Theory of a many-boson system with deformed Heisenberg algebra

We propose to consider nonlinear fluctuations in the theory of liquid $^{4}$He deforming the commutation relations between the generalized coordinates and momenta. Generalized coordinates are coefficients of density fluctuations of Bose particles. The deformation parameter takes into account the effects of three- and four-particle correlations in the behavior of a system. This parameter is defined from the experimental values of the elementary excitation spectrum and the structure factor extrapolated to $T=0$ K. The numerical estimation of the ground state energy and the Bose condensate fraction is made. The elementary excitation spectrum and the potential of interaction between the helium atoms are recovered.


Introduction
The method of collective variables is an effective approach to the study of an N -particle Bose system. In this method, independent variables are the Fourier coefficients of the density fluctuations ρ k of the particle (k is the wave vector). This method was originally proposed by Bohm [1][2][3][4]. It was suggested that from the whole set of an infinite number of variables ρ k , only D N variables are taken, where D is the space dimensionality. To achieve this, the domain of the wave-vector absolute values was restricted by some k c .
An obvious imperfection of this approach is an ambiguous choice of variables. Bogoliubov and Zubarev [5] proposed an approach where the set of values k is not limited. However, the transition from Cartesian coordinates to collective variables ρ k can be provided by the weighting function. The hermitian Hamiltonian of the Bose liquid in the ρ k representation is written as a sum of Hamiltonians of an infinite number of non-interacting harmonic oscillators describing the oscillations of the Bose-liquid density plus a contribution from anharmonicities of these oscillations. Justifications of the feasibility of the method of collective variables are given in many articles using different approaches. Usually, the anharmonic contribution was considered using the perturbation operator [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. The respective results can be brought to the numerical calculations both for model systems and for strongly non-ideal systems like liquid 4 He.
A different approach for the study of the Bose system arose following the introduction of quantum spaces of minimal length. The simplest deformation is that of Kempf being quadratic in generalized momentum [21,22]. The deformed commutation relations can be examined in multidimensional case. Such kind of deformation was applied to a wide variety of quantum mechanical problems, among which we distinguish the eigenvalue problem for D-dimensional isotropic harmonic oscillator [23], threedimensional Dirac oscillator [24], (2+1)-dimensional Dirac equation in a constant magnetic field [25] that were solved exactly. A composite system problem in the deformed space was also considered [26,27]. Some aspects of field theories were investigated in the deformed space, for example: the electromagnetic field [29,30], a photoelectric phenomenon [31], radiation and absorption of photons for deformed field [32], the Casimir effect for the deformed field [33]. An urgent problem is to find a possible generalization of arbitrary one-dimensional Heisenberg algebra with minimal length (or/and momentum) for multidimensional case [28]. Kempf's deformation can be applied not only to the operators of positions and momenta but also to collective variables. The idea to use a deformation of Poisson brackets to explore a Bose system firstly appeared in the article [34]. In paper [35], the Bose particles were represented as a set of q-deformed harmonic oscillators. In paper [36], there was used a representation of deformed creation and annihilation operators with a generalized four-parameter q-algebra. We note that the deformed creation and annihilation operators associate with the non-linear f -oscillator operators. In paper [37], two-parameter deformed Bose gas model was proposed to find the correlation functions of the particle. In a series of papers, the thermodynamics of ideal Bose and Fermi systems [38][39][40] was studied. In paper [41], the phenomenon of the Bose-Einstein condensation of the relativistic ideal Bose gas with deformed commutation relations for positions and momenta operators is investigated.
In our paper, we associate the difficulties connected with anharmonic contribution with deformation of commutation relations between the positions and momenta. Thus, we replace the multimode Hamiltonian of the Bose liquid in ρ k representation by a sum of single-mode Hamiltonians. We assume that this model successfully describes the properties of the Bose system when we make an appropriate choice of the deformation parameter. In this approach, we treat the collective variables as generalized coordinates. Thus, we attempt to take into account the anharmonic contribution described in [42] with deformed Poisson brackets quadratic in generalized momenta. In this paper, we use the deformation function which is quadratic in generalized coordinates. The correctness and efficiency of the proposed approach is shown for the analysis of Bose-systems with developed anharmonisms of the density fluctuations. Such a treatment of anharmonisms lowers the ground state energy if the deformation parameter is negative. It is interesting to note that the energy levels are bounded for such values of the deformation parameter. We make numerical estimations of the analytical results.

Hamiltonian of Bose-liquid in the deformed space of collective variables
Let us consider a system of N spinless Bose particles of mass m and by the coordinates r 1 , . . . , r N which move in the D-dimension space of the volume V . The Hamiltonian of the system readŝ where the first term is the kinetic energy operator,p j = −iħ∇ j is the momentum operator of the j th particle; the second term is the potential energy consisting of a sum of the particle interaction potentials Φ(|r i − r j |). We introduce the collective coordinates representation which takes the form: The Hamiltonian of the Bose liquid (2.1) can be rewritten in the following form [5,6]:

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is the Fourier image of the interaction potential. The operator ∆Ĥ contains all anharmonic terms and in addition contains the term from the specific linear anharmonic by ρ k and quadratic by ∂/∂ρ k : (k 2 1 + · · · + k 2 n )ρ k 1 ... ρ k n . (2.5) The introduced in (2.2) variables ρ k are complex and can be represented as a linear combination of real variables: sin kr j .
Since they are the complex conjugate value ρ * where the generalized momentum operatorP k,µ is conjugate to the generalized coordinateQ k,µ . In the Explicit form of the generalized momentum operator is as follows: We note that the operatorP k,µ has the dimension of action ħ, andQ k,µ is dimensionless. The Hamiltonian of the system (2.3) contains harmonic terms. We suggest that the anharmonicity ∆Ĥ can be taken into account by deformation of the commutation relations between generalized coordinates and momenta:Q where the dimensionless deformation parameter β k depends on the absolute value of the wave-vector. The operatorsQ k,µ andP k ′ ,µ ′ with different index commute. The deformation parameter for 4 He can take negative values. We do not expect a full description of the properties of the Bose liquid Hamiltonian (2.6) with the condition (2.10) but assume that this model describes the behavior of the many-boson systems.

Energy levels and wave functions of a Bose liquid
We find solutions of the stationary Schrödinger equation from harmonic oscillators in the deformed space with Hamiltonian (2.6). We suppose that the deformation is positive. For this purpose, we use the

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Theory of many-boson system with deformed algebra The explicit form of the wave function of the first excited state is as follows: The energy spectrum is quadratic in quantum numbers n k,µ , similarly to the theory of anharmonic oscillator in the case it takes into account the terms proportional to ∼ ħ 2 [6,43]. Now we consider the case β k < 0. Then and we impose the requirement here, the brackets 〈(. . . )〉 denote an average: The uncertainty relation obtained for the deformed algebra (3.9) leads to the fact that minimal uncertainty for the momentum operator is equal to zero. The new canonically conjugated variables (3.1) and a structure of Hamiltonian (3.3) with the changes β k → |β k |, tan (q k,µ β k ) → tanh (q k,µ |β k |) will be the same. Using the factorization method [6] we can write the wave functions of the transformed Hamiltonian for (k, µ) modes: when n 1 : here, In case of deformed algebra (3.9), the quantum numbers are limited by the number n < ν and the energy levels are bounded from above. When ν n, the energy spectrum is continuous. The wave function is also normalized. We note that the integration variable q k,µ runs from ∞ up to −∞ ∞ −∞ ψ n (q k,µ )ψ n (q k,µ ) dq k,µ = 1. (3.14) For example, we get for n = 1: In the non-deformed case (β k = 0), we obtain the energy levels of the Bose liquid in the main approximation [6]. When in equations (3.6) and (3.12), for the wave functions, the quantity β k → 0 and ν → ∞, we obtain: Taking into account the asymptotic formula for the gamma function Γ(ν + a) ∼ 2πe −ν ν ν+a−1/2 , when ν → ∞, we obtain the wave functions of the harmonic oscillator: here, η = q k,µ α k , α k = m k ω k /ħ.

Ground-state energy
If the quantum numbers in equation (3.4) are equal to zero, n k,µ = 0, we obtain the ground-state energy E 0 which can be written after transformations in the form: (4.1) The first two terms recover the ground-state energy in the Bogoliubov approximation [44]. The last term in equation (4.1) allows us to obtain the equation: Having taken into account the obtained equation (4.2) we rewrite the ground-state energy in the thermodynamic limit: The found expressions can be used for the models with exact or perturbative solutions. To verify the validity of the expression (4.3), we might compare it with the results given by an exactly solvable model or a model which allows perturbative consideration. Such a comparison brings about an interesting question concerning the choice of deformation parameter β k .

Elementary excitation spectrum
Let us find an expression for an elementary excitation given by the wave vector q. Suppose that only the quantum number n q,c = 1 and the other quantum numbers n k,µ = 0 when k q, µ c.

Structure factor
The structure factor of the system can be defined as an average of the square of the density fluctuations: Now, we calculate the average (6.1) taking into consideration the fact that the collective variables are generalized ones and using the ground state wave functions (3.6). Since the collective variables are generalized coordinatesQ k,µ from equation (2.7), we find the average (6.1) on the wave functions of the ground-state (3.6): This integral is reduced to the beta function: The parameter ν should be taken from the relation 3.7 and after some transformations we obtain: When β k = 0, we recover the well-know result by Bogoliubov and Zubarev [5], The elementary excitation spectrum (5.1) reads:

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The structure factor can be written as follows: here, F is the free energy of the system, which is equal to the ground-state energy E 0 at T = 0 K. Having performed elementary calculations with equation (4.1) we obtain the result (6.3). Note that the structure factor S k is an analytic function of the deformation parameter β k . This fact can be seen when we calculated S k and make a parameter β k → −|β k | in the wave functions (3.12): This integral is reduced to the beta function: here, the parameter ν is taken from the equation (3.13). Having performed some simplifications we obtain the structure factor from the equation (6.3). We note that condition (3.10) leading to the restrictions on the deformation parameter can be represented in the following form:

Potential and kinetic energy
Having used the structure factor we rewrite the ground-state energy of the many-body system (4.1): here, The mean value for the potential energy can be calculated for the ground state using equations (4.2), (6.3).
After some transformations, we arrive at the relation for the mean value of the potential energy: Let us rewrite the average potential energy in terms of S k and speed of the first sound: (7.4) here,

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Theory of many-boson system with deformed algebra The mean value of the kinetic energy is the average value of the operator (2.6) in case ν k = 0: We calculate the first term in the square brackets on the ground state wave function from the equation (3.6): After simple transformations we get: here, the expression for ν is taken from equation (3.7) and the structure factor from equation (6.3).
The second term in equation (7.6) is the structure factor by definition. Finally, we can write: Taking into consideration equation (6.3) we obtain: This result for the kinetic energy can be obtained as a derivative from the ground state energy with respect to the mass: The sum of expressions (7.4) and (7.11) is the total energy (7.1). Now we find the potential of the interaction between the Bose particles: Φ(r ) = 1 (2π) 3 e ikr ν k dk, (7.14) here, ν k is the Fourier coefficient of the interaction potential between Bose particles represented as a function of the structure factor (6.3): Thus, the interaction potential between the helium atoms reads:

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here, the contribution is ∆Φ(r ) caused by the deformation of the commutation relations:

Momentum distribution
To find the average number of atoms N k with the momentum ħk, k 0 it is sufficient to calculate the variational derivative from the free energy of the system with respect to the free-particle spectrum ħ 2 k 2 /2m. We note that the free energy of the system coincides with the ground state energy when T = 0 K [equation (4.3) by n k,µ = 0]. After simple calculations we get: The expression for the kinetic energy can be represented as a function of average numbers of particles: The average number of atoms N k with the momentum ħk can be obtained from the expression for the mean value of the kinetic energy (8.3). To do so, we take into account the relation (7.10). After simple calculations, we obtain the expression for the average numbers of atoms N k which coincides with the expression (8.1). We estimate the relative number of atoms in the case of their momenta being equal to zero (Bose condensate):

Deformation parameter. Numerical calculations
For the numerical evaluation of the deformation parameter we proceed from the expression of the elementary excitation spectrum (6.5). The value of the structure factor and excitation spectrum are taken from experimental papers [45,46]: We have the values of the deformation parameter for a limited range of wave vectors because the elementary excitation spectrum of the Bose liquid has the ultimate point of completion, and the experimental data of the structure factor are given up to 7.3 Å −1 . However, as k → ∞ the elementary excitation spectrum should be equal to the free-particle spectrum E k → ħ 2 k 2 /2m and the structure factor S k → 1. In figure 1, the deformation parameter β k based on the experimental data for S k and E k is shown. We shall model the deformation parameter by the function where a free parameter will be taken from the extrapolated data for the structure factor of the Bose liquid at T = 0 K [47]:

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The sign of the deformation parameter determines the behavior of the elementary excitation spectrum we have received (6.5). Therefore, β k < 0 because the experimental data for the elementary excitation spectrum of the liquid helium (triangles in figure 3, right panel) is lower than that of the theoretically calculated Feynman's spectrum (circles in figure 3, right panel). The graph of the function (see equation 9.2) is shown in figure 2 (left panel). Note that this choice of the deformation parameter gives a correct behavior in the long-wavelength domain. Function (9.2) can be used to find the physical quantities in the limit of T → 0. The form of the curve for β k (see figure 2) should be such that the elementary excitation spectrum (solid line in  One can offer the model functions satisfying these considerations provided that contributions to the

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basic physical quantities of the system reproduce the results in the post-RPA approximation in the theory of liquid helium-4. With the deformation parameter from (9.2), numerical calculations of the obtained quantities per particle are made. In the thermodynamic limits (V → ∞, N → ∞, N /V = const), 3 dk, the contributions of the physical quantities connected with the deformed commutation relations are estimated. We do not expect a complete agreement between the properties of the Bose liquid within the proposed method and the perturbative results. However, we obtained the results that are consistent with those of the perturbation theory. The ground state energy with the deformation taken into consideration: The numerical value of the first term corresponds to the ground state energy in the Bogoliubov approximation, see [47] (with ρ = 0.0219 Å −3 and speed of the first sound c = 238.2 m/s): E B 0 /N = −5.31 K. Thus, the full energy per particle in the deformation case is E 0 /N = −7.2 K. The experimental result of the energy is E 0 /N = −7.13 K. Note that the linear corrections to the ground state energy over the deformation parameter give a leading contribution. This choice of the deformation parameter (9.2) offers an insignificant amendment to the value of the Bose condensate fraction obtained in the zeroth approximation: In our method, the anharmonic contributions from higher correlations in the Bose-liquid are taken into account within the single-mode approximation. The Bose condensate fraction taking into account the correction ∆N 0 /N in the deformed case [see equation (8.5)] is calculated in the approximation of the one sum over the wave vector. In this approximation, the value N 0 /N is negative.
Nonlinear terms related to the deformation are quadratic in the expression for the interaction potential. This contribution vanishes, whereas the linear correction from the deformation parameter to the elementary excitation spectrum gives a significant contribution at q ≃ 2 Å −1 . This correction does not

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contribute in the long-wavelength limit (figure 3). The behavior of the elementary excitation spectrum in the long-wavelength limit was solved in [48] of the method of two-time temperature Green's functions. At least a two-parametric deformation should be sought for to consider anharmonic terms ∆H in Hamiltonian (2.3) in a proper way. This issue will be a subject of our future studies.