Vortices in quantum droplets: Analogies between boson and fermion systems

The main theme of this review is the many-body physics of vortices in quantum droplets of bosons or fermions in the limit of small particle numbers. Systems of interest include cold atoms in traps as well as electrons confined in quantum dots. When set to rotate, these in principle different quantum systems show remarkable analogies. The topics reviewed include the structure of the finite rotating many-body state, universality of vortex formation and localization of vortices in both bosonic and fermionic systems, and the emergence of particle-vortex composites in the quantum Hall regime. An overview of the computational many-body techniques sets focus on the configuration-interaction and density-functional methods. Studies of quantum droplets with one or several particle components, where vortices as well as coreless vortices may occur, are reviewed, and theoretical as well as experimental challenges are discussed.

Figure 1

Examples of vortices and vortex lattices. Vortices are ubiquitous in both classical and quantum systems: (a) classical whirlpool vortex (7), (b) wake vortex of a passing airplane wing, revealed by colored smoke (NASA Langley Research Center, Figure ID: EL-1996-00130), (c) STM image of an Abrikosov vortex lattice (2) in a type-II superconductor (142), (d) vortex lattice in a rotating Bose-Einstein condensate of ${}^{87}\mathrm{Rb}$ atoms [adapted from 69], (e) cluster of vortices in the calculated electron density of a 24-electron quantum dot, after 285. In (c)–(e), the vortices appear as “holes” in the particle density.

Figure 2

(Color online) Reduced wave function of a two-electron quantum dot in (a) the singlet and (b) the triplet state. The fixed electron is marked by the cross to the right. The contours give the logarithmic electron density of the probing electron and the gray scale illustrates the phase of the wave function. The phase jumps from 0 to $2\pi$ on the line where the scale changes from white to darkest gray. In the singlet state, the electrons have opposite spins and there is no vortex. In the triplet state, the electrons have same spin and a vortex (circle with an arrow in the direction of phase gradient) is attached on top of the fixed electron in accordance with the Pauli principle. Due to fermion antisymmetry the phase changes by $2\pi$ if the probe electron is moved around the fixed electron in this case. From 135.

Figure 3

(Color online) Appearance of vortex structures in the reduced wave function. The details of reduced wave functions for spin-$1∕2$ fermions are shown. The most probable position of the probing particle is to the right; the contours show the magnitude (on a logarithmic scale), and the gray scale shows the phase (darkest $\text{gray}=0$, lightest $\text{gray}=2\pi$). (a) An isolated localized vortex which is not attached to any particle. (b) A Pauli vortex (exchange hole) which is mandated by the wave-function antisymby metry between interchange of indistinguishable fermions. (c) A coreless vortex where the vortex core of spin-down component is filled by a spin-up fermion. (d) A composite of a fermion (with a Pauli vortex) and two additional nodes which are bound to the particle, reminiscent of the Laughlin $\nu =\frac{1}{3}$ state. From 284.

Figure 4

(Color online) Reduced wave functions. (a) A seven-particle MDD state, (b) the MDD state with three holes, and (c) and (d) MDD state with a center-of-mass excitation. The upper panels show the phase and the lower panels the magnitude of the reduced wave function. The bullets mark the fixed particle positions.

Figure 5

Overlap between the fermion ground state and the transformed boson ground state as a function of the total angular momentum, for eight particles. From 45.

Figure 6

Mapping between boson and fermion states. The upper panels show the particle density of 20 (a) bosons and (b) fermions with Coulomb interactions, in the region of three vortices as a function of the radial distance of the droplet center. For bosons, the density of the mapped fermion system is shown as a dashed line. The lower panels show in column (c) the particle-particle pair-correlations determined from the fermion wave functions. The position of the reference point is marked by the arrow at the bottom of the exchange-correlation hole. In column (d) the corresponding vortex-vortex pair correlations are shown.

Figure 7

Transverse absorption images of a Bose-Einstein condensate of ${}^{87}\mathrm{Rb}$, stirred with a laser beam. As the rotation of the trap increases from (a) to (e), a clear vortex pattern evolves. The inset to the right of (b) shows the cross section of the optical density which shows a pronounced minimum at the center. After 214.

Figure 8

Yrast lines and occupancies. Upper panel: Many-body yrast lines for $N=25$ and 50 spinless bosons in a harmonic confinement for angular momenta $2⩽L⩽50$. The inset shows the excitation spectrum for $N=50$ and $L⩽18$, excluding the spurious center-of-mass excitations, see text. From 250. Lower panel: Occupancies of the lowest single-particle states of the harmonic oscillator in the lowest Landau level, for $m=0$ (diamonds), $m=1$ (squares), $m=2$ (circles), and $m=3$ (triangles). Calculations are within the lowest Landau level. After 37.

Figure 9

Equidensity lines of the pair-correlation function $P(\mathbf{r},{\mathbf{r}}_A)$ for $N=40$ spinless bosons at $L=28$, 32, 36, and 40. For clarity, the reference point was located outside the cloud at ${\mathbf{r}}_A=\left(\mathrm{3,0}\right)$. The vortex, which approaches the center from the right with increasing $L$, gives rise to a pronounced minimum in the pair-correlation plots. From 179.

Figure 10

Vortex entry for a spherical bosonic cloud at angular momenta $l=L∕N$. The surfaces of constant density obtained by the Gross-Pitaevskii method are shown. The cloud flattens with increasing angular momentum. From 53.

Figure 11

(Color online) Charge and current densitites. (a) Charge density (grayscale) and current density (arrows) in the maximum density droplet state of a six-electron droplet at magnetic field $B=9\mathrm{T}$ calculated with the density-functional method. The angular momentum is $L=15$, and the density inside the droplet is uniform. The solution also shows an edge current reminiscent of those in quantum Hall states. (b) The single-vortex state in the same droplet at slightly increased magnetic field of $B=11\mathrm{T}$ with $L=21$. It shows a pronounced vortex hole in the middle with a rotating current around it. Adapted from 285.

Figure 12

(Color online) Systematics of boson and fermion ground states. When the external rotation $\Omega$ is gradually increased from zero, a droplet of $N$ particles goes through a series of ground states with increasing angular momentum $L$. Stars mark these $L$ values as a function of $N$ for (a) boson droplets and (b) fermion droplets. Calculations are done with the exact diagonalization method in the lowest Landau-level approximation, and a harmonic confining potential Eq. (4). In the fermion results in (b), the angular momentum of the maximum density droplet $L_{\mathrm{MDD}}$ is subtracted from $L$. The linear $N$ dependence of the first $(N,L)$ combination in bosonic systems (arrow) indicates that the first $L$ above the nonrotating state has a central vortex. Fermionic systems with repulsive interactions show a similar behavior only until $N=12$, where the breakdown mechanism of the MDD clearly changes (arrow), and a nonlocalized node emerges at a finite distance from the center. From 135 and 309.

Figure 13

Occupations of the single-particle (Fock-Darwin) eigenstates with angular momentum $m$ of fermions in a harmonic trap at ground states with $L=L_{\mathrm{MDD}}+13$ for $N=15$ (left) and 25 (right). Since the mean particle distance from the center increases with $m$ the high-$N$ states resemble more edge excitations than central vortex states (cf. Fig. 12). From 309.

Figure 14

(Color online) Yrast line of $N=20$ bosons in a harmonic confinement, obtained by the CI method in the lowest Landau level and for contact interactions between bosonic particles. The inset shows the total angular momentum of the ground state, plotted as a function of $\Omega ∕\omega$. Pair-correlated densities (renormalized in height) are shown for increasing angular momentum per particle, $l=L∕N=0.1$, 1.0, 1.8, and 2.85 (as marked by the triangles). The reference point was chosen at high density for radii of order unity. From 66

Figure 15

(Color online) Vortices in a rotating cloud of bosons. Schematically shown are the vortex holes that penetrate the boson cloud with increasing angular momentum. The lower panel shows the phase of the order parameter, and its density contours. (The black dots indicate the vortex positions.) From 53.

Figure 16

Angular momentum per particle $L∕N$ as a function of the rotational frequency $\Omega ∕\omega$ of the trap. The discontinuities correspond to the transitions between different symmetries. The insets show the surfaces of constant density in a spherical trap for states with two and six vortices. $\gamma ={(2∕\pi )}^{1∕2}aN∕{\sigma }_z$, $a$ the scattering length and ${\sigma }_z$ the axial width of the ground state of a single particle in the trap. From 53.

Figure 17

(Color online) Vortex molecules in a six-electron droplet. Charge density (grayscale) and current density (arrows) show rotating currents around (a) three and (b) four localized vortex cores in density-functional calculations. From 285.

Figure 18

(Color online) Electron density (grayscale) and current density (arrows) in a 24-electron quantum dot calculated with the density-functional method. The solution shows a cluster of 14 localized vortices arranged in two concentric rings. From 285.

Figure 19

(Color online) Reduced wave-function representation of the vortex structure of the model wave function Eq. (48) for bosons and fermions ($N=7$, $n=2$, $K=2$). The fixed particles are shown as light dots, the current field with arrows (logarithmic scale), and the particle density as shades of color (light color corresponding to high density).

Figure 20

(Color online) Current peaks in the electron transport experiments and transitions in the spin-density-functional theory (lines). The dashed lines denote the MDD boundaries and the roman numerals indicate number of vortices in the theory. Adapted from 283; the experimental data are from Fig. 2(b) in 248.

Figure 21

(Color online) Chemical potential of a quantum dot device with $N=30$ (upper panel) and $N=13$ (lower panel) compared to the results from the spin-density-functional theory. The experimental data are from 248. Noise in the experimental data has been reduced by using a Gaussian filter. In the calculations the confining potential is assumed to be parabolic with $\hslash {\omega }_0$ being $4.00\mathrm{meV}$ for $N=13$ and $3.51\mathrm{meV}$ for $N=30$. Finite-size precursors of different quantum Hall states are identified. The roman numbers between the filled triangles indicate the number of vortices inside the electron droplet predicted by the density-functional calculations. The open triangles mark other possible transitions which are beyond the reach of density-functional theory.

Figure 22

(Color online) Pair-correlation functions of four fermions with repulsive Gaussian interactions at four different angular momenta $L=10$, 18, 30, and 42, respectively, showing that localization increases with rotation. The contour plots are in the same scale to demonstrate the expansion due to the rotation. From 239.

Figure 23

Classical orbits and pair-correlation functions of localized electrons in rotating frame (upper panel). At angular momentum $L=24$ the lowest energy state is purely rotational while at $L=25$ a doubly excited rotational state is shown. In the rotating frame the classical motion shows a pseudorotation (middle left) while the pair correlation shows maxima at the classical turning points (middle right).The spectrum (lower panel) compares the exact energies (dots) with those of the classical model (squares). It shows a periodicity of $\Delta L=3$ in angular momentum, which agrees with the localization in a triangular geometry (see upper panel). The horizontal dashed lines indicate the center-of-mass excitations which occur at all angular momenta. From 240.

Figure 24

Electron-electron pair-correlation functions showing localization of four vortices (left) and localization of electrons at the edge of the cloud in the case of one vortex (right). White color means high density, and some constant-density contours are shown. The two most important configurations are given in each case, demonstrating that mixing of single-particle states close to holes leads to hole localization, while correspondingly the mixing of particles localizes particles. The results are calculated with the CI method for 20 particles with angular momenta $L=242$ (left) and $L=202$ (right).

Figure 25

(Color online) Pair-correlation functions for large fermion and boson systems with four vortices. The pair-correlation function of the MDD is displayed for comparison; it only shows the exchange-correlation hole at the reference point.

Figure 26

(Color online) Radial electron densities. Upper panel: Radial electron densities in a harmonic trap $(\omega =1)$ in a six-electron droplet with a central vortex at $L=21$ (in harmonic oscillator units). The exact solution in the LLL is shown by the line, a single-determinant wave function which describes a central vortex is shown by another line, and a single-determinant wave function in the center of mass (CM) transformed coordinates $z_i\to z_i-z_{\mathrm{CM}}$ is shown by the third line. Lower panel: Radial electron densities for central vortex states $L=L_{\mathrm{MDD}}+N$, showing that vortex localization increases with electron number $N$ due to decrease in the center-of-mass motion.

Figure 27

(Color online) Fermion low-energy spectrum for 20 particles. The lowest energy many-particle states as a function of the total angular momentum (yrast states) are connected with lines to guide the eye. A smooth function of angular momentum was subtracted from the energies to emphasize the oscillatory behavior of the yrast line. The periodicity of the oscillation reveals the number of localized vortices as shown schematically. From 276.

Figure 28

(Color online) Fermion yrast spectrum for 20 particles and three vortices (upper panel) and three particles (lower panel) show both the periodicity of $\Delta L=3$ associated with the threefold rotational symmetry of the vortex molecule in the former case and electron molecule in the latter case. From 218.

Figure 29

(Color online) Electron density and current density (arrows) of an elliptically confined six-electron droplet with two localized vortices, calculated by the exact diagonalization method. The confinement strength is $\hslash {\omega }_0=5.93\mathrm{meV}$, the eccentricity of the elliptic confining potential $\delta =1.2$, and the magnetic field is $B=17\mathrm{T}$. Inset: Profile of the electron density at the longest major axis shows fluctuations in the vortex positions, which causes electron density to remain finite at the density minima. Adapted from 287.

Figure 30

(Color online) Angular momentum as a function of the rotational frequency of the parabolic trap with $N=8$ electrons in the lowest Landau level. The unperturbed result (thin line) is comparable to the expectation value of angular momentum in the presence of an added point perturbation which breaks the rotational symmetry (thick solid line). The insets show the densities as shades of color in the perturbed system. The vortices appear as pronounced minima in the density distribution, their number increasing with the trap rotation. Results are calculated with the exact diagonalization method (66).

Figure 31

(Color online) Reduced wave functions of a bosonic five-particle system in a harmonic trap, showing the formation of one and two free vortices in the region of high particle density (marked as circles) at low angular momenta $L=5$ and $L=8$, respectively (left and middle). When the angular momentum increases, two vortices are finally captured by each particle to form a state which is approximated by the bosonic Laughlin state $m=2$ (two concentric circles) at $L=20$ (right). Particle interactions are Coulombic here and the probe particle is at the bottom. After 309.

Figure 32

Schematic phase diagram of the ground states of a bosonic cloud in an anharmonic confinement. From 177.

Figure 33

(Color online) A rotating Bose-Einstein condensate in a mixed state with a giant vortex in the center surrounded by ten single-quantized vortices. The giant vortex is composed of four phase singularities. The left panel shows the particle density (white is high density and black is zero density) and the right panel shows the phase profile. Locations of phase singularities are marked with circles. The large circle marks the ensemble of four phase singularities in the core of the giant vortex. Adapted from 173.

Figure 34

(Color online) A giant vortex in a six-electron quantum dot calculated with the exact diagonalization method. The left panel shows the particle density (black is low density) and current density (arrows), and the right panel shows the reduced wave function, where phase singularities are marked with circles and electron positions with crosses. The giant core in this case comprises three phase singularities. Interactions and fluctuations keep the phase singularities separated. The probe electron is on the bottom right. From 266.

Figure 35

(Color online) The reduced wave functions for (a) the approximate Laughlin state $\nu =\frac{1}{3}$ and (b) the exact $L=30$ ground state for five electrons in a parabolic external potential. The Laughlin state fixes a triple vortex (concentric rings) on each electron position (crosses). In the exact solution there are clusters of three vortices near each electron.

Figure 36

(Color online) A $\nu =\frac{1}{3}$ state of five electrons in a harmonic trap. Top: Electron density from the density-functional method and current density (arrows). The confinement strength is $\hslash {\omega }_0=5\mathrm{meV}$ and the magnetic field is $B=18\mathrm{T}$. Bottom: Reduced wave function for the same state constructed from the Kohn-Sham single-particle states. The probe electron is at the top right.

Figure 37

(Color online) Overlap of a single-determinant wave function with the exact one as a function of the angular momentum increase with respect to the MDD state $\Delta L=L-L_{\mathrm{MDD}}$ for five electrons in parabolic confinement. The higher points (squares) are obtained with a coordinate transformation to the center of mass $z_i\to z_i-z_{\mathrm{CM}}$ and the lower ones (circles) without it. The star at $\Delta L=20$ shows the overlap with the Laughlin $\nu =1∕3$ state. The roman numbers count the vortices inside the electron droplet. From 135.

Figure 38

(Color online) Schematic of the sites of vortices, determined from the reduced wave function of an exact diagonalization for five electrons with angular momentum $L=16$ (six above the MDD, left) and $L=36$ (six above the state with filling factor $1∕3$, right). Fixed electron positions with Pauli vortices, vortices attached to electrons making composite particles, and free vortices inside the electron droplet are denoted by bullets. Free vortices outside the droplet are shown by gray bullets.

Figure 39

(Color online) Densities (left) and phases (right) in a two-component rotating Bose-Einstein condensate, as obtained from the Gross-Pitaevskii equations, for a ratio of atom numbers in the two components of $N_A∕N_B=0.36$ and equal coupling strengths $g_{AA}=g_{AB}=g_{BB}=50\mathrm{a.u.}$ (The densities are cut in one quadrant in order to visualize them for both components in one diagram.) The rotational frequency is $\Omega =0.45$. The upper panel shows a coreless vortex at angular momentum $L=N_A$, where the smaller component $A$ shows a unit vortex at the center, as seen from the phase of ${\Psi }_A$ plotted to the right (from dark to light shading), changing by $2\pi$ when the center is encircled once. The phase singularity is absent in $B$ component. The lower panel shows the case $L=N_B$, where now the larger component encircles the smaller one, filling the unit vortex at the center. The phase singularity consequently now occurs in the order parameter of $B$ component, as shown to the right (from dark to light). From 25.

Figure 40

Schematic of a Skyrmion (top and perspective). The spin tilts from “up” for one component at the center, where one component shows a maximum density filling the vortex in the other component, to “down” toward the edge. From 235.

Figure 41

(Color online) Densities (left) and phases (right) of the Gross-Pitaevskii order parameters in a two-component rotating Bose-Einstein condensate with a coreless vortex with a double phase singularity. For notation see Fig. 39. Adapted data from 25.

Figure 42

(Color online) Angular momentum as a function of the trap rotation frequency (in arbitrary units) for $N=8$ bosons with equal masses and interactions, in a harmonic trap, for equal population $(N_A=N_B=4)$ and unequal population ($N_A=2$ and $N_B=6$). From 24.

Figure 43

(Color online) Density plots of conditional probabilities for a two-component few-boson system in a harmonic trap, with two bosons in component $A$ and six bosons in component $B$. The reference point was chosen in component $B$ at an off-center position close to the maximum of the probability density. Axes are from $(-4,4)$ in atomic units. The scale in the density plots is from zero to maximum. (To increase the visibility, the plot range of the conditional probabilities in the two components was re-scaled to the same constant in all panels.) The charge deficiency of the vortex cores causes deep minima to appear in the pair-correlation functions. Adapted from 24.

Figure 44

(Color online) Mean-field order parameters (left) and phases (right) of a symmetric condensate with $N_A=N_B$, at $L=1.2(N_A+N_B)$, for equal coupling strengths $g_{AA}=g_{AB}=g_{BB}=50$, showing (a) one and (b) two interlaced coreless vortices in the two components. Component $B$ is only shown in a half-plane to make the vortex in the component $A$ visible. Adapted from 25.

Figure 45

(Color online) As in Fig. 43, but for equal components $N_A=N_B=4$ (left panel). The density minima in one component coincide with the density maxima in the other component. This suggests that these states are finite-size precursors of interlaced vortex lattices that occur in the limit of large $N$. The picture becomes much more clear for larger particle numbers, as shown in the right panel for $N=20$ and $L=26$. From 24.

Figure 46

(Color online) Reduced wave functions in a symmetric system of $N_A=N_B=3$ bosons, showing the correlations between phase singularities (circles) with the most probable positions of the particles of opposite species (triangles). This is an indication for the formation of coreless vortices one by one in the system as the angular momentum increases. (a) The nonrotating state, (b) a state with one coreless vortex per particle species, c) two coreless vortices, and (d) three coreless vortices are shown. Note that for identical components $A$ and $B$, the reduced wave functions for the two species are necessarily symmetric, and only one component is shown here. From 284.

Figure 47

(Color online) Angular momentum as a function of the trap rotation frequency (in arbitrary units) for $N=8$ fermions with symmetric and asymmetric component occupations, in Fig. 42. From 24.

Figure 48

(Color online) Pair-correlated densities for fermions, as in Fig. 43, but here for fermions with Coulomb interactions ($N_A=2$ and $N_B=6$). Angular momenta corresponding to the pronounced plateaus in Fig. 47 are shown. Compared to the bosonic case the densities show analogous structures except for an additional exchange hole at the reference point in component B which is reflected also in the component A due to Coulomb repulsion. From 24.

Figure 49

(Color online) Pair-correlated densities for fermions, as above, but for $S_z=0$, i.e., equal components ($N_A=4$ and $N_B=4$). From 24.

Figure 50

(Color online) Reduced wave functions in a two-component system. In a two-component fermion droplet with symmetric occupations $N_A=N_B=3$ the reduced wave function in the lowest Landau level reveals coreless vortices as correlations between phase singularities (circles) with the most probable positions of the particles of opposite spin (triangles). (a) The MDD state with total spin $S=3$ and $S_z=0$, (b) a state with one coreless vortex per particle species, (c) two coreless vortices, and (d) three coreless vortices are shown. This sequence of states is analogous to that of a bosonic system in Fig. 46. Note that vortices of the MDD state are not shown in order to ease the comparison to the bosonic case. From 284.

Figure 51

Partially polarized states beyond the maximum density droplet reconstruction, obtained from a variational Monte Carlo study by 296. The diagrams show the different states of a six-electron quantum dot as a function of the magnetic field and the strength of the Zeeman coupling per spin in the lowest Landau-level (LLL) approximation (upper panel) and including Landau-level mixing (LLM) (lower panel). The states are labeled as $(N_{\uparrow },\Delta L)$, where $N_{\uparrow }$ is the number of electrons with spins parallel to the magnetic field and $\Delta L=L-L_{\mathrm{MDD}}$ is the additional angular momentum with respect to the MDD. The Zeeman coupling strength for GaAs is marked by dashed lines. The confinement strength is $\hslash \omega =5\mathrm{meV}$ and the material parameters are for GaAs, $m^{\ast }∕m_e=0.067$ and ${ϵ}_r=12.4$.

Figure 52

(Color online) Reduced wave function of the $L=24$ fermion state with symmetric occupations $N_A=N_B=3$. The nodal structure closely corresponds to that of the $q=2,p=1$ Halperin wave function with one phase singularity in the component of the probing particle and two phase singularities in the opposite component. The approximate Landau-level filling for the above finite-size system is $\nu \approx 2∕3$, just as for the Halperin state proposed to describe the $\nu =2∕3$ quantum Hall plateau. The symbols in the figure were explained in Fig. 3. From 284.

Figure 53

Density plots of the Gross-Pitaevskii order parameters of the three components $(m=-1,0,1)$ for a $T=1$ ferromagnetic condensate (see text). The calculation was performed for $1.7\times {10}^4$ bosonic atoms of ${}^{87}\mathrm{Rb}$. Length units in the figure are in oscillator lengths. The total angular momentum per particle for the state shown was $L∕N=1.85$, and the rotation frequency in units of the trap frequency was $\Omega =0.17$. White color indicates maximum density. From 221.