Numerical tests of the gauge/gravity duality conjecture
[0.2cm] for D0-branes at finite temperature and finite
Masanori Hanada111 E-mail address : email@example.com, Yoshifumi Hyakutake222 E-mail address : firstname.lastname@example.org, Goro Ishiki333 E-mail address : email@example.com and Jun Nishimura444 E-mail address : firstname.lastname@example.org
Stanford Institute for Theoretical Physics, Stanford University, Stanford, CA 94305, USA
Yukawa Institute for Theoretical Physics, Kyoto University,
Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan
The Hakubi Center for Advanced Research, Kyoto University,
Yoshida Ushinomiyacho, Sakyo-ku, Kyoto 606-8501, Japan
College of Science, Ibaraki University, Bunkyo 2-1-1, Mito, Ibaraki 310-8512, Japan
Center for Integrated Research in Fundamental Science and Engineering (CiRfSE),
Faculty of Pure and Applied Sciences, University of Tsukuba,
Tsukuba, Ibaraki 305-8571, Japan
KEK Theory Center, High Energy Accelerator Research Organization,
1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan
Graduate University for Advanced Studies (SOKENDAI),
1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan
According to the gauge/gravity duality conjecture, the thermodynamics of gauge theory describing D-branes corresponds to that of black branes in superstring theory. We test this conjecture directly in the case of D0-branes by applying Monte Carlo methods to the corresponding gauge theory, which takes the form of the BFSS matrix quantum mechanics. In particular, we take the continuum limit by extrapolating the UV cutoff to infinity. First we perform simulations at large so that string loop corrections can be neglected on the gravity side. Our results for the internal energy exhibit the temperature dependence consistent with the prediction including the corrections. Next we perform simulations at small but at lower temperature so that the corrections can be neglected on the gravity side. Our results are consistent with the prediction including the leading string loop correction, which suggests that the conjecture holds even at finite .
One of the most important directions in theoretical physics is to clarify the quantum nature of gravity, which is crucial in understanding the beginning of our Universe and the final state of a black hole. Superstring theory is considered the most promising candidate of a quantum gravity theory due to its UV finiteness in striking contrast to the conventional field theoretical approach to quantum gravity, in which one faces with nonrenormalizable UV divergences. So far, superstring theory is defined only perturbatively around simple backgrounds such as flat spacetime, and it does not seem to be straightforwardly applicable to the studies of a strongly gravitating spacetime such as the black hole geometry. However, this difficulty has been partly surmounted by the discovery of D-branes . Some extremal black holes were constructed by combining different kinds of D-branes, and the origin of their entropy was understood by counting the microstates of the D-branes . Also there are several proposals for nonperturbative formulations of superstring theory based on super Yang-Mills theory in low dimensions [3, 4, 5, 6].
In fact, it is conjectured that superstring theory on the anti-de Sitter background is dual to four-dimensional U() super Yang-Mills theory , which is realized on a stack of D3-branes. This duality has been generalized in various ways, and it is commonly referred to as the gauge/gravity duality conjecture. The conjecture appears natural considering that D-branes have two descriptions, one from a gravitational viewpoint, and the other from a field theoretical viewpoint. If this conjecture is true even in the presence of quantum effects on the gravity side, the quantum nature of gravity can be studied on a firm ground by investigating the dual gauge theory.
Among various gauge/gravity duality conjectures that have been proposed so far, we are interested in the one that has been studied most intensively, which claims that type II superstring theory in the near horizon limit of the black -brane geometry is equivalent to a maximally supersymmetric Yang-Mills theory in -dimensions [7, 8]. The super Yang-Mills theory is realized on D-branes and it is characterized by the rank of the gauge group and the ’t Hooft coupling . On the gravity side, is the number of black -branes, and the ’t Hooft coupling is written as in terms of the string coupling constant and the string length . The near horizon limit is taken by with and the energy scale of the super Yang-Mills theory kept finite. The above gauge/gravity duality has been tested in detail at and . In this limit, superstring theory is well approximated by supergravity, which makes classical analyses applicable. While the super Yang-Mills theory becomes strongly coupled in this region, one can nevertheless extract the information of BPS states, which are protected by supersymmetry, and confirm the gauge/gravity duality in various ways.
A natural question to ask then is whether the gauge/gravity duality conjecture is valid even for finite or finite , or in a non-supersymmetric setup such as finite temperature. The analyses in these cases are quite difficult, however, because the supergravity approximation is no longer valid on the gravity side and we have to include or corrections. Furthermore, the lack of supersymmetry gives rise to all kinds of nonperturbative corrections to physical quantities on the gauge theory side, which are too hard to handle analytically.
The main purpose of this paper is to test the gauge/gravity duality conjecture for finite and at finite temperature. While it is not possible to calculate finite or corrections in superstring theory in general, the corrections to some physical quantities can be extracted by taking account of higher derivative corrections to supergravity perturbatively. For instance, in the case of D0-branes, the internal energy including the leading corrections has recently been evaluated analytically [9, 10]. On the other hand, nonperturbative studies of the super Yang-Mills theory are possible by performing Monte Carlo simulation. In the case of D0-branes, the super Yang-Mills theory takes the form of matrix quantum mechanics for M theory [3, 11], which can be studied with reasonable amount of computational effort. In fact, several groups have studied this model [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25] and compared the obtained results with the dual gravity predictions. In particular, finite corrections were investigated by Monte Carlo simulation in refs. [16, 18] and more recently in refs. [24, 25], which raised some controversies. In this paper, we first address this issue based on new calculation, which improves our previous analysis [16, 18] by taking the continuum limit.
Then we investigate the corrections by simulating the same system at small . This turns out to be much more difficult than the studies at large because of the instability associated with the flat directions in the potential. The bound state of D0-branes is stable at large , but it becomes only meta-stable at sufficiently low temperature for small . We extract the internal energy of the meta-stable bound states by introducing a cutoff on the extent of the D0-brane distribution, which is chosen in such a way that the obtained result does not depend on it within a certain region. Our results obtained in this way turn out to be consistent with the analytic result obtained on the gravity side including the leading corrections. This suggests that the gauge/gravity duality holds even at finite . In fact, the instability at finite can be understood also on the gravity side. Some of the results are reported briefly in our previous publication .
The rest of this paper is organized as follows. In section 2 we give an overview of the black 0-brane thermodynamics, and discuss how finite and finite corrections appear in the internal energy of the D0-branes. In section 3 we explain how we study the D0-brane quantum mechanics by Monte Carlo simulation. In section 4 we provide numerical tests of the gauge/gravity duality including finite and finite corrections. Section 5 is devoted to a summary and discussions.
2 Brief review on the black 0-brane thermodynamics
In this section we briefly review the thermodynamics of the black 0-brane in type IIA superstring theory, which appears in the gauge/gravity duality we are going to test. In particular, we derive an expression for the quantity that should be compared with the internal energy of the dual gauge theory calculable by Monte Carlo methods. Corresponding to finite and finite corrections on the gauge theory side, we need to consider how the black 0-brane thermodynamics is affected by the higher derivative corrections to the low-energy effective action of type IIA superstring theory.
2.1 The effects of higher derivative corrections on the black 0-brane thermodynamics
In the low energy limit, the scattering amplitudes of strings in type II superstring theory can be reproduced correctly by the type II supergravity action. However, if we go beyond the low energy limit, we have to take into account the effects due to the finite length of strings ( corrections) and the effects of string loops ( corrections). In general, these effects can be extracted by considering the scattering amplitudes of strings associated with a Riemann surface with genus and expanding them with respect to external momenta [26, 27]. The number of external momenta in the expansion raises the power of , whereas the genus gives the power of . The corrections to the type II supergravity action due to these effects are represented by higher derivative terms, and they are organized in the form of a double expansion with respect to and . Below, we discuss some qualitative features of the higher derivative terms in type IIA superstring theory.
Treating the type IIA superstring theory perturbatively with respect to two parameters and the dilaton coupling , one can write its effective action formally as
with an overall coefficient given by . Here represents the terms with mass dimension 2 that appear in the type IIA supergravity action, and represents higher derivative corrections with mass dimension . All these terms are written in terms of the massless fields in the type IIA superstring theory such as the graviton , the dilaton and the R-R 1-form potential . The structure of the higher derivative terms can be determined by explicit calculations of scattering amplitudes, which show that the and terms appear as the leading corrections, respectively, at the tree level and at the one-loop level. It is also known that these terms do not appear from higher loops. Thus we only have terms in eq. (2.1) with for and with for .
The equations of motion that one obtains from the effective action (2.1) can also be expanded in a power series as
omitting the tensor indices for simplicity. Here represents the part obtained from the type IIA supergravity, and represents the part obtained from the high derivative corrections. In order to solve the above equations of motion for the black 0-brane, we make a general ansatz for , and respecting SO(9) rotational symmetry as 
which involves four unknown functions and . The leading behaviors of and are fixed by using the solution of which are asymptotically flat, and they involve two parameters and . As we will see below, and are related to the mass and the charge of the black 0-brane. The subleading terms described by the functions and can be obtained by solving eq. (2.2) order by order. Note that the functions for and can be obtained independently of the other unknowns since they are the leading corrections, respectively, at the tree level and at the one-loop level.
The event horizon , which is defined by , can be obtained perturbatively as . Then the Hawking temperature can be obtained by requiring the absence of conical singularity in the Euclidean geometry at the event horizon as555Here we reserve the variables such as , and without tildes for dimensionless quantities to be defined in (2.14).
where can be determined once the solution is obtained. Since the metric (2.3) is asymptotically flat, the mass of the black 0-brane can be evaluated by using the ADM mass formula and the charge can be calculated by integrating the R-R flux666The integrands for the ADM mass and the R-R charge have corrections due to the higher derivative terms, which vanish at .. They can be written formally as
where has mass dimension .
2.2 Black 0-brane thermodynamics at large
Let us first consider the supergravity approximation, which is valid when the curvature radius is large compared to and the effective coupling is small. Neglecting higher derivative corrections in eqs. (2.4) and (2.7), the temperature and the internal energy of the black 0-brane are obtained as
where and are the parameters of the classical black 0-brane. The extremal limit corresponds to and , where the latter follows from the former using and (2.6). In that limit, the event horizon , the temperature and the internal energy all vanish as long as is kept finite.
Note that the gravitational coupling goes to zero when with the ’t Hooft coupling fixed. This means that the gauge theory on the D0-branes decouples from the bulk gravity. On the other hand, the parameter in (2.9) is proportional to the product of the string tension and the typical length , which represents the gauge boson mass in the gauge theory. Therefore, fixing in the limit corresponds to keeping the energy scale of the dual gauge theory finite. Let us also mention that the limit (2.9) can be rewritten in terms of and , as
Since and , the near horizon limit may be regarded as a special case of the near extremal limit, in which the temperature and the internal energy are kept finite.
Taking the near horizon limit in eq. (2.8), we obtain
where we have defined the dimensionless quantities
Since the curvature radius and the effective coupling around the event horizon are estimated as
where the second term comes from the leading correction with and the third term comes from the next-leading correction with . The absence of follows from some knowledge  on the structure of the effective action (2.1). The explicit values of the non-zero coefficients are not known so far, however.
In fact, the internal energy of the black 0-brane is affected by the Hawking radiation, which has not been taken into account in (2.7). However, the energy loss through the Hawking radiation can be neglected in the near horizon limit, as we show in appendix A. This is reasonable since the near horizon limit implies the near extremal limit as well.
2.3 Black 0-brane thermodynamics at finite
Let us move on to the case with finite . Since the effective coupling given by (2.15) can no longer be neglected, all the higher order terms in eqs. (2.4) and (2.7) remain. Using (2.15), each dimensionless term in (2.4) and (2.7) behaves as
Therefore, the internal energy of the black 0-brane in the near horizon limit is obtained as
The coefficients can be determined, in principle, by solving the equations of motion (2.2) that can be derived from the explicit form of the effective action (2.1). Using some knowledge  on the structure of the effective action (2.1), we have only for and only for , etc..
In general, it is difficult to obtain the higher derivative corrections in the effective action (2.1). There exists one exception, however, which is the leading correction at one loop corresponding to . In this case, the correction can be obtained  by uplifting the black 0-brane solution to the M-wave solution in eleven dimensions, which is purely geometrical. Since higher curvature corrections in eleven dimensions are well-known [29, 30, 31, 32, 33], it is possible to derive the equations of motion and solve them. Using this result, the internal energy including the leading correction can be obtained explicitly as 
(See appendix B for a review on the derivation.)
Thus we find that the internal energy can be expanded with respect to and as
where and correspond to the terms and the terms, respectively, at the tree level, while comes from the terms at the one-loop level.
3 D0-brane quantum mechanics
According to the gauge/gravity duality conjecture, the thermodynamics of the black 0-brane corresponds to that of the gauge theory describing the D0-brane, which takes the form of the BFSS matrix quantum mechanics. In order to investigate the thermodynamics, we use the Euclidean time and compactify it with the periodicity . Then the action of D0-brane quantum mechanics at finite temperature is given by
which can be obtained formally by dimensionally reducing the action of d U() super Yang-Mills theory to dimension. We have introduced and , which are bosonic and fermionic Hermitian matrices, respectively, and the covariant derivative is defined using the gauge field . The bosonic variables obey periodic boundary conditions , , whereas the fermionic variables obey anti-periodic boundary conditions . The matrices in (3.1) act on spinor indices and satisfy the Euclidean Clifford algebra .
The ’t Hooft coupling corresponds to defined in (2.9) on the dual gravity side. Since the coupling constant in the action (3.1) has mass dimension 3, all dimensionful quantities can be measured in units of as we did in (2.14). Note that the expansion (2.20) is valid when and . In particular, the first inequality implies that should be large for fixed temperature . This implies that we need to study the strong coupling dynamics of the D0-brane quantum mechanics in order to test the gauge/gravity duality. For that purpose, we apply Monte Carlo methods analogous to the ones used in lattice QCD.
3.1 Putting the theory on a computer
In order to apply Monte Carlo methods, we have to put the D0-brane quantum mechanics (3.1) on a computer. It is not possible to do it, however, respecting all the maximal supersymmetry generated by 16 supercharges, which the theory has at zero temperature. For instance, if one discretizes the time direction, one cannot maintain all the supersymmetry, since successive supersymmetry transformations induce a translation in time direction, which is broken by the discretization. The lack of exact symmetry in quantum field theories typically necessitates fine tuning in taking the continuum limit due to UV divergences. This does not occur, however, in the present case since the system is just a quantum mechanics, which is UV finite. Here, instead of discretizing the time direction, we expand the functions of in Fourier modes and introduce a mode cutoff after fixing the gauge symmetry . Since the higher Fourier modes omitted in our calculations are suppressed by the kinetic term, the approach to is expected to be fast.
We fix the gauge symmetry by the static diagonal gauge
where are chosen to satisfy the constraint777In actual simulation, we replace the constraint by . This is practically equivalent to (3.3) unless is distributed in the whole region (3.3), which occurs only at very low temperature.
This constraint is needed to fix the symmetry under large gauge transformations888The gauge transformation acts on the gauge field as , where is an unitary matrix which satisfies the periodic boundary condition . After taking the static diagonal gauge, one still has to fix the residual symmetry under large gauge transformations, which correspond to with being integers.. The Faddeev-Popov term associated with this gauge fixing condition is given by
which we add to the action (3.1). The integration measure for is taken to be uniform.
Once we fix the gauge symmetry, we can introduce the momentum cutoff for the Fourier modes of and . Since the bosonic matrices obey periodic boundary conditions, they are expanded as
where , and runs over integers999Note that a large gauge transformation shifts the momentum of the mode as . Therefore, one needs to fix the symmetry under large gauge transformations in order for the momentum cutoff to make sense.. On the other hand, the fermionic matrices , which obey anti-periodic boundary conditions, are expanded as
where runs over half-integers. By using a shorthand notation
the action (3.1) can be expressed as , where
This action is written in terms of a finite number of variables , and , and hence it can be dealt with on a computer. The continuum limit is realized by sending the cutoff to infinity.
The fermionic degrees of freedom are treated in the following way. Note that the fermionic action can be written as , where we have expanded in terms of the generators of as . By integrating out the fermionic variables, the partition function can be written as
where represents the Pfaffian of , which is complex in general and is denoted as . Since Monte Carlo simulation is applicable only when the path integral has a positive semi-definite integrand, we omit the phase factor and define the expectation value of for the phase-quenched model as
Then, the expectation value with respect to the original theory (3.10) is given by
When fluctuates rapidly in the simulation of the phase-quenched model, it is difficult to evaluate (3.12) since both the denominator and the numerator become very small, and the number of configurations needed to obtain the ratio with sufficient accuracy becomes huge. This technical problem is called the sign problem.
In the present system, however, it has been reported that the fluctuation of is strongly suppressed at both high temperature and low temperature, and that it can be neglected throughout the whole temperature region [23, 25]. This can be understood as follows. At high temperature, the high temperature expansion becomes applicable , which implies that the dynamics of and in the Pfaffian becomes perturbative. Therefore, the fluctuation of becomes less important at high temperature. At low temperature, on the other hand, the dynamics is dominated by the low momentum modes, for which the kinetic term (the first term) in (3.9) can be neglected. If we omit the kinetic term, we can show that the Pfaffian becomes real. Thus, the effects of can be neglected also at low temperature, which is supported by some numerical evidence . In the present work, we simply omit the phase factor , and use (3.11) to calculate the VEV of observables. See appendix B of ref.  for the details of our algorithm for Monte Carlo simulation.
Let us also comment on how we treat the zero modes. Note that the constant modes () of the trace part does not appear in the action (3.1). These modes should be omitted in the path integral (3.10). We can extract these modes from a general configuration by
In what follows, we assume that these zero modes are fixed by the constraint for . In Monte Carlo simulation, even if we start from a configuration satisfying the constraint, can become nonzero as the simulation proceeds due to accumulation of round-off errors. We avoid this by making a projection .
3.2 Calculation of the internal energy
The internal energy of the D0-brane quantum mechanics can be calculated using the formula
A peculiar aspect of the D0-brane quantum mechanics is that the action (3.1) has flat directions . These are lifted by quantum corrections in the case of the bosonic model, in which fermionic degrees of freedom are omitted . However, in the supersymmetric model, the flat directions are not lifted by quantum corrections due to supersymmetry. As a result, the supersymmetric model contains scattering states, which form the continuous branch of the energy spectrum, in addition to the normalizable energy eigenstates, which form the discrete branch of the spectrum.101010In ref. , the discrete branch of the spectrum is shown to have a new energy scale proportional to based on the effective Hamiltonian for the relevant degrees of freedom in the flat directions. Based on this observation, the particular power “2.8” of the leading behavior in (2.16) has been understood theoretically on the gauge theory side. See also ref.  for related work on supersymmetric models with 4 and 8 supercharges. Therefore, the path integral (3.10) is ill-defined as it stands, and we need to consider how to make sense out of it.
In the large- limit, the path integral (3.10) becomes well-defined analogously to the well-known example of the model . In this case, we may naturally consider that the well-defined path integral (3.10) actually represents the thermodynamics of the normalizable states only.
The situation becomes subtle at finite . Suppose we prepare an initial state with having sufficiently low energy and let it evolve in time quantum mechanically. It is expected from the Monte Carlo simulation discussed in section 4.2 that the size of the state fluctuates for a while around some finite value depending on the initial state, and eventually starts to diverge. These meta-stable states are linear combinations of normalizable states and scattering states. However, if they are long-lived, we can still think of their thermodynamics by introducing a cutoff , where should be chosen to be the typical size of the meta-stable states for a given energy. This can be achieved in the path integral formalism by replacing the partition function (3.10) by
where is the Heaviside step function and we have defined
It is expected that the internal energy of the cutoff system (3.15) becomes independent of within some region, and the internal energy obtained in that region can be interpreted as the average internal energy of the meta-stable states for the and corresponding to the partition function (3.15). The typical size of the meta-stable states can be identified with the lower end of the region of within which the internal energy is constant. Note that the partition function (3.15) does not represent the thermodynamics of the normalizable states only unlike the case with . Therefore, there is no guarantee that the specific heat corresponding to the internal energy defined in this way becomes positive.
Since the eigenvalues of represent the position of the D0-branes, the meta-stable states can be interpreted as the bound states of D0-branes, which form a black hole. Therefore, it is expected that the internal energy defined above corresponds to the internal energy (2.20) of the black hole. Indeed the black hole is stable in the large- limit, but it becomes meta-stable at finite due to quantum effects corresponding to corrections. This can be seen in (2.19), for instance, where the leading correction makes the specific heat negative at sufficiently low . This instability can be understood physically as caused by the repulsive force acting on a test particle near the event horizon due to the quantum gravity effects at small distances .
4 Numerical tests of the gauge/gravity duality
In this section we provide numerical tests of the gauge/gravity duality including finite and finite corrections, which correspond to the and string loop corrections, respectively, on the gravity side. These corrections are discussed separately in sections 4.1 and 4.2.
4.1 Test including corrections
In this section we provide a test of the gauge/gravity duality in the large- limit. For that purpose, we perform Monte Carlo simulation of the D0-brane quantum mechanics (3.1) at large and compare the results with the prediction (2.16) obtained on the gravity side. As long as , where is some critical value depending on , the instability mentioned in the previous section does not show up practically during Monte Carlo simulation, and we can calculate various observables by taking an average in a straightforward manner. The critical value is found to behave as at , which makes the lower region difficult to study.
|, const. fit||, linear fit|
In ref. , the results obtained by Monte Carlo simulation with and were compared with the prediction including corrections. The numerical data were fitted by an ansatz with and , which is consistent with the prediction from the gravity side. However, a recent paper  repeated the analysis including data points at lower with using a lattice formulation.111111The lattice size was , which roughly corresponds to from the viewpoint of the number of degrees of freedom. The values obtained from the same fit was and . In ref. , a one-parameter fit with the power fixed was also performed, and the coefficient was determined as .
In order to clarify this discrepancy, we improve our previous analysis in ref.  by making extrapolations to . In Fig. 1 (Left) we plot our results obtained for against . We tried both a linear extrapolation and a constant fit. The values at each and the values obtained by the extrapolations are given in Table 1 and 2, respectively. In Fig. 1 (Right), we plot the internal energy obtained by the extrapolations. The new results are consistent with the fit obtained in ref. .
In Fig. 2 we plot the difference between the obtained internal energy and the leading prediction (2.13) from supergravity. The difference is normalized by the leading prediction as and it is plotted against . The leading corrections correspond to a linear behavior towards the origin. Indeed we see a linear behavior for consistent with the fit obtained in ref. . On the other hand, the subleading terms are expected to show up as . The solid line and the dashed line are fits to using the data points within the range and , respectively, obtained by the linear fit. In the latter case, the left-most data point obtained by the constant fit is slightly off the fitting curve. However, this may be due to finite- effects, which become more significant at lower temperature as is suggested from the expansion (2.20). In order to decide which fit is more appropriate, we clearly need more data at lower temperature with larger .
4.2 Test including string loop corrections
In this section we test the gauge/gravity duality including string loop corrections. For that purpose, we need to study the D0-brane quantum mechanics at small such as . As is mentioned at the end of section 3.2, the system has instability at small associated with the flat directions in the action. In order to probe the instability, we define
which represents the extent of the eigenvalue distribution of ’s. In Monte Carlo simulation, we prepare the initial configuration of with small by giving each element a small Gaussian random number. At sufficiently low , we observe that stabilizes as the simulation proceeds, and fluctuates around some value for a while and then starts to diverge. This behavior motivates us to consider the partition function (3.15), where is replaced by for simplicity.121212We consider that this does not make much difference because the fluctuation of as a function of is typically small. What we do in practice is to add the potential term
to the action, where and are some parameters to be chosen appropriately.
Let us define the distribution of by
where the expectation value is taken in the system with the potential (4.2). Figure 3 (Top-Left) shows the distribution obtained from Monte Carlo simulation with , , , where we have set and . There is a clear peak around , which indicates the existence of the meta-stable bound states. The long tail at represents the run-away behavior caused by the instability. In Fig. 3 (Top-Right) we plot the internal energy obtained by averaging only over configurations with for the same set of parameters. We see a clear plateau around , which confirms the argument given in section 3.2. Practically, we define the internal energy of the bound states by the local minimum of in the plateau region.
The extent of the bound state can be identified as the lower end of the plateau region, which we denote as . In practice, we obtain the value of , at which deviates from the local minimum by 5%, and similarly the value of allowing 10% deviation. We use the average of the two values as an estimate of and the difference as an estimate of the ambiguity (“error”). In Fig. 3 (Bottom) we plot thus obtained as a function of for and together with the expectation value obtained from configurations with . We observe that both and increase as is lowered. Note that the quantity at large can be obtained without such a cutoff procedure, and it is a monotonically increasing function of . (See Fig. 2 of ref. , for instance.)
Using the method explained above, we calculate the internal energy of the bound states for various , and . We have studied for , for and for . In Table 3, we present our results for the internal energy obtained at each , and . We make an extrapolation to assuming that finite corrections to the internal energy are given by , from which we extract the value in the continuum limit. This extrapolation is performed using for and for . Figure 4 (Left) shows the case of , . In Fig. 4 (Right), we plot our results for obtained in the continuum limit by extrapolation to . (The explicit values are given in the right most column of Table 3.) The curves in this plot are explained at the end of this section.