Thermodynamics and renormalized quasi-particles in the vicinity of the dilute Bose gas quantum critical point in two dimensions

Thermodynamics and renormalized quasi-particles in the vicinity of the dilute Bose gas quantum critical point in two dimensions

Jan Krieg jkrieg@itp.uni-frankfurt.de Institut für Theoretische Physik, Universität Frankfurt, Max-von-Laue Strasse 1, 60438 Frankfurt, Germany    Dominik Strassel Department of Physics and Research Center Optimas, University of Kaiserslautern, 67663 Kaiserslautern, Germany    Simon Streib Institut für Theoretische Physik, Universität Frankfurt, Max-von-Laue Strasse 1, 60438 Frankfurt, Germany Kavli Institute of NanoScience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands    Sebastian Eggert Department of Physics and Research Center Optimas, University of Kaiserslautern, 67663 Kaiserslautern, Germany    Peter Kopietz Institut für Theoretische Physik, Universität Frankfurt, Max-von-Laue Strasse 1, 60438 Frankfurt, Germany
January 18th, 2017
Abstract

We use the functional renormalization group (FRG) to derive analytical expressions for thermodynamic observables (density, pressure, entropy, and compressibility) as well as for single-particle properties (wavefunction renormalization and effective mass) of interacting bosons in two dimensions as a function of temperature and chemical potential . We focus on the quantum disordered and the quantum critical regime close to the dilute Bose gas quantum critical point. Our approach is based on a truncated vertex expansion of the hierarchy of FRG flow equations and the decoupling of the two-body contact interaction in the particle-particle channel using a suitable Hubbard-Stratonovich transformation. Our analytic FRG results extend previous analytical renormalization group calculations for thermodynamic observables at to finite values of . To confirm the validity of our FRG approach, we have also performed quantum Monte Carlo simulations to obtain the magnetization, the susceptibility, and the correlation length of the two-dimensional spin- quantum model with coupling in a regime where its quantum critical behavior is controlled by the dilute Bose gas quantum critical point. We find that our analytical results describe the Monte Carlo data for rather accurately up to relatively high temperatures .

pacs:
64.60.F-, 05.30.Jp, 75.10.Jm, 75.40.Mg

I Introduction

It is well known Sachdev (2011) that interacting bosons exhibit a quantum critical point (QCP) at vanishing chemical potential and temperature which separates a quantum disordered phase at from a superfluid phase at as sketched in Fig. 1.

Figure 1: (Color online) Schematic phase diagram for a two-dimensional Bose gas with repulsive contact interaction in the plane close to the QCP at . In the quantum disordered regime, , interaction effects are weak and the particle density is exponentially small, while for sufficiently large positive the system is in the superfluid phase with finite density even at zero temperature. This phase is separated from the normal phase by a BKT transition at the temperature as given in the figure, where is a non-universal energy scale.Popov (1972) In the quantum critical regime (red color) a quasi-particle description with free bosons is still valid, but physical quantities exhibit logarithmic corrections.

Of particular interest is the behavior of the system in two dimensions, where the formation of a Bose-Einstein condensate for positive is prevented by the strong thermal fluctuations Mermin and Wagner (1966). However, as has been shown by Popov,Popov (1972) the system nevertheless exhibits a phase transition of the Berezinsky-Kosterlitz-Thouless (BKT) typeBerezinskii (1971, 1972); Kosterlitz and Thouless (1973); Kosterlitz (1974) into a superfluid phase with quasi-long-range order and algebraically decaying correlations. This is closely related to the fact that the upper critical dimension at the QCP is , so that the contact interaction is marginal in the renormalization group sense:Fisher et al. (1989) while directly at the QCP the interaction is renormalized to vanishing strength, the renormalization group flow approaches zero only logarithmically. As a result, at any realistic temperature the collective dynamics of the system is strongly coupled, even though the effective pairwise interaction may be weak.Sachdev and Dunkel (2006) Hence perturbation theory is not applicable at finite and one has to resort to non-perturbative methods to analyze the superfluid as well as the quantum critical regime. The renewed theoretical interest in this model is motivated by a multitude of cold atoms experiments Hadzibabic et al. (2006); Krüger et al. (2007); Cladé et al. (2009); Tung et al. (2010); Hung et al. (2011); Yefsah et al. (2011); Zhang et al. (2012); Ha et al. (2013); Desbuquois et al. (2014); Fletcher et al. (2015) which have explored the phase diagram as well as the BKT transition in two-dimensional Bose gases.

In early theoretical works Popov (1972); Fisher and Hohenberg (1988); Sachdev et al. (1994) the properties of the superfluid phase and the position of the BKT transition were investigated in the extremely dilute limit where the effective dimensionless interaction is renormalized to very small values [see Eq. (26) below]. However, for quantitative calculations this limit may not be realistically reached as was demonstrated by Prokof’ev et al. Prokof’ev et al. (2001); Prokof’ev and Svistunov (2002) since, e.g., the critical density at the BKT transition exhibits a logarithmic dependence of the form

(1)

while the effective interaction in turn only depends logarithmically on the density (due to ). Here is the mass of the bosons and we use units where . The constant appearing in the logarithm in Eq. (1) has been computed numerically using the classical model as ,Prokof’ev et al. (2001) hence the necessary limit where can be neglected is not accessible with available experimental techniques. Nonetheless it is possible to reach relatively small of the order of experimentally, e.g., by tuning the bare interaction of harmonically trapped atoms to small values through a magnetic Feshbach resonance.Hung et al. (2011); Ha et al. (2013); Fletcher et al. (2015) Further theoretical works also increasingly employed numerical methods to complement the asymptotic analysis.Bernardet et al. (2002); Pilati et al. (2005); Sachdev and Dunkel (2006); Rançon and Dupuis (2012); Ceccarelli et al. (2013); Strassel et al. (2015)

Here we are mainly interested in the universal scaling inside the quantum critical regime where . For the special case this has already been investigated analytically by Sachdev et al. Sachdev et al. (1994). Recently Rançon and Dupuis Rançon and Dupuis (2012) expanded on this by using a functional renormalization group (FRG) approach based on a truncated gradient expansion. By solving the truncated FRG flow equations numerically they calculated the scaling of state functions close to the QCP for vanishing and finite chemical potential; they also reconsidered the analytical behavior for and corrected the result for the density in Ref. [Sachdev et al., 1994]. In this work we shall re-examine the universal scaling within the FRG, using an alternative truncation strategy of the formally exact hierarchy of FRG flow equations based on the vertex expansion.Kopietz et al. (2010) This enables us to extend previous analytical results for the universal scaling in the quantum critical regime. We also compare our FRG results to quantum Monte Carlo (QMC) simulations.

The rest of this work is organized as follows: In Sec. II we introduce a Hubbard-Stratonovich transformation to decouple the contact interaction in the particle-particle channel and derive FRG flow equations for the self-energy and the particle-particle susceptibility. We then solve these flow equations approximately in Sec. III and obtain analytical expressions for the pressure, the density, the entropy, the compressibility, and the correlation length which are valid close to the QCP; we also compare these results to experimental as well as to numerical data. In Sec. IV we discuss the spin-1/2 quantum model in two dimensions which can be mapped to the Bose-Hubbard model with infinite onsite interaction, hence showing the same universal scaling towards the QCP as the dilute Bose gas. We present results from QMC simulations of this spin system and compare them to our analytical FRG results. Finally in Sec. V we use the FRG formalism to study the scaling of the effective mass and the wavefunction renormalization; we also compare the FRG result for the wavefunction renormalization with the result from the self-consistent -matrix approximation, which is applicable for and at high temperatures also for . Further technical details can be found in two appendices: in appendix A we present formally exact FRG flow equations for the irreducible vertex functions of interacting bosons, while in appendix B we give some mathematical details on the analytical solution of our truncated FRG flow equations.

Ii FRG flow equations for dilute bosons

We consider a system of interacting bosons with contact two-body interaction and second quantized Hamiltonian

(2)

where annihilates a boson with momentum , energy , and mass . The volume of the system is denoted by and the normalization of the contact two-body interaction with strength has been chosen to simplify the combinatorial factors in the FRG flow equations given in appendix A. We have shifted the momentum labels in Eq. (2) such that can be identified with the conserved total momentum of a pair of incoming or outgoing bosons. This labeling is natural in the dilute limit where the particle-particle channel is the dominant scattering process.

ii.1 Hubbard-Stratonovich transformation in the particle-particle channel

At finite chemical potential and temperature , the Euclidean action associated with the Hamiltonian (2) can be written as

(3)

where the free boson propagator is

(4)

and we have introduced the composite boson fields

(5)

Here and are collective labels for momenta and bosonic Matsubara frequencies, the integration symbols are defined by , and is a complex field associated with the eigenvalues of . Introducing another complex boson field to decouple the interaction by means of a Hubbard-Stratonovich (HS) transformation in the particle-particle channel we obtain

(6)

We have normalized the -field to simplify the combinatorial factors in the exact FRG flow equations given in appendix A. Below we shall refer to the original boson fields and as elementary bosons and to the boson fields and as HS bosons.

ii.2 Truncated FRG flow equations

To set up the FRG, we introduce a sharp cutoff in momentum space for the elementary boson so that the regularized non-interacting propagator is given by

(7)

and the corresponding single-scale propagator is

(8)

where is the cutoff dependent self-energy and we have defined . For our purpose, it is sufficient to use the following ansatz for the generating functional of the irreducible vertices,

(9)

where the energy-momentum labels of the three-legged vertices and correspond to the field types appearing in the superscripts. This ansatz is justified since all four-point and higher order vertices are irrelevant in the RG sense, except for which is marginal. This vertex, however, initially vanishes due to the HS transformation and is only dynamically generated during the flow by either particle-hole processes or by vertices which are irrelevant in the RG sense (see Fig. 13). As our calculations are concerned with the dilute limit, particle-hole processes are suppressed and the relevant physics is captured by particle-particle processes via the three-point vertices and . The exact FRG flow equations for the self-energy and the particle-particle susceptibility as well as for all three- and four-legged vertices of our model are given in appendix A. From the flow equations (74) and (75) for the three-legged vertices we see that within our ansatz (9) for the generating functional these vertices do not flow, so that we can replace them by their initial value

(10)

The exact flow equation for the self-energy given in Eq. (72) then simplifies to

(11)

while the flow equation for the particle-particle susceptibility given in Eq. (73) becomes

(12)

Here we have introduced the flowing propagator

(13)

of the HS boson. We expect these equations to be accurate in the vicinity of the dilute Bose gas QCP where particle-hole scattering processes can be neglected.

Note that if we replace the single-scale propagators in Eqs. (11) and (12) by total derivatives with respect to and ignore the dependence of the particle-particle susceptibility , we can integrate both sides of these equations over to obtain

(14)
(15)

These coupled integral equations are usually called the self-consistent -matrix approximation. For an early application of this method to the dilute Bose gas in two dimensions see Ref. [Stoof and Bijlsma, 1993]. Some of us Streib and Kopietz (2015) have recently used this approximation to study an effective hard-core boson model describing the magnetic properties of the antiferromagnetic material (see also Ref. [Fauseweh et al., 2014] for recently discovered subtleties in this method when applied to hard-core bosons). In Sec. V.1 we shall compare our FRG results for the wavefunction renormalization at to the results obtained from the numerical solution of the integral equations (14) and (15).

Iii Thermodynamics close to the QCP

iii.1 RG flow at the quantum critical point

To begin with, let us briefly recall the renormalization group (RG) flow of the system directly at the QCP in dimensions. Since in this case the equilibrium state of the system corresponds to the vacuum, the elementary propagator is identical to the free propagator , i.e., . According to Eqs. (11) and (12) the flow of the particle-particle susceptibility for vanishing momentum and frequency then simplifies to

(16)

where is the surface area of the -dimensional unit sphere divided by . Defining the dimensionless rescaled interaction

(17)

and switching to the logarithmic scale parameter , where is the ultraviolet cutoff of our theory, we arrive at the well-known exact flow equationFisher and Hohenberg (1988); Sachdev (2011)

(18)

which identifies as the upper critical dimension above which mean field theory is applicable. In the following we will always work at , resulting in logarithmic corrections to the scaling of various observables.

iii.2 Explicit solution of the FRG equations close to the quantum critical point in two dimensions

The above system of FRG flow equations (11) and (12) for the self-energy and the particle-particle susceptibility can be solved approximately by neglecting their momentum and frequency dependence which is justified close to the QCP. In the following we will again use the scaling parameter and the dimensionless interaction

(19)

which is a special case of Eq. (17) for . We also define the effective inverse temperature

(20)

the dimensionless particle-particle susceptibility

(21)

and the effective negative chemical potential over temperature

(22)

where we have introduced the abbreviation

(23)

to simplify the forthcoming equations. With this notation the flow equations (11) and (12) can be written as

(24)
(25)

It turns out that we can analytically solve this system of differential equations approximately for provided we consider the regime close to the QCP and the following dimensionless coupling is sufficiently small,

(26)

where the logarithmic term results from the flow of [cf. Eq. (84)]. Technical details on the analytical solution of Eqs. (24) and (25) are given in appendix B.1. There we show that an approximate analytical solution is possible if either or , where denotes the Lambert W function Corless et al. (1996) which for large arguments can be expanded as

(27)

The condition is thus fulfilled either extremely close to the QCP or for small bare interaction . The regime is of significant practical importance as it allows to probe the asymptotic scaling behavior experimentally by tuning the interaction to small values, since reaching double exponentially low temperatures is not feasible. Actually for small the weak logarithmic dependence on results in an approximate temperature independence of .

As shown in appendix B.1, for the limit can be written as

(28)

which in the special case simplifies to

(29)

On the other hand, for finite and we can expand Eq. (28) as

(30)

which in the quantum disordered regime where results in

(31)

so that the self-energy is exponentially suppressed.

In appendix B.1 we also derive a more general expression,

(32)

which is valid for arbitrary as long as we stay in the normal phase close to the QCP and do not come too close to the BKT phase transition. The reason for the latter constraint is that vortices become increasingly important close to the superfluid transition which eventually leads to a breakdown of the quasi-particle approximation (cf. Fig. 9 at ). Nevertheless, extrapolating (32) into the classical critical regime we can give an estimate for the critical chemical potential at the BKT transition by demanding that , which in the asymptotic limit yields

(33)

Given that our theory is not justified in the classical critical regime, this agrees well with the weak-coupling result Popov (1983); Fisher and Hohenberg (1988)

(34)

where has been obtained numerically as in Ref. [Prokof’ev et al., 2001] and as in Ref. [Rançon and Dupuis, 2012].

iii.3 Thermodynamic state functions

Within our approximation scheme the density at scale is given by

(35)

which corresponds to the particle density of all particles with momentum . The integration can be carried out exactly and we obtain for the physical density

(36)

Analogously the off-diagonal elements of the density matrix in our approximation are

(37)

In the limit of large distances and for we can evaluate the momentum integration analytically,Kopietz et al. (2010)

(38)

where we have introduced the correlation length

(39)

We now define the reduced pressure , the phase-space density , the entropy per particle , the dimensionless compressibility , and the dimensionless correlation length ,

(40a)
(40b)
(40c)
(40d)
(40e)

where is the pressure, is the grand canonical potential, is the entropy, is the compressibility, and the thermal de Broglie wavelength is given by

(41)

Since we have approximated the self-energy by its zero momentum and frequency limit, we can try to incorporate the interaction solely as a shift in the chemical potential, neglecting the renormalized interaction . Thus we compute the state functions for a non-interacting Bose gas and fix the chemical potential such that the particle density coincides with Eq. (36), i.e., . Accordingly we find

(42a)
(42b)
(42c)
(42d)

where is the dilogarithm. Calculating the dimensionless compressibility from Eq. (42b) yields

(43)

We note that these observables do not have a well-defined limit at due to the non-analyticity of the grand canonical potential at the QCP, which separates the zero density ground state at from the finite density superfluid ground state at . For the special case of vanishing chemical potential the above relations have already been obtained by Rançon and Dupuis Rançon and Dupuis (2012) whose results agree with our expressions for .

Note that Eq. (42b) corrects the result for the density at given by Sachdev et al.,Sachdev et al. (1994)

(44)

while we find from Eqs. (42b) and (26)

(45)

We can improve on the calculation of and by directly calculating the grand canonical potential within the FRG formalism, solving the flow equation

(46)

Allowing for first order corrections in the self-energy,

(47)

the flow equation for the reduced pressure reads within our cutoff scheme (see appendix B.2)

(48)

Approximating and results in which agrees with the reduced pressure in Eq. (42a). In contrast, if we solve the flow equation (48) for at with the flowing we find (see appendix B.3)

(49)

Here the leading correction is only half as large as our earlier result in Eq. (42a). This implies that it is necessary to solve the flow equation for the grand canonical potential to obtain the correct leading order scaling behavior for the reduced pressure. The entropy per particle, which at can be derived from the reduced pressure as

(50)

is only affected to subleading order. While the phase-space density can in principle also be calculated from as , it is preferable to use Eq. (36) as it avoids the additional approximations in the computation of the reduced pressure.

In Fig. 2 we compare our results for the thermodynamic state functions at (solid black and dashed blue lines) with empirical data from three ultra-cold atoms experiments (green symbols) which we call Chicago I,Hung et al. (2011) ENS,Yefsah et al. (2011) and Chicago II,Zhang et al. (2012) following the naming introduced in Ref. [Rançon and Dupuis, 2012]. The first two experiments investigated and atoms, respectively, inside a harmonic potential with strong confinement along the -axis, resulting in a quasi two-dimensional system; connection to homogeneous systems was made through the local density approximation. While the 3D scattering length in Ref. [Yefsah et al., 2011] was fixed at  nm (), the Chicago I experiment used a magnetic Feshbach resonance to vary between  nm (). These values of correspond to which is not much larger than unity, in contrast to the assumption in our calculations; nevertheless we find that the agreement between the analytical results and the aforementioned experiments is quite good. On the other hand the measurements from the Chicago II experiment, based on atoms in a two-dimensional optical lattice, differ visibly from our predictions for and which is not surprising given that is relatively large, while the agreement for is still remarkably good.

As a further benchmark we use the results of Rançon and DupuisRançon and Dupuis (2012) (dash-dotted red lines) who numerically computed the state functions within the FRG using a truncated gradient expansion. We find that the plots for the phase-space density essentially agree, while the reduced pressure and the entropy per particle differ for . This indicates the upper boundary of the regime where our asymptotic analysis is valid for these state functions.

Figure 2: (Color online) Comparison of our results for the renormalized state functions at with the numerical results of Rançon and DupuisRançon and Dupuis (2012) (dash-dotted red) as well as with data from three different experiments (green symbols, taken from Fig. 4 in [Rançon and Dupuis, 2012]). The remaining lines correspond to the full analytical expressions in Eqs. (42a) and (42c) on the one hand (dashed blue), and to the improved equations (112), (42b), and (50) on the other hand (solid black), where is always taken from Eq. (28). From top to bottom we show the reduced pressure , the phase-space density , and the entropy per particle as a function of the effective coupling constant as defined in Eq. (26).

Iv Quantum Monte Carlo Simulations of the model

In this section we compare our analytic RG results for the density and the compressibility in the vicinity of the dilute Bose gas quantum critical point derived in Sec. III with numerical results for the two-dimensional quantum model in a magnetic field which we have investigated using QMC simulations with finite-size scaling on lattice sizes from up to spins. We have implemented the stochastic series expansion algorithm Syljuåsen and Sandvik (2002) with directed loop updates and using the so-called Mersenne Twister random number generator.Matsumoto and Nishimura (1998) The Hamiltonian is given in terms of the components of spin- operators localized at the sites of a square lattice,

(51)

where the first sum is over distinct pairs of nearest neighbors on the square lattice with lattice sites and periodic boundary conditions in both directions. Here is the nearest neighbor exchange coupling and the magnetic field is measured in units of energy. The model (51) maps exactly to the two-dimensional Bose-Hubbard model with infinite onsite interaction,Sachdev (2011) i.e., hard-core bosons. To compare both models, we should therefore take the limit of infinite contact interaction () in our boson Hamiltonian (2) so that only the logarithmic term in Eq. (26) survives,

(52)

This limit makes it more challenging to reach the regime where our theoretical results apply as this requires exponentially low temperatures, in contrast to cold atoms experiments where the bare interaction can be tuned to small values (cf. the discussion of Fig. 2). Therefore, numerical simulations on hard-core bosons are suitable to test the limitation of our approximations as we will see below. The quantum critical points of the model at belong to the same universality class as the dilute Bose gas in Eq. (2). Moreover, at the critical fields the bare parameters of the above Hamiltonian (51) can be related to the bare mass and the chemical potential of the dilute Bose gas via and . The magnetization per site in the simulations is directly related to the boson density and the longitudinal spin-susceptibility of the model corresponds to the compressibility of the dilute Bose gas. In what follows we set the lattice spacing to unity.

iv.1 Results for

The magnetization data from the simulations follows a characteristic finite-size scaling of the form , where and are temperature dependent as will be discussed in more detail in Sec. IV.3. Basically the finite-size correlation length decreases with increasing temperature up to some temperature , above which the data becomes largely independent of for the system sizes used. In Fig. 3 we show the QMC results with error bars (black) for the density over temperature as a function of in the thermodynamic limit at the lower critical field , corresponding to a vanishing chemical potential . The solid line (blue) is a fit using Eq. (36) together with Eq. (32) in the limit for hard-core bosons. Keeping both and as fitting parameters we obtain the expected result within a few percent. Setting for simplicity and using only the ultraviolet cutoff as a fitting parameter, we obtain where we have required that the deviation between our analytical result and the QMC data vanishes in the limit (see inset of Fig. 3).

Figure 3: (Color online) Density over temperature as a function of for the model at from QMC simulations (black dots). The blue line represents our analytical prediction in Eqs. (36) and (32) with and . The dashed line (red) is a fit of the QMC data to our leading order result (45) for the density using a larger cutoff and a modified mass . On the top axis we show the corresponding values for and .
Figure 4: (Color online) Compressibility as a function of temperature for the model at from QMC simulations (black dots). For comparison we show as a blue line our analytical prediction from Eqs. (36), (32), and Eq. (40d) with and . The top axis shows the corresponding values for these parameters.

For comparison we also tested the leading order expression for the density given in Eq. (45) using the same cutoff , which predicts to be more than below the QMC data for the temperatures used here. It is possible to use a rather different cutoff and a modified mass to fit the leading order expression in Eq. (45) to the QMC data as shown by the dashed line (red) in Fig. 3. It seems that such a fit compensates higher order corrections by using a modified effective mass and a large value of the cutoff. In turn this means that for most experimental and numerical data the coupling constant is sufficiently small to guarantee that the approximate solution in Eq. (36) is accurate, but is not exponentially small to justify dropping all higher order terms (see -axes in Fig. 3 and Fig. 4). Therefore, a fit to the simple logarithmic behavior in Eq. (45) may yield an incorrect cutoff and mass to compensate different higher order corrections. In the temperature region around the minimum in Fig. 3 the scaling of the density may even appear perfectly linear with temperature, which is consistent with recent results on two-dimensional coupled spin-dimers systems.Strassel et al. (2015)

The corresponding data for the compressibility is shown in Fig. 4. The same form of the finite-size scaling was used, but it should be noted that convergence to the thermodynamic limit requires larger system sizes for this response function. Using Eq. (40d) and the analytic expression in Eq. (43) we find good agreement using again , but a larger cutoff . Note that the effective coupling in Eq. (26) depends only logarithmically on so that the difference in the cutoffs only results in a rather small correction. Nevertheless we find that we cannot choose a single cutoff to fit our analytical results for both the density and the compressibility to the QMC data. We have checked the size-scaling for the particle density as well as for the compressibility very carefully, so that we can be sure that the deviation in the cutoff is not related to any remaining finite-size effects. Instead, the different values of might be due to the fact that the condition , requiring exponentially low temperatures due to the hard-core interaction, is not strictly fulfilled in the accessible temperature regime (see Fig. 3 and Fig. 4). Hence we would expect that our analytical result (32) is only qualitatively correct and we need different values of the cutoff for quantitative agreement with the QMC simulations.

Figure 5: (Color online) QMC results for the model at and of the density over temperature (green solid symbols) and the compressibility (red open symbols). The green solid lines correspond to our analytical prediction for from Eqs. (36) and (32) with , (upper green line), and (lower green line), while the red solid lines correspond to from Eq. (40d) with , (upper red line), and (lower red line).

iv.2 Results for

Next, let us consider the regime of constant, non-zero , corresponding to approaching the QCP diagonally from the left in the --diagram shown in Fig. 1. We investigate a wide range of values for , starting from up to .

The QMC data (solid and open symbols) for the density and the compressibility at are shown in Fig. 5 as a function of temperature, where we compare them to our analytical results in Eqs. (36) and (40d) (solid lines). The fit for the density shows good agreement for and (upper green line), which is consistent with the estimate for from above. For the compressibility we find good agreement using and (upper red line). It should be noted that it is again possible to fit the data to a leading order expansion of our analytical results, but this leads to values for the cutoff which are even larger than in the case of .

Increasing to as shown in Fig. 5 (lower graphs), we see that the magnitude of the density and the compressibility is lowered, but the overall shape is similar. Compared to the case the value of the respective cutoffs decreases, for the density and for the compressibility at , while at the same time the difference between the two cutoffs becomes negligible within errors.

Figure 6: (Color online) QMC results for the model at and of the density over temperature (orange solid symbols) and the compressibility (blue open symbols). The orange solid lines correspond to our analytical prediction for from Eqs. (36) and (32) with and , while the blue solid lines correspond to from Eq. (40d) with and .

Monte Carlo results for and are presented in Fig. 6; with increasing , the temperature region for which we can apply the fit functions is pushed to lower and lower values of , while at the same time the characteristic minimum in at is shifted to lower temperatures as well. Note that also shows a minimum at slightly larger temperatures which does not shift as much with , so that both minima approach each other for . Remarkably, both and take on the same value at a certain crossing temperature of (QMC) or (fits), which is largely independent of .

From the fits of the density the cutoff can be consistently estimated to be in the range for all values of . The data for the compressibility at small gives slightly larger estimates for , indicating that in this case our analytical results for require smaller values of than for the density. This is consistent with Fig. 2 where we find good agreement with Ref. [Rançon and Dupuis, 2012] for in a surprisingly large regime of , while our results for the other observables only agree up to .

iv.3 Correlation length and validity range

As already mentioned before, the size scaling of the magnetization and of the susceptibility has the form

(53)

up to a characteristic temperature , above which the data is mainly independent of so that a simple linear extrapolation suffices. With these fits we can extract a finite-size correlation length from the size scaling of the magnetization, which is shown in Fig. 7 for different as a function of temperature using the rescaling . Within error bars the behavior of is consistent with the divergence in Eq. (39) for all . For comparison we show the analytic result from Eq. (32) for as solid lines for different values of . The correlation length from the finite-size scaling of the Monte Carlo data discussed above shows a similar scaling behavior, although finite-size scaling does not measure the correlation length directly. Choosing the cutoff as consistent with the fits for the density, we observe that the correlation length from QMC finite-size scaling agrees reasonably well with the analytic prediction for up to the rescaling .

Figure 7: (Color online) Comparison of the numerical results for the rescaled finite-size correlation length (symbols) at different values of to our analytical prediction from Eqs. (39) and (32) (solid lines) with and using a cutoff of .

The breakdown of the exponential extrapolation in Eq. (53) defines a temperature , which can be used to identify a region of validity for the continuum description of the lattice model. Above this temperature the correlation length is of order unity so that lattice effects dominate. As can be seen in Fig. 8, this region where the continuum description holds roughly coincides with the parameter range for (red shaded region), which is where our analytic prediction for in (36) can be fitted to the QMC data. The values of the temperatures where the minimum in for a given occurs are also shown and are generally at significantly larger values. Note that these minima in correspond to linear behavior of the density with temperature, which always occurs well above the region of validity. It is remarkable that it is possible to use Eq. (36) to describe the behavior close to the QCP also for finite rather accurately, but as expected smaller temperatures are required for since we then approach the QCP diagonally in the --diagram.

Figure 8: (Color online) Region of validity (red area) where the extrapolation in Eq. (53) works and our analytical predictions from the FRG yield accurate results. For comparison we also show the temperature where exhibits a minimum for a given . The dashed red line is given by for and .

V Quasi-particle properties close to the quantum critical point

Within our FRG approach, it is straightforward to calculate the leading order momentum and frequency dependence of the self-energy in the vicinity of the dilute Bose gas quantum critical point, which we parametrize in terms of the two dimensionless renormalization factors and defined via the expansion (47). can be identified with the usual wavefunction renormalization factor (quasi-particle residue), while determines the effective mass of the bosons.

v.1 Wavefunction renormalization

From the low-energy expansion (47) of the self-energy we see that the RG flow of is determined by

(54)

Although the flow of modifies also the flow equations (24) and (25) for and , we shall ignore this modification since it is not expected to change the leading asymptotics close to the quantum critical point. Neglecting the Bose distribution in the flow of the particle-particle susceptibility we end up with the approximate flow equation (see appendix B.4)

(55)

Integrating this equation we find that is given by

(56)

Although this result is only strictly valid for non-positive chemical potential, we find numerically that it is also qualitatively good for . For vanishing chemical potential scales as

(57)

while in the quantum disordered regime the correction is exponentially small,

(58)

We have verified the validity of these approximate expressions by solving the relevant flow equations numerically, taking the Bose distribution in the flow of into account, which yields very good overall agreement. Some representative results for at various values of are presented in Fig. 9. We find that for the wavefunction renormalization approaches unity in the limit in agreement with the well-known result that at the QCP (cf. Sec. III.1), while for positive chemical potential we have at a finite where the BKT transition takes place.

Finally, we can also explore the regime of validity of the self-consistent -matrix approximation as given in Eqs. (14) and (15), which in this context is expected to be good at high temperatures. On the other hand, our FRG approach should be accurate at low temperatures and it is a priori not clear whether there exists an intermediate temperature regime where both methods are valid. For simplicity we consider only the case and take the limit of infinite bare interaction, corresponding to hard-core bosons. A detailed discussion of the self-consistent -matrix approach to hard-core bosons can be found in Ref. [Streib and Kopietz, 2015], where the spin Hamiltonian for the magnetic insulator was mapped onto a two-dimensional hard-core boson model which was then studied using the self-consistent -matrix approximation.

For a comparison of this method with our FRG approach we use parameters specific to ; in particular, we choose the effective inverse temperature as , where we have fixed the momentum cutoff to an average value of the inverse lattice parameters of . In Fig. 10 we compare the results of both complementary methods; obviously, an intermediate temperature regime where both methods are accurate does not exist, showing the need for an alternative approach in this region. We expect that a numerical solution of the FRG flow equations (11) and (12), retaining the full momentum and frequency dependence of and , should be accurate at low as well as at high temperatures. Another possibility to describe the intermediate temperature regime is to use a lattice FRG scheme along the lines of Ref. [Rançon, 2014].

Figure 9: (Color online) Double logarithmic plot of our analytical result for as given in Eq. (56) versus the dimensionless coupling for different values of . For we can see that the scaling is very close to the scaling at vanishing chemical potential for larger and only begins to differ when is of the order of ; the deviation around is due to the fact that for positive chemical potential we have to use (32) for instead of (28). In the case of positive the wavefunction renormalization then shrinks until it vanishes at the phase transition where [see Eq. (33)].
Figure 10: (Color online) Comparison of our analytical result for the wavefunction renormalization of hard-core bosons in Eq. (56) (black solid line) with numerical computations using the self-consistent -matrix approximation which is expected to be good at high temperatures (blue dots). Note that for our analytic result for becomes exact. The intermediate regime (sketched by the gray dashed line as a simple interpolation between the results) where is below, but not much smaller than unity, is not covered by either method.

v.2 Effective mass

Using Eqs. (47) and (11) we find that the flow equation for the effective mass factor is within our truncation given by

(59)

As before we neglect the momentum dependence in the particle-particle susceptibility as well as the Bose distributions which appear in its flow equation. This allows us to compute analytically to leading order as long as . For we find

(60)

which for and small simplifies to

(61)

In the opposite regime we obtain

(62)

where the coefficient is given by

(63)

Thus at the correction is again exponentially suppressed,

(64)

Representative results of at different values of are shown in Fig. 11, which should be compared with the analogous Fig. 9 for the wavefunction renormalization factor.

Figure 11: (Color online) Double logarithmic plot of our analytical result for as given in Eqs. (60) and (62) versus the dimensionless coupling for different values of . We can see that for the scaling coincides with the curve until is of the order of , where it starts to fall off more rapidly.

Vi Summary and Conclusions

In this paper we have used an FRG approach as well as quantum Monte Carlo simulations to study the dilute Bose gas with contact interaction in two dimensions. From the approximate analytical solution of the FRG flow equations we have been able to obtain explicit analytic results for thermodynamic state functions as well as for quasi-particle properties in the vicinity of the QCP. Our results for the thermodynamics and the wavefunction renormalization are expected to be valid for general in the normal phase save for the classical critical region around the BKT transition, thus extending previous analytic results Sachdev et al. (1994); Rançon and Dupuis (2012) for the thermodynamic observables which considered only the special case . A comparison with experimental data Hung et al. (2011); Yefsah et al. (2011); Zhang et al. (2012) as well as with an alternative FRG approach based on the numerical solution of flow equations obtained within the gradient expansion Rançon and Dupuis (2012) shows good agreement with our expressions for the state functions when the dimensionless effective interaction is sufficiently small (), while the density even agrees up to .

To investigate the validity and the limitations of our FRG approach, we have also studied the spin- quantum model close to the dilute Bose gas QCP using QMC simulations. It turns out that with our FRG approach we can predict the behavior of both density and compressibility even at relatively high temperatures, using only the effective ultraviolet cutoff of the continuum model as a free parameter. In particular we were able to describe the numerical data for negative chemical potential analytically which has not been done before. For both cases of and we could also fit our analytical leading order results to the QMC data; however, this requires a rather large value of the ultraviolet cutoff and a modified bare mass. From the finite-size scaling of the density we have determined the correlation length which is consistent with the analytic predictions within error bars. Moreover, the finite-size scaling also defines a region of validity for the continuum description of the lattice model. Outside this region the exponential behavior in Eq. (53) breaks down and the correlation length is of order unity. Interestingly, in that regime the numerical data always shows a minimum in the density over temperature and in the compressibility as a function of for a given , which is not captured by the continuum description.

Finally, we have computed the wavefunction renormalization factor for hard-core bosons numerically using the self-consistent -matrix approximation which for should be accurate at high temperatures. Comparing the data to our analytic FRG result for which is only valid at very small , we find that an intermediate temperature range where both the truncated FRG and the self-consistent -matrix approximation are accurate unfortunately does not exist.

Acknowledgements.
This work was supported by the SFB Transregio 49 and the Transregio 173 of the Deutsche Forschungsgemeinschaft (DFG) and the Allianz für Hochleistungsrechnen Rheinland-Pfalz (AHRP).

Appendix a Exact Frg Flow Equations

In this appendix we write down exact FRG flow equations for the one-line irreducible vertices of the decoupled bosonic action in (II.1). Therefore we modify the Gaussian propagators of the elementary boson and the Hubbard-Stratonovich boson by inducing a cutoff which suppresses fluctuations with wave-vectors smaller than ,

(65)
(66)

At some large initial value of the cutoff the regularized bare action can be written in the following symmetrized form

(68)

Here the bare values of the symmetrized vertices are

(69)

The exact FRG equations, describing the flow of one-line irreducible vertices of the above theory as we reduce the cutoff, follow from the vertex expansion of the FRG flow equation of the corresponding generating functional.Wetterich (1993); Kopietz et al. (2010) The flowing inverse propagators are of the form

(70)
(71)

where the self-energy of the elementary boson satisfies the following exact flow equation,