Effective-range corrections to the ground-state energy
of the weakly-interacting Bose gas in two dimensions
Abstract
Nonuniversal effects due to leading effective-range corrections are computed for the ground-state energy of the weakly-coupled repulsive Bose gas in two spatial dimensions. Using an effective field theory of contact interactions, these corrections are computed first by considering fluctuations around the mean-field free energy of a system of interacting bosons. This result is then confirmed by an exact calculation in which the energy of a finite number of bosons interacting in a square with period boundary conditions is computed and the thermodynamic limit is explicitly taken.
Introduction. Novel experimental techniques involving trapped ultracold atoms allow the investigation of systems with reduced dimensionality Posazhennikova (2006); Bloch et al. (2008); Yefsah et al. (2011). In particular, experimental studies of dilute atomic gases in two spatial dimensions are able to resolve elements of the equation-of-state, including beyond mean-field effects Hadzibabic et al. (2006); Cladé et al. (2009); Yefsah et al. (2011); Desbuquois et al. (2012). This of course motivates the theoretical study of atomic systems, with their many types of interaction, as the spatial dimensionality is altered. A valuable tool in this context that has enabled model-independent descriptions of both bosonic and fermionic gases is effective field theory (EFT) (for a relevant review see Ref. Kaplan (2005)), which provides a means of systematically improving quantum mechanical descriptions of fundamental properties of atomic gases, including through the addition of nonuniversal modifications, like many-body forces and shape-parameter corrections, to the atomic equation-of-state Andersen (2004); Braaten and Hammer (2006).
This letter focuses on the calculation of effective-range (ER) corrections to the ground-state energy of the weakly-interacting Bose gas in two spatial dimensions. The universal effects in the ground-state energy are known to three non-trivial orders in the weak coupling expansion Schick (1971); Popov (1972); Fisher and Hohenberg (1988); Lieb and Yngvason (2001); Cherny and Shanenko (2001); Mora and Castin (2009); Andersen (2002). In addition there have been several studies of nonuniversal effects Astrakharchik et al. (2010); Salasnich (2017). Here, using the EFT as a starting point, the leading ER corrections to the equation-of-state are computed from the leading fluctuations about the mean-field result. This calculation is a simple extension of the method used in three spatial dimensions to two spatial dimensions Braaten et al. (2001); Andersen (2004). However, the two-dimensional derivation is somewhat more involved because of issues related to renormalization, as will be seen below. Unlike the case of three spatial dimensions, the two-dimensional ground state energy can also be obtained exactly, without resorting to any mean-field arguments Beane (2010). This is achieved by calculating –perturbatively in the weak coupling constant– the energy of bosons in a finite square with periodic boundary conditions, and explicitly taking the thermodynamic limit. This procedure relies on the tractability of the two-dimensional lattice sums: they can be reduced to special functions with known properties. Such a reduction of the three-dimensional lattice sums is not known. It is found that the mean-field and exact results match perfectly, as one would expect.
Effective field theory. The most general Lagrangian, constrained by Galilean invariance, parity and time-reversal invariance, which describes Bosons interacting isotropically at low-energies via an arbitrary finite-range potential is
Here three-body forces and higher-derivative operators have been omitted. Throughout we use units with , and we keep the boson mass, , explicit. In spacetime dimensions, the mass dimensions of the boson field and of the operator coefficients are , .
Using the two-body scattering conventions of Ref. Beane (2010), the ER expansion takes the form
(2) |
where is the phase shift and
(3) |
are the ER parameters, written in terms of the renormalized EFT parameters that are defined using dimensional regularization with Beane (2010). Here is the renormalization scale, is a scale-dependent dimensionless coupling constant, and is the effective range. Unlike the three-dimensional case, in in two spatial dimensions, all of the EFT parameters are scale dependent. The leading beta function in the EFT is
(4) |
which integrates to give the exact renormalization group (RG) evolution equation
(5) |
Free energy from mean-field fluctuations. The technology for computing the range corrections to the free energy of the weakly-coupled Bose gas in the EFT in the case of three spatial dimensions is well known Braaten et al. (2001); Andersen (2004). In that case, in the scheme the EFT coefficients are scale independent. The derivation is somewhat more subtle here due to the non-trivial RG evolution in two spatial dimensions. The mean-field free energy and its leading correction in the absence of ER corrections can be expressed in spacetime dimensions with as
(6) |
where the first term is the mean-field result written in terms of the bare parameter and the second term is related to the sum of the zero-point energies of the quasi-particles. With the renormalization scheme for adopted in Ref. Beane (2010) one easily obtains
(7) |
By Legendre transform one finds the mean-field and leading-loop contributions to the ground-state energy density Schick (1971); Popov (1972); Fisher and Hohenberg (1988); Lieb and Yngvason (2001); Mora and Castin (2009); Andersen (2002)
(8) |
The ER corrections can then be included by shifting the momenta under the square-root in Eq. (6) by where ^{1}^{1}1Note that while is renormalization scale dependent, this dependence is subleading in .
(9) |
The free energy with ER corrections is found to be
(10) |
The shift in momentum requires a change in the renormalization scheme. The corresponding modification of the running of the coupling is:
(11) |
from which it is verified that Eq. (10) is scale independent up to neglected universal corrections. We will see below in the exact calculation how this density-modified RG evolution is reconciled with the two-body evolution equation of Eq. (5).
Expanding to leading order in the effective range gives ^{2}^{2}2It is not necessary to expand in Eq. (10). However, there is little point in keeping the subleading terms as they are expected to be of the same size as shape parameter corrections, which have been neglected but are not generally expected to vanish.
This is the main new result in this letter. Performing the Legendre transform yields the ER contribution to the energy density
(13) |
It is instructive to check that the scaling of this contribution is sensible Braaten et al. (2001); Andersen (2004); Braaten and Nieto (1997). The momentum operator in the mean field scales as as is evident from Eq. (8). Clearly the operator does not contribute at the level of the mean field and therefore ER corrections must arise from a loop. Again from Eq. (8), each loop gives a factor of . And of course scales as and the coefficient scales as . We expect the ER corrections to scale as an insertion of the operator times a loop which gives: , as found above. The cubic dependence on the density implies that the leading ER corrections provide an effective three-body force, albeit one that is highly suppressed at weak coupling. Below we will confirm Eq. (13) by an exact calculation in which no mean field is assumed ^{3}^{3}3Note that the result found here for the leading ER corrections is in disagreement with Ref. Salasnich (2017), which finds an effective-range contribution that is enhanced by a power of , in violation of the basic scaling arguments..
It may prove useful to express current knowledge of the free energy in a more common notation. Using the definition of the ER parameters in Ref. Verhaar et al. (1984); Braaten and Hammer (2006), we identify
(14) |
where is the scattering length and is the effective range. Choosing the renormalization scale so that the logarithms in Eq. (Effective-range corrections to the ground-state energy of the weakly-interacting Bose gas in two dimensions) vanish and, following Ref. Mora and Castin (2009), defining the new coupling
(15) |
then gives current knowledge of the free energy of the weakly-coupled, two-dimensional Bose gas, including the new nonuniversal contribution from range corrections:
(16) |
where the universal contribution calculated (numerically with ) in Ref. Mora and Castin (2009) has been included. It is shown in this reference that Monte Carlo simulations Pilati et al. (2005); Astrakharchik et al. (2009) are able to resolve the universal contribution. It would be interesting to see whether the ER contribution can be similarly detected in numerical simulations.
Two-body energy in a square. We now proceed with an exact calculation of the energy density. In a finite square area with periodic boundary conditions, the energy levels for the two-boson system determine the phase shift through the eigenvalue equation Fiebig et al. (1994); Beane (2010)
(17) |
where
(18) |
Unlike the case of three spatial dimensions, this integer sum is tractable and indeed can be expressed in terms of the digamma function Beane (2010).
Neglecting shape-parameter corrections, the low-energy expansion, Eq. (2), combined with the eigenvalue equation, Eq. (17), gives
(19) |
Using the RG evolution equation, Eq. (5), it follows that
(20) |
where . As the scale of the coupling is fixed to , as the continuum limit is approached, the repulsive theory is at weak coupling. Hence when the two-body interaction is repulsive, the eigenvalue equation, Eq. (20), allows a perturbative expansion of the energy eigenvalues in the coupling . In weak-coupling perturbation theory, the ground-state energy is
(21) | |||
where Beane (2010)
(22) |
Here is Euler’s constant, is the gamma function, is an integer cutoff, and
(23) |
are the Riemann zeta function and Dirichlet beta function, respectively.
Many-body energy in a square. The two-particle energy can be generalized to the -body system at weak coupling using Rayleigh-Schrödinger perturbation theory Beane et al. (2007); Detmold and Savage (2008); Beane (2010) giving the -body ground-state energy
(24) | |||
where =.
Thermodynamic limit. The goal in what follows is to take the thermodynamic limit of Eq. (24), where and are taken to infinity with the density, , held fixed. This limit has been taken explicitly in Ref. Beane (2010) to obtain the density expansion of the universal ground-state energy. Here we will do the same to obtain the leading ER corrections. Several issues should be kept in mind. Firstly, naively, in the thermodynamic limit only the subleading ER correction of survives the thermodynamic limit of . While this is close to the expected scaling, the result depends on the geometric constant . Clearly taking the thermodynamic limit must erase all dependence on the geometry and thus must be be independent of all of the constants. Secondly, the coupling is evaluated at the far infrared scale , and therefore a change of scale to a quantity which is fixed in the thermodynamic limit is necessary. It is straightforward to extend the ER contributions to include the most-singular higher-order contributions giving
(25) | |||
where the dots above correspond to less-singular terms at that order in . Now with , we have
(26) |
where
(27) |
and in Eq. (26) we have neglected subleading terms of within the brackets. Using an integral representation of the binomial coefficient and the representation, Eq. (22), of the two-dimensional lattice sums, this function takes the form
(28) |
where . Noting that
(29) | |||
and using the asymptotic form of the digamma function for large argument gives
(30) |
which clearly removes the geometric constant from the energy, as it must.
In these expressions, the coupling is evaluated at the scale . Therefore, a change of scale is necessary in order to take the thermodynamic limit. Say , where is an arbitrary number. Now note that contains a contribution that scales as which will not vanish in the thermodynamic limit if the coupling runs as in Eq. (5). This singular piece is eliminated only if the running of the coupling is modified so that with , the RG evolution is
(31) |
The form of the extra contribution linear in in the running of the coupling is entirely determined by the requirement that the thermodynamic limit exist: i.e. that the in be cancelled by this extra piece. Furthermore, we know that this linear piece must scale as since it is not present in the two-body case. The necessity of this modification of the RG evolution is of course no surprise as it is clear that the thermodynamic limit of Eq. (31) is equivalent to the density-modified RG evolution of Eq. (11). Now, using Eqs. (26) and (30) and rescaling the coupling using Eq. (31), gives a finite result in the thermodynamic limit
(32) |
in perfect agreement with Eq. (13).
Conclusions. Nonuniversal effects due to a non-vanishing effective range have been computed for the weakly-coupled repulsive Bose gas in two spatial dimensions using two distinct methods, both of which originate in the most general EFT which describes bosons interacting at low-energies via finite-range forces. The first method is a perturbative expansion about a mean field which gives directly the weak-coupling equation of state. The fundamental assumption underlying this method is the presence of the mean field; i.e. that the bosonic field acquires a vacuum expectation value . The second method does not assume a mean field but rather computes the energy of a finite number of bosons in a finite area and then takes the thermodynamic limit. This latter method also gives an energy that is perturbative in the coupling constant, but it is highly singular in the number of bosons. The most singular terms in the series are readily summed to give a result consistent with mean-field theory. The fundamental assumption underlying this method is the existence of the thermodynamic limit. Like the three-dimensional case, the ER corrections are highly suppressed as, in addition to the momentum suppression, they necessarily arise from a loop effect. It would be interesting to verify these new nonuniversal corrections numerically using Monte Carlo simulations.
Acknowledgments. I would like to thank Martin J. Savage for a useful comment on the manuscript. This work was supported in part by the U. S. Department of Energy grant DE-SC001347.
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