Thermal field theory of bosonic gases with finite-range effective interaction

TL;DR

This paper develops an effective field theory for dilute ultracold Bose gases incorporating finite-range interactions, deriving analytical results for the equation of state and stability conditions at zero and finite temperatures.

Contribution

It introduces a beyond-mean-field analytical framework accounting for finite-range effects, extending previous zero-range models and aligning with Monte Carlo simulations.

Findings

Finite-range effects stabilize the Bose gas against quantum fluctuations.

A positive effective range above a critical value removes thermodynamic instability.

Results agree with recent Monte Carlo calculations for hard-sphere bosons.

Abstract

We study a dilute and ultracold Bose gas of interacting atoms by using an effective field theory which takes account finite-range effects of the inter-atomic potential. Within the formalism of functional integration from the grand canonical partition function we derive beyond-mean-field analytical results which depend on both scattering length and effective range of the interaction. In particular, we calculate the equation of state of the bosonic system as a function of these interaction parameters both at zero and finite temperature including one-loop Gaussian fluctuation. In the case of zero-range effective interaction we explicitly show that, due to quantum fluctuations, the bosonic system is thermodynamically stable only for very small values of the gas parameter. We find that a positive effective range above a critical threshold is necessary to remove the thermodynamical instability of the uniform configuration. Remarkably, also for relatively large values of the gas parameter, our finite-range results are in quite good agreement with recent zero-temperature Monte Carlo calculations obtained with hard-sphere bosons.

Equations

$$\mathcal{Z}=\int\mathcal{D}[\psi,\psi^{*}]\exp\bigg{\{}-\frac{S[\psi,\psi^{*}]}{\hbar}\bigg{\}}\;,$$

$$S[\psi ,{\psi }^{\ast }]={\int }_0^{\hslash \beta }d\tau \int d^3\mathbf{r}\mathcal{L}(\psi ,{\psi }^{\ast })$$

$$\Omega =-\frac{1}{\beta }\log \mathcal{Z}.$$

$$\mathcal{L}$$

$$+\frac{1}{2}\int d^3{\mathbf{r}}^{\prime }|\psi ({\mathbf{r}}^{\prime },\tau ){|}^{2}V(|\mathbf{r}-{\mathbf{r}}^{\prime }|)|\psi (\mathbf{r},\tau ){|}^2,$$

$${\tilde{V}}_{p,0}(q)=g_0,$$

$$\mathcal{L}=\psi^{*}(\mathbf{r},\tau)\bigg{[}\hbar\frac{\partial}{\partial\tau}-\frac{\hbar^{2}\nabla^{2}}{2m}-\mu\bigg{]}\psi(\mathbf{r},\tau)+\frac{1}{2}g_{0}|\psi(\mathbf{r},\tau)|^{4}\;,$$

$$\psi (\mathbf{r},\tau )={\psi }_0+\eta (\mathbf{r},\tau )$$

$$S_{g}[{\tilde{\eta}},{\tilde{\eta}}^{*}]={1\over 2}\sum_{Q}({\tilde{\eta}}^{*}(Q),{\tilde{\eta}}(-Q))\ {\bf M}(Q)\left(\begin{array}[]{c}{\tilde{\eta}}(Q)\\ {\tilde{\eta}}^{*}(-Q)\end{array}\right)\;$$

$$\mathbf{M}(Q)=\beta \left(\begin{array}{cc}-i\hslash {\omega }_n+\frac{{\hslash }^2{q}^2}{2m}-\mu +2g_0{\psi }_0^2& g_0{\psi }_0^2\\ g_0{\psi }_0^2& i\hslash {\omega }_n+\frac{{\hslash }^2{q}^2}{2m}-\mu +2g_0{\psi }_0^2\end{array}\right).$$

$${\Omega }_g$$

$$E_{\mathbf{q}}(\mu ,{\psi }_0)=\sqrt{{\left(\frac{{\hslash }^2{q}^2}{2m}-\mu +2g_0{\psi }_0^2\right)}^2-g_0^2{\psi }_0^4}.$$

$$\frac{1}{2\beta }\sum _{n=-\infty }^{+\infty }\ln [{\beta }^2({\hslash }^2{\omega }_n^2+E_{\mathbf{q}}^2)]=\frac{E_{\mathbf{q}}}{2}+\frac{1}{\beta }\ln (1-e^{-\beta E_{\mathbf{q}}}),$$

$$\Omega (\mu ,{\psi }_0)={\Omega }_0(\mu ,{\psi }_0)+{\Omega }_g(\mu ,{\psi }_0)+{\Omega }_g^{(T)}(\mu ,{\psi }_0),$$

$${\Omega }_0(\mu ,{\psi }_0)=\left(-\mu {\psi }_0^2+\frac{1}{2}{g}_0{\psi }_0^4\right)L^3$$

$${\Omega }_g^{(0)}(\mu ,{\psi }_0)=\frac{1}{2}\sum _{\mathbf{q}}{E}_{\mathbf{q}}(\mu ,{\psi }_0)$$

$${\Omega }_g^{(T)}(\mu ,{\psi }_0)=\frac{1}{\beta }\sum _{\mathbf{q}}\ln \left(1-e^{-\beta E_{\mathbf{q}}(\mu ,{\psi }_0)}\right)$$

$$\frac{\partial {\Omega }_0(\mu ,{\psi }_0)}{\partial {\psi }_0}=0,$$

$${\psi }_0(\mu )=\sqrt{\frac{\mu }{g_0}}$$

$$E_{\mathbf{q}}(\mu )=\sqrt{\frac{{\hslash }^2{q}^2}{2m}\left(\frac{{\hslash }^2{q}^2}{2m}+2\mu \right)}.$$

$$\Omega_{g}^{(0)}(\mu)={8\over 15\pi^{2}}\bigg{(}{m\over\hbar^{2}}\bigg{)}^{3/2}\mu^{5/2}\;.$$

$$P(\mu )=-\frac{\Omega (\mu )}{L^3}.$$

$$P(\mu)={\mu^{2}\over 2\,g_{0}}-{8\over 15\pi^{2}}\bigg{(}{m\over\hbar^{2}}\bigg{)}^{3/2}\mu^{5/2}\;,$$

$$\frac{{\partial }^{2}P(\mu )}{\partial {\mu }^2}>0.$$

$$n(\mu )=-\frac{\partial \Omega (\mu )}{\partial \mu },$$

$$n(\mu )=\frac{\mu }{g_0}-\frac{4}{3{\pi }^2}{\left(\frac{m}{{\hslash }^2}\right)}^{3/2}{\mu }^{3/2}.$$

$$\mu (n)$$

$${\tilde{V}}_{p,2}(q)=g_0+g_2{q}^2.$$

$$g_0=\tilde{V}(0)=\int d^3\mathbf{r}V(r)$$

$$g_2=\frac{1}{2}{\tilde{V}}^{\prime \prime }(0)=-\frac{1}{6}\int d^3\mathbf{r}{r}^{2}V(r)$$

$$V_{p}(r)=g_{0}\,\delta(r)-{g_{2}\over 2}\bigg{[}{\overleftarrow{\nabla}}^{2}\,\delta(r)+\delta(r)\,{\overrightarrow{\nabla}}^{2}\bigg{]}\;.$$

Full text

I Introduction

The experimental achievement of Bose-Einstein condensation with dilute and ultracold alkali-metal atoms anderson1995 ; bradley1995 ; davis1995 has triggered many theoretical investigations of weakly-interacting Bose gases. The key theoretical tool for the description of these bosonic systems at zero temperature is the Gross-Pitaevskii equation gross , where the nonlocal inter-atomic interaction is approximated by a local contact interaction characterized by only one physical parameter, the s-wave scattering length $a_{s}$. Also quantum and thermal fluctuations have been analyzed within this contact approximation of the inter-atomic potential (see for instance shi1998 ; dalfovo1999 ; leggett2001 ; andersen ; schakel ; sala-review ), which can be interpreted as an zero-range effective field theory (EFT) of the dilute Bose gas andersen ; braaten ; roth . In the last years the EFT approach to the Bose gas has been extended including finite-range effects due to the effective range $r_{s}$ of the interaction potential. It has been shown that this approach gives a modified Gross-Pitaevskii equation sala-old ; gao ; boris ; pethick ; zinner ; ketterle ; sgarlata for the condensate and non universal effects for quantum fluctuations at zero temperature andersen ; braaten ; roth ; gao .

In this paper, by using finite-temperature functional integration schakel ; atland , we show that with a zero-range repulsive interaction the uniform configuration of the Bose gas becomes thermodynamically unstable at large values of the gas parameter due to Gaussian quantum fluctuations. This puzzling problem is solved by including the effects of the effective range $r_{s}$ of the interaction potential. In particular, we derive the beyond-mean-field (one-loop, Gaussian) equation of state of the bosonic system at zero temperature. This equation of state (see also andersen ; braaten ) depends on both the scattering length $a_{s}$ and the effective range $r_{s}$ of the interaction. We prove that it is necessary a positive effective range $r_{s}$ above a critical threshold ($r_{s}/a_{s}>0.25$) to remove the thermodynamical instability of the uniform configuration for large values of the gas parameter. In the case of a hard-core interaction potential, where $r_{s}/a_{s}=2/3$, our EFT results for the ground-state energy are in quite good agreement with zero-temperature Monte Carlo calculations pigs . Finally, we include thermal fluctuations and obtain analytically a finite-temperature equation of state which is reliable at low temperatures.

II Partition function of the system

In the study of the interacting Bose gas we adopt the path integral formalism, where bosonic atoms are described by a complex field $\psi(\mathbf{r},\tau)$. Within this framework, all the relevant thermodynamical properties of the system can be computed starting from the grand canonical partition function ${\cal Z}$ at finite temperature atland

[TABLE]

where

[TABLE]

is the Euclidean action and $\beta\equiv 1/(k_{B}T)$ with $k_{B}$ being the Boltzmann’s constant. The grand potential $\Omega$, as a function of the chemical potential $\mu$ and the temperature $T$, can be obtained by atland

[TABLE]

By working in the grand canonical ensemble the nonlocal Lagrangian density of interacting identical bosons is given by

[TABLE]

where $V(|\mathbf{r}-\mathbf{r^{\prime}}|)$ is the spherically-symmetric two-body interaction potential between bosons.

III Zero-range effective potential

In order to get analytical results, several authors showed that it is worthwhile to replace the inter-atomic potential with a pseudo-potential which must reproduce low-energy scattering properties and energy shifts of the original one roth . Dealing with ultracold and dilute atoms, the usual, and most simple, scheme consists in replacing $V(r)$ with the zero-range Fermi pseudo-potential $V_{p,0}(r)=g_{0}\,\delta^{(3)}(r)$, where $\delta^{(3)}(r)$ is the Dirac delta function. Clearly, the Fourier transform ${\tilde{V}}_{p,0}(q)$ of $V_{p,0}(r)$ reads

[TABLE]

where $g_{0}$ is given, from scattering theory, by $g_{0}={4\pi\hbar^{2}}a_{s}/m$ with $a_{s}$ the s-wave scattering length andersen ; dalfovo1999 . Consequently, the Lagrangian density becomes

[TABLE]

The mean-field (saddle-point) plus Gaussian (one-loop) approximation is obtained setting

[TABLE]

and expanding the action $S[\psi,\psi^{*}]$ of Eq. (2) around the uniform and constant $\psi_{0}$ up to quadratic (Gaussian) order in $\eta({\bf r},\tau)$ and $\eta^{*}({\bf r},\tau)$. We find that, in the momentum space, the Gaussian contribution of quantum fluctuation is described by

[TABLE]

where $Q=({\bf q},i\omega_{n})$ is the $3+1$ vector denoting the momenta ${\bf q}$ and bosonic Matsubara frequencies $\omega_{n}=2\pi n/(\beta\hbar)$. The matrix ${\bf M}(Q)$ is the inverse fluctuation propagator, given by the following

[TABLE]

Integrating over the bosonic fields ${\tilde{\eta}}(Q)$ and ${\tilde{\eta}}^{*}(Q)$ we obtain the Gaussian grand potential

[TABLE]

where $E_{\bf q}$ is the dispersion relation:

[TABLE]

The sum over bosonic Matsubara frequencies gives andersen

[TABLE]

and, in this way, taking into account Eq.(6) one finds the grand potential andersen ; schakel ; sala-review

[TABLE]

where

[TABLE]

is the mean-field contribution (assuming a real $\psi_{0}$) with $L^{3}$ the volume of the system,

[TABLE]

is the zero-point energy of bosonic excitations, i.e. the zero-temperature contribution of quantum Gaussian fluctuations, while

[TABLE]

takes into account thermal Gaussian fluctuations.

Imposing the crucial condition

[TABLE]

one gets

[TABLE]

and the well-known Bogoliubov spectrum of collective excitations

[TABLE]

Remarkably, the spectrum is now gapless as required by the Goldstone theorem. Now, one can replace $\psi_{0}$ with its classical value given by Eq. (18) and, after removing the ultraviolet divergence (for a detailed review see sala-review ) in $\Omega_{g}^{(0)}(\mu)=\Omega_{g}^{(0)}(\mu,\psi_{0}(\mu))$, the zero temperature grand potential becomes

[TABLE]

The pressure $P(\mu)$ of the system is simply related to the grand potential $\Omega(\mu)$ by the formula

[TABLE]

At zero temperature the beyond-mean-field pressure $P(\mu)$ as a function of the chemical potential $\mu$ is then given by

[TABLE]

and it agrees with the result derived in 1960 by Lee and Yang lee1960 within the framework of quantum statistical mechanics. Eq. (22) has been also obtained in haugset by using of a functional approach. It is important to stress that the pressure $P$ given by Eq. (22) becomes negative, i.e. unphysical, for a large value of the chemical potential $\mu$. Actually, the uniform configuration is thermodynamically stable if and only if

[TABLE]

Eq. (22) implies thermodynamical instability for $\mu>\mu_{c}=\pi(\hbar^{2}/(2m))^{3/4}/\sqrt{2g_{0}}$.

The zero-temperature number density $n$ can be derived from the grand potential by using this thermodynamic formula

[TABLE]

which gives

[TABLE]

From this equation one can easily numerically determine $\mu$ as a function of $na_{s}^{3}$. The result is the solid line of Fig. 1. The plot clearly shows that in the absence of finite-range corrections the chemical potential has two branches and there are no solutions above a critical value of the gas parameter $na_{s}^{3}$. This result, that is full consistent with the discussion of Eq. (23), means that for $na_{s}^{3}>0.004$ the uniform configuration does not exist anymore.

We observe that a simple analytical result is obtained from Eq. (25) by using a perturbative expansion where $g_{0}n\ll\mu$. In this way one gets

[TABLE]

The first term of this chemical potential was derived by Bogoliubov bogoliubov while the second term is the one deduced by Lee, Huang and Yang yang . However, it is extremely important to remind that Eq. (26) is obtained assuming a very small gas parameter $na_{s}^{3}$ (parturbative scheme above mean-field plus Gaussian results) and it cannot be used when the system is thermodynamically unstable according to the general equation Eq. (25). Close to the instability the contribution of Gaussian fluctuations to the grand potential, given by Eq. (20), becomes of the same order of the mean-field term; this signals that quantum fluctuations are strong enough to destabilize the uniform configuration. The critical value $(na_{s}^{3})_{c}\simeq 0.004$ specifies the upper threshold of applicability of the zero-range Gaussian theory and, consequently, of Eq. (26).

We shall now show that the inclusion of a effective-range interaction in the thermodynamics can remove the instability by assuring a positive sign of the second derivative of the pressure.

IV Finite-range effective potential

An improvement of the contact (zero-range) approximation can be achieved by replacing the interaction potential ${\tilde{V}}({\bf q})$ with the finite-range pseudo-potential

[TABLE]

The relation with the true inter-atomic potential appearing in Eq. (4) is given by the following relations

[TABLE]

and

[TABLE]

where $\tilde{V}(\mathbf{q})=\int d^{3}\mathbf{r}\exp(i\mathbf{q}\cdot\mathbf{r})V(r)$. It has been shown gao ; roth that, in real space, the pseudo-potential in (27) is given by

[TABLE]

The connection with experimental quantities such as the s-wave scattering length $a_{s}$ and the s-wave effective range $r_{s}$ can be established by requiring the matching between the expansion parameters of Eq. (27) and the ones obtained by a more general pseudo-potential $V_{p}(q)$. Several authors andersen ; braaten ; roth adopt

[TABLE]

which depends on the s-wave phase shift $\delta_{0}(q)$. Notice that $\delta_{0}(q)$ is related to the scattering length $a_{s}$ and the effective range $r_{s}$ by the equation

[TABLE]

One can expand $\tilde{V}_{p}(q)$ and $\delta_{0}(q)$, respectively Eq. (31) and Eq. (32), up to the second order for small $q$. In this way, one finds that the coupling constants $g_{0}$ and $g_{2}$ are related to the physical parameters $a_{s}$ and $r_{s}$ according to

[TABLE]

which is a well-known relation, and

[TABLE]

By using the pseudo-potential in Eq. (27) the nonlocal Lagrangian density given by Eq. (4) becomes

[TABLE]

This is the finite-range effective field theory (EFT) we shall use in the remaining part of the paper. Gaussian (one-loop) results of Eq. (35) have been obtained in Refs. braaten ; andersen ; roth , but mainly in the perturbative regime within one-loop calculations and at zero temperature. Here we explicitly prove that, working in regions where the gas parameter is small, the inclusion of the effective-range term can remove the instability of the zero-range theory. Moreover, we compare our EFT calculatons with Monte Carlo data and analyze also finite-temperature effects.

By using the Lagrangian density derived in Eq. (35) in the Eq. (2), the stationary Gross-Pitaevskii equation for the space-dependent field $\psi_{0}(\mathbf{r})$ can be derived by means of the saddle-point approximation

[TABLE]

which leads to

[TABLE]

Eq. (37) was derived for the first time in gao for a Bose gas under external confinement. In this case, simulations based on Eq.(37) leads to a better agreement with Quantum Monte Carlo datas concerning, for example, the grounde-state energy gao . Moreover, by using Eq.(37) it is possible to study, also in the absence of an external confining potential, how space dependent topological solutions, such as vortex and solitons, are affected by this effective-range expansion sgarlata .

As in the previous section, in the remaining part of the paper we adopt the shift of Eq. (7) and expand the action $S[\psi,\psi^{*}]$ of Eq. (2) around a uniform and stationary $\psi_{0}$ up to quadratic (Gaussian) order in $\eta({\bf r},\tau)$ and $\eta^{*}({\bf r},\tau)$, but now using (35) instead of (6). Formally, we find again Eqs. (13), (14), (15), (16) for the grand potential $\Omega$ but now the dispersion relation reads

[TABLE]

By using Eqs. (17) and (18) to remove the dependence on $\psi_{0}$ in $E_{\bf q}$ we obtain

[TABLE]

where

[TABLE]

takes into account finite range effects of the inter-atomic potential.

IV.1 Zero-temperature results

The zero-temperature Gaussian grand potential $\Omega_{g}^{(0)}(\mu)$, given by Eq. (15), is ultraviolet divergent with $E_{q}(\mu)$ given by Eq. (39). However, this divergence can be regularized with dimensional regularization thooft . For a recent review of this and other regularization methods applied to the dilute and ultracold atomic systems one can see sala-review . In this way we find

[TABLE]

The zero-temperature number density $n$ is obtained by using the number equation (24) with $\Omega$ given by Eq. (13) with Eqs. (14), (15) and (41). We obtain

[TABLE]

From this equation we determine $\mu$ as a function of $na_{s}^{3}$ for different values of $\chi$ at fixed density $n$. The results are shown in Fig. 1, where $\alpha$ is proportional to the ratio $g_{2}/g_{0}$ in adimensional units, namely

[TABLE]

As previously discussed, Fig. (1) clearly shows that in the absence of finite-range corrections ($\alpha=0$) the chemical potential $\mu$ versus $n$ has no solutions above the critical value $0.004$ of the gas parameter $na_{s}^{3}$. This problem is indeed solved by using a positive value of $\alpha$ larger than about $0.25$, while for $\alpha<0$ the problem gets worse. The different behaviour of numerical solutions of Eq. (42) for $\alpha\neq 0$ compared to Eq. (25) (equivalent to $\alpha=0$ case) can be understood thanks to modified dependence from $\mu$ of Eq. (41): for $\alpha\gtrsim 0.25$ the finite-range Gaussian correction never becomes of the same order, or bigger, of the mean field term. Differently from the zero-range case, the finite-range correction manages to control the growth of fluctuations and it stabilizes the system, as highlighted by the dashed and dashed-dotted lines in Fig. (1): for these values of $\alpha$ there is no critical value of the gas parameter.

It is interesting to compare it with Quantum Monte Carlo (QMC) data obtained for a Bose gas of hard spheres by using the path-integral ground-state method pigs , as done in Fig. 2. In the case of hard spheres the s-wave scattering length $a_{s}$ and effective range $r_{s}$ are related by $r_{s}/a_{s}=2/3$ (see, for instance, gao ; pethick ; sgarlata ). The adimensional parameter $\alpha$, Eq. (43), of our theory must be $\alpha=4/3$ in order to model the hard-sphere Bose gas. The figure shows that the Monte Carlo data (filled circles) are reproduced reasonably well by our EFT with $\alpha=4/3$ (solid line) also for quite large values of the gas parameter $na_{s}^{3}$. So, by including the finite-range Gaussian corrections in Eq. (42), we are able to recover the zero-range results, which is reliable at very low values $na_{s}^{3}$ and surely not above $na_{s}^{3}\simeq 0.004$, but we can reproduce the QMC datas for a more dense system. The range of applicability of our finite-range theory is of the order shown in the horizontal axis of Fig. (2).

QMC simulations are performed with a hard-core potential which has a finite range. Our theory, at a Gaussian level, introduces an effective range $r_{s}$ whose relation with the scattering length $a_{s}$ of the hard-core potential is $r_{s}=(2/3)a_{s}$. In the case of the hard-core potential, at fixed scattering length $a_{s}$ (and consequently at fixed effective range $r_{s}$), by increasing the gas parameter $na_{s}^{3}$ one increases the density $n$ and, at the same time, the average distance $d\simeq n^{-1/3}$ between atoms reduces. In this way, $d$ becomes comparable with the effective range $r_{s}$ and finite-range effects of the inter-atomic potential are then sizable.

We stress that, at two-loop level (next-to-Gaussian), the correction to energy density proportional to $na_{s}^{3}$ is characterized by an unknown coefficient which cannot be specified in the zero-range framework nieto . This coefficient can be expressed in terms of the s-wave scattering length and a three-body coupling not easy to determine experimentally. It is remarkable that, at least for alkali atoms, second-order quantum corrections are proportional to the logarithm of an additional length scale fixed by the van der Waals interaction.

IV.2 Finite-temperature results

The finite-temperature one-loop contribution to the equation of state is obtained from $\Omega_{g}^{(T)}(\mu)$, which can be written as

[TABLE]

Introducing the variable $x=\beta E_{q}(\mu)$ we get

[TABLE]

where $q(x,\mu)$ is given by

[TABLE]

Expanding this expression at low temperature $T$ we find

[TABLE]

and the finite-temperature contribution $n_{g}^{(g)}$ to the total number density $n$ reads

[TABLE]

Thus, within the perturbative expansion approach ($|\mu-g_{n}|\ll 1$) previously discussed, the finite-temperature equation of state reads

[TABLE]

where $\mu_{0}(n)+\mu_{g}^{(0)}(n)$ is given by Eq. (26) and

[TABLE]

Notice that Eq. (49), with Eqs. (26) and (50), generalizes the old familiar result obtained in 1958 by Lee and Yang lee .

V Conclusions

We have used a finite-temperature Gaussian (one-loop) functional integration to obtain the equation of state for a dilute and ultracold gas of bosons with uniform number density $n$, taking into account both the scattering length $a_{s}$ and the effective range $r_{s}$ of the inter-atomic interaction. The divergent zero-point energy of the system has been regularized by performing dimensional regularization. Our analytical results at zero and finite temperature, which are non trivial generalizations of old but familiar formulas bogoliubov ; lee1960 ; yang ; lee depending only on the scattering length $a_{s}$, are in quite good agreement with recent Monte Carlo calculations pigs also for relatively large values of the gas parameter $na_{s}^{3}$. As discussed by Braaten, Hammer and Hermans braaten , the Gaussian grand potential can be improved taking into account also two-loop corrections but, in this case, a three-body interaction is needed to regularize the divergent two-loop grand potential braaten ; nieto . On this respect a self-consistent derivation of an effective field theory with a three-body term starting from a nonlocal two-body interaction potential is still missing and it surely deserves a deep investigation.

Acknowledgments

The authors acknowledge for partial support Ministero Istruzione Universita Ricerca (PRIN Project ”Collective Quantum Phenomena: from Strongly-Correlated Systems to Quantum Simulators”). The authors thank Prof. Flavio Toigo for many enlightening discussions.