Bootstrapping Matrix Quantum Mechanics
BBootstrapping Matrix Quantum Mechanics
Xizhi Han, Sean A. Hartnoll, Jorrit Kruthoff
Department of Physics, Stanford University,Stanford, CA 94305-4060, USA
Abstract
Recent work has shown that large N (multi-) matrix integrals can be solved nu-merically by imposing positivity constraints on higher point correlation functions [1].We have generalized this method to obtain the spectrum and simple expectation valuesof large N , gauged matrix quantum mechanics. In our approach, operator expectationvalues are related through conditions such as (cid:104) [ H, O ] (cid:105) = (cid:104) G O(cid:105) = 0 . Here H is the Hamil-tonian, G the generator of SU ( N ) gauge transformations and O an arbitrary operator.Bounds on the energy and expectation values of short operators are then obtained frompositivity constraints on the expectation values of certain longer operators. We firstdemonstrate how this method efficiently solves the conventional quantum anharmonicoscillator. We then reproduce the known solution of large N single matrix quantummechanics. Finally, we present new results on the ground state of large N two matrixquantum mechanics. a r X i v : . [ h e p - t h ] A p r ontents N collective field solution 17 Large N matrices are at the heart of the holographic emergence of semiclassical, gravitatingspacetime geometry [2]. Matrix quantum mechanics theories focus our attention on theessence of this phenomenon; unlike higher dimensional large N Yang-Mills theories, matrixquantum mechanics has no built in locality to start with, and thus space must emerge inits entirety. The simplest such theory is the single matrix quantum mechanics descriptionof two dimensional string theory [3], while the richest are the maximally supersymmetricmulti-matrix theories of BFSS [4] and BMN [5]. There are many theories in between, withvarying numbers of matrices and degrees of supersymmetry [6]. Thus far, only the singlematrix quantum mechanics has proved solvable at large N [7].Nonzero temperature Monte Carlo studies of large N multi-matrix quantum mechanicalsystems have successfully captured aspects of a known dual spacetime in supersymmet-ric theories [8–11]. Substantial Monte Carlo studies have also been performed for nonzerotemperature bosonic multi-matrix theories, e.g. [12, 13]. However, recent work increasinglysuggests that the quantum structure of holographic quantum states — revealed for instancein their entanglement [14–17] — plays a central role in the emergence of space. It thereforebehooves us to find methods suitable for studying the zero temperature quantum states ofmulti-matrix quantum mechanics directly. Progress was made recently in this direction byusing a neural network variational wavefunction [18]. Here we describe a different approach.Our work is directly inspired by a recent beautiful paper by Lin [1]. That paper studied2arge N matrix integrals, which is an easier problem than large N quantum mechanics butshares important features. Lin showed how relatively simple positivity constraints on higherpoint correlation functions could be combined with the large N loop equations to efficientlyproduce strong numerical constraints on correlation functions of matrix integrals. In thefollowing we will show how this methodology can be adapted to the quantum mechanicalproblem. The positivity constraints will be essentially the same, augmented to include mo-mentum as well as position operators in the case of matrix theories, while certain operatoridentities in energy eigenstates (or more general Gibbs states) will play the role of the loopequations.In §2 we consider a warm-up case of a conventional quantum mechanical anharmonicoscillator. The results are in Fig. 1, showing that the ground and first excited state energies E and expectation values (cid:104) x (cid:105) can be strongly constrained with little numerical effort. Thiscase is the closest to the matrix integral results of [1], because expectation values of operatorsdepending on the momentum can be solved for explicitly and a recursion relation (9) canbe found that determines all expectation values (cid:104) x t (cid:105) in terms of just (cid:104) x (cid:105) and E .In §3 we use the bootstrap methods to solve a large N one matrix quantum mechanics.Here the momentum operators cannot be eliminated explicitly, and we do not use a closedform recursion relation for all expectation values. However, we find that the energy andexpectation values of short operators can be efficiently constrained by applying positivityconstraints to a matrix generated by all strings of operators of length ≤ L . There are L such strings, leading to a matrix with L entries. The results in Fig. 2 show that theknown analytic results for the ground state energy E and expectation value (cid:104) tr X (cid:105) can bereproduced to very high accuracy already with L = 3 . We also constrain (cid:104) tr X (cid:105) in excitedstates with energy E , which is continuous in the large N limit, as shown in Fig. 3.In §4 we turn to a two matrix quantum mechanics. The methodology is the same as for theone matrix case, except that now there will be L matrix elements to consider. This modelis not solvable, so we have corroborated our numerical results with a Born-Oppenheimerapproximation, that gives rigorous lower and upper bounds on the ground state energy. Theresults in Fig. 4 show that the minimal energy allowed by the bootstrap equations at L = 4 is within the narrow region allowed by the Born-Oppenheimer bounds, and quite close tothe lower bound, suggesting that the numerics is close to convergence onto the true groundstate energy. That figure also shows the bootstrap results for the expectation values (cid:104) tr X (cid:105) and (cid:104) tr[ X, Y ] (cid:105) . 3 The quantum anharmonic oscillator
The essence and effectiveness of our bootstrap method can be demonstrated in a toy exampleof a quantum anharmonic oscillator, with Hamiltonian H = p + x + gx . (1)Here [ p, x ] = − i . We will use positivity constraints on expectation values to obtain theresults shown in Fig. 1 below for the energy E and expectation value (cid:104) x (cid:105) of the groundstate and first excited state. In later sections we will generalize this approach to the quantummechanics of matrices.The first step is to relate the expectation values of different operators. We will obtainthe recursion relation (9) below. For expectation values in energy eigenstates, and for anyoperator O , (cid:104) [ H, O ] (cid:105) = 0 . (2)For example, let O = xp . Equation (2) then gives the Virial theorem, (cid:104) p (cid:105) = (cid:104) x + 4 gx (cid:105) .This relation links the expectation values of the operators p , x and x . Following from theVirial theorem, the energy of the eigenstate is E = 2 (cid:104) x (cid:105) + 3 g (cid:104) x (cid:105) . (3)More systematically, take O = x s and O = x t p in (2) for integers s, t ≥ . Commuting theoperators x, p with the identity [ p, x r ] = − irx r − gives the relations s (cid:104) x s − p (cid:105) − is ( s − (cid:104) x s − (cid:105) = 0 , (4) t (cid:104) x t − p (cid:105) − it ( t − (cid:104) x t − p (cid:105) = 2 (cid:104) x t +1 (cid:105) + 4 g (cid:104) x t +3 (cid:105) . (5)Eliminating the terms with a single p operator, we arrive at the relation t (cid:104) x t − p (cid:105) + t ( t − t − (cid:104) x t − (cid:105) − (cid:104) x t +1 (cid:105) − g (cid:104) x t +3 (cid:105) = 0 . (6)A powerful identity in this single particle example is that (cid:104)O H (cid:105) = E (cid:104)O(cid:105) , (7)which is a strengthened version of (2). We emphasize (2) instead of (7) because, as we willsee later, (7) becomes less useful in the matrix case. Nonetheless, in the present anharmonicoscillator case, we can set O = x t − in (7) to obtain (cid:104) x t − p (cid:105) = E (cid:104) x t − (cid:105) − (cid:104) x t +1 (cid:105) − g (cid:104) x t +3 (cid:105) . (8)4lugging (8) into (6) gives a recursive relation between expectation values of powers of x : tE (cid:104) x t − (cid:105) + t ( t − t − (cid:104) x t − (cid:105) − t + 1) (cid:104) x t +1 (cid:105) − g ( t + 2) (cid:104) x t +3 (cid:105) = 0 , (9)where E is given by (3). Also we know that (cid:104) x (cid:105) = 1 and (cid:104) x t (cid:105) = 0 if t is odd, so allexpectation values of x t can be computed from E and (cid:104) x (cid:105) with (9).With the recursion relation (9) at hand we move onto the second step. We wish to solvefor E and (cid:104) x (cid:105) , the only two unknown variables, by bootstrapping. This step works as in [1]. + + Figure 1: Bootstrap allowed region (shaded) for the anharmonic oscillator (1) with g = 1 .Upper plot: the allowed region for ( E, (cid:104) x (cid:105) ) near the ground state solution (marked by thered cross) for different sizes of the bootstrap matrix K = 7 , , ; lower plot: the allowedregion near the first excited state. 5he basic positivity constraint is that (cid:104)O † O(cid:105) ≥ , ∀O = K (cid:88) i =0 c i x i , (10)which means that the matrix M of size ( K + 1) × ( K + 1) , M ij = (cid:104) x i + j (cid:105) , should be positivesemidefinite. The constraint becomes stronger as we increase K , thus enlarging the spaceof trial operators. For a given K and test values of E and (cid:104) x (cid:105) , the matrix elements M ij can be computed using the recursion relation (9). The bootstrap consists in scanning overthese test values, computing the eigenvalues of the matrix M , and thereby determining ifpositivity excludes the test values as inconsistent.The bootstrap result is shown in Fig. 1. Even for moderate K the values of E and (cid:104) x (cid:105) can be determined quite accurately. We also see that the region of allowed values splits intoa discrete set of islands, corresponding to the discrete spectrum of the Hamiltonian. (Thespectrum will be continuous later in the infinite- N matrix models we consider.) Both thespectrum and observables at different energies can be extracted in this way, but higher-orderoperators or higher-energy states will require more constraints to be computed accurately. The bootstrap method discussed in the anharmonic oscillator example can be generalizedto matrix quantum mechanics at N = ∞ . To start, we consider the single-matrix quantummechanics with Hamiltonian: H = tr P + tr X + gN tr X , (11)where P and X are N -by- N Hermitian matrices with quantum commutators [ P ij , X kl ] = − iδ il δ jk . The theory (11) can be mapped onto the quantum mechanics of N free fermionsand is hence solvable [7]. In this section we will show how positivity constraints reproducethe known solution. The matching is shown in Figs. 2 and 3 below.To apply the bootstrap ideas, we relate operator expectation values by symmetries. Inthe following, denote (cid:104)O(cid:105) = tr ρ O . If the state ρ commutes with the Hamiltonian then (cid:104) [ H, O ] (cid:105) = 0 , ∀O . (12)Such a state ρ could be a pure energy eigenstate or a mixed thermal state, for example.Similar to the single-particle case in the previous section, the Virial theorem is derived from(11) by choosing O = tr XP : (cid:104) [ H, tr XP ] (cid:105) = 0 ⇒ (cid:104) tr P (cid:105) = 2 (cid:104) tr X (cid:105) + 4 gN (cid:104) tr X (cid:105) . (13)6he Hamiltonian (11) also has an SU ( N ) -adjoint symmetry generated by the matrix G = i [ X, P ] +
N I, (14)where the extra identity piece ensures that (cid:104) tr G (cid:105) = 0 , with the operator ordering [ X, P ] = XP − P X in (14). We are interested in gauged matrix quantum mechanics, so that physicalstates must be invariant under this symmetry. Thus in particular, N (cid:88) i,j =1 (cid:104) G ij O ji (cid:105) = 0 , ∀O ij . (15)The simplest implication of this constraint is as follows. From (cid:104) tr G (cid:105) = 0 we have (cid:104) tr XP (cid:105) − (cid:104) tr P X (cid:105) = iN , (16)and on the other hand because (cid:104) [ H, tr X ] (cid:105) = 0 , (cid:104) tr XP (cid:105) + (cid:104) tr P X (cid:105) = 0 ⇒ (cid:104) tr XP (cid:105) = −(cid:104) tr P X (cid:105) = iN . (17)From symmetries we have determined the expectation values of tr XP and tr P X .Cyclicity of the trace gives another set of relations between operators. However, notethat commuting quantum operators may be necessary in applying the cyclic formula. Forexample, as an operator equation, tr XP = tr P X + i tr I tr P + i tr P tr P + i tr P tr I. (18)We can further use large- N factorization to replace expectation values of multi-trace oper-ators by products of single-trace operators. Therefore, to leading order in N → ∞ , (cid:104) tr XP (cid:105) = (cid:104) tr P X (cid:105) + 2 iN (cid:104) tr P (cid:105) + i (cid:104) tr P (cid:105)(cid:104) tr P (cid:105) . (19)Equations (12), (15), cyclicity of the trace, and reality conditions (cid:104)O † (cid:105) = (cid:104)O(cid:105) ∗ generate allrelations between expectation values that we will use for the bootstrap.As a mini-bootstrap example, consider trial operators I, X, X and P . From the condition(10), the following bootstrap matrix should be positive semidefinite: I X X PI (cid:104) tr I (cid:105) (cid:104) tr X (cid:105) X (cid:104) tr X (cid:105) (cid:104) tr X (cid:105) X (cid:104) tr X (cid:105) (cid:104) tr XP (cid:105) P (cid:104) tr P X (cid:105) (cid:104) tr P (cid:105) (20)7nlike in the single-particle case of the previous section, here we are considering operatorsbuilt from both X and P . We will not be able to explicitly eliminate P in favor of the energyin the matrix case. This is because only tr P appears in the Hamiltonian, but the matrix P can appear in many other combinations. We also note that the expectation value for anodd number of matrices vanishes. Positivity of (20) tells us that (cid:104) tr X (cid:105) ≥ , N (cid:104) tr X (cid:105) ≥ (cid:104) tr X (cid:105) , (cid:104) tr X (cid:105) (cid:18) (cid:104) tr X (cid:105) + 2 gN (cid:104) tr X (cid:105) (cid:19) ≥ N , (21)where equations (13) and (17) are used. The inequalities (21) are the bootstrap constraints onexpectation values in this simple example. When g = 0 , (cid:104) tr X (cid:105) = N and (cid:104) tr X (cid:105) = N ,so the last inequality in (21) is saturated and the other two are not.The bootstrap constraint will be stronger if we include more trial operators. We can dothe following. Firstly, take all possible strings of X and P that have length ≤ L . Giventhese operators, write down the matrix analogous to (20). This matrix must be positivesemidefinite. Secondly, we regard each entry in the bootstrap matrix as a variable (whichis the expectation value of a single-trace operator with length ≤ L ), and write down theequalities between them following from (12), (15) (taking O in those equations to be allstrings of matrices of length ≤ L , any constraints generating operators outside of the trialsubset are discarded), cyclicity of the trace (an example is shown in (19)), (cid:104)O † (cid:105) = (cid:104)O(cid:105) ∗ andthat the expectation value of an odd number of matrices vanishes. Thirdly, for numericalefficiency, although not strictly necessary, we solve the equalities between observables thatare linear (mostly from (12) and (15); note that using cyclicity of the trace and large- N factorization might introduce terms quadratic in expectation values) to obtain a reduced setof linearly free variables. In this way we obtain a number of constraints (quadratic cyclicityand bootstrap positivity constraints) on a reduced set of variables.Unlike in the single-particle case, we do not necessarily require that the state is anenergy eigenstate and the energy E does not appear explicitly in the bootstrap constraints.At infinite N the matrix quantum mechanics has a continuous spectrum and therefore we donot expect to see separated islands corresponding to excited states. The bootstrap constraintsallow for a continuum of energies, as they must. We therefore proceed to use gradient descentto minimize the energy in the allowed region of expectation values. In this way we obtain alower bound on the ground state energy of the theory. The result is a lower bound becausefor any number of constraints, certainly the true ground state energy is allowed, and hencethe true ground state energy is above the minimum that we find. In Fig. 2 we observe thatthe lower bound is very close to the true ground state value, already for L = 3 , and other8 Figure 2: One matrix quantum mechanics bootstrap for the Hamiltonian (11). L is the max-imal length of trial operators (we have offset the values of g used for L = 3 to make the plotclearer). Upper: The markers show the minimal energies allowed by the bootstrap positiv-ity constraints, in comparison with the exact ground state solution. Lower: the expectationvalues of tr X , for the minimal energy parameters found in the upper plot.observables, such as (cid:104) tr X (cid:105) , are also solved accurately.Compared to the single-particle example of the previous section, in the matrix quantummechanics case we cannot determine all expectation values from a finite subset. However thebootstrap constraints seem still to be sufficiently restrictive to solve for few-matrix expec-tation values. The uncertainties on our result can be obtained by minimizing or maximizingthe allowed expectation values while keeping the energy fixed — in this way we determine apossible window for the observable at any given energy. In Fig. 3 this window is seen to besmall near the lowest energies, and shrinks as we increase the number of constraints. Thesefacts are consistent with the bootstrap converging towards the correct ground state energyas L is increased, as evidenced already in Fig. 2.9 .5 2.0 2.5 3.00.350.400.450.500.55 Figure 3: Bounds on (cid:104) tr X (cid:105) for excited energies in the one matrix quantum mechanics (11)with g = 1 . The values of (cid:104) tr X (cid:105) allowed by the bootstrap constraints at fixed energies areshown as shaded. L is the maximal length of trial operators. The dashed curve shows thethermal expectation values for comparison, which should be allowed by bootstrap. One matrix quantum mechanics are tractable analytically as one can always diagonalizethe matrix. This is not the case for multi-matrix quantum mechanics, where the matricesare not simultaneously diagonalizable in typical states of interest. In this section we willillustrate how bootstrap methods can successfully be used for such theories, focussing on arelatively simple two-matrix quantum mechanics with a global O (2) symmetry (in additionto the large N gauge symmetry). The Hamiltonian is given by H = tr (cid:0) P X + P Y + m ( X + Y ) − g [ X, Y ] (cid:1) , (22)with X and Y being N -by- N Hermitian matrices, with conjugate momenta P X and P Y , and m and g coupling constants. This theory is not exactly solvable. An early discussion of themassless ( m = 0 ) limit of the theory is [19]. More generally, by rescaling the matrices we seethat physical quantities can only depend on the ratio g/m .We will impose rotational invariance to obtain even more relations between observablesin the two-matrix case. We expect the ground state to be rotationally invariant. The Hamil-tonian (22) has an SO (2) symmetry generated by S = tr( XP Y − Y P X ) . (23)Similar to previous discussions, for states ρ with [ S, ρ ] = 0 , including eigenstates of S , (cid:104) [ S, O ] (cid:105) = 0 , ∀O . (24)10hus in the two matrix quantum mechanics, equations (12), (15), (24), cyclicity of the trace,and reality relations (cid:104)O † (cid:105) = (cid:104)O(cid:105) ∗ will be used to generate all equations between expectationvalues that we will use for the bootstrap. The bootstrap then proceeds in exactly the sameway as for the case of a single matrix. The results for the ground state energy, (cid:104) tr X + tr Y (cid:105) and (cid:104) tr[ X, Y ] (cid:105) are shown in figure 4.In order to corroborate the accuracy of the L = 4 results, we obtain rigorous upper andlower bounds on the true ground state energy using a trial Born-Oppenheimer wavefunction.We see in figure 4 that the L = 4 bootstrap results indeed lie within a narrow window allowedby these bounds. This suggests that the bootstrap results are close to convergence. We nowbriefly describe the trial wavefunction, with details given in the appendices A and B.The SU ( N ) gauge invariance allows us to diagonalize one of the two matrices, say X .Let the eigenvalues be x i . The Hamiltonian for the entries y ij of the remaining matrix isa sum of harmonic oscillators, with frequencies ω ij = m + g ( x i − x j ) . We can thereforewrite down a Born-Oppenheimer wavefunction in which these oscillators are placed in theirground state: Ψ( X, Y ) = ψ ( x i ) N (cid:89) i,j =1 (2 ω ij /π ) / e − ω ij | y ij | . (25)This ansatz can be expected to describe the actual ground state wavefunction if the ω ij become sufficiently large, so that the off-diagonal components of Y are ‘fast’ compared tothe eigenvalues x i . This will not be the case here. However, as we recall in the appendix,Born-Oppenheimer wavefunctions lead to both upper and lower bounds on the ground stateenergy. The upper bound follows from treating the wavefunction as a variational ansatz.The lower bound is obtained by finding the ground state of the eigenvalues in an effectivepotential due to the zero point energy of the y ij oscillators. In Fig. 4 we see that thebounds following from the wavefunction (25) turn out to be remarkably tight. This providesa nontrivial check on the bootstrap results.The wavefunction (25) is not rotationally invariant, but this is unimportant for boundingthe ground state energy. Furthermore, rotational invariance is restored by acting on thewavefunction with the generator S . The advantage of the form (25) is that computing theenergy reduces to a single-matrix large N eigenvalue problem, as we now describe.Both the variational upper bound and the effective potential lower bound amount tominimizing an effective Hamiltonian for the eigenvalues. These effective Hamiltonians, H var and H BO respectively, are derived in appendix A. The quantum mechanics of eigenvaluescan be solved using well-established collective field methods [20]. We give details in appendix11 Figure 4: Minimal energy configuration in the bootstrap allowed region for L = 3 , . Thegray dashed curves are rigorous lower and upper bounds of the ground state energy fromthe Born-Oppenheimer approximation, as discussed in the text. In the plots we have set m = 1 . The numerical uncertainty, estimated from different choices of the accuracy goal,the optimization step size, the initialization and the regularization parameters, is within thescale of the markers. 12. In particular, as N → ∞ the eigenvalue distribution ρ ( x ) = (cid:80) i δ ( x − x i ) must minimizecertain functionals, so that the true ground state energy E is bounded by min ρ E BO [ ρ ] ≤ E ≤ min ρ E var [ ρ ] . (26)In the appendices we show that E BO [ ρ ] = (cid:90) dxρ ( x ) (cid:18) π ρ ( x ) + m x (cid:19) + (cid:90) dxdyρ ( x ) ρ ( y ) ω ( x, y ) , (27)with ω ( x, y ) = (cid:112) m + g ( x − y ) , while E var [ ρ ] = E BO [ ρ ] + (cid:90) dxdydzρ ( x ) ρ ( y ) ρ ( z ) ( ω ( x, z ) − ω ( y, z )) ω ( x, z ) ω ( y, z )( x − y ) . (28)It is straightforward to perform the minimizations in (26) numerically by discretizing ρ ( x ) .This leads to the results for the energy shown in Fig. 4.From the results in Fig. 4 one can verify that the ratio N tr[ X, Y ] / (tr X ) tends to anonzero constant at large N g . This means that the matrices do not become commuting inthis limit. This can be constrasted with the analogous two matrix integral, with no time,that does become commuting at large N g [21]. This is consistent with the fact that thetwo matrix integral diverges in the massless limit [22,23], as the eigenvalues spread far apartalong the classically flat directions of the potential due to commuting matrices, while themassless matrix quantum mechanics still has a discrete spectrum of normalizable states [24].That is, the flat directions are lifted quantum mechanically. In summary, we have introduced a systematic numerical method to obtain energies andexpectation values of large N matrix quantum mechanics states. The method involves es-tablishing relationships between expectation values and then imposing positivity of a certainmatrix of expectation values, in the spirit of [1]. In Fig. 2 we see that the known analyticresults for one matrix large N quantum mechanics are readily reproduced. In Fig. 4 we haveobtained new results for the ground state energy and expectation values of a two matrixlarge N quantum mechanics.There are many natural extensions to our work. The extension to more matrices shouldbe possible with increased computing power or perhaps by optimizing the algorithm. Look-ing at supersymmetric states in supersymmetric theories may allow for stronger relationshipsbetween expectation values, using the supersymmetry generators. Both more matrices and13upersymmetry will of course be necessary to tackle the full blown BFSS and BMN theories.Finally, extensions to Gibbs states (or, to high energy eigenstates) may allow nonzero tem-perature quantum physics to be accessed with our bootstrap methods. The main challengehere will be to understand the density of states. This could give an alternative probe of thethermal phase transitions studied via Monte Carlo in e.g. [12, 13], as well as a new windowonto black hole microstates. Acknowledgements
This work arose from discussions with Edward Mazenc and Daniel Ranard, who also collab-orated on the early stages of the project. JK is supported by the Simons Foundation. SAHis partially supported by DOE award de-sc0018134 and by a Simons Investigator award.
A Born-Oppenheimer wavefunction and effective Hamiltonian
This appendix gives details of computations involving a Born-Oppenheimer wavefunction forthe two matrix quantum mechanics (22). The role of this wavefunction is to give a lower andan upper bound on the actual ground state energy. This gives a check on the accuracy of ournumerical bootstrap in this case. The results of this appendix are the effective Hamiltonians(40) and (41) for the eigenvalues of one of the two matrices. These will be solved in thefollowing appendix B, giving the upper and lower bounds respectively.The wavefunction that we are searching for is a complex function Ψ( X, Y ) of Hermitianmatrices X and Y . The state should be SU ( N ) gauge invariant and hence for any unitarymatrix W ∈ SU ( N ) , Ψ( X, Y ) = Ψ(
W XW − , W Y W − ) . (29)It will be convenient to parametrize such a state with the following set of variables: a diagonalreal matrix x i , a Hermitian matrix y ij and a unitary matrix U ∈ SU ( N ) , such that X = U diag( x i ) U − , Y = U yU − . (30)In these variables we can write down the following Born-Oppenheimer ansatz, in which the y ij oscillators are put in their ground state for a fixed configuration of eigenvalues x i : Ψ( X, Y ) = ψ ( x i ) φ ( x i , y ij ) , φ ( x i , y ij ) = N (cid:89) i,j =1 (2 ω ij /π ) / e − ω ij | y ij | , (31)14ith ω ij = m + g ( x i − x j ) . Equation (31) defines a gauge invariant wavefunction byspecifying its values on the gauge slice where X is diagonal. However, we should check that(31) is well-defined because (30) does not uniquely determine x i and y ij as a function of X and Y . Indeed, there is a residual U (1) N − gauge symmetry after fixing X to be diagonal: ifwe choose U = diag(exp iθ i ) in (30), X = diag( x i ) but Y ij = y ij exp i ( θ i − θ j ) . Because (31)is invariant under this residual gauge symmetry as well, Ψ( X, Y ) in (31) is well-defined.To obtain a variational upper bound, we wish to find an effective Hamiltonian for the‘slow’ x i degrees of freedom that calculates the expectation value of the full Hamiltonian(22) in the variational state (31). The expectation value of the Hamiltonian in the state Ψ consists of a kinetic part and a potential part: (cid:104) Ψ | H | Ψ (cid:105) = (cid:90) dXdY Ψ ∗ ( X, Y )( H kin + H pot )Ψ( X, Y ) . (32)We discuss these in turn. The kinetic energy is (cid:104) Ψ | H kin | Ψ (cid:105) = N (cid:88) i,j =1 (cid:90) dXdY (cid:32)(cid:12)(cid:12)(cid:12)(cid:12) ∂ Ψ( X, Y ) ∂X ij (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) ∂ Ψ( X, Y ) ∂Y ij (cid:12)(cid:12)(cid:12)(cid:12) (cid:33) . (33)Here ∂/∂X ij = ( ∂/∂ Re X ij − i∂/∂ Im X ij ) are complex derivatives because the matrices areHermitian. Because the kinetic energy operator is also gauge invariant, the integrand in (33)is constant along gauge orbits. So it suffices to evaluate it on the gauge slice where U in (30)is the identity. Then by the chain rule and (30), at U = I , ∂ Ψ ∂x i = ∂ Ψ ∂X ii , ∂ Ψ ∂y ij = ∂ Ψ ∂Y ij , (34)and ∂ Ψ ∂U ij = ( x j − x i ) ∂ Ψ ∂X ij + N (cid:88) m,n =1 ( δ im y jn − δ jn y mi ) ∂ Ψ ∂Y mn . (35)Because Ψ is gauge invariant as in (29), ∂ Ψ /∂U = 0 so for i (cid:54) = j , ∂ Ψ ∂X ij = 1 x i − x j N (cid:88) m,n =1 ( δ im y jn − δ jn y mi ) ∂ Ψ ∂y mn . 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