A trial wave function Ψ(1, 2,..., N ) of an N electron system can always be written as the product of an antisymmetric Fermion factor F { z ij } = Π i<j z ij , and a symmetric correlation factor G { z ij }. F results from the Pauli principle, and G is caused by Coulomb interactions. One can represent G diagrammatically [1] by distributing N points on the circumference of a circle, and drawing appropriate lines representing correlation factors (cfs) z ij between pairs. Here, of course, z ij = z i – z j , where z i is the complex coordinate of the i th electron. Laughlin correlations for the ν = 1/3 filled incompressible quantum liquid (IQL) state contain two cfs connecting each pair ( i,j ). For the Moore-Read state of the half-filled excited Landau level (LL), with ν = 2 + 1/2, the even value of N for the half-filled LL is partitioned into two subsets A and B , each containing N /2 electrons [2]. For any one partition ( A,B ), the contribution to G is given by G AB = Π i<j∈A z ij 2 Π k<ι∈B z kι 2 . The full G is equal to the symmetric sum of contributions G AB over all possible partitions of N into two subsets of equal size. For Jain states at filling factor ν = p/q < 1/2, the value of the single particle angular momentum ι satisfies the equation 2 ι = ν -1 N - C ν , with C ν = q + 1 - p . The values of (2 ι, N ) define the function space of G { z ij } , which must satisfy a number of conditions. For example, the highest power of any z i cannot exceed 2 ι + 1 - N . In addition, the value of the total angular momentum L of the lowest correlated state must satisfy the equation L = ( N /2)(2 ι +1 - N ) - K G , where K G is the degree of the homogeneous polynomial generated by G . Knowing the values of L for IQL states (and for states containing a few quasielectrons or a few quasiholes) from Jain's mean field CF picture allows one to determine K G . The dependence of the pair pseudopotential V(L 2 ) on pair angular momentum L 2 suggests a small number of correlation diagrams for a given value of the total angular momentum L . Correlation diagrams and correlation functions for the Jain state at ν = 2/5 and for the Moore-Read states will be presented as examples. The generalizations of the method of selecting G from small to larger systems will be discussed.
A trial wave function Ψ(1, 2,..., N ) of an N electron system can always be written as the product of an antisymmetric Fermion factor F { z ij } = Π i<j z ij , and a symmetric correlation factor G { z ij }. F results from the Pauli principle, and G is caused by Coulomb interactions. One can represent G diagrammatically [1] by distributing N points on the circumference of a circle, and drawing appropriate lines representing correlation factors (cfs) z ij between pairs. Here, of course, z ij = z i – z j , where z i is the complex coordinate of the i th electron. Laughlin correlations for the ν = 1/3 filled incompressible quantum liquid (IQL) state contain two cfs connecting each pair ( i,j ). For the Moore-Read state of the half-filled excited Landau level (LL), with ν = 2 + 1/2, the even value of N for the half-filled LL is partitioned into two subsets A and B , each containing N /2 electrons [2]. For any one partition ( A,B ), the contribution to G is given by G AB = Π i<j∈A z ij 2 Π k<ι∈B z kι 2 . The full G is equal to the symmetric sum of contributions G AB over all possible partitions of N into two subsets of equal size. For Jain states at filling factor ν = p/q < 1/2, the value of the single particle angular momentum ι satisfies the equation 2 ι = ν -1 N - C ν , with C ν = q + 1 - p . The values of (2 ι, N ) define the function space of G { z ij } , which must satisfy a number of conditions. For example, the highest power of any z i cannot exceed 2 ι + 1 - N . In addition, the value of the total angular momentum L of the lowest correlated state must satisfy the equation L = ( N /2)(2 ι +1 - N ) - K G , where K G is the degree of the homogeneous polynomial generated by G . Knowing the values of L for IQL states (and for states containing a few quasielectrons or a few quasiholes) from Jain's mean field CF picture allows one to determine K G . The dependence of the pair pseudopotential V(L 2 ) on pair angular momentum L 2 suggests a small number of correlation diagrams for a given value of the total angular momentum L . Correlation diagrams and correlation functions for the Jain state at ν = 2/5 and for the Moore-Read states will be presented as examples. The generalizations of the method of selecting G from small to larger systems will be discussed.
Motivated by the structure of the quasiparticle wavefunction in the composite fermion (CF) theory for fractional quantum Hall filling factor [Formula: see text] (m odd), I consider a suitable quasiparticle operator in differential form, as a modified form of Laughlin's quasiparticle operator, that reproduces quasiparticle wave function as predicted in the CF theory, without a priori assumption of the presence of CFs. I further consider the conjugate of this operator as quasihole operator for obtaining a novel quasihole wave function for 1/m state. Each of these wave functions is interpreted as expelling an electron into a different Hilbert subspace from the original Hilbert space of the Laughlin condensate while still maintaining its correlation (although changed) with the electrons in the condensate such that the expelled electron behaves as a CF with respect to the electrons in the condensate. With this interpretation, I show that the ground state wavefunctions for general states at filling fractions [Formula: see text], respectively, can be constructed as coherent superposition of n coupled Laughlin condensates and their 'conjugates', formed at different Hilbert subspaces. The corresponding wave functions, specially surprising for [Formula: see text] sequence of states, are identical with those proposed in the theory of noninteracting CFs. The states which were considered as fractional quantum Hall effect of interacting CFs, can also be treated in the same footing as for the prominent sequences of states describing as the coupled condensates among which one is a non-Laughlin condensate in a different Hilbert subspace. Further, I predict that the half filling of the lowest Landau level is a quantum critical point for phase transition between two topologically distinct phases each corresponding to a family of states: one consists of large number of coupled Laughlin condensates of filling factor 1/3 and the other corresponds to large number of coupled conjugate Laughlin condensates of filling factor 1, which may be distinguished, respectively, by the absence and presence of upstream edge modes.
In no wave function of the type (1) [product of single determinants for each spin] is there a statistical correlation between the positions of the electrons with antiparallel spin. The purpose of the aforementioned generalization of (1) is to allow for such correlations. This will lead to an improvement of the wave function and, therefore, to a lowering of the energy value. E. Wigner, Phys. Rev. 46, 1002 (1934) Summary Although the exact many-body wavefunction cannot be written down in general, we do know of some of its properties. For example, there are differences between the wavefunctions of insulators and metals and the cusp condition gives the behavior as any two charges approach each other. In this chapter we also discuss approximate wavefunctions, ways to judge their accuracy and how to include electronic correlation. Examples of many-body wavefunctions are the Slater–Jastrow (pair product) wavefunction and its generalization to pairing and backflow wavefunctions. In other places in this book, we argue that it is not necessary to look at the many-body wavefunctions explicitly because they are unwieldy; the one- and two-body correlation functions discussed in Ch. 5 are sufficient to determine the energy and give information on the excitation spectra. However, these correlation functions do not always contain all information of interest. In principle, the ground-state energy of a many-electron system is a functional of the density, but the very derivation of the theorem invokes the many-body wavefunction, as expressed explicitly in Eq. (4.16). The effects of antisymmetry are manifest in the correlation functions, but antisymmetry is most simply viewed as a property of the wavefunction; electronic correlation is fundamentally a result of properties of the many-body wavefunction. Studying many-body wavefunctions provides a very useful, different point of view of many-body physics from the approaches based on correlation functions. Many of the most important discoveries in physics have come about by understanding the nature of wavefunctions, such as the Laughlin wavefunction for the fractional quantum Hall effect, the BCS wavefunction for superconductors, p-wave pairing in superfluid 3 He, and the Heitler– London approach for molecular binding. The role of Berry's phases has brought out the importance of the phase of the wavefunction in determining properties of quantum systems. This has led to new understanding of the classification of insulators, metals, superconductors, vortices, and other states of condensed matter.
Two-and three-body correlation functions (number of pairs or triplets vs. relative angular momentum) of electrons or Laughlin quasielectrons (i.e., composite fermions in their first excited Landau level) are studied numerically in several fractional quantum Hall liquids.It is shown directly that the νe = 4/11 liquid (corresponding to a ν = 1/3 filling of composite fermions in their first excited Landau level) is a paired state of quasielectrons, hence interpreted as a condensate of "second-generation" quasiholes of Moore-Read ν = 1/2 state of composite fermions.
The electron correlations near the half-filling of the lowest and excited Landau levels (LL's) are studied using numerical diagonalization. It is shown that in the low-lying states electrons avoid pair states with relative angular momenta R corresponding to positive anharmonicity of the interaction pseudopotential V(R). In the lowest LL, the superharmonic behavior of V(R) causes Laughlin correlations (avoiding pairs with R=1) and the Laughlin-Jain series of incompressible ground states. In the first excited LL, V(R) is harmonic at short range and a different series of incompressible states results. Similar correlations occur in the paired Moore-Read $\ensuremath{\nu}=\frac{5}{2}$ state and in the $\ensuremath{\nu}=\frac{7}{3}$ and 38 states, all having small total parentage from R=1 and 3 and large parentage from R=5. The $\ensuremath{\nu}=\frac{7}{3}$ and 38 states are different from Laughlin $\ensuremath{\nu}=\frac{1}{3}$ and 32 states and, in finite systems, occur at a different LL degeneracy (flux). The series of Laughlin-correlated states of electron pairs at $\ensuremath{\nu}{=2+2/(q}_{2}+2)=\frac{8}{3},$ 25,512, and 37 is proposed, although only in the $\ensuremath{\nu}=\frac{5}{2}$ state pairing has been confirmed numerically. In the second excited LL, V(R) is subharmonic at short range and (near the half-filling) the electrons group into spatially separated larger $\ensuremath{\nu}=1$ droplets to minimize the number of strongly repulsive pair states at R=3 and 5.
Symétries et corrélations dans les gaz quantiques fortement interagissants à une dimension L’objectif principal de cette thèse est l’étude théorique de mélanges quantiques fortement interagissants à une dimension et soumis à un potentiel externe harmonique. De tels systèmes fortement corrélés peuvent être réalisés et testés dans des expériences d’atomes ultrafroids. Leurs propriétés de symétrie par permutation non triviales sont étudiées, ainsi que leurs effets sur les corrélations. Exploitant une solution exacte pour des interactions fortes, nous extrayons des propriétés générales des corrélations encodées dans la matrice densité à un corps et dans les distributions des impulsions associées, dans les mélanges fermioniques et de Bose-Fermi. En particulier, nous obtenons des résultats substantiels sur le comportement à courtes distances, et donc les queues à haute impulsions, qui suivent des lois en k^−4 typiques. Les poids de ces queues, dénotés contacts de Tan, sont liés à de nombreuses propriétés thermodynamiques des systèmes telles que les corrélations à deux corps, la dérivée de l’énergie par rapport à la longueur de diffusion unidimensionnelle, ou le facteur de structure statique. Nous montrons que ces contacts universels de Tan permettent également de caractériser la symétrie spatiale des systèmes, et constituent donc une connexion profonde entre les corrélations et les symétries. En outre, la symétrie d’échange est extraite en utilisant une méthode de théorie des groupes, à savoir la méthode de la somme des classes (class-sum method en anglais), qui provient à l’origine de la physique nucléaire. De plus, nous montrons que ces systèmes suivent une version généralisée du fameux théorème de Lieb-Mattis. Souhaitant rendre nos résultats aussi pertinents expérimentalement que possible, nous dérivons des lois d’échelle pour le contact de Tan en fonction de l’interaction, de la température et du confinement transverse. Ces lois présentent des effets intéressants liés aux fortes corrélations et à la dimensionnalité.