The advent of computer era has opened the possibility to perform large scale

Transcription

1 Chapter 2 Density Matrix Renormalization Group 2.1 Introduction The advent of computer era has opened the possibility to perform large scale numerical simulations of the quantum many-body systems and thus revealing completely new perspectives in the field of condensed matter theory [100]. Indeed, together with the analytic approaches, numerical techniques provide lot of information and details, otherwise inaccessible. However, the simulation of a quantum mechanical system is a very difficult task. It is because of the number of parameters required to represent the quantum state is large. This value usually grow exponentially with the number of constituents of the system due to the corresponding exponential growth of its Hilbert space. This exponential scaling drastically reduces the possibility of a direct simulation of many-body quantum systems. In order to overcome this 12

2 13 limitation many numerical tools have been developed, which include Monte Carlo techniques [101], efficient Hamiltonian digitalization methods such as Lanczos [102] and Davidson [103] procedures and real space renormalization group method [104]. Each of these methods has its own limitations, for example sign problem for fermion models in Monte Carlo, small system size in exact digitalization method, large error in real space renormalization group methods. The Density Matrix Renormalization Group (DMRG) was developed by S. R. White in 1992 [68, 69, 105] to remove some of these limitations and since then it has been proved to be a very powerful method for low dimensional interacting systems. Its remarkable accuracy can be seen in the spin-1 Heisenberg chain; where for a system of hundreds of sites, a precision of for the ground state energy has been achieved [106]. Since then it has been applied to a great variety of systems and problems including spin chains and ladders [107], fermionic [108, 109] and bosonic [34, 75, 110, 111] systems, disordered models [32, 67], impurities [112, 113], molecules [114], to list a few. The details of these can be seen in the recent review by Schollwock [70] and by Hallberg [71]. The global idea of DMRG may be stated in a single sentence: When splitting a system into blocks, these should not be isolated [115]. White [69] realized that, in a real space renormalization group calculation, the lowest energy states of the isolated blocks need not be the best building

3 14 bricks to construct the big block. It is necessary for the block to be related to its environment so as it may choose the best states. Density Matrix Renormalization Group method provides the best method of choosing these best states when going from smaller block to bigger block. This method has since developed into one of the most accurate methods for obtaining not only the ground-state properties and but also the low-lying excited states of onedimensional (d = 1) quantum systems [70, 116, 117]. Extensions to higher dimensions [118] and finite temperature are active areas of research [119]. DMRG algorithms can be classified into two categories viz. Infinite system DMRG and Finite Size DMRG (FS-DMRG). Infinite DMRG can be applied for the quantum system when there is no disorder in the system for which FS-DMRG algorithm has been proved to be better. For the sake of completeness we describe these methods below. The following part of this chapter is organized in the following manner. In the Section (2.2) we briefly discuss the real space renormalization group method and its limitations. Section (2.3) discusses about methods of overcoming these limitations. The details of the Infinite Size Density Matrix Renormalization Method and the Finite Size Density Matrix method are given in Sections (2.4) and (2.5) respectively. The Section(2.6) we describe the physical quantities which are obtained from DMRG method described above and apply this method to pure Bose-Hubbard model.

4 2.2 Real-Space Renormalization Group Method 15 Historically Density Matrix Renormalization Group (DMRG) method has its origin in the analysis of the failure of the real-space renormalization group (RSRG) methods to yield quantitatively acceptable results for the low-energy properties of quantum many-body problems [69]. Since most of the DMRG algorithm can be formulated in the standard real space renormalization group language, we briefly discuss the same below. The tensorial structure of the Hilbert space of a composite system leads to an exponential growth of its dimension and so is the resources needed for the simulation with the number of the system constituents. However, if one is interested in the ground state properties of a one-dimensional system, the number of parameters is limited for noncritical systems or grows polynomially for a critical one. This implies that it is possible to rewrite the state of the system in a more efficient way, i.e., it can be described by using a number of coefficients which is much smaller than the dimension of Hilbert space. Equivalently, a strategy to simulate ground state properties of a system is to consider only a relevant subset of states of the full Hilbert space. This idea is reminiscent of the renormalization group (RG) introduced by Wilson [104]. In the real space renormalization group procedure one typically begins

5 with a small part of a quantum system, called a block, (represented by B, of size L, living on an M - dimensional Hilbert space), and a Hamiltonian which describes the interaction between two identical blocks. Then one projects the composite 2-block system (of size 2L) representation (whose dimension M2) onto the subspace spanned by the M lowest energy eigenstates, thus obtaining a new M-dimensional basis. This procedure is then iteratively repeated, until the desired system size is reached. Below we list the iterative steps of RSRG method Standard Real-Space Renormalization Group algorithm for a 1D quantum system 16 The iterative procedure of the standard real-space renormalization group algorithm contains the following steps: Step 1. Consider a block B of length L (usually L = 2 for the first iteration). Let the basis of this block be represented by WO with p 1,2,, M, where M is the dimension of the Hilbert space of the block B. For example, by taking L = 2, M = 4 and 16, respectively, for spin 1/2 and Hubbard Hamiltonian. We denote B I with represents a single site.

6 17 Step 2. Consider two such blocks B+B' as shown below and its Hamiltonian is given by HBB' = HB HB/ + Interaction between B and B'. (2.1) The HBB' matrix has a dimension of M 2 x M2. BB' Step 3. Diagonalize HBB' and obtain the M eigenvectors of the lowest eigenvalues. Let us represent them by la > with a = 1,2,3,, M 2. i.e. la) = E (2.2) /fir Step 4. Change basis of block B + B' from Iµ/, pr) to la), keeping only the lowest M states, and corresponding Hamiltonian become Hgnew =OHBB/0 1- Step 5. Replace B with Bnew. Thus after one iteration I,u/) is replaced by la) and truncated the Hilbert's space from M 2 to M. Thus BB' B Step 6. Go to Step 1 and continue with the iteration. Renormalization group was successfully applied for the Kondo prob- lem [104], but fails in the description of the strongly interacting systems.

7 18 This failure is due to the procedure followed to increase the system size and to the criterion used to select the representative states of the renormalized block [69, 104]: indeed decimation procedure of the Hilbert space is based on the assumption that the ground state of the entire system will essentially be composed of energetically low-lying states living on smaller subsystems (the forming blocks) which is not always true. A simple counter-example is given by a free particle in a box: the ground state of the system with length 2L has no nodes, whereas any combination of states of two boxes with length L will have node in the middle, thus resulting in higher energy. A convenient strategy to solve this RG breakdown is the following: before choosing the states to be retained for a finite-size block, it is first embedded in some environment that mimics the thermodynamic limit of the system. This is the new key ingredient of the DMRG algorithm. As discussed below, system growth is then slowed down with the number of the algorithm's iterations: from the exponentially fast growth Wilson's procedure to the DMRG linear growth [69, 105]. 2.3 Density-Matrix Approach The Density Matrix Renormalization Group algorithm receives its name from one of the tools, i.e., density matrix, incorporated into it. It is arguable whether the density matrix is the key ingredient of the method, but history

8 19 has made to be considered to be so. The reason for which the density matrix appears-is.the necessity of "fitting": it is necessary to find out which are the block states which reproduce most accurately a chosen global state, which shall be termed the target state. According to some authors, density matrices are the most fundamental way to describe quantum mechanical systems. According to Feynman [121] density matrices are necessary because systems must always be separated from their environment in order to be analyzed. This comment was the one which made White [69, 105] to use them as the foundation of his RG formulation. The superblock method forms the basis of density matrix approach suggested by White and Noack [69]. In the superblock method, one diagonalizes a larger system composed of three or more blocks which includes the two blocks, say BB' which are used to form B,,. The wave function for the superblock are projected onto blocks BB', and these projected states of BB' are kept for the next iteration. For a single particle wave function, this projection is single valued and trivial. The superblock method works quite well in a single particle model [69], with the accuracy increasing rapidly with the number of extra blocks used. However for a many-particle wave function, the "projection" of a wave function onto BB' is many valued, and, in fact, a single many-particle state for the entire lattice generally "projects" onto a complete set of block states. However some of these states are most

9 20 important than others; the density matrix tells us which states are the most important. To understand why density matrix is used to choose the states which are to be kept, consider the following argument by analogy. For an isolated block at finite temperature, the probability that the block is in eigenstates a of the block Hamiltonian is 'proportional to its Boltzmann weight exp(- 13Ea ), 13 = 1/k BT, kb being the Boltzmann's constant. The Boltzmann weight is an eigenvalue of the density matrix exp(--ohb ), and an eigenstates of the Hamiltonian is also an eigenstates of the density matrix. Since lowest energy corresponds to highest probability in the Boltzmann's weight, we can view the standard RG approach as choosing the M most probable eigenstates to represent the block given the assumption that the block is isolated. However in reality the block is not isolated, the density matrix is not exp( OHB), and eigenstates of the block Hamiltonian are not eigenstates of the block's density matrix. For a system which is strongly coupled to the outside universe, it is much more appropriate to use the eigenstates of the density matrix to describe the system rather than the eigenstates of the system's Hamiltonian. Thus a natural generalization of the standard approach is to choose to keep the M most probable eigenstates of the block density matrix. In particular, it can be shown that keeping the most probable eigenstates of the density matrix gives the most accurate representation of the state of

10 21 the system as a whole, i.e. the block plus rest of the lattice [68). Assume that a superblock is diagonalized and one particular state, known as target state, kb) is obtained. Let li), with i = 1, 2,, I be a complete set of states of BB' (the system) and 1j), j = 1, 2,, J be the states of the rest of the superblock, i.e.,the "universe". So kb) = E (2.3) For simplicity assume that oij is real. To define a procedure for producing a set of states of the system lua), a = 1,..., m, with lua) = Ei u'li), which are optimal for representing exactly if I > m. An accurate expansion for > can be constructed in the following form, >= > 1./ > (2.4) a,j In other words one has to minimize the following term, s > - 17-T (2.5) by varying over all ct,,,,j and lua >, subject to < ualua' >= Without loss of generality one can write, >= E aale > Iv- > (2.6) a where v7 =< jlva >= Naac,j, with N chosen to set Ei ItT1 2 = 1. In matrix

11 22 notation one can write, s = (02, - E aau7v7) a=1 2 (2.7) and S to be minimized over all eta, va, and a a, given the specified value of m. The solution to this minimization problem is known from linear algebra [68]. By considering as a rectangular matrix, the solution is produced by the singular value decomposition [68, 120] of 0, = UDV T (2.8) where U and D are I x I matrices, V is an I x J matrix (where j = 1, J, and J > 1), U is orthogonal, V is column orthogonal and the diagonal matrix D contains the singular values of 0. ua, va and aa which minimize S are given as follows; the m largest-magnitude diagonal elements of D are the aa, and the corresponding columns of U and V are the u and v. These optimal states ua are also eigenvectors of the reduced density matrix of the system as part of the universe. This reduced density matrix for the system depends on the state of the universe, which in this case is a pure state 10 >. The density matrix for system is given by, pii = EY i lkiej (2.9) These gives, p = UD2UT (2.10)

12 23 i.e. U diagonalizes p. The eigenvalues of p are wa = 4, and the optimal states u are the eigenstates of p with the largest eigenvalues. Each w a represents the probability of the system being in the state ua, with Ea wa = 1. The deviation of P, = Erna_ l wa from unity measures the accuracy of the truncation to m states Density-Matrix Algorithms A density-matrix algorithm is defined mainly by the form of the superblock and the manner in which the blocks are enlarged (such as by doubling the block, B' = BB, or by adding a single site, B' = B + *), and by the choice of superblock eigenstates i.e. target states, used in constructing the density matrix. An eigenstate of the superblock Hamiltonian used for the determination of density matrix is called the target state. The most efficient algorithms use only a single target state in constructing the density matrix. By targeting only one state, the block states are more specialized for representing that state. The important characteristic of a density matrix algorithm is that the accuracy increases with the number of states kept in the density matrix m. The coefficient governing the increase of accuracy with m is largest with a single target state used for the construction of the density matrix. Superblock can be constructed in different ways. Generally it is enlarged by adding two sites, rather than by adding a block as done in standard R,SRG

13 approach (See Section (2.2)). We construct superblock for a system of L sites with open boundary as shown below; r L Here Be represents left block composed of a - 1 sites. Similarly L represents right block composed of 4 1 sites. The represents a single site. In every iteration of the DMRG algorithm, the left (right) block is up- dated with adding a new site to it, thus increasing its size by one. i.e. BQ and L L _ 1 Br L The standard DMRG algorithm can be used in two different ways viz infinite size method and finite size method, which are briefly discussed below. 2.4 Infinite size density-matrix renormalization group method In the first step of the infinite size density-matrix renormalization group method, one starts with a chain of four sites. The superblock configuration will then be Bi. Here 141. and.13;. both represent a single site. We construct the Hamiltonian for this superblock and diagonalize it to obtain the energy and the wave function of the target state. The Davidson algorithm [103] is used for the diagonalization of the hamiltonian matrix

14 which is highly sparse. Using this target state, the density matrix for the new left (right) block is obtained. The new left and right block is, respectively 25.B1 and 41: The density matrix for the left (right) block is then diagonalized to obtain its eigenvalues and the eigenstates. The new left (right) block is represented by the eigenstates corresponds to largest m eigenvalues of the density matrix of the left (right) block. This is the main idea behind the density matrix renormalization group method. The effective hamiltonian is formed for the new left and right blocks, which has now two sites each. With this new left and right blocks, the new superblock, with length L = 6 is B2 13?. This iteration is continued till desired length is achieved. At each iteration step, both blocks size is increased by one site and the total length of the chain is increased by 2 sites. The infinite size method is used to find ground state properties of infinite chain of pure system. Each step of the iteration pushes the ends of the chain farther from the two sites at the center. After many steps, each block approximately represents one-half of an infinite chain.

15 2.4.1 Graphical presentation of Infinite Size DMRG Method 26 I I rol L= 4 L= 6. I L = 8 (continued till the required length is reached) Algorithm For Infinite size DMRG method 1. represent a site. The basis of single site is represented by la) with a = 1, 2,, n where n is the number of states per site. 2. BL 1-1 and B, represents left and right block respectively. The basis of the left and the right blocks are represented by IN) and Iµ,.) 7.) respec- tively. Here {pi, p,r} = 1, 2,, m where m is the number of states in the left and right blocks. In the first iteration, L = 4 and the left and right blocks have one site each. i.e. m = n in the first iteration. 1, Br 1 B, So the 3. Consider the superblock configuration L _i.b.1 L _ BT 1. The

16 27 basis states for this superbiock is written as jilt at O r >, where at and a, represent the basis for the single sites close to left and right block respectively. The dimension of the Hilbert space of this superblock is m 2 x n2. However, considering fixed density of bosons in the lattice, this dimension can be further reduced. Only states with fixed density of bosons needed to be considered because the number operator commutes with the Hamiltonian. 4. The Hamiltonian of the superbiock H = HLB H LS HRS + HRB HLB,LS HLS,RS HRS,RB Here first four terms represent Hamiltonian of left block(lb), left single site(ls), right single site(rs) and right block(rb) respectively. The last three terms represent the interaction between LB-LS, LS-RS and RS-RB respectively. 5. The Hamiltonian matrix Hij is highly sparse. Diagonalize this Hamiltonian to obtain the energy EL(N) and wavefunction RoLN) of the target state. Note that L is the length of the system and N is the number of particle in this system. Thus WOLN) = Ecili >= E al ar (2.11) ar

17 6. Construct the Density Matrix of new left block which is represented by 28 B 2 whose states can be represented by lkitat) The density matrix for the left block is calculated using Eq. (2.11). ni c*c la la r 11 r. Crr (2.12) Similarly the density matrix for the new right block, represented by Br* -1, is given by P(artir)1(Trvr) Cillatarttr CiltatTrur Mat ( 2. 13) 7. Diagonalize p( No., ),(,0.1) to obtain its eigenvalues co c, 1 k and eigenstates laik ) 1,4 E oatati,a/0-0, (2.14) Mat where k = 1, 2,, m x n. Note that the dimension of the new left block is m x n. Similarly we diagonalize the density matrix for the right block p'(.0., A, ),(7.,,,,, ) to obtain its eigenvalues war k and eigenstates 1c4) lark E > (2.15) anur 8. The eigenvalues of density matrix obey the condition 7 a t(r) Wat(r) = 1 and (.4.) l(r) a > 0. k 9. The new left block is represented by the eigenstates of the density matrix lalk ) with k = 1, 2, m corresponding to m largest c.o c, / k. In other

18 29 words we truncate the new left block with states m x n to m states. This truncation procedure is the most important part of the DMRG algorithm. Similarly we represent the new right block by 14) with k = 1, 2, m corresponding to m largest wok. 10. Go to step 3. and continue with the iteration. 2.5 Finite-Size Density Matrix Renormalization Group method The Infinite Size Density-Matrix Renormalization Group (DMRG) method has proven to be very useful in the studies of pure one-dimensional quantum systems [70, 71]. Open boundary conditions are normally preferred for such calculations since the loss of accuracy with increasing system size is much less than system with periodic boundary conditions. However, this approach has many serious draw back applying it to disorder system. In the infinitesystem DMRG method outlined above the left and right-block bases are not optimized in the following sense: The DMRG estimate for the target-state energy, at the step when the length of the system is L, is not as close to the exact value of the target-state energy for this system size as it can be. It has been found that the FS-DMRG method overcomes this problem [69, 68, 75]. In addition if one is interested in the scaling behavior of various quantities, it is necessary to use Finite-Size Density-Matrix Renormalization Group(FS-

19 DMRG) as discussed below [75]. The conventional FS-DMRG method consists of the following two steps: The infinite-system density-matrix renormalization group method (DMRG), where we start with a system with four sites, add two sites at each step of the iteration, and continue till we obtain a system with the desired number L of sites as discussed in the Sec. (2.4). 2. The finite-system method in which the system size L is held fixed, but the energy of a target state is improved iteratively by a sweeping procedure, described below, till convergence is obtained. First we use the infinite-system DMRG iterations to build up the system to a certain desired size, say L. The L-site superblock configuration is now given by 13, -1 B7* -1. In the next step of the FS-DMRG method, the superblock L configuration E17.2, which clearly keeps the system size fixed at L, is used. This step is called sweeping in the right direction since it increases (decreases) the size of the left (right) block by one site. For this superblock the system is L 1, the universe is B 2, the associated density matrix can be found, and from its most significant states the new effective Hamiltonian for the left block, with (2 + 1) sites, is obtained. We sweep again, in this way, to obtain a left block with (1 + 2) sites and so on till the left block has (L 3) sites and the right block has 1 site so that, along

20 31 with the two sites in between these blocks, the system still has size L; or, if a pre-assigned convergence criterion for the target-state energy is satisfied, this sweeping can be terminated earlier. Note that, in these sweeping steps, for the right block we need to BL-3, which we have already obtained in earlier steps of the infinite-system DMRG. Next we sweep leftward: the size of the left (right) block decreases (increases) by one site at each step. Furthermore, in each of the right- and left- sweeping steps, the energy of the target state decreases systematically till it converges (we use a six-figure convergence criterion in our calculations) Graphical presentation of Finite Size DMRG Method Let us assume we want system size L = 12. First using infinite size DMRG, build a system of L = 12 as represented below. Infinite size DMRG I a I I I

21 After reaching the desired system size (in this case L = 12), we begin finite size sweeping as given below. 32 I I I Finite size sweeping I I I I I I I I I I I 2.6 Method of calculation In the last two sections we had described the algorithm of both infinite size DMRG and FS-DMRG methods. This session we apply this method to the pure Bose-Hubbard model given by

22 33 = t E (alai + Eni (ni <i, j> i 1). (2.16) The number of possible states per site n in the case of Bose- Hubbard model is infinite since there can be any number of bosons on given a site. In a practical DMRG calculation, however, we must restrict the number n = nmax. The smaller the interaction parameters U in the model (2.16), the larger must be nmax. We have taken nmax = 4 and is sufficient for the values of U considered here; we have checked in representative cases that our results do not change significantly if nmax = 5. Using the FS-DMRG method described in the previous sections (2.4) and (2.5) the ground state energy EL(N) and its ground state wave functions i'/poln) of model (2.16) on a chain of length L having N bosons are obtained. From them we calculate a number physically relevant quantities as given below. These quantities are used to distinguish different phases possible in the model (2.16). The single particle energy gap GL for a system of size L is defined as = (2.17) where At (pi) is the energy required to add (remove) one boson for a system of length L; i.e., = EL(N + 1) EL (N) (2.18)

23 = EL(N) E L (N 1). (2.19) 34 The single particle gap of model (2.16) for two values of on-site interactions, U = 2 and 8 and for density p = NIL = 1 is given in Fig It is obvious from this figure that the single particle energy gap vanishes in the thermodynamic limit (L > oo) for U = 2 and remain finite for U = 8. This implies a phase transition from gapless superfluid to finite gaped Mott insulator phase as on-site interaction is increased /L 0.25 Figure 2.1: Single particle energy gap CI, as a function of 1/L of model (2.16) for U = 2 and 8 with density p = 1. The critical value of on-site interaction UG, for the SF to. MI transition is best obtained from finite size analysis of single particle energy gap [75] where plots of LG L versus U for different values of length L will coalesce in

24 the superfluid region of the phase diagram. Such a figure is given in Fig. 2.2 and the critical on-site interaction UG, fz L= fr o- ********* I 5 I I 8 I U Figure 2.2: Finite size scaling of single particle energy gap LGL versus U for different lengths showing SF to MI quantum phase transition. The plots of LG L coalesce below U. < Uc 3.4. The correlation function that characterizes the SF phase is rs,f(r) (PoLNI atoarlooln), (2.20) where I b, al,n) is the ground-state wavefunction of the system with size L and N bosons. The associated correlation length can be obtained from the second moment of this correlation function, namely, (sf = Er r2r1f(r)] 1/2 (r) (2.21)

25 36 Note that (f,f is the correlation length for SF ordering in a system of size L; it remains finite so long as L < oo. In the superfluid phase the correlation function II,F(r) shows power law decay when plotted against r which is as follows, risf 1 r -K/2 (2.22) where K is Luttinger Liquid parameter [62, 63]. For onsite interaction U = 2 the power law decay of the SF correlation is given in Fig Plots similar to this is used to obtain the Luttinger Liquid parameter K and the variation of K as a function of onsite interaction is given in Fig The Bosonization study of model (2.16) [62, 63] predict the SF to MI transition at K = 1/2 which has been verified in Fig Another way of showing the SF-MI transition in the model (2.16) is from the study of finite size scaling of correlation length defined by Eq. (2.21). The SF correlation function decay exponentially in the Mott insulator phase due to the existence of finite gap in its energy spectrum and the correlation length 1/GL. Thus finite size scaling of correlation length behaves similar to that of gap and such plots are given in Fig. 2.5 showing the coalescence of L/(15! for different lengths L below U = Uc =, 3.4 which is consistent with finite size scaling of LG L as given in Fig Other physical quantities which we calculate include the expectation value

26 r Figure 2.3: Variation of SF correlation function rsf(r) with respect to r for U = 2. The continuous line is the fit r s-e(r) ti r-1 /2 which gives the LL parameter K = 0.33 ± N E. aims N U Figure 2.4: Variation of Luttinger Liquid parameter K with respect to the onsite interaction U for model (2.16).

27 L= U Figure 2.5: Finite size scaling of correlation length LAE versus U for different lengths showing SF to MI quantum phase transition. The plots of Lgi, coalesce below U < Uc ti 3.4. of number operator (n i) which measures the local density and is defined as, (ni) = (,bolninilooln) (2.23) Using this expectation value, we calculate CDW order parameter OMDW = - E(-1)z[(n%) - (2.24) This order parameter is finite in density wave phase but zero in superfluid and Mott insulator phases. We can also distinguish superfluid phase from Mott insulator or charge density wave phases using compressibility defined as change in the density

28 39 with respect to change in the chemical potential, i.e., 8n (2.25) where the chemical potential is define as the change in the ground state energy with respect to change in the total number of bosons, the is obtained from the relation SE 5N (2.26) U= Pa OOOOOOOOOOOOOOOOOOO OOOOOO OOOOOOOOO Figure 2.6: Boson density p and compressibility is; as a function of chemical potential,u for U = 2. In Fig. 2.6 density p and compressibility lc are plotted as function of chemical potential for onsite interaction U = 2. Finite value of compressibility for all values of 1.1, for this value of onsite interaction suggest that the

29 U= , Figure 2.7: Boson density p and compressibility is as a function of chemical potentialµ for U = 8. system is always in the superfluid phase. However, as we increase the onsite interaction U > Uc, Mott insulator phase exist for some values of chemical potential where the compressibility vanishes. For example in Fig. 2.7 we have plotted both p and n for various values of chemical potential,u. It is obvious that for 1.45 < µ < 4.23, the compressibility is zero and corresponding value of density p = 1. This region has density pinned to one. The value ofµ at upper and lower knees, respectively, are equal to A+ and bi-, the energy require for adding and removing one particle from the ground state (see Eq From the calculation of p+ and tr we obtain the phase diagram of pure Bose-Hubbard model in the µ U plane and is given in Fig. 2.8(a).

30 41 The Mott insulator phase has finite gap and is given in Fig. 2.8(b). 7 6: 5: 3: 2: 1: 0 (a) 4 8 SF MI U 10 Figure 2.8: (a) Phase diagram of Bose-Hubbard model (2.16) for p = 1 and (b) the variation of gap G = 11+,u in the Mott insulator phase.