Two exactly soluble quantum models of planar gas

Abhijeet Joshi¹, Vivek M. Vyas¹ and Prasanta K. Panigrahi^2,3

¹ Department of Applied Physics, Indian Institute of Information Technology Vadodara, Sector 28, Gandhinagar 382028, India
² Centre for Quantum Science and Technology, Siksha ‘O’ Anusandhan University, Bhubaneswar, Odisha 751030, India
³ Department of Physical Sciences, Indian Institute of Science Education & Research (IISER) Kolkata, Mohanpur, Nadia 741 246, India

Abstract. Here we study two quantum models depicting gas of point particles defined in two spatial dimensions. The first model consists of a gas formed by point particles interacting pairwise via repulsive delta function potential, which physically captures the zero range contact interactions between the particles. While the second model consists of an additional harmonic trapping potential apart from the pairwise contact interactions. We show that both of these models are exactly soluble, in the sense that, there exists an exact invertible transformation mapping them to a collection of free particles or decoupled oscillators. It is found that the occurrence of $\mathfrak{sl}(2,\mathbb{R})$ algebra plays a critical role in the construction of these maps.

1. Introduction

In the study of interacting many body quantum systems [1–3], the interacting point particle system with contact interactions have a special place [4–6]. This is due to the fact that such a system closely resembles interesting physical scenario of an imperfect gas, which manifest in many diverse contexts like behaviour of electrons in condensed matter systems and ultracold atomic gases [7–9]. It is physically sound to consider the contact interactions between the particles of zero range, mathematically represented by delta function interparticle potential, when one is interested in the macroscopic dynamics of the system. This is particularly relevant when the interparticle interaction range is several orders of magnitude smaller than the typical dimensions wherein the system exists.

In the 1960s it was found that when such a quantum gas system is defined in one spatial dimension, it is integrable and yields an exact solution [10, 11]. Several interesting mathematical developments eventually followed from the study of this system, like Yang-Baxter equation [12], quantum groups [13], Bethe ansatz [6] and so on [3, 5, 14]. However not much has been known about this system in higher dimensions, in particular, about the aspects of integrability and solubility.

In the present work, we consider two quantum models of nonrelativistic point particles in two spatial dimensions, interacting pairwise via a repulsive delta function potential, without and with an external harmonic trapping potential. Such models are of physical significance since they capture the behaviour of two dimensional cold atomic systems, and electron/hole gases in planar condensed matter systems [7, 8, 15, 16]. The model with a harmonic trap was studied by Pitaevskii and Rosch to find the breathing mode, which was later on experimentally confirmed in cold atomic systems [9]. Here we find that both these models are exactly soluble in the sense that there exists an exact invertible map from these models to a collection of free noninteracting particles or to decoupled isotropic oscillators. Thus the spectrum and eigenstates of both these models are exactly determined. Furthermore these models are also found to be Liouville integrable, which means that they admit enough independently conserved quantities as many degrees of freedom there are in the system. The occurrence of $\mathfrak{sl}(2,\mathbb{R})$ in various forms in both these models is found to play a critical role for their solubility.

2. Two Dimensional Quantum Gas

We are interested in the quantum dynamics of $N$ two dimensional nonrelativistic point particles in two spatial dimensions. Working in Heisenberg picture, we promote the classical canonical variables $(\vec{r}_j,\vec{p}_j)$ to self adjoint Hilbert space operators obeying the Dirac commutation relations [17, 18]:

[x_j, p_{xk}] = i\hbar\delta_{jk}, [y_j, p_{yk}] = i\hbar\delta_{jk}

We consider that the particles have mass $m$ and are interacting with one another with pair-wise repulsive contact interaction $V_{int}$ . The Hamiltonian depicting such a system reads:

H = K + V = \sum_{j=1}^{N} \frac{\vec{p}_j\cdot\vec{p}_j}{2m} + \sum_{\langle ij\rangle} V_{int}(\vec{r}_i,\vec{r}_j) (1)

Here the potential energy term $V$ arises due to summation over all the pairs of particles. The repulsive contact interaction potential, which is a function of distance between any two particles, explicitly reads:

V_int(r_i, r_j) = V_int(|r_i - r_j|) = g δ(x_i - x_j) δ(y_i - y_j) (2)

where $g$ is the positive real coupling constant. In the next section we shall also work with a harmonic potential term given by:

U = \sum_{j=1}^{N} \frac{m\Omega^2}{2}\vec{r}_j\cdot\vec{r}_j (3)

In this system one can define the generator of the scale transformations or dilations $D$ as [19]:

D = \frac{1}{2}\sum_{j=1}^{N}(\vec{r}_j\cdot\vec{p}_j + \vec{p}_j\cdot\vec{r}_j) (4)

It is very interesting to note that:

\frac{d}{dt}U = \Omega^2 D = \frac{1}{i\hbar}[U, H]

From the canonical commutation relations one immediately finds these important commutators:

[D, K] = 2i\hbar K, [D, V] = 2i\hbar V, [D, U] = -2i\hbar U, [U, K] = i\hbar\Omega^2 D (5)

Using these commutation relations, one can define the raising/lowering operators $T_{\pm}$ and $T_3$ operator as: $T_{+} = \frac{1}{\hbar\Omega}K$ , $T_{-} = \frac{1}{\hbar\Omega}U$ and $T_3 = \frac{D}{2\hbar}$ , which form the $\mathfrak{sl}(2,\mathbb{R}) \equiv \mathfrak{su}(1,1) \equiv \mathfrak{so}(2,1)$ algebra [20, 21]:

[T_3, T_\pm] = \pm i T_\pm, [T_+, T_-] = -2i T_3 (6)

Importantly one also has yet another $\mathfrak{sl}(2,\mathbb{R})$ algebra in the system, which is defined by the generators: $T'_{+} = \frac{1}{\hbar\Omega}(K+V)$ , $T'_{-} \equiv T_{-} = \frac{1}{\hbar\Omega}U$ and $T'_3 \equiv T_3 = \frac{D}{2\hbar}$ such that:

[T'_3, T'_\pm] = \pm i T'_\pm, [T'_+, T'_-] = -2i T'_3 (7)

Employing the Baker-Campbell-Hausdorff formula [20, 21] leads us to an interesting identity:

e^{i\frac{\pi}{2}(T_+ + T_-)} T_+ e^{-i\frac{\pi}{2}(T_+ + T_-)} = T_-

Using this identity for both the algebras, we see that $T'_{+}$ can be transformed to $T_{+}$ as:

e^{-i\frac{\pi}{2}(T_+ + T_-)} e^{i\frac{\pi}{2}(T'_+ + T'_-)} T'_+ e^{-i\frac{\pi}{2}(T'_+ + T'_-)} e^{i\frac{\pi}{2}(T_+ + T_-)} = T_+ (8)

Recollecting the definitions of the generators, one learns that there essentially exists a unitary map from the Hamiltonian $H = K + V$ describing the gas (see Eq. (1)) to Hamiltonian $K$ describing free noninteracting particles as:

\mathcal{U}^\dagger (K+V) \mathcal{U} = K (9)

where the unitary operator $\mathcal{U}$ is:

\mathcal{U} = e^{-i\frac{\pi}{2}(T'_+ + T'_-)} e^{i\frac{\pi}{2}(T_+ + T_-)} (10)

Thus the system of two dimensional gas with repulsive delta function interaction is unitarily equivalent to the system of free particles devoid of any interparticle interactions.

Before we proceed further let us note some key points:

The spectrum of the interacting gas is same as that of collection of free particles
The excited states, whose energy is identical to free particles, represent the quasiparticle modes of the interacting gas
The excited states of the interacting system can be explicitly constructed from free particle states by the action of the unitary operator $\mathcal{U}$
These results hold for both fermions and bosons since the above derivation makes no reference to the particle statistics

In fact the interacting quantum system defined by $H$ is also Liouville integrable, in the sense that it admits $2N$ conserved quantities in involution with each other. This can be easily seen by noting that the single particle kinetic terms $K_j = \frac{\vec{p}_j\cdot\vec{p}_j}{2m}$ and angular momenta $L^z_j = x_j p_{yj} - y_j p_{xj}$ commute with the free system Hamiltonian $K$ : $[K, K_j] = 0$ , and $[K, L^z_j] = 0$ . From here using Eq. (9) it follows that:

[H, \mathcal{U}K_j\mathcal{U}^\dagger] = 0 and [H, \mathcal{U}L^z_j\mathcal{U}^\dagger] = 0 (11)

And since $[K_j, L^z_k] = 0$ , one has $2N$ conserved quantities $\mathcal{U}K_j\mathcal{U}^{\dagger}$ and $\mathcal{U}L^z_j\mathcal{U}^{\dagger}$ which are also mutually commuting. In fact it is also possible to translate these commutators and the unitary map to classical setup suitably using the Dirac correspondence principle [22, 23], and establish the Liouville integrability of the corresponding classical model.

3. Two Dimensional Quantum Gas in a Trap

Let us now consider that we are given the quantum gas system, as discussed in the earlier section, but now subjected to the harmonic trapping potential (3). The Hamiltonian depicting such a system reads:

H_{osc} = K + V + U = \sum_j \frac{\vec{p}_j\cdot\vec{p}_j}{2m} + \sum_{\langle ij\rangle} V_{int}(\vec{r}_i,\vec{r}_j) + \sum_j \frac{m\Omega^2}{2}\vec{r}_j\cdot\vec{r}_j (12)

This model was studied by Pitaevskii and Rosch [9] to find the breathing modes. It was noted that the system admits an $\mathfrak{sl}(2,\mathbb{R}) \cong \mathfrak{so}(2,1)$ algebra structure defined by self adjoint Cartesian generators: $T'_1 = \frac{1}{\hbar\Omega}(K+V-U)$ , $T'_2 = \frac{1}{2\hbar}D$ and $T'_3 = \frac{1}{\hbar\Omega}(K+V+U)$ , obeying the commutators:

[T'_1, T'_2] = -i T'_3, [T'_2, T'_3] = i T'_1, [T'_3, T'_1] = i T'_2 (13)

We also find that there is yet another way of defining $\mathfrak{sl}(2,\mathbb{R})$ algebra in terms of self adjoint Cartesian generators: $T_1 = \frac{1}{\hbar\Omega}(K-U)$ , $T_2 \equiv T'_2 = \frac{D}{2\hbar}$ , and $T_3 = \frac{1}{\hbar\Omega}(K+U)$ obeying the commutation relations:

[T_1, T_2] = -i T_3, [T_2, T_3] = i T_1, [T_3, T_1] = i T_2 (14)

Employing the Baker-Campbell-Hausdorff formula, it is straightforward to see that:

e^{-\frac{\pi}{2}T_1} T_3 e^{\frac{\pi}{2}T_1} = i T_2 (15)

Using this identity for both the algebras we get a similarity map connecting the two generators $T_3$ and $T'_3$ :

e^{\frac{\pi}{2}T'_1} e^{-\frac{\pi}{2}T_1} T_3 e^{\frac{\pi}{2}T_1} e^{-\frac{\pi}{2}T'_1} = T'_3 (16)

Recollecting the definitions of these generators, we find that we have actually constructed a similarity map connecting the two quantum systems:

e^{\frac{\pi}{2}T'_1} e^{-\frac{\pi}{2}T_1} (K+U) e^{\frac{\pi}{2}T_1} e^{-\frac{\pi}{2}T'_1} = K + V + U (17)

or more explicitly:

\sum_j \frac{\vec{p}_j\cdot\vec{p}_j}{2m} + \sum_{\langle ij\rangle} V_{int} + \sum_j \frac{m\Omega^2}{2}\vec{r}_j\cdot\vec{r}_j = W^{-1} \sum_j (\frac{\vec{p}_j\cdot\vec{p}_j}{2m} + \frac{m\Omega^2}{2}\vec{r}_j\cdot\vec{r}_j) W (18)

where the invertible operator $W = e^{\frac{\pi}{2}T_1}\, e^{-\frac{\pi}{2}T'_1}$ .

This shows that the system of two dimensional gas with repulsive delta function interaction in a harmonic trap is equivalent to the system of decoupled two dimensional isotropic oscillators via similarity transformation $W$ .

This remarkable result leads to several interesting points:

The spectrum of the interacting gas defined by $H_{osc} = K + V + U$ is same as that of $K + U$ , that of decoupled oscillators
These results hold for both fermions and bosons
The action of $W^{-1}$ on decoupled oscillator states would yield the eigenstates of $H_{osc}$

One can also see that the interacting quantum system defined by $H_{osc}$ is also Liouville integrable. This can be seen by noting that the single oscillator Hamiltonians $H_j = \frac{\vec{p}_j\cdot\vec{p}_j}{2m} + \frac{m\Omega^2}{2}\vec{r}_j\cdot\vec{r}_j$ and angular momenta $L^z_j = x_j p_{yj} - y_j p_{xj}$ commute with the Hamiltonian $K + U$ : $[K+U, H_j] = 0$ , and $[K+U, L^z_j] = 0$ . Now from Eq. (18) it follows that:

[H_{osc}, W^{-1}H_jW] = 0, [H_{osc}, W^{-1}L^z_jW] = 0

Owing to fact that $[H_j, L^z_k] = 0$ , one thus has $2N$ conserved quantities $W^{-1}H_jW$ and $W^{-1}L^z_jW$ which are mutually commuting.

4. Conclusion

In this work, we have studied two quantum models depicting gas of nonrelativistic point particles with zero range contact interaction, captured by the repulsive delta function potential, in two spatial dimension. In the first model, the gas is not subjected to any external trapping potential, while in the second model we consider that the gas is trapped by an external harmonic trap. We find that the first model is exactly soluble by explicitly unitarily mapping it to the system of noninteracting free particles. The second model is shown to be exactly soluble via an explicit similarity mapping to a system of decoupled isotropic oscillators. It should be possible to construct and show that the corresponding classical versions of both these models are also Liouville integrable.

References

N. Gurappa and P. K. Panigrahi, Equivalence of the Calogero-Sutherland model to free harmonic oscillators, Physical Review B 59, R2490 (1999).
A. P. Polychronakos, The physics and mathematics of Calogero particles, Journal of Physics A: Mathematical and General 39, 12793 (2006).
H.-P. Eckle, Models of Quantum Matter: A First Course on Integrability and the Bethe Ansatz (Oxford University Press, 2019).
E. Gutkin, Integrable systems with delta-potential, Duke Mathematical Journal 49, 1 (1982).
E. H. Lieb and D. C. Mattis, Mathematical physics in one dimension: Exactly soluble models of interacting particles (Academic Press, 2013).
M. Gaudin, The Bethe Wavefunction (Cambridge University Press, 2014).
F. Dalfovo, S. Giorgini, L. P. Pitaevskii and S. Stringari, Theory of Bose-Einstein condensation in trapped gases, Reviews of Modern Physics 71, 463 (1999).
C. J. Pethick and H. Smith, Bose-Einstein condensation in dilute gases, Vol. 2 (Cambridge, 2002).
L. Pitaevskii and A. Rosch, Breathing modes and hidden symmetry of trapped atoms in two dimensions, Physical Review A 55, R853 (1997).
E. H. Lieb and W. Liniger, Exact analysis of an interacting Bose gas. I. The general solution and the ground state, Physical Review 130, 1605 (1963).
E. H. Lieb, Exact analysis of an interacting Bose gas. II. The excitation spectrum, Physical Review 130, 1616 (1963).
M. Jimbo, Introduction to the Yang-Baxter equation, International Journal of Modern Physics A 4, 3759 (1989).
M. Chaichian and A. Demichev, Introduction to quantum groups (World Scientific, 1996).
Z. Ristivojevic, The Lieb-Liniger model, Journal of Physics A: Mathematical and Theoretical (to be published) (2026).
I. Bloch, J. Dalibard and W. Zwerger, Many-body physics with ultracold gases, Rev. Mod. Phys. 80, 885 (2008).
K. Martiyanov, V. Makhalov and A. Turlapov, Observation of a two-dimensional Fermi gas of atoms, Phys. Rev. Lett. 105, 030404 (2010).
K. Gottfried and T.-M. Yan, Quantum mechanics: fundamentals, Vol. 2 (Springer, 2004).
L. I. Schiff, Quantum Mechanics (McGraw-Hill, 1968).
R. Jackiw, Diverse topics in theoretical and mathematical physics (World Scientific, 1995).
N. Mukunda and S. Chaturvedi, Continuous groups for physicists (Cambridge University Press, 2023).
R. Raczka and A. O. Barut, Theory of group representations and applications (World Scientific Publishing Company, 1986).
E. C. G. Sudarshan and N. Mukunda, Classical dynamics: A modern perspective (World Scientific, 1974).
G. Esposito, G. Marmo and E. C. G. Sudarshan, From classical to quantum mechanics: An introduction to the formalism, foundations and applications (Cambridge University Press, 2004).