Weakly interacting Bose gas in a random environment
Zero temperature properties of a dilute weakly interacting -dimensional Bose gas in a random potential are studied. We calculate geometrical and energetic characteristics of the localized state of a gas confined in a large box or in a harmonic trap. Different regimes of the localized state are found depending on the ratio of two characteristic length scales of the disorder, the Larkin length and the disorder correlation length. Repulsing bosons confined in a large box with average density well below a critical value are trapped in deep potential wells of extension much smaller than distance between them. Tunneling between these wells is exponentially small. The ground state of such a gas is a random singlet with no long-range phase correlation For repulsion between particles overcomes the disorder and the gas transits from the localized to a coherent superfluid state. The critical density is calculated in terms of the disorder parameters and the interaction strength. For atoms in traps four different regimes are found, only one of it is superfluid. The theory is extended to lower (1 and 2) dimensions. Its quantitative predictions can be checked in experiments with ultracold atomic gases and other Bose-systems.
Bose-Einstein condensation (BEC) is one of the key quantum phenomena which shaped the physics of the 20th century. In BEC a macroscopic part of the atoms is in the state with lowest energy. Well known examples are the superfluid phase of He and laser cooled atoms in magnetic or optical traps, which are a playground for the demonstration of generic quantum phenomena like the appearance of quantized vortices, the Josephson effect etc. Ketterle (); Leggett (); Dalfovo ().
A more complicated situation arises if these phenomena are studied in a random environment like in porous media. Experimentally, superfluidity was shown to persist in random media like Vycor or aerogel glass Reppy92 () which contain pores filling from up to of the material. Partial coverage of the pores leads to a reduction of the transition temperature: the transition temperature ranks from for full-pore Vycor to about for low coverage films in Vycor. Finally, for very low coverage superfluidity corresponding to less than monolayers the superfluidity vanishes completely even at . One must presume that the helium is strongly localized there. The remaining Helium atoms were considered to form an ”inert” layer which screens the random potential once the density of the Helium atoms is again increased.
The experimental investigations on BEC in disordered environments concentrated on critical behavior of the superfluid density and specific heat at decreasing coverage of the porous media Reppy92 (). More recently, Bose-Einstein condensates have been studied in optical traps which superpose a harmonic and a random potential Lye+05 (); Schulte+05 (); Fallani+07 (); Lugan+07 (); Sanchez+07 (); Hulet+08 (). Most of the papers concentrate on the possibility of Anderson localization in effectively one-dimensional condensates. In contrast to random potentials in crystals, the random speckle potential has a typical correlations length of the order of which is about of the same order as the transverse extension of the atomic cloud. The longitudinal extension is about to times larger Lye+05 (). In most of the experiments, the expanding density profile after switching off the harmonic potential was analyzed. For weak random potential Lye et al. Lye+05 () found essentially the density profile corresponding to the Thomas-Fermi theory. For larger values of the random potential these authors found a stripe-like pattern of about spacing and damped dipole oscillations. For very strong disorder the atoms are localized in the minima of the random potential. Chen et al. Hulet+08 () also studied in situ and found fragmentation of the BEC. In some papers the superposition of a periodic plus a random potential was considered to enrich the number of possible phases Schulte+05 (); Fallani+07 (). Schulte et al. Schulte+05 () found features in the expanding condensate like pronounced fringes and changes in the axial size of the central peak which they trace back to the random potential.
Theoretical investigations of Bose gases or fluids in a disordered environment focused on three issues:
(i) The critical behavior at the normal to superfluid transition in
the vicinity of the transition temperature
Rasolt+84 (); Weichman+86 (); Weichman-Fisher86 (), where denotes the boson
density. For fixed disorder strength decreases with decreasing
boson density and eventually vanishes for .
(ii) The description of zero-temperature quantum phase transition from the superfluid to a non-superfluid (glassy) phase taking place at the critical value of the boson density Ma+86 (); Giamarchi (); Fisher (); Weichman07 ().
(iii) The microscopic description of the superfluid density (and the condensate) in a disordered environment in the region of comparatively high average density of the gas and Lee-Gunn (); Huang (); Vinokur (); Pitaevskii (); Pelster (); Graham ().
The problem of the Bose gas in the random environment attracted so much attention because it connects two central ideas of the condensed matter theory: the Bose condensation and localization. Especially the problem of the interplay of the interaction and disorder is intriguing. So far this problem was studied for Fermi gas altshuler-aronov (); finkelstein (), but only in the case of a weak disorder it could be done rigorously. Therefore the delocalization induced by interaction remains beyond the frameworks of the theory, though most of theorists are sure that it does exist. In the case of the Bose gas the situation is even more involved since there is no Pauli principle which delocalizes the ideal Fermi gas in 3 dimensions. Our study shows that the interaction delocalizes the Bose gas at some critical density.
Since in this paper we restrict ourselves mainly to zero temperature, below we give
a brief review only of the works mentioned in (ii) and
(ii) The quantum phase transition, taking place at zero temperature and , exhibits in general a critical behavior different from that observed at finite . This is because in the quantum domain static and dynamic critical behavior are intertwined. This quantum phase transition was considered by Ma, Halperin and Lee Ma+86 () who mapped the problem onto a dimensional classical theory with disorder completely correlated in time. Their description gives rise to new critical exponents which were calculated by -expansion around dimensions. This work has been criticized by Fisher et al. Fisher (). One conclusion of the work of Fisher et al. Fisher () is that the dynamic critical exponent should be equal to the dimension of the system in all space dimensions. This result has been disputed recently Weichman07 (). Indeed, quantum Monte Carlo studies show for two-dimensional system, in discrepancy with the relation Priya06 ().
Some arguments of the work by Fisher et al. Fisher () rely on the treatment of Giamarchi and Schulz Giamarchi () who found the superfluid to insulator transition in one dimension. This transition happens at arbitrary disorder by increasing the interaction to a sufficiently large value. The fact that the transition into the non-superfluid phase happens at increasing interaction seems paradoxical, but it can be explained by the increase of quantum fluctuations of the phase. As we will show, for weak interaction there is a second transition resulting from the competition between the disorder and the interaction. We believe that namely this transition has its counterparts in higher dimension and is observed in experiments with cooled gases.
(iii) The fate of the inert layer and deeply localized (Lifshitz) states was discussed by Lee and Gunn Lee-Gunn () without specific predictions to be checked by experiments. More recently the reduction of the superfluid and the condensate density deeply in the superfluid phase was considered microscopically by Huang and Meng Huang (). These authors used the Bogoliubov transformation and found a decrease of the superfluid density and of the condensate if disorder is taken into account. The corrections to both quantities are proportional to in our terms. Thus this approach is restricted to weak disorder and . These results have been confirmed and extended by other authors Pitaevskii (); Vinokur (); Pelster (); Graham+Pelster (); Graham (). All these works used the approximation of weak disorder. The extrapolation of their results to the range of strong disorder made in Huang () was illegitimate.
Zhou Zhou () argued that the transition to the superfluid phase is of the percolation type. Lugan et al. Lugan+07 () considered a BEC in a special potential which is random in one and parabolic in the two remaining directions. The disorder is bounded from below but not from above. Using the numerical solution of the Gross-Pitaevskii equation as well as a mean field like approach they identified three different regimes: the Lifshitz glass, the fragmented BEC and the non-fragmented BEC. Because of the difference in the models (our potential includes randomness in all directions) a direct comparison with our results derived below is not possible, except for the one-dimensional case when the harmonic trap potential is negligible. Only in this case, and using our characteristics of the random potential, it can be shown that their approach gives results for the fragmented BEC comparable to ours. However we do not see a difference between the Lifshitz glass and the fragmented BEC. Moreover, Lugan et al. Lugan+07 () did not obtain important geometrical and energetic characteristics of the fragmented state vs. the average density of particles as we did based on a simple physical picture of the fragmentation.
More recent papers considered the dynamic consequences of Anderson localization in BEC and do not directly overlap with our work Paul+07 (); Lugan+07.2 (); Skipetrov08 (). Using arguments similar to those used in a published work by two of the authors Nattermann08 (), Shklovskii Shklovskii () independently and practically simultaneously arrived at the same conclusions about the critical value of density at which the transition between an insulating and the superfluid phase takes place, but he did not give the description of the localized phase.
The aim of this paper is to develop an alternative approach to the quantum phase transition starting from deeply localized state. We deliver a detailed study of the deeply localized state of the Bose gas in a random potential. Such a state appears at sufficiently low average density. We show, that in this region the Bose-Einstein condensate decays into remote fragments of small sizes. We give a rather transparent qualitative and quantitative geometrical description of the localized state. We show that, at a critical density which we express in terms of the disorder characteristics and interaction strength, the increasing tunneling of particles between fragments leads to transition from the random singlet state to the coherent superfluid. The localized state can exist in several different regimes. These regimes as well as their quantitative characteristics strongly depend on the ratio of two basic length scales characterizing the disorder: the Larkin length and the correlation length . We extended our consideration to a practically important situation of the Bose gas confined by a harmonic trap. In this case the gas forms a cloud which size is determined by the energy minimization. We demonstrate that all regimes which appear for the gas in the box appear also in the trap driven by the number of particles and other parameters. Our consideration is based on theory of deeply localized single-particle states in an uncorrelated random potential given in seminal works by I. Lifshitz IMLifshitz (), Zittartz and Langer zittartz (), and Halperin and Lax halperin-lax (). Their predecessors Keldysh and Proshko Keldish () and Kane Kane () studied electron density of state in a smooth random potential having in mind semiconductors. These works were extended and detailized in References Bonch_a (); Bonch_b (); Andreev (); Sh_Efros (); cardy (); John-Stephen (); thirumalei (). In this work we propose a modest extension and simplification of their arguments to find some geometrical and physical characteristics of these state necessary for our purposes. On the other hand, our theory can be considered as a generalization of their instanton-type theory incorporating the self-consistent field of the interacting particles.
We consider the ground state of the system at zero temperature. It is not always reachable since the system may be frozen in a metastable state. It is therefore important that nevertheless the ground state occurs to be reachable in a sufficiently wide range of parameters. We also demonstrate that properties of the localized states of the Bose gas are rich enough and that they are available for the experimental study.
Our article is organized as follows. In the next section we characterize the random potential. In Section III we consider the single-particle deeply localized states in 3 dimensions for the uncorrelated random potential obeying the Gaussian distribution. In Section IV we consider how the dilute Bose gas fills the deep fluctuation potential wells in this case. In Section V the transition from the random singlet to the coherent superfluid state is considered. Section VI extends all these results to the case of strongly correlated random potential. In Section VII we consider the Bose gas subjected to simultaneous action of a harmonic trap and random potential. Section VIII represents extension of these results to lower dimensions 1 and 2. The discussion of our results and conclusions are left for Section IX.
Ii Description of the disorder
The disorder will be represented by a random potential with zero average at each point of the space and obeying the Gaussian distribution:
where is the volume of an infinitesimal cell; is the correlation function or correlator of the random potential:
and is the inverse correlator, defined by equation:
The simplest random potential with zero correlation length has a like correlator:
We will call it uncorrelated random potential. The corresponding distribution function is the product of independent distribution functions at each point of space:
More general Gaussian random potential has a finite correlation length . The simplest realization of the random Gaussian potential with the finite correlation length is the Ornstein-Zernike correlator:
The corresponding probability distribution reads:
The disorder correlator may be regular at coincident coordinates . Then it has the following form:
where is the average quadratic fluctuation of the random field at a point. The function which determines the shape of the correlation function is normalized by the condition . In this case the Fourier-component of the inverse correlator is a growing function of the wave-vector. As a consequence, is not well defined. Instead we can use the probability in the momentum space:
Simple examples of non-singular correlators are the Lorenz distribution with and and the Gaussian distribution with exp and . In all these cases the scale of energy is established by and the scale of length is .
Iii Single-particle levels in an uncorrelated random potential
In this section we study the properties of single-particle states in the uncorrelated random potential defined by Eqs. (4, 5). This is a modest extension of the well-known works by I.M. Lifshitz IMLifshitz (), Zittartz and Langer zittartz () and Halperin and Lax halperin-lax (). Some simplifications allow us to find more detailed information about the distribution functions of the energy, sizes, distances between the states and the tunneling amplitudes between them. We are interested in a statistical description of the spectrum and wave functions of Schrödinger equation in the random potential :
Its energy levels in a finite volume are functionals of the potential . The only characteristic of the random potential together with world constants and establishes the scale of length:
which will be called Larkin length in analogy with the scale found in the Larkin’s work Larkin-70 () for an elastic medium in a random field. We will show that sets the scale of the extension of the deeply localized states with large by modulus negative energy. For delocalized states with large positive energy the same value with precision of numerical factor is the mean free path. The scale of energy can be obtained from the Larkin length in an obvious way:
Further we will work in a rather rough approximation similar to that used by Larkin and Ovchinnikov LO () and Imry and Ma imry-ma (). However, we start with a rigorous statement of the problem which gives a clue for our further estimates. The most easily calculable value is the density of state which can be written as a path integral (see the cited works IMLifshitz (),zittartz (),halperin-lax ()):
where is the spectrum of eigenvalues of the Schrödinger equation (10) in the volume .
In a large 3d volume the states with energy are delocalized, whereas the states with negative energy sufficiently large by modulus and are strongly localized. The threshold of localization is a positive energy of the order of . Note that this is correct only for 3d systems. As it was conjectured in the work by Abrahams et al. abrahams (), all single-particle states in 1 and 2d systems are localized. Near the threshold the wave function has very complicated fractal structure mirlin (), feigel (). In the interval between and the transition from fractal to strongly localized states evolves. The latter are supported by rare fluctuations of the random potential, which form a potential well sufficiently deep to have the negative energy as its only bound state. The problem of the deep random levels was considered by already mentioned authors IMLifshitz (), zittartz (), halperin-lax () about 40 years ago. We reproduce some of their results and extend them to find the probability distribution of the levels with energy less than some (), the distances between such states and the tunneling amplitude between them.
As it is clearly seen from equations (1,13), the main exponential factor in the density of state can be found by minimization of the integral at a fixed value of the energy level , which is a functional of the random potential . The latter can be determined as a minimum of the energy over the wave function:
Thus, we need to minimize a functional:
over and . Here is a Lagrangian factor. The minimization over leads to Schrödinger equation (10), whereas the minimization over results in a relation between and :
Thus, equation (10) turns into the Ginzburg-Landau equation. For our purpose the most important consequence of the relationship (16) is that the fluctuation potential well has the same characteristic linear size as the wave function . It is clear that the maximum probability requires the bound state with the fixed energy to be the only bound state in the potential well. Otherwise, at the same energy, we need a deeper well, i.e. larger . For the same reason the fluctuation well must have the spherical shape. Let the radius of the well is . Then the depth of the well can be estimated as and the energy level in it differs by a factor of the order of 1/2: . The exponential factor in the density of state reads:
where we redefined the Larkin length incorporating the factor in it. Here and henceforth we perform calculations for 3d systems. The results for other dimensions will be derived later. The result (17) is valid provided the number in the exponent is large, i.e. and .
Let us consider the probability or the part of volume occupied by the wells with the radius less than or the energy less than . It is obvious that contains an exponential factor (17) and some preexponent. Since is a dimensionless value, the preexponent must be a function of the dimensionless ratio :
The function must be much slower function of its argument than the exponent. Most naturally it is a power function with a critical exponent . It can be extracted from the work by J. Cardy cardy (): for the uncorrelated disorder. We will see that it is inessential for further conclusions. By knowledge of we can calculate the number of the wells with the radius less than per unit volume, i.e. the density of such wells d-w (). In order to do that the unit volume must be divided into the cells of the volume . Each cell can contain or do not contain the fluctuation, but fluctuations with the centers approaching each to other at a distance less than must be considered as one asymmetric potential well. The number of such cells in unit volume is . Thus:
The average distance between the wells with the radius less than reads:
This equation shows that the distance between the wells is significantly larger than the size of the wells. The situation is depicted in Fig. 2.
Finally we find the tunneling factor for two typical wells with the energy levels of the same order or the radius of the same order of magnitude. It is given by the semiclassical expression , where the integral path connects two wells. By the order of magnitude and the length of the path of integration is . Thus, . Finally we find:
At or , the distances between the optimal potential wells become of the same order of magnitude as their size . Simultaneously the tunneling amplitude between the wells becomes of the order of 1. The potential wells percolate and tunneling is not small, but the states still are not propagating due to the Anderson localization anderson ().
Iv Bose gas in a large box with an uncorrelated random potential
In the ground state of an ideal Bose gas in a large box with the Gaussian random potential all particles are located at the deepest fluctuation level. In the box of the cubic shape with the side the deepest level which occurs with probability of the order of 1 has the radius determined by equation: , i.e. . Note that the prefactor introduces a correction to the denominator of the order of which can be neglected. The corresponding energy is . Such a state is highly non-ergodic since the location and the depth of the deepest level strongly depends on a specific realization of the disordered potential. Therefore the average energy per particle and other properties averaged over the ensemble has nothing in common with the properties of a specific sample. Even an infinitely small interaction immediately changes the situation: the system becomes ergodic in the thermodynamic limit, i.e. when first the size of the system grows to infinity and then the interaction goes to zero. For example, the energy per particle in sufficiently large volume coincides with its average over the ensemble. The reason of such a sharp change is that, at any small but finite interaction, the particles can not more fill one well since their repulsion finally overcomes the attraction to the potential well. They will be redistributed over multiple wells. Since the distribution of wells in different parts of sufficiently large volume passes all possible random configurations with proper ensemble probabilities, the ergodicity is established. In this section we find how the interacting particles eventually fills deep single-particle levels simultaneously changing their size and shape by their self-consistent field. Similar idea was considered in the work by Lugan et al.Lugan+07.2 (). However, in this work the consideration was done at fixed chemical potential instead of fixed number of particles. Besides, they did not perform geometrical analysis and calculation of the tunneling amplitude for single-particle states. Therefore, their results do not contain our explicit expressions for the geometrical structure and tunneling of the localized states as well as the analysis of the transition line.
In a real experiment the Bose gas may be quenched in a metastable state depending on the cooling rate and other non-thermodynamic factors. This is what M.P.A. Fisher et al. Fisher () call the Bose glass. Such a state is also possible in the case of weakly repulsive Bose gas. It looks plausible that the Bose glass whose metastable state has energy close to that of the ground state has also close to the ground state statistical characteristics, but this question needs more thorough study. However, as it will be demonstrated later, in the case of cooled alcali atoms the tunneling amplitude still remains large enough to ensure the relaxation to the equilibrium state in 1010. Our further estimates relate to the real ground state.
The weakly repulsive Bose gas in a random potential is described by the well-known Hamiltonian:
where is the secondary quantized wave function; the positive coupling constant is associated with the scattering amplitude by the relationshipLLQM ():
As in the Bogoliubov’s theory bogoliubov () and its extension by Belyaev belyaev (), we assume that the gas criterion is satisfied. Here is the average particle density and is their total number. The purpose of the following consideration is to find how, with the growth of the number of particles, they eventually fill the fluctuation potential wells. Implicitly our considerations takes in account the change of the optimal potential wells due to the interaction.
Let particles with the average density of particles fill all potential wells with the radii less than in the ground state. The average number of particles per well is
The local density inside the well of the linear size of is . The gain of energy per particle due to the random potential is ; the repulsion energy due to interaction is equal to , where we used Eq. (23) and the well-known relation for an effective potential field induced by a gas of scatterers LLQM (). Minimizing the total energy per particle over and employing Eqs. (19) and (24), we find the value of corresponding to the minimum of energy at fixed with the logarithmic precision:
where the critical density is defined as follows:
The factor in Eq. (19) as well as the next approximation to the solution (25) leads to the corrections of the type which can be neglected. The distances between the filled wells according to Eq. (20) are:
This distance strongly exceeds the average size of the potential well (25) at . At the same condition the chemical potential of atoms can be estimated as:
The tunneling amplitude between two wells separated by a typical distance can be found by employing the single particle result (21):
Thus, the Bose gas at is fragmented into a multiple clusters of small size separated by much larger distances and containing about particles each. The amplitude of tunneling between the wells is exponentially small, so that the number of particles in each cluster is well defined. The phase is therefore completely uncertain. Such a state is a singlet with inhomogeneously distributed particles, a random singlet: the ground state is non-degenerate.
The previous approach is valid if . From Eqs. (19, 24, 26) we find . Thus, the disorder must be weak enough to satisfy the constraint . But it is not sufficient. Additionally the average density must be large enough. Namely, it must satisfy strong inequalities . At smaller , only few particles appear in a potential well. At , the size of filled potential wells approaches . Most of the filled wells contain 2 particles, but only a small part of the total number of wells of this size (per unit volume) is filled, approximately . The consistency of our approximation requires that the gas parameter remains small inside an optimal potential well. The density inside the well is equal to . Therefore, the interaction is weak inside the well if .
The compressibility is finite as expected for the Bose glass phase Fisher (). In the absence of interaction the probability that two single-particle eigenstates localized in different potential wells have the same energy is zero. Thus the zero temperature conductivity vanishes in the thermodynamic limit. If the interaction is switched on, we can consider its effect as splitting of the energy levels by an amount of the order . For the wave functions are still localized and the conductivity at zero temperature vanishes. Thus, the Bose glass phase is insulating. At finite temperatures however the energy gap between different localized states can be bridged by thermal activation leading to the Mott variable range hopping (VRH) Efros (). Below we apply the VRH argument to our situation. The tunneling probability between two wells, each of extension , separated by the distance , is (see analogous derivation in section 3). Together with the activation factor it gives the hopping probability between the two wells equal to
where is the energy difference between the two distant wells. There is with high probability at least one localized state in the interval of energy provided the condition is satisfied. Employing the derived dependencies , and and maximizing the probability (30) over , we find the hopping conductivity
where C is a constant or slow function of . In a similar way it is possible to calculate the non-linear field-dependent conductivity.
V Transition region
Though our results are valid only for , it follows from them that at the overlapping of different occupied wells becomes large and the tunneling amplitudes reach the value of the order of 1. Therefore, the phase coherence in different wells grows until at some critical value of density which we roughly identify with the global coherence is established. Thus, at , the quantum phase transition from localized singlet to the superfluid state proceeds. As usual, the order parameter is the average phase factor . The superfluid state bears clear traces of disorder: its superfluid density is inhomogeneous at characteristic scale . It can be treated as a disordered superfluid. An alternative argument goes as follows. The condition can be rewritten in the form where denotes the superfluid healing length. The latter describes the range over which the superfluid order parameter, when changed by a local perturbation (e.g. by a wall or in the center of a vortex), regains its bulk value. Since denotes the scale at which the disorder becomes relevant, the condition simply means that the superfluid order parameter cannot adapt to the rapid variation of the disorder which happens at the scale and superfluidity is destroyed. In the opposite case , i.e. this is not longer the case and superfluidity is restored.
At the energy of the repulsion becomes much larger than the characteristic disorder energy . Then the gas becomes almost homogeneous with precision of small parameter . In this case the approach by Huang and Meng Huang () based on the homogeneous zero-approximation ground state and refined by Giorgini et al. Pitaevskii () and by Lopatin and Vinokur Vinokur () is justified. They demonstrated that the superfluid density is not equal to the total density as it is without disorder, but remains close to the total density. Thus, the superfluidity definitely strives in the limit of large density. We conjecture that vanishes at the same quantum phase transition point at which the coherence disappears.
The small corrections found in the cited work Huang () can be estimated by the order of magnitude without long calculations. Indeed, the disorder Hamiltonian reads:
where is Fourier transform of the random field . It produces the change of energy in the second order of the perturbation theory. The change of energy per particle can be estimated as follows:
The delta-like correlation function of the random potential (4) implies that for any . The excitation energy can be estimated as in the effective range of integration after subtraction of the contribution of disorder to the energy of the normal state. Collecting all these factors, we find:
With precision of a numerical coefficient 0.85 this result coincides with that obtained by Huang and Meng (Equation (7) of their work Huang () gives the value of energy per unit volume; it must be divided by to find the energy of the ground state per particle). In terms of the Larkin length and energy and critical density introduced earlier (see Eqs. (11, 12, 26)) the correction to the energy of the ground state per particle can be expressed as
The change of superfluid and condensate densities can be found from their initial values (one exactly and another approximately equal to the total density ) by multiplication to a small factor proportional to , where .
In discussing critical phenomena it is crucial to determine the critical (marginal) dimensions. In order to do that we need some results which will be derived later. We will show that the critical density at the transition between the superfluid and the Bose glass phase in dimensions reads
where and denote the -dimensional Larkin and healing length, respectively. For strong disorder and strong interaction the natural limits of the validity of equation (36) seem to be and , i.e. the healing length and the Larkin length must be much larger than the mean spacing between the bosons.
For weak disorder, the Larkin length diverges at . Hence in dimensions disorder has to overcome a threshold value to destroy superfluidity even in the limit of weak interaction. Thus is an upper critical dimension of the problem as has been discussed already in Fisher (). A lower critical dimension is . This can be seen from a calculation of (35) in one dimension which is logarithmically infrared divergent reflecting strong quantum fluctuation. Giamarchi and Schulz Giamarchi () in a seminal paper found a quantum phase transition in 1d Bose gas in disordered environment from a superfluid to an insulating phase for increasing interaction parameter (compare Appendix). Since then this transition has been considered as the genuine transition between these two phases (see e.g. Fisher ()). However, at this transition strongly violating the gas condition . Thus, we conclude that the transition found in Giamarchi () is not the one-dimensional equivalent of the transition taking place at . The phase portrait obtained in this work is reproduced in Fig. 8. It follows from this figure that there are two different phase transitions at small disorder: one at large value of dimensionless coupling constant and another at small . The large part of the phase diagram beyond a small vicinity of the critical point , was obtained by a speculation rather than a rigorous treatment. Therefore, the question about the existence of two different quantum phase transition remains open.
Critical behavior near the quantum transition was studied by Halperin, Lee and MaMa+86 () and by M.P.A. Fisher et al.Fisher (). In the latter work the scaling consideration was developed and even the critical exponents were found. However, we are not sure that they are valid for our problem since, according to previous remark, they are associated with the strong-coupling critical point. Secondly, even if there is only one critical point, the marginal dimensionality is 1+1 and it is very doubtful that exact critical exponent can be found in the case 3+1.
Vi Correlated disorder
In this section we consider properties of the weakly interacting Bose gas in a big box with a correlated random potential. The properties of correlated random potential were already described in the second half of Section II (see the text related to Eqs. (8 -9)). Here, following the line of consideration developed in Sections III-V, we study first the size and distribution of single-particle states and tunneling between them and how they are modified by interaction. We start with the single-particle states.
vi.1 Single-particle states in a correlated random potential
Keldysh and Proshko Keldish (), and Kane Kane () were the first to find the electron density of states in a semiclassical random potential. They proved that, in contrast to the uncorrelated disorder for which , the density of state in a correlated disorder is a Gaussian function of energy , where is a numerical constant. Independently but later the same result was found in Bonch_a (); Bonch_b (); Andreev (). Shklovskii and Efros Sh_Efros () derived the same result employing the instanton approach. John and Stephen John-Stephen () found the dependence of the preexponential factor on energy. Their result was confirmed by Thirumalai thirumalei () by a different method. Below we rederive their result and find new important characteristics of optimal fluctuation wells in the case when the correlation length is large enough. It is intuitively clear that, at , the density of state and other characteristics of the spectrum and optional states only slightly differ from their values for uncorrelated random potential. At numerical coefficients will be different from those for uncorrelated disorder, but with this reservation still our semiquantitative description is valid. Therefore, the most interesting is the situation with . Such a disorder we call strongly correlated. In what follows we assume the strongly correlated disorder . The Larkin length for the correlated disorder reads:
It is convenient to introduce another characteristic length . For strongly correlated disorder . The Larkin length is associated with by the following relation:
To find the exponent in the density of deeply localized states it is necessary to minimize the functional at fixed energy of the quantum state in the potential . The condition of minimum is reduced to a following relationship between the optimal fluctuation potential and the wave function :
analogues to Eq. (16) for the uncorrelated disorder. This equation shows that the characteristic size of is always equal to . Indeed, it is correct if the characteristic size of the wave function less or equal to . In the opposite case the characteristic size of the potential would be much larger than and, respectively, the probability of such configuration would be much smaller. Deep levels have negative energy , much larger by modulus than the energy scale of the random potential . Since the optimal fluctuation potential well must have the level , its depth is not less than . The fundamental difference between the optimal potential wells in the cases of uncorrelated and strongly correlated disorder is that the former contains only one level, whereas the latter contains many levels as shown in Fig 3. Indeed, the number of levels in the optimal well is not less than . Therefore, the level is located close to the bottom of the optimal potential well. Its depth with high precision is equal to and its shape is determined by the function from Eq. (8). If the correlation function is non-singular, then . The exponent in Eq. (1) can be calculated exactly leading to the result for the probability to find the level with energy less than (with precision of a prefactor):
In a more accurate treatment one needs to take in account that the level can be not the ground state in the optimal well, but an excited state. Since even excited states are close to the bottom, the spectrum is the same as for a spherically symmetric quantum oscillator with the frequency . Therefore, the depth of the well is not exactly equal to . It depends on what excited state occupies the particle:
The second term in the depth of well determined by this equation is much less than the first one. After summation over all from 0 to infinity we find the corrected expression for the probability :
The ratio which appears in Eq. (42) by the order of magnitude is equal to the product of two ratios . The first factor is large, the second one is small and their product varies from small to large values always remaining larger than . Nevertheless, a strong enhancement of probability up to a factor in comparison to the initial expression (40) is reached at decreasing modulus of energy. Summarizing, we conclude that the participation of excited states may change a comparatively slow-varying prefactor leaving the principal exponent (18) unchanged.
Next we analyze the case of a singular correlator using the example of the Ornstein-Zernike distribution. The relationship (39) is valid for singular correlators implying that the radial size of the optimal potential well is equal to the correlation length . We use the functional (7) to estimate the exponent in the probability . The two terms in the exponent (7) proportional to and are equal. With the precision of a numerical constant the potential is equal to . The integration leads to in accordance with the earlier results John-Stephen (), thirumalei (). Further we incorporate the constant into the definition of and write the equation for the probability in all cases as:
For simplicity we put the prefactor to be equal to 1.
The density of potential wells containing deep levels lower than for strongly correlated random potential reads:
and the average distance between such wells is
The tunneling amplitude between such wells can be found in semiclassical approximation:
It is always very small, but the direct tunneling between wells does not play an important role since the particles remain almost classical even near the mobility edge in contrast to the uncorrelated disorder. According to the Shklovsky estimate Shklovskii (), the percolation of wells happens at . The characteristic De-Broglie wavelength at this energy is much less than for strongly correlated disorder. Therefore, the particles propagate according to classical mechanics. The mobility edge energy is negative and not far from the percolation threshold.
vi.2 Weakly repulsive Bose gas in strongly correlated disordered potential
In this section we consider the Bose gas with the average density . The gas partly fills all potential wells with the depth less than . Let the particles fill a typical well up to the radius . If the correlator is regular at , the following equation gives a rough estimate of energy per particle:
The first term in this expression is the gain of energy due to the random potential, the second term is the energy of interaction. The quadratic term in the brackets corresponds to the oscillator-like potential at small . Minimizing over and the ratio , one arrives at following results:
The critical density is defined as follows:
All results are valid at . According to Eq. (49), the ratio is small. This fact justifies the approximation of quadratic potential in Eq. (47). It allows to estimate the characteristic momentum of particles or the width of their distribution over momentum:
Two filled wells are separated by a typical distance much larger than the size of the well:
With this value of it is ready to estimate the tunneling coefficient:
The number of particles in each typical well can be estimated as . It is large if .
These results conclude the description of localized states for strongly correlated disorder. At energies between and the motion of particle is strongly disordered, but it becomes more and more free when energy approaches and exceeds . At , the random potential can be considered as a perturbation. The quantum phase transition from the localized random singlet state to a superfluid proceeds at . Note that the critical density in the presence of strongly correlated disorder does not depend on the correlation length .
Vii Bose gas in a trap with disordered potential
In this section we consider deeply localized states in a harmonic trap supplied with a Gaussian random potential. The Hamiltonian of the system has an additional term: the potential energy of Bose particles in the trap:
where the harmonic potential of the trap generally has a form:
In this section we consider only the isotropic trap with . Strongly anisotropic traps can be considered as a system with reduced dimension 2 or 1 and will be considered in the next section. A new scale of length associated with the trap is the well-known oscillator length .
A principal difference between the Bose gas in a box and in a trap is that in the latter case the gas forms a cloud whose size is determined by energy minimization at a fixed number of particles, whereas in the former case the size of the box and, therefore, its average density is fixed. Nevertheless, we will see that all phase states of the Bose gas in the box appear when this gas is placed into a harmonic trap, but their appearance is regulated by the total number of particles in the gas instead of their average density.
vii.1 Trap with uncorrelated disorder
Four competing parts of a Bose-particle energy are: kinetic energy, the confining potential energy of the trap, the repulsion from other particles and the energy of the random potential. Two of them, the interaction with the trap and the random potential tend to confine and localize the particle. Let the gas form a single cloud of the radius . The rough estimate of the two confining energy contributions can be done as follows Nattermann08 ()
The disorder energy for the same cloud can be estimated as
The latter estimate can be justified as follows. The cloud chooses such a position that the random potential energy is negative. A characteristic fluctuation of energy in a volume is
Substituting into the last expression , we arrive at Eq. (57). Comparing the two confining contributions, we conclude that, at , the influence of disorder is negligible. The remaining two terms of energy: kinetic energy
and the repulsion energy per particle:
are deconfining. The kinetic energy dominates at . In the opposite
case the repulsion energy dominates. Below we analyze several
1 Weak disorder: . Depending on the relative value of the interaction one should distinguish two different situations:
1a. Weak interaction: . In this case the interaction can be neglected. Minimizing too remaining terms, the kinetic energy and energy of the trap, we find . Physically it means that all particles are condensed at the oscillator ground state.
1b. Strong interaction: . Neglecting kinetic energy and minimizing remaining energy of traps plus the repulsion energy, one finds the result known as Thomas-Fermi approximation Stringari (): .
More interesting is the case of strong disorder.
2. Strong disorder: . Again we consider several different limiting cases depending on relative strength of disorder, interaction and the trap potential.
2a. Weak interaction: . In this range of variables the non-ergodic phase is realized. Since interaction is negligible, the particles find a random potential well with the deepest level and fall into it. Let such a well can be found at a distance from the trap center. Its depth typically is about . This gain of energy must be not less than the loss of the trap energy . A typical value of appears when both this energies have the same order of magnitude. Then . A typical size of the well is .
2b. Moderate interaction: . In this case the ergodicity is restored. Our experience with the gas in a box prompts that the gas cloud is split into fragments each occupying a random potential well from very small size till some size depending on . From Eq. (28) the typical disorder energy per particle is . It becomes equal to the trap energy at a distance (see the definition of the parameter below in Eq. (60)). Therefore, the average density is . The ratio plays the decisive role: the state of the Bose gas is fragmented and strongly localized when it is large; the transition to delocalized superfluid state proceeds when this ratio becomes 1. With precision of a numerical factor the ratio is equal to a new dimensional parameter:
Strongly localized fragmented state is realized at . The transition proceeds at . The superfluid phase emerges at . The regime diagram is shown in Fig. 4.regime () Note the counter-intuitive dependence of the cloud size on the number of particles: the cloud slightly contracts with increasing number of particles. It happens because the number of particles in each fragment increases more rapidly with the average density than the number of fragments.
vii.2 Trap with strongly correlated disorder
Following the line of development accepted in the previous subsection, we start with the case of weak disorder.
1. Weak disorder: or . The value in the left-hand side of this interaction is the characteristic variation of the random potential on the minimal of two linear sizes: oscillator length or correlation length . In this case the random potential only slightly changes the state of the gas in the absence of disorder (see previous subsection). In what follows we assume for simplicity that .
2. Strong disorder: or . As in the previous subsection we consider different relative strength of the interaction and disorder.
2a. Weak interaction: . The interaction is negligible. Therefore, all particles find the deepest well in the trap and fall into it. The typical distance from the center of the trap to the deepest well is and its typical depth is . In this range of parameters the system is non-ergodic.
2b. Moderate interaction: . In this case the particles are distributed among multiple potential wells. The system is ergodic. The average depth of the well is , where
is the dimensionless parameter analogues to that introduced in the previous subsection. It represents the ratio . Until is large the ground state of the Bose gas is deeply localized random singlet. The transition to the superfluid state proceeds at . The total linear size of the cloud is . Inside a fluctuation potential which has a typical size particles occupy a smaller sphere of the radius . The phase state (regime) of the system depends on three dimensionless parameters: , and . The regime diagram is shown on Fig. 5.
Viii Lower dimensions
In the experiments with cooled gases the trap is mostly realized as a rotation ellipsoid with a large aspect ratio. Therefore, and also for comparison with exact 1d solution Giamarchi () it is reasonable to extend our theory for lower dimensions .
viii.1 Uncorrelated disorder
We start with the extension of Eq. (11) connecting the Larkin length and the constant for the uncorrelated disorder:
following from the dimension consideration. The characteristic energy reads:
The extension of the exponential law (17) for the probability to find the deep potential well of the size not larger than is:
where the Larkin length differs from that given by definition (62) by the factor , in which is the volume of the sphere with the radius 1 in dimensional space. As earlier is a power-like function of the ratio at : with for and for cardy (). The extension of the Eq. (19) for the density of the fluctuation potential wells with the radius not exceeding looks in dimensions as follows:
and the average distance between such wells is
The tunneling amplitude between nearest wells has a characteristic value
Next we consider the filling of the potential wells by the diluted Bose-gas of the density . Again, let the gas fills all the wells up to the radius . The average number of particles per well is:
and the density of particles in the well is
The repulsion energy per particle is as before , but the connection between the coupling constant and the scattering amplitude depends on dimensionality of the space: