υ y ) (cid:1) = M (cid:90) dτ r (cid:16) ˙ θ + sin θ ˙ φ (cid:17) , (2.17)where in the right-hand side we have expressed the answer in terms of x , but nowwritten in spherical coordinates. The higher order terms can be similarly computedand they contain even powers of time derivatives of the angular variables divided byone less power of r . Denoting the characteristic frequency for some particular motion of θ and φ by ω c , the condition that there is a small derivative expansion is: ω c (cid:28) r . (2.18)For r near the minimum of the effective potential, we have ω c (cid:28) M . Hence, for large M there is a parametrically large range of frequencies allowing for a small velocityexpansion. In appendix B we consider a modified vector model where the leading kinetic piece is (2.17). β r min Fig. 2: Plot of value of r minimizing V eff ( β ) vs. 10 × β for M = 70. Notice thatthe value stays close to 2 M = 140 all the way down to β ∼ /M . As was previously noted, the original Grassmann system contains a large number ofhigh energy, i.e. J = 0, states at large M . On the other hand the ground state energyis E g = − M ( M − Z [ β ] = Tr e − β ˆ H at large β is dominated by the ground states and goes as:lim β →∞ Z [ β ] = ( M + 1) e M ( M − β , (2.19)whereas at small β we have simply the dimension of the Hilbert space:lim β → Z [ β ] = 2 M . (2.20)The transition between these two behaviors occurs at β ∼ /M .We now consider the finite temperature partition function as a Euclidean pathintegral over x . We must integrate out the Grassmann numbers with anti-periodicboundary conditions along the thermal circle. In analogy to previous calculations,we can compute the thermal effective potential. What changes is that the ω -integralsare replaced by sums over the thermal frequencies ω n = 2 π ( n + 1 / /β with n ∈ Z .The thermal effective potential thus becomes: V eff ( β ) = − Mβ (cid:88) n ∈ Z log (cid:0) ω n + r (cid:1) + r − Mβ log cosh rβ r . (2.21)As before, the sum has been regulated by differentiating with respect to r .For large β , the minimum of V eff is at r = 2 M as for the zero temperature9nalysis. We can find the critical point for r in a large β expansion. To first order: r = 2 M (cid:16) − e − Mβ + . . . (cid:17) . (2.22)From this we see the tendency of r to decrease upon increasing the temperature. Atsmall β , we can Taylor expand: V eff ( β ) = r − β M r + O ( β ) . (2.23)We see that for β (cid:46) /M the thermal potential is minimized at r = 0. In figure weshow a plot for the values of r minimizing V eff ( β ) as we vary β .When r is near zero, we can no longer assume that the kinetic contributionsare small and thus our analysis breaks down. This as an indication that the hightemperature phase does not have a reliable small velocity description in terms of x .Instead, the correct description requires taking into account the full set of Grassmanndegrees of freedom. So far we have introduced the variable x as a convenient integration variable tocapture correlations in the vacuum state and thermal properties. Here we wouldlike to point out that in a fixed large angular momentum sector, there is some moresignificance to x .Following Bloch, we define a collection of coherent states built from the state | v (cid:105) , which has the lowest angular momentum in the z -direction and hence is also aminimal energy state. In other words | v (cid:105) = (cid:81) A ¯ ψ A | (cid:105) . We can act on | v (cid:105) with thespin raising operator ˆJ + = ˆJ x + i ˆJ y to generate states in the maximally spinningmultiplet, | ¯ z (cid:105) = 1(1 + z ¯ z ) M/ e ¯ z ˆJ + | v (cid:105) , z ∈ C . (2.24)These states are not orthogonal, but they constitute an over-complete basis of theHilbert space of the maximally spinning multiplet, (cid:104) w | ¯ z (cid:105) = (1 + w ¯ z ) M (1 + w ¯ w ) M/ (1 + z ¯ z ) M/ , (cid:90) d z M + 1 π (1 + z ¯ z ) | ¯ z (cid:105) (cid:104) z | = I . (2.25)The purpose of these states is to describe, with minimal uncertainty, points on the S of spin directions. Indeed, the angular momentum expectation value defines a point10n S – through the stereographic projection – with decreasing uncertainty in thelarge M limit J a ≡ (cid:104) z | ˆJ a | ¯ z (cid:105) = M | z | ) (cid:0) z + ¯ z, i (¯ z − z ) , | z | − (cid:1) , (2.26) (cid:104) z | ( ˆJ a − J a ) | ¯ z (cid:105)(cid:104) z | ˆJ a | ¯ z (cid:105) = 2 M .
One may ask about transition amplitude between two such states: (cid:104) z N | e − iT ˆ H | ¯ z (cid:105) forsome given Hamiltonian ˆ H built out of the ˆ J a . The result is [23, 24]: (cid:104) z N | e − iT ˆ H | ¯ z (cid:105) = (cid:90) D z D ¯ z ( M + 1) π (1 + z ¯ z ) e iS ( z, ¯ z ) , (2.27)with S = i M (cid:90) dt ( z ˙¯ z − ˙ z ¯ z )1 + z ¯ z − (cid:90) dt H ( z, ¯ z ) , (2.28)where H ( z, ¯ z ) ≡ (cid:104) z | ˆ H | ¯ z (cid:105) . The boundary conditions are z ( T ) = z N and ¯ z (0) = z .For our particular choice of Hamiltonian, H ( z, ¯ z ) = − M ( M − z can be viewed as a complex coordinate parameterizing a two-dimensionalphase space. From the linear velocity piece in (2.28) we note that the phase space iscurved and compact, with K¨ahler metric: ds = 2 M dzd ¯ z (1 + z ¯ z ) . (2.29)This is the Fubini-Study metric on CP ∼ = S , and we occasionally refer to it as theBloch sphere. The symplectic form is given by the K¨ahler form and the large M limit plays the role of the small Planck constant limit. Time evolution of a function A ( z, ¯ z ) in the emergent classical phase space is governed by the Poisson bracket, i.e.˙ A ( z, ¯ z ) = { A ( z, ¯ z ) , H ( z, ¯ z ) } p.b. = i M − (1 + z ¯ z ) ( ∂ ¯ z H∂ z A − ∂ ¯ z A∂ z H ). The SU (2)symmetry of the original Grassmann model acts on z as: z → ( αz + β )( γz + δ ) − , (cid:32) α βγ δ (cid:33) · (cid:32) α βγ δ (cid:33) † = I × . (2.30)Since the classical phase space has finite volume, we recover the fact that the under-lying system has a finite number of ground states. The complex coordinate ( z, ¯ z ) can11ig. 3: Schematic plot of classical and nearby trajectories on the Bloch sphere forsome H ( z, ¯ z ), contributing to the path integral (2.28). At large M the classicaltrajectory dominates.be related to the spherical coordinates introduced in (2.11) by identifying the expec-tation value (2.26) with the bosonic variable x introduced in the previous section.The stereographic projection then gives z = e iφ cot θ/
2. With this identification, thelinear velocity term in (2.28) becomes precisely the one found in (2.16). Thus, we seethat certain transition amplitudes are captured by a real time path integral betweendifferent points localized on an S . This allows for physical interpretation of the ( θ, φ )coordinates as real time degrees of freedom, rather than merely integration variables.We can quantize this low energy effective theory to leading order in the velocityexpansion. This becomes the quantum mechanics of an electrically charged particlewith unit charge. Its motion is confined to a unit sphere in the presence of a magneticmonopole of strength M/ M the groundstates are given by the M lowest Landau levels, each with energy E g = − M for ourchoice of Hamiltonian. Due to the Dirac quantization condition, we recover that M must be an integer.We have seen how certain low energy features in the original Grassmann theoryare described in the language of the effective bosonic degree of freedom x . Instead of12aximally spinning states built out of anti-commuting creation operators, we havelowest Landau levels of a charged particle. The energies (at least in the the low energyregime) are registered by the absolute value of x . We have observed the breakdown ofthe bosonic effective theory at high temperatures. Certain features were particular toour model. But others such as the presence of linear velocity terms and the absenceof a kinetic term for r may be general features of a larger class of models. At thispoint we proceed to generalize these observations to the case where we have a matrixworth of Grassmann degrees of freedom. The goal of this section is to analyze a matrix version of the vector model studiedabove. Given that the model is more complicated, we will not be able to attain asexplicit a description, however we will uncover and generalize several of the featuresfound in the vector model.
Our degrees of freedom are now 2
M N complex rectangular Grassmann matrices, ¯ ψ αiA and ψ αAi , with A = 1 , . . . , M and i = 1 , . . . , N . As before, α is an SU (2) spinor index.The dimension of the Hilbert space now becomes 2 NM . The Grassmann elementsobey the anti-commutation relations { ψ αAi , ¯ ψ βjB } = δ αβ δ ij δ AB .We will focus on the following action: S = (cid:90) dt i ¯ ψ iA ∂ t ψ Ai + g ( ¯ ψ iA σ a ψ Aj )( ¯ ψ jB σ a ψ Bi ) . (3.1)When N = 1, the above action reduces to the one analyzed in the previous section.The model exhibits a U ( M ) × SU ( N ) × SU (2) global symmetry. The SU (2) acts bysimultaneously rotating all the Grassmann elements. The capitalized index of ( ¯ ψ αiA ) ψ αAi transforms in the (anti-)fundamental representation of U ( M ) whereas the lowercase index transforms in the (anti-)fundamental of SU ( N ).The Hamiltonian of the model is given by:ˆ H = − g (cid:88) i,j,A,B : ¯ ψ iA σψ Aj ¯ ψ jB σψ Bi : (3.2) We have and will continue to suppress the SU (2) spinor index in ψ αAi to avoid cluttering ofindices.
13f we view the A index as a lattice site, our system describes SU (2) spin-spin inter-actions of the spin-1/2 fermions. But now the fermions are labeled by an additionalquantum number, the color index i = 1 , , . . . , N , which can be exchanged throughthe interaction. Since interactions between all lattice sites have the same strength,the model exhibits no notion of spatial locality.We will analyze g > g = 1. Unlikethe vector case previously studied, the combinatorial problem of finding the exactspectrum of ˆ H seems to be rather difficult and we have not solved it. Instead, we willtry to extract information about the low energy sector of the theory by going to aneffective description in terms of bosonic matrices. Before doing so, we will establishsome further properties about the operator algebra. U (2 N ) operator algebra The analogues of the spin operators ˆJ a = (cid:80) A ¯ ψ A σ a ψ A / U ( M ) invariant N × N spin matrix operators: ˆS aij = (cid:80) A ( ¯ ψ iA σ a ψ Aj ) / SU (2), as well as in the adjoint of the SU ( N ). Introducing an additional operator ˆS ij = (cid:80) A ( ¯ ψ iA σ ψ Aj ) /
2, with σ the 2 × ˆS aij , ˆS bkl ] = 12 δ ab (cid:16) δ kj ˆS il − δ il ˆS kj (cid:17) + i (cid:15) abc (cid:16) δ kj ˆS ail + δ il ˆS bkj (cid:17) , (3.3)[ ˆS ij , ˆS akl ] = 12 (cid:16) δ kj ˆS ail − δ il ˆS akj (cid:17) , (3.4)[ ˆS ij , ˆS kl ] = 12 (cid:16) δ kj ˆS il − δ il ˆS kj (cid:17) . (3.5)The N diagonal components of the ˆS aij generate N copies of the usual su (2) algebra.The above operators can be arranged in a 2 N × N Hermitian matrix σ µαβ ⊗ ˆS µij (with µ = { , x, y, z } summed over) and hence they generate a u (2 N ) algebra. Theyact as ψ αAi → ψ αAi G αβij and ¯ ψ αiA → ( G αβij ) − ¯ ψ βjB with G αβij = e iλ αβij ∈ U (2 N ) and λ αβij = λ µij σ µαβ the elements of a 2 N × N Hermitian matrix.The U (2 N ) symmetry manifestly commutes with the U ( M ) group and preservesthe anti-commutation relations between the ψ αAi and ¯ ψ αiA . Our Hamiltonian (3.2) doesnot commute with the full U (2 N ) but rather the U ( N ) diagonal subgroup generatedby the ˆS ij . When N = 1, the U (2 N ) algebra becomes nothing more than the global SU (2) symmetry of the vector model, which not only commutes with the U ( M ) global14ymmetry but also with the Hamiltonian. We introduce three N × N Hermitian bosonic matrices Σ aij = ( Σ xij , Σ yij , Σ zij ). Inanalogy with the vector case, we introduce them as auxiliary variables which aregiven on-shell by Σ aij = 2 ˆS aij . Upon integrating out the ψ αAi , the generating functionof vacuum correlations of ψ and ¯ ψ can be expressed as a Euclidean path integral overthe Σ ij : Z [ ξ αAi , ¯ ξ αiA ] = (cid:90) D Σ e M Tr log( − ∂ τ + R ) − tr (cid:82) dτ Σ · Σ e (cid:82) dτ ¯ ξ αiA ( − ∂ τ + R ) − ij,αβ ξ βAj . (3.6)We have defined R ≡ Σ x ⊗ σ x + Σ y ⊗ σ y + Σ z ⊗ σ z . We also denote the full functionaltrace by ‘Tr’ and reserve the ‘tr’ symbol for the ordinary matrix trace. It followsfrom this definition that tr R = 0. The global SU ( N ) symmetry acts as Σ → U Σ U † .Also, Σ transforms as in the three-dimensional (vector) representation of the global SU (2) symmetry group. We can also write down the generating function for vacuumcorrelations of the composite spin-matrix operator ˆS aij . These are computed by thecorrelation functions of Σ ij itself: Z [ J aij ] = (cid:90) D Σ e M Tr log( − ∂ τ + R ) − tr (cid:82) dτ Σ · Σ e tr (cid:82) dτ J · Σ − tr (cid:82) dτ J · J , (3.7)where J aij are sources for the ˆS aij . It is worth noting that, unlike the N = 1 case, the ˆS aij no longer commute with the Hamiltonian and thus non-trivial time correlationsamongst them may exist.We now proceed to study the validity and properties of the ‘small velocity’ ex-pansion of det ( − ∂ τ + R ) = exp [Tr log ( − ∂ τ + R )]. Since R is a 2 N × N Hermitianmatrix, we can diagonalize it as U † R U = λ with λ = diag [ λ , . . . , λ N ] , U ∈ U (2 N )and λ n ∈ R . Note that due to the tracelessness of R , not all λ n can have the samesign. Similar to the N = 1 case, in the diagonal R frame, we can write the functionaldeterminant as: Tr log ( − ∂ τ + R ) = Tr log (cid:16) − ∂ τ − U † ˙ U + λ (cid:17) . (3.8)With the above expression we can again use the time reparameterization symmetry τ → f ( τ ) , λ n ( τ ) → f (cid:48) ( τ ) λ n ( f ( τ )) , U ( τ ) → U ( f ( τ )) , (3.9)15o see that the effective action will be independent of ˙ λ n , analogous to how the vectormodel is independent of ˙ r . Using the propagator: G ( ω ) = diag (cid:104) ( − iω + λ ) − , . . . , ( − iω + λ N ) − (cid:105) , (3.10)we can expand the logarithm in powers of the Hermitian matrix υ = iU † ˙ U . Eachterm in the expansion will be endowed with a U (2 N ) symmetry taking U † ˙ U → Λ † (cid:16) U † ˙ U (cid:17) Λ and λ → Λ † λ Λ with Λ ∈ U (2 N ).The linear velocity contribution to the effective action is: S (1) kin = − iM tr (cid:90) dω π G ( ω ) ˜ υ (0) = − i M (cid:88) m sgn( λ m ) (cid:90) dτ (cid:104) i U † ˙ U (cid:105) mm . (3.11)The ˜ υ ( l ) is the Fourier transform of υ at frequency l . To define the above ω -integralwe have put a cutoff at large ω , performed the exact integration and then taken thelarge cutoff limit. The kinetic piece containing two time derivatives in U ( τ ) is givenby: S (2) kin = − M (cid:90) dω dl (2 π ) G ( ω ) ˜ υ ( l ) G ( ω ) ˜ υ ( − l )= M (cid:88) n,m (cid:90) dτ (cid:104) i U † ˙ U (cid:105) nm Λ mn (cid:104) i U † ˙ U (cid:105) mn , (3.12)with Λ mn = 1 / | λ m − λ n | and the sum running only over the pairs ( n, m ) for which λ n and λ m have opposite signs. The reason why only pairs of λ m with opposite signappear in the sum is that the integral appearing in (3.12): I mn = (cid:90) dω π − iω + λ m ) 1( − iω + λ n ) (3.13)vanishes whenever λ n and λ m have the same sign. It is interesting to note thatthe effective kinetic piece of the theory, and hence what we mean by the dynamicalcontent, depends on the particular distribution of eigenvalues λ n .Having obtained expressions for the first few velocity dependent terms in theeffective action, we can estimate when the low velocity expansion is valid. Denotingthe characteristic frequency for some motion as ω c , then in order for S (1) kin to be largecompared to S (2) kin one requires: ω c (cid:28) λ n N . (3.14)16he factor of N stems from the fact that S (2) kin has an additional matrix index to besummed over that was not present in the vector model previously studied. In whatfollows we will see that the effective potential is minimized for λ m ∼ M . Thus, inthe limit M (cid:29) N , we can have a large range of allowed ω c (in units where g = 1).If instead M does not scale with N and we take the large N limit, the window ofallowed ω c shrinks to zero.Since the global symmetry group of the theory, for our choice of Hamiltonian, is not the full U (2 N ), the situation is not as simple as the N = 1 case. For instance,the Σ measure in the path integral is not U (2 N ) invariant. Moreover, it is in generalcomplicated to quantify how the Σ matrices are encoded in the λ n eigenvalues and U matrices. In what follows we express several parts of the effective action directlyin terms of the Σ . We would now like to focus on the effective potential V eff for Σ . In order to computethis we can take Σ to be time independent. V eff must respect the SU ( N ) × SU (2)symmetries. For instance it can contain a piece which is the trace of a function ofthe SU (2) invariant matrix Σ · Σ . Moreover, when the Σ are diagonal (or when theyall commute with each other), it must reproduce N copies of the potential (2.15) wefound in the vector model. Finally, the piece of V eff originating from the functionaldeterminant must scale linearly in Σ . We can write a general expression by notingthat: det N × N ( − iω + R ) = N (cid:89) n =1 ( − iω + λ n ) , (3.15)is the characteristic polynomial for matrix R with eigenvalues λ n . We must also takethe product over all ω , a procedure which must be regulated. For each λ n , we canexpress the product over the ω as the exponential of an integral over the logarithm:12 (cid:90) dω π log (cid:0) ω + λ n (cid:1) = | λ n | . (3.16)17o define the above integral, we have subtracted the integral of log( ω ). Puttingthings together: V eff = − M N (cid:88) n =1 | λ n | + 14 tr Σ · Σ = − M √ R + 14 tr Σ · Σ . (3.17)As expected, V eff is invariant under both the SU ( N ) and SU (2) global symmetries.It is instructive to write the 2 N × N matrix R explicitly: R = (cid:32) Σ · Σ − i [ Σ x , Σ y ] [ Σ z , Σ x + i Σ y ] − [ Σ z , Σ x − i Σ y ] Σ · Σ + i [ Σ x , Σ y ] (cid:33) . (3.18)From the above expression, it immediately follows that tr R = 2 tr Σ · Σ . However,this does not imply that tr √ R = 2 tr √ Σ · Σ unless all the Σ commute amongsteach other. Thus, we see how the commutator interaction enters the potential. If ithappens that the Σ are almost commuting, we can perform a matrix Taylor expansionof tr √ R , which to leading order gives: − M √ R ≈ − M tr √ Σ · Σ + M
16 tr( Σ · Σ ) − / i [ Σ a , Σ b ]( Σ · Σ ) − i [ Σ a , Σ b ]+ . . . (3.19)The indices ( a, b ) run over all distinct pairs of ( x, y, z ), thus rendering the expression SO (3) invariant. Since the Hermitian matrix Σ · Σ has positive eigenvalues, andthe commutator i [ Σ a , Σ b ] is Hermitean, we see that non-zero commutations costpotential energy. Thus, at least locally the potential (3.17) is minimized when the Σ mutually commute (which means, in turn, that we can mutually diagonalize the Σ ). In this approximation, we can estimate the minimum value of V eff as the firstterm in the expansion (3.19). The problem we want to solve becomes a saddle pointapproximation of the following matrix integral for M (cid:29) N : Z [ Σ ] = (cid:90) d Σ x d Σ y d Σ z e M tr √ Σ · Σ − tr Σ · Σ / . (3.20)In order to obtain the saddle point equation for the eigenvalues, we first introducea delta function δ ( ρ − Σ · Σ ) and integrate out the Σ , such that we remain with anintegral over the N × N Hermitian ρ matrix. Upon diagonalizing ρ , and including the One may be concerned about the discontinuity of the first derivative at λ n = 0. However, theexpression agrees with what we expect of the determinant (cid:81) ω (1 + λ n /ω ). Namely, it should equalone when λ n = 0, it should be symmetric under λ n → − λ n and have an exponent linear in λ n .Moreover, one can check that at any non-zero temperature T for which ω → πT ( n + 1 /
2) with n ∈ Z , the kink at λ n = 0 smoothens out. ρ i ≥
0. Itis convenient at this point to rescale ρ i = M ˜ ρ i . We find: V eff [ ˜ ρ i ] = − (cid:88) j (cid:54) = i log | ˜ ρ i − ˜ ρ j | − M (cid:88) i (cid:18)(cid:112) ˜ ρ i − ˜ ρ i NM log ˜ ρ i (cid:19) , (3.21)up to an additive constant of order N log M . The log ˜ ρ i contribution comes fromthe measure of the path integral: there is a Jacobian when changing variables fromthe Σ matrices to the ρ matrix. The saddle point equation governing the eigenvaluesis: N (cid:88) j (cid:54) = i ρ i − ˜ ρ j = − N ˜ ρ i − M (cid:18) √ ˜ ρ i − (cid:19) . (3.22)To leading order in a large M expansion (taking M to be much larger than N ) wecan consider ˜ ρ i to be peaked around ˜ ρ i ∼
4. Expanding about ˜ ρ i = 4 + δ i for small δ i , and keeping the leading term only, we have: N (cid:88) j (cid:54) = i δ i − δ j = M δ i . (3.23)For large N , the above eigenvalue equation is solved by the Wigner semicircle dis-tribution [5] and has compact support in the interval ( √ N /M ) × [ − , ρ i ≈ M with a width of order √ N M . We can approximate the ground state energy to be V ( min ) eff ≈ − M N . It would be interesting to study subleading corrections, due to therepulsion of eigenvalues from the Vandermonde, but we will not do so here.There is a slightly more efficient way to see the above. Using the propertytr R = 2 tr Σ · Σ we can write the effective potential (3.17) completely in termsof the eigenvalues of R as: V eff = 12 N (cid:88) n =1 (cid:18) − M | λ n | + λ n (cid:19) . (3.24)Again, at least in the limit M (cid:29) N where we can ignore the effects of the matrixmeasure, we find V ( min ) eff ≈ − M N as before.We now proceed to study the kinetic contribution linear in velocity. We are considering here the situation where both M and N are large but M (cid:29) N . .2.2 Linear velocity term We consider the linear velocity term for the matrix model. The simplest case occurswhen the Σ ij matrix is diagonal, i.e. Σ ij = x i δ ij with i = 1 , . . . , N . In this case, wesimply find a sum of N terms (one for each x i ) each identical with the vector case.Each will have their own M + 1 lowest Landau levels. Generally, however, the Σ a will not be mutually diagonalizable. Inspired by the expression (2.28), we claim thatthe linear velocity term is given by: S (1) kin = i M (cid:90) dt (cid:20) ˙ Z † (cid:16) I + ZZ † (cid:17) − Z − Z † (cid:16) I + ZZ † (cid:17) − ˙ Z (cid:21) , (3.25)where Z ij is a complex N × N matrix. The stereographic map (2.26) relating z to apoint on the Bloch sphere is generalized to: Σ x + i Σ y ≡ M Z ( I + Z † Z ) − , (3.26) Σ x − i Σ y ≡ M Z † ( I + ZZ † ) − , (3.27) Σ z ≡ M (cid:20) I − (cid:16) I + ZZ † (cid:17) − − (cid:16) I + Z † Z (cid:17) − (cid:21) . (3.28)In order to verify that Σ a = ( Σ a ) † it is useful to take advantage of identities suchas: ( I + ZZ † ) − Z = Z ( I + Z † Z ) − . Naturally, when N = 1 our expression (3.25)reduces to the expression (2.28). It is also time reparameterization invariant under τ → f ( τ ) and Z ij ( τ ) → Z ij ( f ( τ )). Moreover, our expression is invariant under theglobal SU ( N ), under which Z → Λ Z Λ † , with Λ ∈ SU ( N ). In fact, as we shall see inthe next subsection, (3.25) invariant under a larger group U (2 N ) acting as: Z → ( AZ + B )( CZ + D ) − , (cid:32) A BC D (cid:33) · (cid:32) A BC D (cid:33) † = I N × N . (3.29)where A , B , C and D are N × N matrices. The U (2 N ) invariance is in agreement withour observation that terms stemming from the functional determinant (3.8) exhibit a U (2 N ) symmetry. This generalizes the SU (2) symmetry (2.30) that is present in the N = 1 case. Recall that in the N = 1 case, the linear velocity term only dependedon two of the three variables in x . Analogously, our expression (3.25) only dependson 2 N of the 3 N variables in the three Hermitian matrices Σ a .20 .3 Berezin coherent states As in the vector case, the matrix action (3.25) can stem from a curved phase spaceendowed with a K¨ahler structure. These compact K¨ahler manifolds were studiedextensively by Berezin [26]. The K¨ahler metric is given by: ds = M tr dZ (cid:16) I + ZZ † (cid:17) − dZ † (cid:16) I + Z † Z (cid:17) − , (3.30)where c is a normalization constant. The K¨ahler potential is given by: K = M log (cid:16) I + ZZ † (cid:17) . (3.31)This potential transforms under the U (2 N ) isometry (3.29) as K → K − M log det( Z † C † + D † ) − M log det( CZ + D ) , (3.32)leaving the metric (3.30) invariant. It is the natural generalization of the N = 1 case.More precisely, what Berezin shows [26] is that there exist a collection of coherentstates, analogous to the Bloch coherent states, parameterized by a complex matrix Z ij . Explicitly: | Z † ij (cid:105) = e Z † ij ˆS + ji det( I + Z † Z ) M/ | v (cid:105) , ˆS ± ij = ˆS xij ± i ˆS yij , (3.33)where the state | v (cid:105) is the state annihilated by all ψ Ai and ¯ ψ iA operators. It can beexpressed as | v (cid:105) = (cid:81) A,i ¯ ψ iA | (cid:105) , where | (cid:105) is the state that is annihilated by all the ψ αAi operators. Consequently | v (cid:105) is annihilated by ˆ S − ij . The overlap between two Berezincoherent states is given by: (cid:104) W ij | Z † ij (cid:105) = det (cid:0) I + W Z † (cid:1) M det ( I + W † W ) M/ det ( I + Z † Z ) M/ . (3.34)At large M the quantum evolution of a certain class of U ( M ) invariant operatorsin the Grassmann theory becomes approximately classical with an emergent curvedphase space [26] , the geometry of which is described by the K¨ahler metric (3.30). Therole of large M becomes that of the small Planck constant. The classical Hamiltonian,governing the time evolution of functions on the emergent phase space, is given by H [ Z, Z † ] = (cid:104) Z | ˆ H | Z † (cid:105) . The volume of the emergent classical phase space computesthe number of quantum states obtained upon quantizing it. The number of quantum21tates was computed in [27]. The result reads:dim H K = N (cid:89) j =1 Γ[ N + M + j ]Γ[ j ]Γ[ N + j ]Γ[ M + j ] . (3.35)We can study the behavior of dim H K in various limits. When N (cid:29) M (cid:29) H K ∼ MN to leading order. Thus in this limit, the dimension of the effectiveHilbert space closely approximates the full Hilbert space of the original Grassmannsystem. For M (cid:29) N (cid:29) H K ∼ M N . Finally, for M = αN where α is fixed in the large N limit, we have:log dim H K = f ( α ) N + . . . (3.36)with: f ( α ) = 12 (cid:0) α log( α ) − α + 1) log( α + 1) + ( α + 2) log( α + 2) − (cid:1) . (3.37)Notice that in the limit α → f ( α ) ∼ α log 2 for which log dim H K ∼ N M log 2.Similarly, in the α → ∞ limit, f ( α ) ∼ log α for which log dim H K ∼ N log M . Asshown in the appendix, (3.35) is precisely the number of states we would obtain inthe Grassmann matrix model, had we gauged the U ( M ) global symmetry. This is tobe expected. The full space of U ( M ) invariant states can be built by acting with afunction of the U ( M ) invariant operator ˆS + ij on the state | v (cid:105) (which is itself defined tobe U ( M ) invariant by a suitable choice of the normal ordering constant in the U ( M )generators). In the vector case, the Hamiltonian ˆ H (2.3) we studied was constant along the Blochtwo-sphere given that all the Bloch coherent states had the same total angular mo-mentum. In this regard our matrix model differs from the vector case. Given ourHamiltonian operator (3.2), the Hamiltonian H [ Z, Z † ] ≡ (cid:104) Z | ˆ H | Z † (cid:105) governing timeevolution on the emergent classical phase space is found to be: H [ Z, Z † ] = − N M + M tr ( S ) , (3.38)22o leading order in M . We have defined: S ≡ (cid:20)(cid:16) I + ZZ † (cid:17) − − (cid:16) I + Z † Z (cid:17) − (cid:21) . (3.39)Notice that H [ Z, Z † ] is invariant under Z → U ZU † where U ∈ SU ( N ). Moreover,the Hamiltonian H [ Z, Z † ] is minimized when Z and Z † commute, where it takes thevalue E min = − N M . Consequently, the state | v (cid:105) is one of these minimal energystates. This agrees with our analysis of the effective potential in section 3.2.1, wherethe minimum was also found to be − N M in the large M limit. When Z and Z † commute they can be mutually diagonalized and the K¨ahler metric becomes N copiesof CP , i.e. one Bloch sphere for each eigenvalue. Furthermore, as was found in theanalysis of section 3.2.1, the commutator of Z and Z † costs energy. Nevertheless, sincethe Z can be continuously deformed, there is a rich low energy sector continuouslyconnected to the ground states given by almost commuting complex matrices.Given the kinetic term and the Hamiltonian on phase space, following Berezin[26], we can write down the real time path integral for transition amplitudes betweencoherent states | Z † i (cid:105) and (cid:104) Z f | . It reads: A fi = (cid:90) D µ [ Z, Z † ] exp (cid:18) M (cid:90) T − T dt (cid:104) ˙ Z ( I + Z † Z ) − Z † − h.c. (cid:105) − i (cid:90) T − T dtH [ Z, Z † ] (cid:19) , (3.40)with boundary conditions Z † [ − T ] = Z † i and Z [ T ] = Z f . The measure factor is givenby: D µ [ Z, Z † ] ≡ N D Z D Z † det ( I + ZZ † ) N . (3.41)The normalization constant N ensures that Tr I = (cid:82) dµ [ Z, Z † ] = dim H K . It can becomputed by use of the Selberg integral S N (1 , M + 1 ,
1) [31].Consider finally the following rescaling Z = M − / ˜ Z , with ˜ Z fixed in the large M limit, and in addition M (cid:29) N . To leading order in the large M expansion, the pathintegral becomes: A fi = (cid:90) D ˜ Z D ˜ Z † exp (cid:20)
12 tr (cid:90) T − T dt (cid:16) ˙˜ Z ˜ Z † − h.c. (cid:17) − i tr (cid:90) T − T dt [ ˜ Z, ˜ Z † ] (cid:21) . (3.42)This limit is a small fluctuation limit in which the geometry of the curved phase spacebecomes flat and the Hamiltonian boils down to the trace of the square of the com-23utator. Naturally, in the N = 1 case, no such commutator arises, and the rescalinglimit simply describes motion in a small flat patch of the full CP .Thus, we generalize several of the features observed in the vector model to the matrixmodel. As before, there is an emergent classical phase space endowed with a K¨ahlermetric, a low velocity expansion of a bosonic Hermitian matrix model in a suitablelarge M regime and a large number of low energy states. Given the appearance of abosonic matrix model, we can wonder about a holographic interpretation at large N .We end with some speculative remarks on this question. We have discussed systems with a finite dimensional Hilbert space, whose constituentsare a large number of spin-1/2 fermions. For certain collections of states, we haveseen how the systems we have considered exhibit an emergent classical phase spaceparameterized by complex coordinates. The phase space is endowed with a K¨ahlermetric which in the simplest case is nothing more than the round two-sphere. Moregenerally, it is a complex matrix generalization thereof. In the vector case, the size ofthe Bloch sphere (2.29) scales as the logarithm of the dimension of the Hilbert space.The specific Hamiltonian we considered, commutes with the total angular momen-tum operator. Consequently, transition amplitudes between different Bloch coherentstates lie on a Bloch sphere of fixed size. One manifestation of this is that the param-eter r acquires no time derivatives in the effective action. More generally, one mightimagine Hamiltonians with matrix elements connecting Hilbert spaces with differenttotal angular momenta. In such a case, one might consider an additional directiongiven by the size of the two-sphere, such that in a suitable large M limit, the lowenergy degrees of freedom are parameterized by coordinates in a three-dimensionalball. So long as the dimension of the Hilbert space remains finite, there is still acap on the maximal size of the two-sphere. A natural matrix generalization of theparameter r is given by the trace of the Hermitian matrix √ Σ · Σ . Unlike the vectorcase, transitions between different values of tr √ Σ · Σ are possible within the space ofBerezin coherent states. In other words, the K¨ahler metric of the emergent classicalphase space does not constrain Σ · Σ (which is a now a function of Z and Z † ) to takea specific value.Holographically, large N matrix models might be associated with a gravitationaltheory. For the quantum mechanical model [7] dual to the ten-dimensional geometry24ear a collection of N D0-branes, one has nine N × N Hermitian bosonic matrices X Iij and their Fermionic superpartners. The index I is an SO (9) index, corresponding tothe rotational symmetry of the eight-sphere in the near horizon of a stack of N D0-branes in type IIA string theory. The indices i and j run from 1 to N . The Hilbertspace is infinite dimensional and there are states with indefinitely high energy. Inthese models, the emergent radial direction has been argued to be captured by theenergy scale. At high energies, the quantum mechanics is weakly coupled. Onemanifestation of this, from the bulk viewpoint, is that the size (in the string frame)of the eight-sphere shrinks indefinitely at large radial distances, eventually leading toa stringy geometry.Consider now a system where the spectrum is capped, as occurs in the deepinfrared of a CFT living on a spatial sphere (due to the curvature coupling of thefields). In such a situation we expect the emergent sphere to cap off. This is indeedwhat happens in global anti-de Sitter space where the sphere at fixed r and t smoothlycaps off in the deep interior. Consider now the geometry of the static patch of four-dimensional de Sitter space: ds = − dt (1 − r ) + dr (1 − r ) + r d Ω . (4.1)Notice that the size of the two-sphere resides on a finite interval. It smoothly caps offat r = 0 and is largest at r = 1 where the cosmological horizon resides. If, somehow, r was an emergent holographic direction related to the energy scale [28], then it wouldseem we have to cap the spectrum both in the infrared as well as the ultraviolet.This would indicate a holographic quantum mechanical dual with a finite numberof states [15, 16, 17, 18, 19, 20], so long as the spectrum is discrete. If moreover werequire the holographic model to have a matrix-quantum mechanical sector describedby ordinary bosonic matrices, perhaps the systems we have considered above arenatural candidates. We postpone the examination of this proposal and the relationto other approaches of de Sitter holography (for an overview see [29]) to future work. Acknowledgements
It is a pleasure to thank Tom Banks, Chris Beem, Umut Gursoy, Sean Hartnoll, JuanMaldacena, Nati Seiberg, Douglas Stanford, Herman Verlinde, and especially Diego Recall the metric of global AdS d +2 is given by ds = − dt (1 + r ) + dr (1 + r ) − + r d Ω d . As r → d -sphere caps off smoothly. A Counting U ( M ) gauge invariant states In this appendix we present the derivation of the formula for the dimension of theHilbert space of two complex Grassmann matrices χ iA and θ iA with indices rangingfrom i = 1 , . . . , N and A = 1 , . . . , M .Therefore we consider the action: S = (cid:90) dt (cid:2) ¯ χ iA i D t χ iB + ¯ θ iA i ¯ D t θ iB − (cid:0) m ¯ χ iA χ iA + m ¯ θ iA θ iA (cid:1)(cid:3) , (A.1)with D t = ∂ t + iA t and ¯ D t = ∂ t − iA t . The gauge field A t = A δt T δ is a Hermitian M × M matrix, with T δ the M generators of U ( M ). The Grassmann matricestransform in the (anti-)fundamental representation of U ( M ) (we pick χ A , ¯ θ A in thefundamental). We consider the case with m > m >
0. From the Poissonbrackets originating from the above action we obtain the anti-commutation relationsof fermionic creation/annihilation operators: { χ iA , ¯ χ jB } = δ AB δ ij , { θ iA , ¯ θ jB } = δ AB δ ij . (A.2)Integrating out the gauge field gives us M constraints: δA t : ¯ χ A T δAB χ B − ¯ θ A T δAB θ B = 0 , ∀ δ = 1 , , . . . , M (A.3)We define the vacuum state | (cid:105) of the theory to be annihilated by all χ and θ operators.Note that it obeys the gauge constraint and is thus gauge invariant. Moreover, actingwith gauge invariant operators always increases the energy, hence | (cid:105) is unique.We wish to find the thermal partition function and extract the entropy S ( T ) atinfinite temperature. We can then use the fact that lim T →∞ S ( T ) = log dim H tofind the dimension of the Hilbert space with a U ( M ) singlet constraint imposed. Inthe absence of the gauge field A t , we would have dim H = 2 NM .26 .1 Euclidean path integral We can compute the thermal partition function as a Euclidean path integral. Wickrotate time t → − iτ such that S E = (cid:90) β dτ (cid:2) ¯ χ iA D τ χ iB + ¯ θ iA ¯ D τ θ iB + m ¯ χ iA χ iA + m ¯ θ iA θ iA (cid:3) . (A.4)The Grassmann variables obey anti-periodic boundary conditions around the thermalcircle. The Euclidean path integral of interest is: Z [ β ] = (cid:90) D A τ D χ D ¯ χ D θ D ¯ θ e − S E . (A.5)The gauge transformations acting on A τ are given by A τ → U A τ U † + i∂ τ U · U † . Dueto the non-contractible thermal circle, we can only fix the gauge up to the holonomyaround the thermal circle [30]. The Fadeev-Popov procedure in doing so gives usthe following action for the (time independent upon gauge fixing) eigenvalues of A τ which we denote α A : (cid:90) M (cid:89) A =1 dα A (cid:32) (cid:89) A
In this appendix we briefly mention a slight modification of the vector model consid-ered in the main body of the text. The degrees of freedom are given by two sets of28 complex fermion spinors { ψ αA , θ αA } . We consider the following Euclidean action: S E = (cid:90) dτ ¯ ψ αA ∂ τ ψ αA + ¯ θ αA ∂ τ θ αA − (cid:16) ¯ ψ αA σ αβ ψ βA − ¯ θ αA σ αβ θ βA (cid:17) . (B.1)Following the procedure outlined in the main text, we end up with an effective actionfor a bosonic three-vector x : S eff = M Tr log ( − ∂ τ + x · σ ) + M Tr log ( − ∂ τ − x · σ ) + 14 (cid:90) dτ r . (B.2)Performing a small velocity expansion one realizes that the term linear in velocityin fact cancels. This is due to the relative sign in front of x in the two functionaldeterminants in (B.2). Thus the leading term in the velocity expansion is: S (2) kin = M (cid:90) dτ r (cid:16) ˙ θ + sin θ ˙ φ (cid:17) . (B.3)The reason for the cancellation is that this model has a Hamiltonian given by thedifference in angular momentum. The ground state is given by the configurationwhere the two angular momenta, whose operators are given by ˆ J = ¯ ψ A σψ A / J = ¯ θ A σθ A /
2, are anti-aligned. In the language of the charged particle on the two-sphere, it is as if we have added a positron on top of the electron, thus canceling theeffect of the Lorentz force, leaving an ordinary kinetic term for the bound neutralparticle. The configuration space is still parameterized by the angles on a two-sphere.The mass of the neutral particle is twice that of the original one, explaining the 1 / / M we have a controlled low velocityexpansion. At high energies, the two angular momenta can fluctuate independentlyand this simple picture is lost. A similar modification can be made for the matrixmodel. References [1] V. A. Kazakov, “Ising model on a dynamical planar random lattice: Exact solu-tion,” Phys. Lett. A , 140 (1986).[2] J. M. Maldacena, “The Large N limit of superconformal field theories and super-gravity,” Int. J. Theor. Phys. , 1113 (1999) [Adv. Theor. Math. Phys. , 231(1998)] [hep-th/9711200]. 293] T. Banks, W. Fischler, S. H. Shenker and L. Susskind, “M theory as a matrixmodel: A Conjecture,” Phys. Rev. D , 5112 (1997) [hep-th/9610043].[4] J. McGreevy and H. L. Verlinde, “Strings from tachyons: The c=1 matrixreloaded,” JHEP , 054 (2003) [hep-th/0304224].[5] E. Brezin, C. Itzykson, G. Parisi and J. B. Zuber, “Planar Diagrams,” Commun.Math. Phys. , 35 (1978).[6] I. R. Klebanov, “String theory in two-dimensions,” In *Trieste 1991, Proceedings,String theory and quantum gravity ’91* 30-101 and Princeton Univ. - PUPT-1271(91/07,rec.Oct.) 72 p [hep-th/9108019].[7] N. Itzhaki, J. M. Maldacena, J. Sonnenschein and S. Yankielowicz, “Supergravityand the large N limit of theories with sixteen supercharges,” Phys. Rev. D ,046004 (1998) [hep-th/9802042].[8] E. Marinari and G. Parisi, “The Supersymmetric One-dimensional String,” Phys.Lett. B , 375 (1990).[9] Y. Makeenko and K. Zarembo, “Adjoint fermion matrix models,” Nucl. Phys. B , 237 (1994) [hep-th/9309012].[10] G. W. Semenoff and R. J. Szabo, “Fermionic matrix models,” Int. J. Mod. Phys.A , 2135 (1997) [hep-th/9605140].[11] L. D. Paniak and R. J. Szabo, “Fermionic quantum gravity,” Nucl. Phys. B ,671 (2001) [hep-th/0005128].[12] L. F. Cugliandolo, J. Kurchan, G. Parisi, and F. Ritort, “Matrix Models asSolvable Glass Models,” Phys. Rev. Lett. 74, 1012 (1995). [cond-mat/9407086].[13] D. Anninos, S. A. Hartnoll, L. Huijse and V. L. Martin, “Large N matrices froma nonlocal spin system,” arXiv:1412.1092 [hep-th].[14] D. Berenstein, “A Matrix model for a quantum Hall droplet withmanifest particle-hole symmetry,” Phys. Rev. D , 085001 (2005)doi:10.1103/PhysRevD.71.085001 [hep-th/0409115].[15] T. Banks, B. Fiol and A. Morisse, “Towards a quantum theory of de Sitterspace,” JHEP , 004 (2006) [hep-th/0609062].3016] M. K. Parikh and E. P. Verlinde, “De Sitter holography with a finite number ofstates,” JHEP , 054 (2005) [hep-th/0410227].[17] X. Dong, B. Horn, E. Silverstein and G. Torroba, “Micromanaging de Sitterholography,” Class. Quant. Grav. , 245020 (2010) [arXiv:1005.5403 [hep-th]].[18] M. Li, “Matrix model for de Sitter,” JHEP , 005 (2002) doi:10.1088/1126-6708/2002/04/005 [hep-th/0106184].[19] A. Volovich, “Discreteness in deSitter space and quantization of Kahler mani-folds,” hep-th/0101176.[20] J. J. Heckman and H. Verlinde, “Instantons, Twistors, and Emergent Gravity,”arXiv:1112.5210 [hep-th].[21] A. Abanov “WZW term in quantum mechanics: single spin,”http://felix.physics.sunysb.edu/ abanov/Teaching/Spring2009/Notes/abanov-cp07-upload.pdf[22] N. Karchev, “Path integral representation for spin systems,” arXiv:1211.4509[cond-mat.str-el].[23] F. D. M. Haldane, Phys.Lett.,A 93, 454 (1983).[24] F. D. M. Haldane, Phys.Rev.Lett.,50, 1153 (1983).[25] I. J. R. Aitchison, “Berry Phases, Magnetic Monopoles and Wess-Zumino Termsor How the Skyrmion Got Its Spin,” Acta Phys. Polon. B , 207 (1987).[26] F. A. Berezin, “Models of Gross-Neveu Type as Quantization of Classical Me-chanics With Nonlinear Phase Space,” Commun. Math. Phys. , 131 (1978).[27] D. Das, S. R. Das, A. Jevicki and Q. Ye, “Bi-local Construction of Sp(2N)/dSHigher Spin Correspondence,” JHEP , 107 (2013) [arXiv:1205.5776 [hep-th]].[28] D. Anninos, S. A. Hartnoll and D. M. Hofman, “Static Patch Solipsism: Confor-mal Symmetry of the de Sitter Worldline,” Class. Quant. Grav. , 075002 (2012)[arXiv:1109.4942 [hep-th]].[29] D. Anninos, “De Sitter Musings,” Int. J. Mod. Phys. A , 1230013 (2012)doi:10.1142/S0217751X1230013X [arXiv:1205.3855 [hep-th]].3130] O. Aharony, J. Marsano, S. Minwalla, K. Papadodimas and M. Van Raamsdonk,“The Hagedorn - deconfinement phase transition in weakly coupled large N gaugetheories,” Adv. Theor. Math. Phys.8