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Large N renormalization group study of the commensurate dirty boson problem

(Short Description) We use a large N renormalization group method to study a model of interacting boson system with a quenched random potential.

One of the reasons why this problem is very challenging is that there is no sensible noninteracting limit for disordered bosons. That is, the zero interaction limit is pathological in the sense that the bosons will condense into the lowest localized state around a small region.
For the metal-insulator transition of electrons, which has been understood better, disorder alone or interaction alone can induce localization of electrons and drive the electrons to the Anderson insulator (AI) or the Mott insultor (MI) state [11]. The interplay between these two effects has been studied during the last decade but there are still open problems [11].
The dirty boson problem has direct experimental realization on 4 He in Vycor glass and in other porous media [12,13]. This disordered interacting boson problem may be used to understand the superconductor-insulator transition in the disordered thin films [14] and short coherence length superconductors [15]. Recently Wen and Wu [16] showed that the superfluid-Mott insulator transition of bosons with the Chern-Simons gauge field can describe the transition between quantum Hall (QH) states in the absence of disorder. Therefore, the dirty boson problem with the Chern-Simons gauge field is intimately to the quantum Hall transitions in the presence of random impurities. The QH-MI transition for the pure system is also studied by Chen et al. [17] in which they studied fermions with the Chern-Simons gauge field.
Following Fisher et al. [1,2], we can write the Hamiltonian of the interacting lattice bosons in a random on-site potential as wheren i = b † i b i and b † i is the boson creation operator at the site i. µ is the average chemical potential that fixes the boson density and δµ i is the random on-site potential with zero average. J ij is the hopping matrix element and J 0 = j J ij . In order to study the critical phenamena of the system, it is convenient to find the effective field theory. We will summarize the approach of Fisher et al. [1,2]. First the off-site hoping term in H 1 is decoupled by introducing the Hubbard-Stratanovich field Ψ i , then the resulting action can be expanded in terms of Ψ i . Since Ψ i is linearly related to b i for small b i , the field Ψ i can be identified as a superfluid (SF) order parameter. It was shown [2] that the effective field theoretic action is given by where w i , u i are random functions of i with zero average.
It is established [2] that SF-MI transition of the pure system has two uninversality classes, the commensurate case and the incommensurate case. In the commensurate case, the superfluid density commensurates with a periodic potential. In this case, SF-MI transition is described by a tricritical point which belongs to the universality class of (d + 1) dimensional XY model with the dynamical exponent z = 1. In the incommensurate case, the SF-MI transition happens on a line in thew −ū plane. It is argued [2] that the generic SF-MI transition should be the latter case ( with z = 2 at the transition ) rather than the former case.
One natural question is that how disorder affects these two different SF-MI transitions.
The destruction of superfluid in the presence of disorder brought the concept of the Bose glass (BG) phase [4,18] in which bosons are localized by disorder. In seminal papers, Fisher and coworkers [1,2] suggested a scaling theory of SF-BG transition. They argued that superfluid-insulator tansition should occur through the Bose glass phase and the generic transition should be described by the action (2) withū + u i = 0 which does not have space-time isotropy. In the scaling theory, they postulated that the compressibility is totally due to the phonon mode and one of the main result of this assumption is that the dynamical exponent z = d. The simulation of the quantum rotor model [3] and some renormalization group calculation [7] partially supported this picture although a recent quantum Monte-Carlo calculation [9] contradicts to these results. The earlier work of Ma, Halperin and Lee (MHL) [4] was reexamined and the importance of the term that is linear in ω is emphasized. However, it was also argued that MHL theory may apply to the possible direct SF-MI transition at the commensurate case ( particle-hole symmetric case ).
We can see that the dirty boson effective action has the strict particle-hole symmetry ifū + u i = 0. It was also argued that the general commensurate case corresponds to the weaker particle-hole symmetric case [2], i.e.,ū = 0 but u i = 0. Are the transitions of particle-hole symmetric and asymmetric cases in the same universality class ? Originally Fisher and coworkers [1,2] prefered that even arbitrarily weak disorder will induce the Bose glass phase for both of the incommensurate and commensurate cases ( Fig.1 (a)).
However, numerical calculations in [8] have not revealed the intervening Bose glass phase in the commensurate case, although the superfliud-insulator transition indeed occur through the Bose glass phase in the incommensurate case ( Fig.1 (b)). Singh and Rokhsar [5] performed a real space renormalization group analysis for the commensurate case and they found the direct transition from SF to MI when the disorder is sufficiently weak ; the Bose glass is found beyond a threshold ( Fig.1 (a) or Fig.1 (b) depending on the impurity strength). Zhang and Ma [6] considered hard core bosons with disorder. In this real space renormalization group analysis of a quantum spin-1 2 XY model with transverse random field ( the hard core boson model is mapped to this model), they concluded that commensurate and incommensurate cases are in the same universality class and the SFinsulator transition occurs alwalys from the Bose glass phase ( Fig.1 (a)). The conventionl renormalization group calculation by Weichman and Kim [7] in which they used double dimensional expansion around d = 4 partially supported the original picture [1,2] for the general dirty boson problem although some technical problems exist.
In this paper, we are going to study a large N generalization of the original action.
By doing 1 N expansion, we can treat the interaction non-pertubatively in the coupling constant. Both types of disorder are assumed to be weak and we do the pertubation in the strength of two types of disorder. The large N generalized action of the original dirty boson model in the Eucledian space is given by where i = 1, · · · , N and N = 1 corresponds to the original model. u(x) and w(x) are gaussian random functions of x with zero mean and their variances are given by means the random average.
Whenū+u(x) =w+w(x) = 0, (3) describes the muticritical point of SF-MI transition of the pure boson system. Near d = 2, both ofū andw terms are strongly relevant. In this paper, we will takeū =w = 0 and study the RG flow of U 0 , W 0 and v 0 . The physical meaning of settingū = 0 is the following. We tune the chemical potential µ to make the average boson density alwalys commensurate with the lattice. Thus, we will call (3) with u = 0 the commensurate dirty boson theory.
Our RG calculations are done at the critical point and with the renormalized mass term w = 0 in the course of the renormalization. Since the effect of φ 4 term is calculated exactly at each order of 1/N , the coupling constant g 0 is not renormalized. Following Ref. [19] and using dimensional regularization, we move an infrared scale µ in the renormalized theory with fixed bare parameters to obtain the RG flow of the renormalized parameters.
Introducing dimensionless measures of disorder, where a = 1 2π , b = 128 π , c = 64 π and l is the logarithmic measure of the RG flow. This is the central result of this paper.
Looking at ǫ ≥ 0 case, we can immediately see that there is only the pure fixed point which is given byŨ * = 0,W * = 0. At this trivial fixed point,Ũ is irrelevant andW is relevant so that the RG flow goes to the strong randomness regime where our RG breaks down. However, for ǫ < 0, there are three fixed points. All of them have at least one relevant direction in theŨ −W plane. Thus, includingū andw, these fixed points in the original theory (3) has at least three relevant directions. Therefore, in both cases, the direct SF-MI transition in theW −Ũ plane is unlikely to occur due to the absence of the weak random fixed point with two or less relevant directions. The superfluid insulator transition is alwalys governed by a strong random fixed point which cannot be reached by weak randomness expansion. More details of the RG flow will be discussed later.
The organization of this paper is as follows. In section II, we consider a restricted model in which u(x) = 0 and show the basic formalism we used. Here, we also examine some possible effects of long range interactions. The RG calculation for the generic commensurate dirty boson problem up to (1/N ) 0 order is presented and the RG equation is calculated in section III. we also discuss about the results. In section IV, 1/N correction due to φ 4 interaction is considered. In section V, we summarize and conclude this paper.

II. RENORMALIZATION GROUP ANALYSIS FOR THE STRONG PARTICLE-HOLE SYMMETRIC MODEL
In this section, we consider a rather restricted model in which we set u(x) = 0. This corresponds to the strong particle-hole symmetric case in the sense that more general model requires the particle-hole symmetry only in the average sense, i.e., u(x) = 0 and allows the local breaking of the symmetry at each site. We would like to consider more general The λ = 0 case corresponds to the usual short range interaction.
However, it should be mentioned that these are not the true long range interactions of bosons because the true one should be But the long range interaction between densities will induce the long range interaction in the φ 4 term.
In order to perform the RG calculation in the critical theory, we need to introduce an arbitrary infrared mass parameter µ [19]. We will use the dimensional regularization method to calculate relevant divergent diagrams. The renormalization of the theory is given by the renormalization of two point Γ (2) and four point Γ (4) vertices. We will take the inverse of the full propagator as a two point vertex and the two-boson scattering amplitude as a four point vertex function. The relation between the bare theory and the renormalized theory is given by where Γ (N) bare and Γ (N) represent the bare and the renormalized vertices respectively. We found that appropriate renormalization condition for Γ (2) can be chosen as Renormalization condition for the scattering amplitude will be discussed later. Also, following standard procedure, we require the independence of the bare theory with respect to µ, where Λ is the mass parameter of the bare theory.
Let us start with the evaluation of the self energy to the (1/N ) 0 order. The (1/N ) 0 order self energy diagram is given by Fig.2 (a). The polarization bubble Π 1 (q, ω = 0) in .
Assuming 3 + λ > d, the diagram in Fig.2 (c) can be approximated as Using this result, we can evaluate the self energy as where v 0 q ≫ ω is assumed. The bare two point vertex Γ Note that c 2 and c 3 diverge at d = 2 + 2λ. Therefore, let us try ǫ = d − 2 − 2λ expansion in order to handle these divergences. Also, for convenience, let us introduce dimensionless the renormalized theory. Adding appropriate counter terms to cancel the 1 ǫ divergences from c 2 and c 3 and using the renormalization conditions (5) (6), we get the following wherec 2 = ǫc 2 andc 3 = ǫc 3 . From µ independence of the bare parameters, we get Now we are going to renormalize the four point function Γ (4) . We will take the scattering amplitude of two bosons at q 1 = q 4 = q, q 2 = q 3 = 0, ω 1 = ω 2 = ω 3 = ω 4 = ω as our Γ (4) . First of all, let us calculate Γ (4) bare in the bare theory. Γ (4) bare in the ( 1 N ) 0 order is the sum of Γ (4) bare,1 , Γ (4) bare,2 , Γ (4) bare,3 , Γ bare,4 which are shown in Fig.3 (a), (b), (c), (d) respectively. Let us introduce a renormalized polarization bubleΠ 1 , a new polarization bubble Π 2 and a vertex V 1 which are given by the diagrams in Fig.3 (e), (f) and (g). Γ The bubbleΠ 1 which is renormalized by the self energy correction is given bỹ Evaluation of Π 2 (q, ω = 0) is a little bit long task and the result is where We found that λ = 0 and λ = 0 cases should be treated separately. First, for λ = 0 case, it is found that the constant e does not diverge as ǫ → 0 so that it can be dropped in the final RG equations. Now, let us look at the case of λ = 0. The strategy is that we investigate the most divergent contributions in various limits and add up all of the contributions.
Since the most divergent contribution comes from x 2 → 0 or y 2 → 0 limit in Eq.(17) and we want just this contribution, we can set x 2 = 0 or y 2 = 0 inside the square bracket in Eq. (17). We found that both ways give the same answer. Here we will take x 2 = 0 limit inside the square bracket and multiply by 2. The equation (17) becomes where only the leading divergence is taken in the second equation. The evaluation of the remaining integrals is straightforward and the result is V 1 (q 1 = q 3 = q, q 2 = 0, ω 1 = ω 3 = ω, ω 2 = 0) can be evaluated similarly as where v 0 q ≫ ω is assumed. Note that d 2 = 0 for the short range interaction (λ = 0).
Putting together all of these results, we can get the following Γ (4) bare (q 1 = q 4 = q, q 2 = q 3 = 0, ω 1 = ω 2 = ω 3 = ω 4 = ω), The renormalization condition for Γ (4) is taken as If appropriate counter terms were added to cancel the 1/ǫ divergences, the following equation can be obtained whereẽ = ǫe andd 1 = ǫd 1 . Using the fact that the bare parameters are fixed, as we change µ, we get the following equation forW Let b be the reduction factor for the momentum scale from µ to µ/b, then the RG equations forW and v up to ( 1 N ) 0 order is where l = lnb is the logarithmic measure of the RG flow.c 2 andc 3 are given bỹ Now, the RG equations for 0 < λ < 1/2 up to ( 1 N ) 0 order is where the following coefficients are used: For the usual short range interactions λ = 0, where were used.ẽ = ǫe is the order of ǫ for the short range interaction and is dropped in the RG equation. and slightly larger than two. For d < 2, the pure fixed point becomes stable. There is also an unstable fixed point which is given byW * = |ǫ|π/128. The dynamical exponent at the unstable fixed point is z = 1 + |ǫ|/2.

III. RENORMALIZATION GROUP ANALYSIS FOR THE COMMENSURATE DIRTY BOSON MODEL
Now we are going to study the generic model (3) withū =w = 0. Here, we consider the usual short range interaction λ = 0. The disorder characterized by u(x) in Eq. (3) will be considered in addition to the w(x) type disorder. This means that we have to consider more diagrams that are generated by this new disorder. The additional self energy correction due to u(x) type disorder is given by the diagram in Fig.4 The new bare two point vertex Γ It is clear that we need to do ǫ = d −2 expansion in order to handle the divergences. Let us introduce additional dimensionless measures of the disorderŨ in the bare and the renormalized theory. Adding appropriate counter terms and using (5) (6) (7), we can again obtain the following equations for Z and v There can be 5 additional diagrams that contribute to Γ (4) bare . Let us identify Γ bare,9 as the diagrams in Fig.5 (a), (b), (c), (d), (e) respectively. First of all, let us evaluate the new vertex V 2 of Fig.6 (a) where v 0 q ≫ ω is assumed. The bubble Π 3 of Fig.6 (b) is given by where f = 1 2 is a convergent integral. We can see that the bubble Π 3 is smaller by the factor ǫ ( putting d = 2 + ǫ ) than Π 1 and Π 2 so that the contribution from Γ bare,9 is higher order in ǫ and we can neglect it.
Using the calculated V 2 , we can evaluate Γ (4) bare as where Γ (4) bare,w = where d 2 = 0 for the short range interaction is used.
We take the follwoing renormalization condition for Γ (4) = Γ Adding appropriate counter terms and using the renomalization condition, we obtain the follwoing equations Using µ independence of the bare parameters, we obtain the following equations forW wherec 4 = ǫc 4 and . For small ǫ, the coefficients of these equations are given by We can ignoreẽ which is the order of ǫ. Now the RG equations forW ,Ũ and v are given where l is agian the logarithmic measure of the RG flow. The RG flow is drawn in Fig.7 (a) for ǫ ≥ 0, (b) for ǫ < 0.
Among these three fixed points, (W * ,Ũ * ) = (0, 2π|ǫ|) is a non-trivial fixed point that has the least number (one) of relevant directions. However, we need to fine tune the strength of the w(x) type disorder in order to reach this fixed point. For 2 ≤ d < 3 (ǫ ≥ 0), there is only the trivial fixed point (W * ,Ũ * ) = (0, 0). At this fixed point,Ũ is relevant andW is irrelevant so that the RG flow goes to the strong randomness regime where our RG scheme breaks down. Therefore, in both cases, there is no stable non-trivial random fixed point near the pure XY fixed point. This means that the SF-insulator transition should be discribed by possible strong random fixed points. The conclusion we can deduce from these results is that, near d = 2, the direct SF-MI transition is unlikely to happen in the wholew −ū plane even for the weak disorder. This result essentially support the original picture of Fisher et al. [1,2] (Fig.1 (a)) that the SF-insulator transition should alwalys occur from the Bose glass phase rather than from the Mott insulator even for the commensurate case.

IV. 1/N CORRECTION
In this section, we are going to investigate the effects of the 1/N correction to the RG equation. The 1/N correction due to the φ 4 interaction in the previous commensurate dirty boson model will be considered. The most important 1/N correction that can affect the RG equation enters in the coefficient of theW term in Eq.(44). This 1/N correction can be read off from the scaling dimensions of φ † φ at the pure fixed point [20]: where As a result, the critical dimension forW is changed from 2 to 2 − 2η. If the convention of ǫ = d − 2 was taken, the RG equation becomes Note that the density ρ has no anormalous dimension [16] and the RG equation forŨ is not affected at this order. Now there can be three possible RG flows that depend on the dimensionality of the system and N . For ǫ = d − 2 ≥ 0, there is only one fixed point which is the unstable trivial fixed point (W * ,Ũ * ) = (0, 0). The RG flow is given by Fig.7 (a) and it flows to the strong disorder regime. For ǫ < 0 and |ǫ| < 2η, there can be two fixed points which are (W * ,Ũ * ) = (0, 0), (0, 2π|ǫ|) with the dynamical exponents z = 1, 1 + |ǫ|. But all of them are unstable and the RG flow is given by Fig.8. On the other hand, for ǫ < 0 and |ǫ| > 2η, there is one more fixed point which is given by (W * ,Ũ * ) = ( π 128 (|ǫ| − 2η), 0) with z = 1 + 1 2 (|ǫ| − 2η). All of the three fixed points are still unstable and the RG flow goes to the strong disorder regime. The RG flow is again given by Fig.7 (b). Therefore, after the inclusion of the 1/N correction, there is no stable weak random fixed point. The direct SF-MI transition is unlikely in the wholew −ū plane.

V. SUMMARY AND CONCLUSION
We study a large N generalization of the commensurate dirty boson problem. The 1 N expansion allows us to treat interaction effects properly. On the other hand, the disorder is assumed to be weak and the pertubation in the strength of the randomness is performed.
In order to understand the behaviors of bosons in this model, we need two types of disorder, i.e., the random coefficient of the density term φ † ∂ 0 φ, u(x), and the random coeffient of the quadratic term φ † φ, w(x).
For a restricted model with u(x) = 0 which has an additional particle-hole symmetry, we introduce more general long range interactions V (q) = g 0 /q λ , (0 ≤ λ < 1/2) ( which is not a genuine long range interaction as mentioned in Sec. II ). The critical dimension for w(x) type disorder is found to be d c = 2 + 2λ and we performed ǫ = d − 2 − 2λ expansion.
It is found that λ = 0 case and λ = 0 case show different behaviors and they are not continuously related in the ǫ = d − d c expansion.
First, let us look at the case of λ = 0. For d > d c , the pure fixed point is unstable, but there is a stable fixed point which governs the transition. For d ≤ d c , the pure fixed point is stable and the disorder is irrelevant. Therefore, we expect the direct SF-MI transition.
For d ≤ d c , the critical point is the same as that for the pure system. For d > d c , SF-MI transition is described by a new non-trivial fixed point.
For the short range interaction (λ = 0), the disorder is relevant for d ≥ 2 and irrelevant for d < 2. There is only the unstable pure fixed point and the RG flow goes to the strong disorder regime for 2 ≤ d < 3. For d < 2, the pure fixed point becomes stable and there is also an unstable non-trivial fixed point. Therefore, we expect a direct SF-MI transition for d < 2 but none for 2 ≤ d < 3.
We consider the usual short range interaction for the general case of u(x) = 0 but u = 0 (the commensurate dirty boson problem). The critical dimension of both types of disorder is found to be d = 2 (at (1/N ) 0 order) and we performed ǫ = d − 2 expansion (d < 3). For d < 2, we have three fixed points and they have at least one relevant direction in theW −Ũ plane. In the case of 2 ≤ d < 3, the RG flow is governed by the w(x) type disorder and it flows to the strong disorder regime which cannot be reached by the pertubation in the strength of the randomness. Therefore, we expect that the direct SF-MI transition is unlikely to happen near two dimensions.
The effects of the 1/N correction are considered. For the commensurate dirty boson model, the 1/N correction due to the φ 4 interaction does not change the qualitative features of the problem. There is still no stable weak random fixed point for d ∼ 2 and the direct SF-MI transition is unlikely.