{"id":139766,"date":"2025-08-12T13:30:20","date_gmt":"2025-08-12T13:30:20","guid":{"rendered":"https:\/\/www.europesays.com\/us\/139766\/"},"modified":"2025-08-12T13:30:20","modified_gmt":"2025-08-12T13:30:20","slug":"exact-collective-occupancies-of-the-moshinsky-model-in-two-dimensional-geometry","status":"publish","type":"post","link":"https:\/\/www.europesays.com\/us\/139766\/","title":{"rendered":"Exact collective occupancies of the Moshinsky model in two-dimensional geometry"},"content":{"rendered":"

We consider a system of N bosons in a two-dimensional isotropic harmonic potential, interacting with a force proportional to the square of the distance between the particles. The Hamiltonian of this system has the form11<\/a><\/p>\n

$$\\begin{aligned} \\mathscr {H} = \\sum _{i=1}^N\\left[ -{\\hslash ^2\\over 2m}\\nabla ^{2}_{\\textbf{r}_{i}} + {m \\omega ^2\\textbf{r}^2_{i}\\over 2}+\\sum _{j=i+1}\\Lambda |\\textbf{r}_i-\\textbf{r}_j|^2\\right] , \\end{aligned}$$<\/p>\n

\n (1)\n <\/p>\n

which can be transformed into a dimensionless form using the spatial coordinates expressed in \\(\\sqrt{\\hslash \/m\\omega }\\), the interaction parameter \\(\\Lambda\\) in \\(m \\omega ^2\\) and the energy in \\(\\hslash \/ \\omega\\).<\/p>\n

One-particle reduced density matrix<\/p>\n

The integral representation of the one-particle reduced density matrix is as follows<\/p>\n

$$\\begin{aligned} \\rho (\\textbf{r},\\textbf{r}^{‘})=\\int \\Psi ^{*}(\\textbf{r},\\textbf{r}_{2},…,\\textbf{r}_{N})\\Psi (\\textbf{r}^{‘},\\textbf{r}_{2},…,\\textbf{r}_{N})d\\textbf{r}_{2}…d\\textbf{r}_{N}. \\end{aligned}$$<\/p>\n

\n (2)\n <\/p>\n

Their eigenvectors \\(u_{k}(\\textbf{r})\\) (natural orbitals) and eigenvalues \\(\\lambda _{k}\\) (occupancies) are determined by the following integral eigenproblem<\/p>\n

$$\\begin{aligned} \\int \\rho (\\textbf{r},\\textbf{r}^{‘})u_{k}(\\textbf{r}^{‘})d\\textbf{r}^{‘}=\\lambda _{k}u_{k}(\\textbf{r}).\\end{aligned}$$<\/p>\n

\n (3)\n <\/p>\n

These eigenvalues can be interpreted as expansion coefficients in the Schmidt decomposition, while the corresponding eigenfunctions represent the natural orbitals of the system<\/p>\n

$$\\begin{aligned} \\rho (\\textbf{r},\\textbf{r}^{‘})=\\sum _{k}\\lambda _{k}u^{*}_{k}(\\textbf{r}^{‘})u_{k}(\\textbf{r}).\\end{aligned}$$<\/p>\n

\n (4)\n <\/p>\n

The eigenvalues \\(\\lambda _{k}\\) are widely employed to quantify correlation effects in many-body systems and to distinguish between various quantum phases, including condensed and fragmented states. In the system under consideration, the one-particle reduced density matrix can be derived analytically in Cartesian coordinates19<\/a> and is conveniently expressed in polar coordinates as follows<\/p>\n

$$\\begin{aligned} \\rho (\\textbf{r},\\textbf{r}^{‘})=A \\exp \\left( -{B\\over 2}(r^2+{r^{‘}}^{2})+{C\\over 2} r r^{‘}\\textrm{cos}(\\varphi -\\varphi ^{‘})\\right) , \\end{aligned}$$<\/p>\n

\n (5)\n <\/p>\n

with the normalization constant<\/p>\n

$$\\begin{aligned} A=\\left( 2\\pi \\int _0^{\\infty } e^{-Br^2+\\frac{C}{2} r^2} r dr\\right) ^{-1} = {\\omega \\over \\pi \\gamma }, \\end{aligned}$$<\/p>\n

\n (6)\n <\/p>\n

where<\/p>\n

$$\\begin{aligned} B={\\omega \\over \\gamma }+{C\\over 2}, \\ \\ \\ \\ \\ \\ \\ \\ C=\\left( {1-\\omega \\over N}\\right) ^2{(N-1)\\over \\gamma }, \\end{aligned}$$<\/p>\n

\n (7)\n <\/p>\n

and<\/p>\n

$$\\begin{aligned} \\gamma ={(N-1+\\omega )\\over N}, \\ \\ \\ \\ \\ \\ \\ \\ \\omega =\\sqrt{1+2\\Lambda N}. \\end{aligned}$$<\/p>\n

\n (8)\n <\/p>\n

Diagonal representation of the one-particle reduced density matrix<\/p>\n

The objective of this study is to derive the Schmidt decomposition of Eq. (5<\/a>) in polar coordinates, providing a diagonal representation of the one-particle reduced density matrix. To this end, we first expand Eq. (5<\/a>) in a Fourier-Lagrange series.<\/p>\n

$$\\begin{aligned} \\rho (\\textbf{r},\\textbf{r}^{‘})={\\rho _{0}(r, r^{‘})\\over {2\\pi }}+\\sum _{l=1}{ \\rho _{l}(r,r^{‘}) \\textrm{cos}[l(\\varphi -\\varphi ^{‘})]\\over {\\pi }}, \\end{aligned}$$<\/p>\n

\n (9)\n <\/p>\n

where the l-partial wave component is given by the following integral<\/p>\n

$$\\begin{aligned} \\rho _{l}(r, r^{‘})=\\int _{0}^{2\\pi } \\rho (\\textbf{r},\\textbf{r}^{‘}) \\textrm{cos}(l\\theta ) d\\theta ,\\end{aligned}$$<\/p>\n

\n (10)\n <\/p>\n

wherein \\(\\theta =\\varphi -\\varphi ^{‘}\\), which results in<\/p>\n

$$\\begin{aligned} \\rho _{l}(r, r^{‘})=2A\\pi \\exp \\left( -{B\\over 2}(r^2+{r^{‘}}^2)\\right) I_{l}\\left( {C rr^{‘}\\over 2}\\right) ,\\end{aligned}$$<\/p>\n

\n (11)\n <\/p>\n

where we have employed an integral formula for the modified Bessel function of the first kind \\(I_{l}(z)\\), that is23<\/a><\/p>\n

$$\\begin{aligned} \\int _{0}^{2\\pi }d\\theta \\exp \\left( z \\cos (\\theta ) \\right) \\cos (l \\theta )=2\\pi I_{l}(z). \\end{aligned}$$<\/p>\n

\n (12)\n <\/p>\n

A detailed analysis indicates that the Schmidt decomposition of the l-partial component, denoted as \\(\\rho _{l}\\) in Eq. (11<\/a>), can be derived by direct comparison with the Hardy-Hille formula24<\/a><\/p>\n

$$\\begin{aligned} \\exp \\left( -{\\left( {1\\over 2}+{t\\over 1-t} \\right) (x+y)} \\right) I_{\\alpha }\\left( {2 \\sqrt{ xyt}\\over 1-t}\\right) = \\sum _{n=0} {n! t^{n+\\frac{\\alpha }{2}}(1-t)\\over (n+\\alpha )!}(xy)^{\\frac{\\alpha }{2}}\\exp \\left( -{(x+y)\\over 2} \\right) L_{n}^\\alpha (x)L_{n}^{\\alpha }(y), \\end{aligned}$$<\/p>\n

\n (13)\n <\/p>\n

where \\(L_{n}^{\\alpha }(x)\\) is the generalized Laguerre polynonomial and<\/p>\n

$$\\begin{aligned} t=-1+{4B\\over C^2}\\left( 2B-\\sqrt{4B^2-C^2}\\right) . \\end{aligned}$$<\/p>\n

\n (14)\n <\/p>\n

By substituting \\(x = z^2r^2\\) and \\(y = z^2{r^{‘}}^2\\) into the formula (13<\/a>), where<\/p>\n

$$\\begin{aligned} z={(4B^2-C^2)^{1\\over 4}\\over \\sqrt{2}},\\end{aligned}$$<\/p>\n

\n (15)\n <\/p>\n

we derive the matching condition<\/p>\n

$$\\begin{aligned} B=\\left( 1 + 2t(1 – t)^{-1}\\right) z^2, \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ C=4\\sqrt{t}(1 – t)^{-1}z^2,\\end{aligned}$$<\/p>\n

\n (16)\n <\/p>\n

which, together with the requirement that Schmidt orbitals be normalized, yields<\/p>\n

$$\\begin{aligned} \\rho _{l}(r,r^{‘})=\\sum _{n} \\lambda _{n,l}v_{n,l}(r^{‘}) v_{n,l}(r),\\end{aligned}$$<\/p>\n

\n (17)\n <\/p>\n

with<\/p>\n

$$\\begin{aligned} v_{nl}(r)=\\sqrt{{2 n! z^2 \\over (n+|l|)!}}(z r)^{|l|}e^{-z^2r^2\/2}L_{n}^{|l|}(z^2 r^2), \\end{aligned}$$<\/p>\n

\n (18)\n <\/p>\n

and<\/p>\n

$$\\begin{aligned} \\lambda _{nl}={A\\pi t^{M}(1-t)\\over z^{2}}, \\qquad \\text {with} \\qquad M=n+\\frac{|l|}{2}.\\end{aligned}$$<\/p>\n

\n (19)\n <\/p>\n

Using the above expressions and the identity \\(\\cos (l\\theta ) = (\\exp (i l\\theta ) + \\exp (-i l\\theta ))\/2\\), where i is an imaginary unit, we can represent the 1-RDM as follows<\/p>\n

$$\\begin{aligned} \\rho (\\textbf{r},\\textbf{r}^{‘})=\\sum _{n=0}^{\\infty } \\sum _{l=-\\infty }^{\\infty } \\lambda _{n,l}u^{*}_{n,l}(\\textbf{r}^{‘})u_{n,l}(\\textbf{r}), \\end{aligned}$$<\/p>\n

\n (20)\n <\/p>\n

where<\/p>\n

$$\\begin{aligned} u^{}_{n,l}(\\textbf{r})={v_{n,l}(r)}{e^{i l\\varphi }\\over \\sqrt{2\\pi }}. \\end{aligned}$$<\/p>\n

\n (21)\n <\/p>\n

The orbitals (21<\/a>) form an orthonormal basis set<\/p>\n

$$\\begin{aligned} \\int _{0}^{2\\pi } \\int _{0}^{\\infty }d\\varphi dr [r u^{*}_{n,l}(\\textbf{r})u^{}_{n^{‘}l^{‘}}(\\textbf{r})]=\\delta _{nn^{‘}}\\delta _{ll^{‘}},\\end{aligned}$$<\/p>\n

\n (22)\n <\/p>\n

and they can be recognized as the natural orbitals of the occupancies (19<\/a>). The values of \\(\\lambda _{n,l}\\) are \\((2M+1)\\)-fold degenerate indicating that for given M, there exist exactly \\((2M+1)\\) orthogonal orbitals corresponding to \\(\\lambda _{n,l}\\). We note that while the Schmidt decomposition uniquely determines the set of non-zero occupation numbers, the decomposition is not unique. Alternative forms of Schmidt orbitals can be constructed for the degenerate points \\(\\lambda _{n,l} = \\lambda _{n|l|}\\) as follows25<\/a>,26<\/a><\/p>\n

$$\\begin{aligned} \\chi _{n,l}(\\textbf{r})= {\\left\\{ \\begin{array}{ll} \\frac{1}{\\sqrt{2}} (u_{n,-l}(\\textbf{r})+u_{n,l}(\\textbf{r})), & \\text {for} \\ \\ \\ l\\ne 0\\\\ \\\\ u_{n,0}(\\textbf{r}), & \\text {for} \\ \\ \\ l=0 \\end{array}\\right. } \\qquad \\qquad \\xi _{n,l}(\\textbf{r}) = {\\left\\{ \\begin{array}{ll} \\frac{i}{\\sqrt{2}}(u_{n,-l}(\\textbf{r})-u_{n,l}(\\textbf{r})) , & \\text {for} \\ \\ \\ l\\ne 0\\\\ \\\\ 0, & \\text {for} \\ \\ \\ l=0 \\end{array}\\right. }. \\end{aligned}$$<\/p>\n

\n (23)\n <\/p>\n

The family \\(\\{\\chi _{n,l}, \\xi _{n,l}\\}\\) forms a complete and orthonormal set. In terms of the orbitals (23<\/a>), the 1-RDM takes the form<\/p>\n

$$\\begin{aligned} \\rho (\\textbf{r},\\textbf{r}^{‘})=\\sum _{l,n=0}^{\\infty } \\lambda _{n,l} \\left( \\chi _{n,l}(\\textbf{r})\\chi _{n,l}(\\mathbf {r’})+\\xi _{n,l}(\\textbf{r})\\xi _{n,l}(\\mathbf {r’})\\right) . \\end{aligned}$$<\/p>\n

\n (24)\n <\/p>\n

Collective occupancies, participation, and correlation<\/p>\n

There are many different quantities for studying the properties of many-body states. In the case of isotropic symmetry, one of them is the collective occupancy, defined as the sum of occupancies with angular momentum l27<\/a><\/p>\n

$$\\begin{aligned} \\eta _l= \\sum _n \\lambda _{n,l}=\\frac{n_{l}}{N}, \\end{aligned}$$<\/p>\n

\n (25)\n <\/p>\n

where \\(n_{l}\\) denotes the number of particles with that angular momentum. This quantity determines a fraction of particles with a given l. For the present system, the collective occupancy is given by the geometric series (\\(0) and can thus be accurately determined in closed analytical form<\/p>\n

$$\\begin{aligned} \\eta _l= \\frac{ \\pi A t^{\\frac{|l|}{2}}}{z^2}, \\end{aligned}$$<\/p>\n

\n (26)\n <\/p>\n

which gives us a unique opportunity to examine its behavior in the full interaction regime and the general case of N particles. Another closely related quantity is participation, defined as<\/p>\n

$$\\begin{aligned} K_{\\eta }= \\left( \\sum _{l} \\eta _l^2 \\right) ^{-1},\\end{aligned}$$<\/p>\n

\n (27)\n <\/p>\n

which measures the effective number of collective occupancies, i.e., the number of l fragments that contribute significantly. The collective occupancies and this tool offer insight into correlations in the single-particle angular momentum domain. For the same reason as for collective occupancy, this measure can be obtained analytically.<\/p>\n

$$\\begin{aligned} K_{\\eta }=\\frac{z^4(1-t)}{\\pi ^2 A^2 (1+t)}.\\end{aligned}$$<\/p>\n

\n (28)\n <\/p>\n

By contrast, participation<\/p>\n

$$\\begin{aligned} K=\\left( \\sum _{n,l} \\lambda _{n,l}^2 \\right) ^{-1} =\\frac{z^4}{\\pi ^2 A^2} \\end{aligned}$$<\/p>\n

\n (29)\n <\/p>\n

quantifies the effective number of natural orbitals and serves as an indicator of the degree of single-particle correlations28<\/a>. More specifically, it assesses how a subsystem containing one particle correlates with a subsystem composed of the remaining particles.<\/p>\n

Fig. 1<\/b>\"figure<\/a><\/p>\n

Graphs (a<\/b>) and (b<\/b>) show the behaviors of collective occupancies as functions of interaction strength \\(\\Lambda\\) for two different particle numbers \\(N=2\\) and \\(N=500\\), respectively. The dashed lines represent the approximation results (30<\/a>). The strength of the interaction \\(\\Lambda\\) is expressed in units of \\(m \\omega ^2\\). The graphs (c<\/b>) and (d<\/b>) illustrate the corresponding results for the participation \\(K_{\\eta }\\), together with its approximation obtained from the expansion to infinity, \\(\\Lambda \\rightarrow \\infty\\): \\(K_{\\eta } \\approx 2\\beta ^{-1}(N)\\Lambda ^{1\/4}\\) (dashed lines).<\/p>\n","protected":false},"excerpt":{"rendered":"We consider a system of N bosons in a two-dimensional isotropic harmonic potential, interacting with a force proportional…\n","protected":false},"author":3,"featured_media":139767,"comment_status":"","ping_status":"","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[25],"tags":[10046,10047,492,13632,8068,159,6458,67,132,68],"class_list":{"0":"post-139766","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-physics","8":"tag-humanities-and-social-sciences","9":"tag-multidisciplinary","10":"tag-physics","11":"tag-quantum-information","12":"tag-quantum-mechanics","13":"tag-science","14":"tag-theoretical-physics","15":"tag-united-states","16":"tag-unitedstates","17":"tag-us"},"share_on_mastodon":{"url":"https:\/\/pubeurope.com\/@us\/115016042201893189","error":""},"_links":{"self":[{"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/posts\/139766","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/comments?post=139766"}],"version-history":[{"count":0,"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/posts\/139766\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/media\/139767"}],"wp:attachment":[{"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/media?parent=139766"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/categories?post=139766"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/tags?post=139766"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}