Quantum Rings for Beginners: Energy Spectra and Persistent Currents S

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Physica E 21 (2004) 1–35 www.elsevier.com/locate/physe Review Quantum rings for beginners: energy spectra and persistent currents S. Viefersa;∗, P. Koskinenb, P. Singha Deoc, M. Manninenb

aDepartment of Physics, University of Oslo, P.O. Box 1048 Blindern, N-0316 Oslo, Norway bDepartment of Physics, University of Jyvaskyl$ a,$ FIN-40014 Jyvaskyl$ a,$ Finland cS.N. Bose National Centre for Basic Sciences, JD Block, Sector III, Salt Lake City, Kolkata 98, India Received 30 April 2003; accepted 12 August 2003

Abstract

Theoretical approaches to one-dimensional and quasi-one-dimensional quantum rings with a few electrons are reviewed. Discrete Hubbard-type models and continuum models are shown to give similar results governed by the special features of the one-dimensionality. The energy spectrum of the many-body states can be described by a rotation–vibration spectrum of a “Wigner molecule” of “localized” electrons, combined with the spin-state determined from an e7ective antiferromagnetic Heisenberg Hamiltonian. The persistent current as a function of the magnetic 8ux through the ring shows periodic oscillations arising from the “rigid rotation” of the electron ring. For polarized electrons the periodicity of the oscillations is always the

8ux quantum 0. For nonpolarized electrons the periodicity depends on the strength of the e7ective Heisenberg coupling and changes from 0 ÿrst to 0=2 and eventually to 0=N when the ring gets narrower. ? 2003 Elsevier B.V. All rights reserved.

PACS: 73.23.Ra; 73.21.Hb; 75.10.Pq; 71.10.Pm

Keywords: Quantum ring; Persistent current; Electron localization

Contents 9.3. Bethe ansatz ...... 14 9.3.1. Ground state of the t-model ...... 14 1. Introduction ...... 2 9.3.2. Higher bands ...... 15 2. Experimental situation ...... 3 9.4. Vibrational bands ...... 16 2 3. Strictly 1D-ring with noninteracting spinless electrons ... 4 9.5. 1=dij interactions ...... 17 4. Localization of noninteracting spinless fermions in 1D .. 5 9.6. Finite U ...... 18 5. Classical interacting electrons in a strictly 1D ring...... 7 10. Quasi-1D-continuum rings: Exact CI method and 6. E7ect of a magnetic 8ux—persistent currents ...... 8 e7ective Hamiltonian ...... 20 7. Noninteracting particles with spin ...... 10 11. Exact diagonalization: ÿnite magnetic ÿeld ...... 21 8. Noninteracting electrons in a lattice ...... 11 11.1. Flux inside the ring ...... 21 9. Interacting electrons on a lattice: the Hubbard model ... 11 11.2. Homogeneous magnetic ÿeld, no Zeeman splitting . 23 9.1. Model and exact diagonalization ...... 11 12. Periodicity of persistent current ...... 24 9.2. The t-model ...... 12 12.1. Strictly 1D rings: Spectrum of rigid rotation ...... 24 12.2. Periodicity change in quasi-1D rings ...... 25 13. Other many-body approaches for quasi-1D continuum rings 27 ∗ Corresponding author. 13.1. Quantum Monte Carlo ...... 27 E-mail address: [email protected] (S. Viefers). 13.2. Local density approximation ...... 28

14. Relationto Luttingerliquid ...... 28 possibility is to assume a ring-shaped smooth external 15. Pair correlation ...... 29 conÿnement where interacting electrons move. In this 16. Interactionofthe spinwith the magneticÿeld: the continuum model the electron–electron interaction is Zeemane7ect ...... 29 usually the normal Coulomb interaction (e2=4 r). 17. E7ect of animpurity ...... 30 0 18. Summary ...... 32 In the case of a small number of electrons (typically N¡10) the many-particle problem is well deÿned in Acknowledgements ...... 33 both models and can be solved with numerical diago- References ...... 33 nalization techniques for a desired number of lowest energy states. Other many-particle methods have also been applied for studying the electronic structure of quantum 1. Introduction rings. The discrete rings can be solved by using the Bethe ansatz [10,11], which becomes powerful espe- The observationof the Aharonov–Bohmoscilla- cially in the case of an inÿnitely strong contact inter- tions [1] and persistent current [2] in small conducting action(the so-called t-model). In the case of contin- rings, on one hand, and the recent experimental devel- uum rings, quantum Monte Carlo methods and density opments in manufacturing quantum dots [3] and rings functional methods have been used (see Section 13). [4] with only a few electrons, on the other, have made The purpose of this paper is to give anintroductory quantum rings an ever increasing topic of experimen- review to the many-particle properties of rings with a tal research and a new playground for many-particle few electrons. Our aim is not to give a comprehensive theory in quasi-one-dimensional systems. review of all the vast literature published. The main Many properties of the quantum rings can be ex- emphasis is to clarify the relations between di7erent plained with single-electron theory, which in a strictly methods and to point out general features of the elec- one-dimensional (1D) system is naturally very sim- tronic structures of the rings and their origins. Most ple. On the contrary, the many-particle fermion prob- of the results we show inÿgures are our owncom- lem in1D systems is surprisinglycomplicated due putations made for this review. However, we want to to enhanced importance of the Pauli exclusion princi- stress that most of the phenomena shown have been ple. In general, correlations are always strong, leading published before (inmanycase by several authors) to non-fermionic quasiparticles as low energy excita- and we will give reference to earlier work. tions. It is then customary to say that strictly 1D sys- We take anapproach where we analysethe tems are not “Fermi liquids” but “Luttinger liquids” many-body excitation spectrum and its relation to with speciÿc collective excitations (for reviews see the single particle spectrum and electron localization Refs. [5–8]). along the ring. We will show that, irrespective of The many-particle approach normally used in study- the model, the excitationspectrum innarrowrings ing the properties of Luttinger liquids starts by assum- can be understood as a rotation–vibration spectrum ing an inÿnitely long strictly one-dimensional system. of localized electrons. The e7ect of the magnetic 8ux In small ÿnite rings a direct diagonalization of the penetrating the ring is also studied as the change of many-body Hamiltonian, using numerical techniques the spectrum as a function of the 8ux. This is used to and a suitable basis, is possible and can provide more analyze the periodicity of the persistent current as a direct informationonthemany-particlestates. Two function of the 8ux. Again, it is shown that similar di7erent theoretical models have been used for the ÿ- results are obtained with the discrete and continuum nite rings. In a discrete model the ring is assumed models. to consist of L discrete lattice sites (or atoms) with Throughout this paper we use the term “spinless N electrons,which canhop from site to site. Inthis electrons” to describe a system of completely polar- case the electron–electron interaction is usually as- ized electronsystem, i.e. the many-particlestate hav- sumed to be e7ective only when the electrons are at ing maximum total spin and its z-component. We use the same site. The many-particle Hamiltonian is then lower case letters to describe single particle proper- a Hubbard Hamiltonian [9] or its extension. Another ties and capital letters for many-particle properties S. Viefers et al. / Physica E 21 (2004) 1–35 3

(e.g. m and M for angular momenta). We use terms of nanoscopic rings containing only a few electrons like “rotational” and “vibrational” states quite loosely [4,13,14]. At the same time, many of the experi- for describing excitations, which in certain limiting ments still study mesoscopic rings with hundreds of cases have exactly those meanings. When we talk electrons. Methods of forming such rings include about the rotational state we will use the terminology lithographic methods for forming individual rings of the nuclear physics and call the lowest energy state ona semiconductorsurface. The spectroscopic tech- of a given angular momentum an yrast state. niques are based on the tunneling current through the Experimentally, the study of the spectra of quantum ring or capacitance spectroscopy. Another possibility ringis still inanearly state of animpressive devel- is to create a large number of self-organized rings opment. It is not yet the time to make detailed com- on a substrate. The large number of rings allows for parisonbetweenthe theory andexperiments.Never- observationof direct optical absorption. theless, we will give inSection 2 a short overview of One of the hallmarks in this ÿeld of research has the experimental situation. beenthe experimentalobservationof the Aharonov– We thenattempt to review the theory of quantum Bohm e7ect [1] or, equivalently, persistent currents. rings in a logical and pedagogical way, starting with One of the main challenges in order to observe this the simplest case of noninteracting spinless fermions purely quantum mechanical e7ect, has been to en- (Sections 3 and 4) and classical interacting electrons sure phase coherence along the circumference of the (Section 5), then introducing the e7ect of magnetic ring. Early experiments in the eighties and nineties re- 8ux (Section 6) and spin (Section 7). Lattice models ported observations of Aharonov–Bohm oscillations are presented in Sections 8 and 9, followed by numer- and persistent currents in metallic (Au or Cu) rings ical approaches (Sections 10, 11, 13). The periodi- [15–17] and in loops in GaAs heterojunctions, i.e. city properties of the many-body spectrum is discussed two-dimensional electron gas [18–24]. A related ef- inSection 12. We also brie8y discuss the relationof fect which has received recent theoretical [25–29]and these previous approaches to the Luttinger liquid for- experimental [30–33] attention, is the occurrence of a malism (Section 14), and introduce pair correlation spinBerry phase [ 34] in conducting mesoscopic rings. functions as a tool to study the internal structure of the The simplest example of this topological e7ect 1 is 1 many-electron state (Section 15). Most of the review the phase picked up by a spin 2 which follows adia- will deal with rings where the external magnetic 8ux batically an inhomogeneous magnetic ÿeld; the Berry penetrates the ring in such a way that the magnetic phase is then proportional to the solid angle subtended ÿeld is zero at the perimeter of the ring (Aharonov– by the magnetic ÿeld it goes through. It has been Bohm 8ux) and the ring is free from impurities. Nev- shown[ 27] that in1D rings,a Berry phase may arise ertheless, inSections 16 and 17 we will give short due to spin–orbit interactions. The ÿrst experimental overviews of the e7ects of the Zeeman splitting and evidence of Berry’s phase in quantum rings was re- impurities onthe excitationspectrum. ported in1999 by Morpurgo et al. [ 30] who inter- Several interesting aspects of quantum rings will preted the splittingof certainpeaks inFourier spectra have to be neglected in this paper. Among these are of AB oscillations as being due to this e7ect. Very the exciting possibilities of observing experimentally recently, Yang et al. [33] observed beating patterns the spinBerry phase (see, however, Section 2 for a in the Aharonov–Bohm conductance oscillations of brief experimental overview) and, even more exoti- singly connected rings; these results are interpreted as cally, fractional statistics (anyons) [12]. an interference e7ect due to the spin Berry phase. There is by now a vast literature on experiments on quantum rings, and in the remainder of this section we 2. Experimental situation will just discuss a few selected papers which report measurements of the many-body spectra, as this is the Sincethe mid-1980s, there has beenanimpres- maintopic of this review. sive experimental development towards smaller and smaller 2D quantum rings; with the most recent 1 Infact, the Aharonov–Bohmphase canbe regarded as a special techniques one has reached the true quantum limit case of the Berry phase. 4 S. Viefers et al. / Physica E 21 (2004) 1–35

Fuhrer et al. [35] used anatomic force microscope angle ’ to oxidize a quantum ring structure on the surface 2 2 ˝ 9 im’ of a AlGaAs-GaAs heterostructure. By measuring the H = − ; m(’)=e ; (1) 2m R2 9 ’2 conductance through the dot they could resolve the e so-called additionenergyspectrum (see e.g. Ref. [ 36]) where R is the ring radius, me the electronmass (or ef- as a function of the magnetic ÿeld. The energy levels fective mass) and m˝ is the angular momentum. (Note in the ring and their oscillation as a function of the that the direction of the angular momentum axis is al- magnetic ÿeld could be explained with a single particle ways ÿxed in a 2D structure). The corresponding en- picture assuming small nonspherical disturbation for ergy eigenvalues are the ring. The ring had about 200 electrons. ˝2m2 = : (2) m 2 Lorke et al. [4,14,37] have succeeded to produce 2meR a few electron quantum rings from self-assembled We will ÿrst study spinless electrons, or a polarised InAs dots on GaAs using suitable heat treating. They electronsystem, where each electronhas a S = ↑.In used far-infrared (FIR) transmission spectroscopy and z the noninteracting case the many-body state is a single capacitance–voltage spectroscopy to study the ground Slater determinant. The total angular momentum is and excited many-body states. The ring radius was es- N timated to be R0 = 14 nm and the conÿning potential M = mi (3) strength ˝!0 = 12 MeV (the conÿning potential is as- 1 ∗ 2 2 i sumed to be 2 m !0(r − R0) ). Lorke et al. were able to study the limit of one and two electrons in the ring. and the total energy The experimental ÿndings were in consistence with a N E = : (4) single electron picture. mi Warburtonet al. [ 38] studied the photoluminiscence i from self-assembled InAs quantum rings at zero mag- The lowest energy state for a given angular momen- netic ÿeld. By using a confocal microscope and tak- tum, or the yrast state, is obtained by occupying single ing advantage of the fact that each ring had unique particle states consecutively (next to each other). This charging voltages they were able to measure the pho- is due to the upwards curvature of m. We will denote toluminiscence from a single quantum ring. The pho- states consisting of single particle states with consec- toluminiscence spectra show the e7ect of the Hund’s utive angular momenta, say from m0 to m0 + N − 1, rule of favoring parallel spins and other details of the as “compact states”. These states have energy spectra, for example the singlet–triplet splitting. ˝2 The periodicity = h=e of the persistent current E = Nm2 +(N 2 − N)m 0 CS 2m R2 0 0 predicted for normal metal (not superconducting) e rings has been observed several times for semicon- 1 + (2N 3 − 3N 2 + N) (5) ductor quantum rings [18–21,23]. Also observations 6 of other periodicities, or higher harmonics, have been and the angular momentum reported [24,39] but they have beeninterpretedas ef- fects of the nonperfectness of the rings. Very recently, N(N − 1) M = Nm0 + : (6) the ÿrst observation of the inherent 0=N periodicity, 2 which should appear in perfect very narrow rings, It is interesting to note that while the single parti- was reported [40]. cle energy increases with the angular momentum as 2 2 2 ˝ m =2meR , the lowest many-body energy increases, 2 2 2 inthe limit of large M,as˝ M =2(Nme)R . Anyrast 3. Strictly 1D-ring with noninteracting spinless state of the ring with N electrons corresponds then to electrons a single particle with mass Nme. This is our ÿrst notion of “rigid rotation” of the quantum state. The single particle Hamiltonian of an electron The structure of the yrast states is illustrated in a strictly 1D ring depends only on the polar schematically inFig. 1, and the actual energy levels S. Viefers et al. / Physica E 21 (2004) 1–35 5

a state, we refer to interparticle correlations which canbe seenby usinga rotatingframe, as discussed by Maksym inthe case of quantumdots [ 41].) This follows from the notion that N Slater Slater M+N ({’i}) = exp i ’i M ({’i}); (7) i where is any integer. This kind of change of the total angular momentum corresponds to a rigid rotation of the state and naturally leads to the above mentioned result that the N-electronsystem rotates like a single particle with mass Nme. Moreover, correlation Slater Slater functions are the same for both M+N and M since they are derived from the square ||2. Note that Fig. 1. Conÿgurations of the many-body states of eight noninter- the minima of the yrast spectra occur at angular mo- acting electrons in a ring for spinless electrons. A and A give menta M = N if N is odd and at angular momenta local minima in the yrast line, while B and C correspond to M = N + N=2ifN is even. “vibrational excitations”.

4. Localization of noninteracting spinless fermions

50 in 1D

45 Classical noninteracting particles do not have phase 40 transitions and are always in a gas phase: There is 35 nocorrelationbetweenthe particles. Inthe quantum 30 mechanical case, due to the Pauli exclusion principle, 25 one-dimensional spinless fermions behave very dif- E(M) 20 ferently: Two electrons cannot be at the same point.

15 This means that noninteracting spinless fermions are

10 identical with fermions interacting with an inÿnitely

5 strong delta-function interaction. The requirement that the wave function has to go to zero at points where 0 0 5 10 15 20 M electrons meet, increases the kinetic energy proportional to 1=d2 where d is the average distance between Fig. 2. The lowest total energy of eight noninteracting electrons the electrons (this is analogous to the kinetic energy of as a function of the total angular momentum, i.e. the yrast line. a single particle in a one-dimensional potential box). Black dots: spinless electrons; open circles: electrons with spin. The local minima are connected with dashed lines. The requirement of the wave function being zero in the contact points has the classical analogy that the electrons cannot pass each other [7]. The pressure of the kinetic energy scales as an interparticle energy of 2 2 as a function of the angular momentum are shown the form 1=[R (’i − ’j) ], leading to the interesting inFig. 2. Naturally, the compact states give local result that the energy spectrum of the particles inter- minima of the yrast spectrum. In what follows, the acting with the -function interaction agrees with that most important property of the many-particle states of particles with 1=r2 interaction. In fact, the model is that the internal structure of the state does not of 1D particles with a 1=r2 interaction, the so-called change when the angular momentum is increased by Calogero–Sutherland model [42,43], is exactly solv- a multiple of N. (By the term “internal structure” of able. We will returnto this inSection 9.5. 6 S. Viefers et al. / Physica E 21 (2004) 1–35

We will now demonstrate that the noncompact states may infact be regarded as vibrational states. The simplest case is that with two electrons. The Slater determinant is (omitting normalization)

im1’1 im2’2 im1’2 im2’1 m1m2 (’1;’2)=e e − e e : (8) The square of the amplitude of this wave function can be writtenas

| (’ ;’ )|2 = 4 sin2[ 1 (RmR’)]; (9) m1m2 1 2 2 where Rm=m1−m2 and R’=’2−’1. This means that the maxima of this wave function occur at points R’= (1+2n) =Rm where n is aninteger.The compact ground state has m1 = 0 and m2 = 1 occupied, i.e. Rm = 1 while all noncompact states have Rm¿1. This means that between 0 and 2 there will be one maximum for the ground state, the two electrons being at the opposite side of the ring. For the noncompact states there are two or more maxima between0 and2 and the wave functions will resemble excited states of a harmonic oscillator, i.e. vibrational states (actually inthe two-electroncase they are exactly those of a single particle in a 1D potential box). Let us now generalize this analysis to a general case with N electrons. To this end we determine the 0.0 A 0.6 square of the many-body wave function, |({’ })|2, i Fig. 3. The correlation of di7erent quantum states of noninteracting in terms of the normal modes of classical harmonic electronswith the electronlocalizationat the sites determinedby vibrations. The equilibrium positions of classical par- the classical harmonic vibrational modes, as a function of the am- 0 ticles are ’j =2 j=N. The displaced positions of the plitude of the atomic displacement from the classical equilibrium particles corresponding to a normal mode are position, .i.e. |({’})|2 as a function of A of Eq. (10). The panels from top to bottom correspond to quantum states (↑↑↑↑↑↑↑↑), ’ = ’0 + A sin i − 1 2 =N ; (10) (↑↑↑↑↑↑↑ ◦ ↑), (↑↑↑↑↑↑ ◦ ↑↑), (↑↑↑↑↑ ◦ ↑↑↑), (↑↑↑↑ ◦ ↑↑↑↑), i i 2 respectively, where ◦ refers to an empty angular momentum state. The solid, long-dashed, short-dashed, and dotted lines correspond where is aninteger(1 :::N=2) and A the amplitude to = 1, 2, 3, and 4, respectively. of the oscillation. 2 Fig. 3 shows |({’i })| for the di7erent yrast states of a ring with 8 electrons, as a function of the amplitude A of the classical oscillation. In the case of a maximum |2| of the corresponding ÿgure. Clearly, compact state (denoted by A in Fig. 1) ||2 decreases there seems to be a correspondence between the dif- rapidly with increasing A for all , as seeninthe up- ferent yrast states of noninteracting spinless electrons permost panel. This means that particles appear as be- and vibrational modes of classical interacting parti- ing localized at the sites of classical particles with a cles ina ring. repulsive interaction. For the noncompact yrast states The conclusionofthis sectionis that the yrast (B and C in Fig. 1 and so on) the maximum of ||2 spectrum of noninteracting spinless particles can is reached with a ÿnite value of A so that = 1 corre- be understood as rotational vibrational spectrum sponds to state B, =2 corresponds to C, etc. The insets of classical particles with a repulsive interaction: of the ÿgure show the positions of the electrons at the The compact states are purely rotational states, and S. Viefers et al. / Physica E 21 (2004) 1–35 7 the noncompact states correspond to vibrational excitations.

5. Classical interacting electrons in a strictly 1D ring

A ÿnite number of classical interacting particles in a strictly 1D ring will have discrete vibrational frequencies, which, after quantization, will give the quantum mechanical vibrational energies ˝!.Weas- sume a monotonic repulsive pairwise potential energy V (r) betweenthe particles, where r is the direct inter-particle distance. The potential energy can be writtenas 1 N ’ − ’ E = V 2R sin i j ; (11) 2 2 i=j Fig. 4. Vibrational modes of 40 classical particles in a 1D ring. where R is the radius of the ring and ’i the position Di7erent symbols correspond to di7erent forms of the repulsive (angle) of particle i. For any pair potential V it is interactionbetweentheparticles, as indicatedinthe ÿgure. The straightforward to solve (numerically) the vibrational solid line is the result of a nearest neighbor harmonic model. The energies are scaled so that the highest vibrational energy for each modes. In the case of a short range potential, reach- potential is one. ing only to the nearest neighbors, one recovers the text-book example of acoustic modes of an inÿnite 1D lattice [44] (now only discrete wave vectors are allowed due to the ÿnite length 2 R) gives the energy spectrum ˝! = C sin ; (12) E = Erot(M)+Evib({n}) N ˝2M 2 = + n ˝! ; (13) where C is a constant (proportional to the velocity of 2NmR2 sound) and aninteger. Fig. 4 shows the classical vibrational energies for √ where M is the (total) angular momentum and n the di7erent pair potentials −ln(r), 1= r,1=r,1=r2,and number of phonons . 1=r7. Naturally, when the range of the potential gets Fig. 5 shows the rotational vibrational spectrum shorter the vibrational energies approach that of the derived from Eq. (13) for eight particles with 1=r2 nearest neighbor interaction, Eq. (12), also shown. interaction. It agrees exactly with that calculated Note that the vibrational energies of the Coulomb in- quantum mechanically for electrons with (an inÿnite) teraction(1 =r) do not di7er markedly from those of -function interaction (see Section 9). It is important 1=r2-interaction. The latter has the special property to notice that since the vibrational states are fairly that the energies agree exactly with the quantum me- independent of the interparticle interaction, the quan- chanical energies of noninteracting spinless fermions, tum mechanical spectrum close to the yrast line is as discussed inSection 9.5. expected to be qualitatively the same irrespective of Inadditionto the vibrationalenergy,the classical the interaction. system can have rotational energy determined by the Quantum mechanics plays an important role, how- 1 2 ˙ 2 2 angular momentum: Erot = 2 NmR Â (NmR being ever, when the Pauli exclusion principle is consid- the moment of inertia and Â˙ the angular velocity). ered. The requirement of the antisymmetry of the total Quantization of the vibrational and rotational energies wave function restricts what spin-assignments can be 8 S. Viefers et al. / Physica E 21 (2004) 1–35

100

70 E(M)

40 0 5 10 15 20 M

Fig. 5. Low-energy spectrum of eight electrons interacting with inÿnitely strong -function interaction. The black dots indicate the state with maximum spin (“spinless electrons”). White circles give the energies of other spin states. The spectrum is identical with the rotation–vibration spectrum of eight particles interacting with 1=r2-interaction.

 combined with a certain rotation–vibration state. For  B0r  if r 6 rc; 2 example, for a completely polarized ring (maximum A = (14) ’ 2 spin) only certain rotational vibrational states are al-  B0r  c = if r¿r; lowed, as showninFig. 5. Group theory canbe used 2r 2 r c to analyze the possible spin-assignments [45,46]ina 2 similar way as done in rotating molecules [47,48]. which gives a 8ux = rc B0 penetrating the ring in Inthe case of bosons(with S = 0) the total such a way that the ÿeld is constant inside rc and zero wave function has to be symmetric, and the allowed outside. If the ringis inthe ÿeld-free region( R¿rc) rotation–vibration states are exactly the same as for the electron states depend only on the total 8ux pen- spin-1/2-particles in the fully polarized state. Fig. 5 is etrating the ring. then also a “general” result for eight bosons in a quan- The solutions of the single particle SchrTodinger tum ring (assuming the interaction to be repulsive). equation of an inÿnitely narrow ring, 1 i˝ 9 e 2 − − (’)= (’); (15) 2me R 9 ’ 2 R 6. E&ect of a magnetic 'ux—persistent currents can be still written as 8ux-independent plane waves We consider a magnetic 8ux going through the m = exp(im’), but the corresponding single particle quantum ring in such a way that the magnetic ÿeld is energies now depend on the 8ux as zero at the radius of the ring. This can be modeled, ˝2 2 (m; )= m − : (16) for example, by choosing the vector potential to be (in 2m R2 circular cylindrical coordinates) e 0 The many-body wave function is still the same Slater Ar = Az =0; determinant as without the ÿeld, while the total energy S. Viefers et al. / Physica E 21 (2004) 1–35 9 becomes [1]. Inparticular, if anelectronmoves alonga closed ˝2M path around the 8ux tube, the Aharonov–Bohm phase E(M; )=E(M; 0) − 2 becomes meR 0 −ie −ie ˝2N 2 R& = ∇& dl = A dl + : (17) ˝ ˝ 2m R2 2 R 2 R e 0 −ie −ie The e7ect of increasing 8ux is not to change the level = B ds = : (23) structure but to tilt the spectrum, say of Fig. 2, such ˝ R2 ˝ that the global minimum of the total energy jumps In other words, the boundary condition has now from one compact state to the next compact state. Note changed. While the original wave function satisÿes that Eq. (17) is true also for interacting electrons in periodic boundary conditions, (’)= (’ +2 ), a strictly one-dimensional ring. This follows from the for the new wave function we have the condi- fact that any good angular momentum state can be tion (’ +2 )= (’) exp(−ie=˝), i.e. “twisted written as a linear combination of Slater determinants boundary conditions”. Note that this boundary of noninteracting states. For each of these, the last condition naturally leads to periodic eigenvalues two terms are the same, while E(M; 0) depends on the (m)= (m + = ), inthe same way as the Bloch interactions. 0 conditionforelectronstates ina periodic lattice leads Alternatively, one may perform a unitary trans- to the periodicity of the eigenenergies in the recipro- formationto obtaina descriptionof the system in cal lattice [44]. terms of a ÿeld-free Hamiltonian, but with multival- Spectrum (16) is clearly periodic in8ux with period ued wavefunctions (“twisted boundary conditions”). .Anygiven eigenstate = exp(i(m − = )’), Let us choose a gauge A = ∇& (which ensures 0 m 0 however, will have its angular momentum eigenvalue that B = ∇×A is zero at the ring; in a strictly shifted by one as the 8ux is changed by one 8ux quan- one-dimensional ring one may write &(’)=’=(2 ) tum. and consider the unitary transformation The persistent current of a quantum ring can be → = U ; (18) writteningeneralas 9F H → H = UHU −1; (19) I()=− ; (24) 9 where the operator U is deÿned as where F is the free energy of system. To illustrate this U =e−ie=˝ A dl =e−ie=˝&: (20) inthe simplest possible case, considerthe SchrTodinger equationfor a one-electronring, Obviously, the eigenspectrum will be conserved un- ˝2 der this transformation—if H = E , then H = − D2 (’)=E (’); (25) UHU −1U = UH = E . It is easy to show that the 2m m m m e7ect of this transformationontheHamiltonianisto where cancel out the gauge ÿeld, i.e. 1 9 ie −i˝ 9 −i˝ 9 D = − A : (26) U U −1 = + eA ; (21) R 9 ’ ˝ ’ R 9 ’ R 9 ’ ’ Multiplying both sides of Eq. (25)by ∗ from the left and the Hamiltonian takes the ÿeld-free form m and integrating along the circumference of the ring ˝2 92 H = − : (22) one gets 2m R2 9 ’2 e 1 2 Meanwhile, the wave function now picks up a phase E = − R d’ ∗(’)D2 (’): (27) m 2m m m (−ie=˝)& when moving along a given path, even e 0 though it moves ina regionwhere the magneticÿeld Using expression (14) for the gauge ÿeld and taking is zero. This is the so-called Aharonov–Bohm e7ect the derivative with respect to 8ux one obtains, after 10 S. Viefers et al. / Physica E 21 (2004) 1–35 anintegrationbyparts, 9E 1 ie˝ m = 9 2 R 2me 2 ∗ ∗ ∗ R d’[ mD m − mD m ] (28) 0 1 2 = − d’j(’) ≡−Im; (29) 2 0 where we have identiÿed the RHS with the 1D current (density) associated with the state m. (Note that j(’) has to be independent of the angle ’.) Obvi- ously, the same argument applies for a noninteracting many-body system where the contributions from dif- ferent angular momentum states are simply summed. The same is true in the presence of interactions due to the fact that all 8ux dependence is in the kinetic energy term of the many-body Hamiltonian (see dis- cussionafter Eq. ( 17)). Due to the periodicity of the energy spectrum, the persistent current will be a periodic function of the Fig. 6. Conÿgurations of the many-body states of eight nonin- 8ux. In the case of noninteracting spinless electrons, teracting electrons with spin in a ring. A, A, and D give local the period is , owing to the fact that the minimum minima in the yrast line, while B and C correspond to “vibrational 0 excitations”. energy for any 8ux corresponds to a compact state. As will be seeninSection 12, the electron–electron interactions can change the periodicity to 0=2orto 0=N. are deeper. In the case of noninteracting electrons with spinthere is notsuch a clear relationto classical rotation–vibrationspectrumas inthe case 7. Noninteracting particles with spin of spinless noninteracting case discussed above. One could consider spin-up and spin-down states The spindegree of freedom allows two elec- separately as spinless systems which do not in- trons for each single particle orbital. The yrast teract with each other. However, considering the states of the many-body spectrum are still con- complete many-body spectrum one should no- sisting of compact or nearly compact states, but tice that the energy of state B in Fig. 6 is de- now for each spin component as shown in Fig. generate with the state where the spin-up system 6. The corresponding yrast spectrum for 8 elec- has ÿve electrons and the spin-down system only trons is shown in Fig. 2 incomparisonto the three. spectrum of spinless electrons. The total spin of The persistent current of the noninteracting the states A (inFig. 6)isS = 0 while for all system with the spinis againa periodic func- other states have either S = 0 (singlet) or S =1 tionof the 8ux. The period is 0 like inthe (triplet) as caneasily be deducted from Fig. case of spinless electrons. When the ÿeld is in- 6 by considering the possible ways to arrange creased, the ground state shifts from a compact the Sz-components in the orbitals with only one state (of the kind A in Fig. 6) to the next sim- electron. ilar state A’, i.e. the angular momentum shifts The yrast spectrum now consists of downward with N. At the transition point the three lev- cusps at angular momenta M = nN=2, but those els, A, D and A’, are actually degenerate, but minima corresponding to S = 0 compact states the D-states do not change the periodicity of the S. Viefers et al. / Physica E 21 (2004) 1–35 11 persistent current. However, the amplitude of the we choose the tight-binding parameters as 0 =2t and 2 2 persistent current is only half of that of the spinless t = ˝ =2mea . In this case the quantum number m gets electrons. the meaning of the orbital angular momentum. (This So far we have considered only even numbers of equivalence of the tight-binding model and free elec- electrons. In the case of an odd number of electrons, tron model is valid for simple lattices of any dimension the spinless case is very similar to that of even num- and can be derived also by discretizing the Laplace op- bers, the only di7erence being a phase shift of 0=2 erator of the free particle SchrTodinger equation [50]). of the periodic oscillations. However, in the case of The many-body state is again a simple Slater de- electrons with spin, the odd number of electrons has terminant of the single particle states with the total ane7ect also onthe amplitude of the persistentcur- energy rent: For a large number of electrons the amplitude N 2 for odd numbers of electrons is exactly twice that of E = N − 2t cos m ; (32) 0 L j even numbers of electrons, as ÿrst shown by Loss and j=1 Goldbart [49]. This e7ect is sometimes referred to as a parity e7ect. where the selection of the “angular momenta” mj is restricted by the Pauli exclusionprinciple.For example, in the polarized case (spinless electrons) all mj must be di7erent, and the ground state is obtained with 8. Noninteracting electrons in a lattice a compact state where the mj’s are consecutive integers as inthe case of free electrons.Similarly, we can Instead of a continuum ring, we will now consider identify “vibrational states” by making a hole in the noninteracting electrons in a strictly 1D ring with a compact state as will be demonstrated in more detail strong periodic potential. In this case the standard in the following sections. solid state physics approach is the tight-binding model where the Hamiltonianmatrixcanbe writtenas 9. Interacting electrons on a lattice: the Hubbard 0 if j = i; Hij = (30) model −t if j = i ± 1; 9.1. Model and exact diagonalization where the diagonal terms describe the energy level at a lattice site and the o7-diagonal terms describe the hop- A much studied approach to quantum rings and per- ping between the levels. This simplest tight-binding sistent currents is the Hubbard model [51–58]. It de- model assumes one bound state per lattice site and is scribes electrons on a discrete lattice with the freedom oftencalled the H uckelT model or CNDO-model (com- to hop between lattice sites, and the Coulomb inter- plete neglect of di7erential overlap). action is represented by an on-site repulsion. An in- Assuming the ring, with radius R, to have L lattice teresting feature of the Hubbard ring is that, despite sites, the problem of solving the eigenvalues becomes being a strongly correlated electron system, it can be a text-book example of 1D band structure [44], and solved exactly. For a small number of particles the the singleelectroneigenvaluescanbe writtenas solutioncanbe foundby direct diagonalizationofthe 2 Hamiltonian matrix as discussed ÿrst. For any num- (k)= − 2t cos(ka)= − 2t cos m ; (31) 0 0 L ber of particles another solution technique, the Bethe ansatz, can be used. This technique is most suitable where a is the lattice constant, k the wave vector and in the limit of inÿnite U (so-called t-model) and will m aninteger.The last step follows from the facts that be addressed in the next subsection. The Hamilto- the lattice constant is a =2 R=L, and in a ÿnite ring nian describing the Hubbard model for an N-electron k will have only discrete values k = m=R. Notice that ring with L lattice sites, in the presence of a magnetic inthe large L limit, m=L → 0, the energy spectrum 8ux , = =0 piercing the ring, can be derived with is equivalent with that of free electrons, Eq. (2), if the help of the unitary transformation introduced in 12 S. Viefers et al. / Physica E 21 (2004) 1–35

Section 6 (the 8ux dependence was ÿrst derived by described by the simpler “t-model” Hamiltonian Peierls [59]): [62,63] N −i2 ,=L † i2 ,=L † Ht = PHkinP; (34) H = −t (e ci+1;-ci;- +e ci;-ci+1;-) i=1 - where Hkin is the kinetic (hopping) term of (33)and N P denotes a projection operator which eliminates all + U nˆi↑nˆi↓: (33) states with doubly occupied sites. It canbe shown i=1 [62] that this projected Hamiltonian is equivalent † to a tight-binding Hamiltonian describing spinless Here, the operator c (ci;-) creates (annihilates) an i;- fermions. Moreover, going to large but ÿnite U, electronwith spin - at site i;ˆn =c† c is the number i- i;- i;- near half ÿlling, the Hubbard model reduces to the operator for spin-- electrons at site i. The ÿrst part so-called t − J model, i.e. the t-model (34) plus a of (33) describes the hopping of electrons between Heisenberg term J S · S [63]. Inother words neighboring sites (“kinetic term”) while the last part i i i+1 the translational and spin degrees of freedom get gives the repulsionbetweenelectronsoccupyingthe decoupled. We will returnto the t − J model in same site. We will set the hopping parameter t = 1 for Section 9.6. simplicity. Inthe nextsubsectionwe will describe how so- For small numbers of electrons N and lattice sites lutions of the t-model can be constructed using the L the Hubbard Hamiltoniancanbesolved exactly Bethe ansatz. However, for a small number of elec- by diagonalization of the Hamiltonian matrix. We trons and lattice sites (N and L) the direct diagonaliza- use anoccupationnumberbasis (see e.g. Ref. [ 60]) tion has the advantage of giving all the eigenvalues at | = |n ;n :::n ; n :::n where the / /1↑ /2↑ /L↑ /1↓ /L↑ once and some information of the many-body state is z-component of the total spin is ÿxed, i.e. n =N i /i↑ ↑ more transparent. The results shown for small discrete and n = N (N = N + N ). Taking matrix i /i↓ ↓ ↑ ↓ rings are obtained either with diagonalization tech- elements |H| gives us a matrix with dimen- / / niques or using the Bethe ansatz solution. It should be sion( L )( L ). The eigenvalues of this matrix are N↑ N↓ stressed that both methods are exact and give the same the exact many-body energy levels of the Hubbard results. Hamiltonian. For a given total spin S, the energy We will ÿrst study four electrons (with spin) in a eigenvalues do not depend on Sz (there is no Zeeman lattice having at least one empty site. Fig. 7 shows the splitting since we assume the magnetic ÿeld to be energy spectrum for L = 5 and 8. The right-hand pan- nonzero only inside the ring). In the case of an even els show the energies as a function of the magnetic N we canchoose Sz =0,orN↑ = N↓ = N=2, and the 8ux piercing the ring, while the left-hand side shows diagonalization of the Hamiltonian will give us all the lowest energies at zero 8ux, analyzed with respect possible eigenvalues (for odd N we take Sz =1=2). to the total angular momentum, for the di7erent “vi- The easiest way to solve the total spinof a given brational bands” in the spectra. The following gen- energystate is thento repeat the computationwith eral features should be noted: The energy (and thus all possible values of Sz and look at the degeneracies. the persistent current) has a periodicity 0=N. Inpar- (Note that the matrix dimension is largest for Sz =0 ticular, as ÿrst discussed by Kusmartsev [53]andby and thus solving the matrix for all Sz ¿ 0 takes less Yu and Fowler [54], inthe groundstate there are N computationthansolvingthe Sz = 0 case). cusps in every 8ux period; increasing the 8ux makes the ground state jump from one angular momentum 9.2. The t-model state to the next. (A more general discussion of the periodicity of the groundstate will be giveninSection Inthe followingwe will ÿrst focus onthe strongre- 12.) Each individual state, onthe other hand,is a har- pulsionlimit U→∞, which was ÿrst discussed nearly monic (cosine) function of the 8ux, with periodicity 20 years ago [61]; we shall returnto ÿnite U ef- L0, inaccordancewith the periodicity of the Hamil- fects later. Inthis limit, the system is equivalently tonian. Thus, the energy of each individual state has S. Viefers et al. / Physica E 21 (2004) 1–35 13

Fig. 7. Energy spectra of rings of four electrons in ÿve (upper panel) and eight (lower panel) sites, calculated with the t-model Hamiltonian. The right hand side shows the energy levels as a function of the 8ux (in units of 0). Onthe left, the zero 8ux energylevels are shown as a function of the angular momentum M (black dots). Note that the L = 8 case shows several vibrational bands. Only negative energy levels are shown since the spectrum is symmetric with respect to zero energy.

the form, up to anoverall phase shift, ever, the above generic 8ux dependence of the lowest 2 p states of the spectra is obtained even for a single free E ∼ cos − − (35) site (i.e. L = 5 for N = 4). For L¿N + 1, there are L N 0 several “energy bands” consisting of states with dif- with p an integer. Obviously, the (kinetic) energy col- ferent amplitude; the number of bands increases with lapses to zero inthe case of half ÿlling, L=N, as inthis the number of empty sites. As will be discussed below case there is no freedom to hop. Surprisingly, how- (Section 9.4), the lowest energy state at any given 8ux 14 S. Viefers et al. / Physica E 21 (2004) 1–35 corresponds to the rotational band without vibrational half-integer. The total (canonical) angular momentum energy, while the higher bands can be interpreted as corresponding to the quantum numbers is given by the vibrational states of the system. N N↑ It should be noted that each individual energy level M = Ij + J/: (39) inFig. 7 still has a spindegeneracy:Two or more j=1 /=1 states with di7erent total spin S belong to each energy level. The magnetic ÿeld cannot separate these since In the general case of a ÿnite U the nonlinear Bethe we have neglected the possible Zeeman splitting. A equations turn out to be diWcult to solve numerically, ÿnite U will separate the states belonging to di7erent at least for some quantum numbers [67]. For small total spinas will be showninSection 9.6, but will still systems (N and L small) it is easier to ÿnd all eigenvalues by a direct diagonalization of the Hamiltonian leave the degeneracy due to the Sz. matrix as explained in the previous section. 9.3. Bethe ansatz In the limit of inÿnite U, however, the Bethe ansatz solutionbecomes particularly simple. Inthis case the The Hubbard model in1D caninfact be solved quantum numbers kj ∈ [0; 2 are simply givenby [53,54] exactly, interms of the Bethe ansatz,as shownby Lieb and Wu in 1968 [11]. The corresponding solution 2 p kj = Ij − + − (40) for a Hubbard ring in the presence of an Aharonov– L N 0 Bohm 8ux has later beendiscussed by a numberof with authors [53,54,56,64–66]. Bethe ansatz solutions can be used to construct not only the ground state but N↑ in fact the entire spectrum presented in the previous p = − J/: (41) section. According to this method, the energy of a /=1 givenmany-bodystate may be writtenas So a givensolutionis constructedfor a ÿxed N↑, i.e. a N ÿxed value of the z-component of the spin. The quan- E = −2 cos kj; (36) tum numbers Ij and J/ are related to the charge- and j=1 spindegrees of freedom, respectively. Inthe case we shall consider, even N, the Ij:s are integers (half-odd where the numbers kj can be found by solving the set of Bethe equations integers) and the J/:s are half-odd integers (integers) if N↑ is even(odd). Inpractice, all eigenvaluescan Lkj =2 Ij − be found by letting p run over all integers and choosing all possible sets of Ij:s with the restriction Imax − N↑ −1 Imin ¡L. − 2 tan [4(sin kj − 4ÿ)=U]; (37) ÿ=1 9.3.1. Ground state of the t-model N In analogy to the noninteracting case, the states 2 tan−1[4(4 − sin k )=U] / j forming the outermost band (see Fig. 7), i.e. states j=1 which become the ground state at some value of the N↑ 8ux, are compact states in the quantum numbers {Ij}. −1 =2 J/ + 2 tan [2(4/ − 4ÿ)=U]: (38) At zero 8ux, these are of course just the yrast states ÿ=1 (see left-hand panels in Fig. 7). For example, if N and M are both even, the quantum numbers {I } are con- For obtaining the whole energy spectrum, the un- j secutive integers, I = −N=2; −N=2+1;:::;N=2 − 1. known (complex) constants k and 4 have to be j j ÿ The corresponding total energy, Eq. (36), as function solved for di7erent quantum numbers I and J , which j / of 8ux, becomes [54] are restricted to be integers or half-integers depending on the numbers N and N↑. For example, if N 2 p E0 = −Em cos − − + Dc ; (42) and N↑ both are even, Ij must be an integer and Jj a L 0 N S. Viefers et al. / Physica E 21 (2004) 1–35 15 where Dc =(Imax + Imin)=2and for ÿxed 8ux, all energy levels of the lowest band, corresponding to di7erent p. sin(N =L) E =2 : (43) m sin( =L) 9.3.2. Higher bands The integer p should be chosen, for given 8ux, such Generally, excitations can be constructed within the as to minimize the energy. Of course, there are in Bethe ansatz by introducing holes inthe groundstate general several ways of choosing the “spin quantum distribution of the quantum numbers Ij and J/ [66]. numbers” J/ to give the same sum, leading to a large Naturally, these states are related to the “noncompact” degeneracy of the state. This has to do with the fact states of noninteracting fermions in a strictly 1D ring, that spinexcitationsare massless inthe limit of the discussed inSection 3. The higher bands in our spectra t-model; for ÿnite U this degeneracy would be lifted. correspond to charge excitations, i.e. holes in the Ij Note that the energy collapses to zero for half ÿlling, (J/-excitations do not cost any energy in the inÿnite i.e. when L = N. U limit), and this reproduces exactly all the energies As a simple illustration, consider the example N =4 obtained numerically in the previous section. We will at zero 8ux. Then, Ij = −2; −1; 0; 1, Dc = −1=2and illustrate this procedure with a few examples for N =4, M = 2. The lowest possible “excitation” in the charge 2 p 1 3 cos k = cos − − − quantum numbers Ij is lifting the topmost one by one j L N 2 2 step, j 2 p 1 1 {Ij} = −2; −1; 0; 2; (48) + cos − − − (44) L N 2 2 and the corresponding kj as giveninEq. ( 40). The resulting energy is 2 p 1 1 +cos − − + L N 2 2 E1 = −2 cos kj (49) 2 p 1 3 j +cos − − + (45) L N 2 2 4 2 = −2 1+2cos + cos L L 3 =2 cos + cos 2 p L L ×cos − − L N 0 2 p 1 ×cos + (46) 2 2 p L N 2 − 2 sin sin − + − : (50) L L N 0 sin(4 =L) 2 p 1 = cos + : (47) This again gives a band of states which are cosine sin( =L) L N 2 functions of the 8ux, with period L0. Di7erentiating E wrt. / ≡ 2 =L(−p=N − = ), one ÿnds the “am- It is easy to check that the “amplitudes”, −2 sin(4 =L)= 1 0 plitude” of this band as sin( =L) agree with those inFig. 7 for L = 5 and 8. Note that the choice of the “central value” D as 4 2 c E = −2 1 + 2cos + cos cos / anoverall phase shift makes the sum symmetric; one 1;min L L min obtains pairs, cos[2 =L(−p=N + D ± 5)], so that, c 2 whenwritingout the sum, all sin(2 5=L)-terms cancel. −2 sin sin / (51) L min This also implies that shifting all the Ij:s by the same amount, does not alter the total energy (42); it only with changes the overall phase shift Dc. The integer p is related to the angular momentum M, see Eq. (39). −1 sin(2 =L) /min = tan : (52) This construction gives not only the ground state but, 1 + 2 cos(4 =L) + cos(2 =L) 16 S. Viefers et al. / Physica E 21 (2004) 1–35

Table 1 Total weight of the most important conÿgurations of the many-body states of di7erent vibrational states for the t-model ring with L =8 and N =4

Conÿguration nw0 w1 w2 ◦•◦•◦•◦• 8 0.1248 0.0000 0.0000 ◦•◦••◦•◦ 32 0.1824 0.2136 0.0624 ◦••◦◦••◦ 16 0.0624 0.0000 0.1256

The conÿguration is shown as ÿlled and empty circles indicating whether or not there is a electron in the corresponding site. The second column shows the number of such states (not including the degeneracy coming from the spin conÿguration). The last three columns show the total weights of these states for the vibrational ground state (w0) and the two vibrational bands. The total weights are the same for all rotational states and di7erent spin states belonging to the same vibrational mode.

Note that at zero 8ux, the lowest state of this band they no longer “see” the delta function interaction), in may have anenergy larger than E1;min—the nearest a similar way as the simple tight-binding model ap- minimum may occur at a ÿnite 8ux. This is because p proaches to the continuum model when the number has to be an integer, and (LN=2 )/min is not generally of lattice sites increases (see Section 8). One way of aninteger(except for some special values of L). illustrating that this is indeed the case, is to examine The simplest example for four particles is L =6. the ratios Inthis case / = =3, i.e. the energy of the state min En − E0 (48) has a minimum at zero 8ux with p=N = −1and Rn ≡ ; (54) √ E1 − E0 E1;min = −cos( =3) − 3 sin( =3) = −2. This method is easily generalized to construct where Ei is the (minimum) energy of the ith excited higher bands. For example, the second excitation band band. In the case of noninteracting, spinless particles the energy, in units of ˝2=(2m R2), of an N-particle corresponds to, for four particles, {Ij} = −2; −1; 1; 2 e with energy state at 8ux , = =0 is givenby 2 4 N E2 = −4 cos + cos 2 L L E = (mi − ,) ; (55) i=1 2 p ×cos + : (53) where mi are the single particle angular momenta and L N 0 the total angular momentum is M = mi. The mini- The higher (inner) bands constructed in this way mum inenergyof this state occurs at , = M=N and is may be interpreted as corresponding to vibrational ex- N 2 2 givenby Emin= i=1 mi −M =N. As discussed inSec- citations. As an example, we have examined, for the tion 3, the ground state corresponds to ÿlling consecu- N =4,L=8 solutions of the t-model, which electron– tive angular momentum states, e.g. mi =0; 1;:::;N−1. hole conÿgurations have the largest amplitude (in the A set of N=2 excited states is constructed by creat- many-body wavefunction) in each of the three lowest ing one-hole excitations, the lowest one being mi = bands. Table 1 shows the weights of the most impor- 0; 1;:::;N − 2;N. (Note that anoverall shift of all tant basis states for the ground state and for the two the angular momenta does not change the minimum lowest vibrational states. These are consistent with the energy—it just occurs at a di7erent 8ux.) Computing classical motionof electronsinthe ÿrst vibrational the ratios Rn as deÿned in Eq. (54)oneÿnds modes. n(N − n) Rn ≡ : (56) 9.4. Vibrational bands N − 1 As we shall see, these one-hole excitations in a sense Inthe limit L →∞, i.e. inÿnitely many sites, the correspond to fundamental phonon excitations. More- t-model is expected to correspond to a system of non- over, it is easy to show that two-hole excitations lead interacting (spinless) particles in the continuum (as to ratios (inthe limit N →∞) which are twice those S. Viefers et al. / Physica E 21 (2004) 1–35 17 inEq. ( 56), thus corresponding to two-phonon exci- where 5 =(4 − 4)=4. In other words, the energy and tations, etc. total angular momentum take the same form as in Now, the same ratios canbe recovered from the the noninteracting case, with the interactions absorbed Bethe ansatz solution of the t-model, as givenby in the shift of the quasi-momenta. For each nonin- Eqs. (36)and(40): Constructing the set of excitations teracting many-body state characterized by a set of which correspond to one hole in the quantum numbers fermionic quantum numbers {mj} there is a corre- Ij, minimizing the energy as described in the previ- sponding state with quantum numbers {pj}. Note that ous section, and taking the limit L →∞, one again P = j pj = j mj = M. Now consider the Suther- gets Eq. (56). This shows that the energy bands of land model in the presence of an Aharonov–Bohm 8ux the Hubbard model reduce to those of noninteracting piercing the ring. The energy of a given state can then particles in the limit of inÿnite repulsion and inÿnite be writtenas number of sites. (Of course the correspondence be- 2 tweenthe t-model and free particles can also be seen E = (pj − ,) (59) j using a similar argument as in Section 8: For the maximum spinstate, −p = J = 0 inEq. ( 39) so that / and, as in the noninteracting case, is minimized for the integers I correspond just to the single particle j , = M=N, angular momenta.) E = p2 − M 2=N: (60) 9.5. 1=d2 interactions min j ij j

There is another type of interaction in one dimen- We want to compute the shift in energy of a sionwhich leads to the same set of excitationbands many-body state as compared to the noninteracting 2 as above, namely Vij =1=dij, where dij is the distance one, which then will be used to compute the ratios Rn. betweenparticles i and j. Models in1D with this type Inserting the solution (58) into Eq. (59) one ÿnds, up of interactionareknownasthe Calogero- (ona line) to an overall constant, or Sutherland model (on a circle) [42,43] and have beenstudied extensivelyover the past three decades. RE(5)=Emin(5) − Emin(5 =0) What makes the Calogero (or Sutherland) model spe-   cial, is that it mimics as closely as possible a system N   of free particles. Consider the N-particle Sutherland = 5 jmj − (N +1)M : (61) Hamiltonian, j=1 N 2 2 Again constructing the set of one-hole excitations (in HS = − 9 =9xi mj) as before, one easily ÿnds the di@erence between i=1 the shift of the nth excitationandthe shift inthe ground 1 state, +24(4 − 1) ; (57) 2 (2 sin((xi − xj)=2)) i¡j RE(n)(5) − RE(0)(5)=5(n(N − n)): (62) where we have set the radius of the circle to 1. The complete excitationspectrum of this model can From this it immediately follows that the ratios Rn for be found exactly in terms of the asymptotic Bethe the Sutherland model are the same as in the nonin- ansatz (see e.g. Ref. [68] and references therein) teracting case, see Eq. (56). Inaddition,recall Sec- and is remarkably simple. The total energy and an- tion 5 where we performed a calculationof the nor- gular momentumof a givenstate canbe writtenas mal modes for a set of classical particles ona ring 2 2 E = j pj and P = j pj, respectively, where the with 1=d repulsion. In a semiclassical picture, the quasi-momenta pn are related to the free angular corresponding frequencies !j were theninterpretedas momenta m by the eigenfrequencies of the many-body problem, with j N +1 energies ˝! . Using these to compute the ratios R , p = m + 5 j − ; (58) j n j j 2 one again obtains the same expression as above. This 18 S. Viefers et al. / Physica E 21 (2004) 1–35

Table 2 Energy states of a Heisenberg ring with four electrons, sorted according to the spin S and total angular momentum M quantum numbers

SSz ME 00 2−2J 00 0−J 1 −1; 0; 100 1 −1; 0; 110 1 −1; 0; 130 2 −2; −1; 0; 1; 22 J

M is determined with the help of the symmetry of the state.

that the energy of the higher groups of levels increase as E ≈ U and 2U. Fig. 8. Energy levels as a function of U for a four electron The high-U limit of the lowest group of states can Hubbard ring with four sites. be explained with the so-called tJ -model. Inthe limit of large U the half-ÿlled Hubbard model canbe ap- proximated as (see Ref. [63] and references therein) illustrates that the “single-particle excitations”, both in the noninteracting case and in the Sutherland model, J N 1 H = H + S · S − ; (63) can indeed be interpreted as corresponding to vibra- tJ t 2 i j 4 tional modes. (Cf. the discussion of the free particle i=j case inSection 9.) where Ht denotes the Hamiltonian of the inÿnite U 2 limit (t model), J =4t =U and Si is a spinopera- 9.6. Finite U tor (S =1=2). Inthe case of a half-ÿlled bandthe t-model gives only one state (with zero energy). In the The Hubbard model with ÿnite U canstill be solved large U limit all the low energystates canthenbe de- with the Bethe ansatz but now it requires numerical scribed with an antiferromagnetic Heisenberg Hamil- solution of the set of nonlinear equations (37). Inthe tonian which separates the di7erent spin states. The case of small number of electrons and sites a direct case with four electrons is especially easy to solve numerical diagononalization of the Hubbard Hamilto- (see exercise 30.3. of Ref. [44]), leading to energy nian, Eq. (33), is infact easier. The results shownin levels showninTable 2. The table also shows the or- this sectionhave beencomputed with the direct nu- bital angular momentum determined from the symme- merical diagonalization. As an example case we use try properties of the Heisenberg state. againthe 4 electronring. Inthe case of a large U the magnetic 8ux does not Fig. 8 shows the energy spectrum for four electrons have any e7ect if N =L due to the fact that the electron in four sites as a function of on-site interaction U. motion is strongly hindered. The situation changes if The results start from the noninteracting case (with empty sites are added. Fig. 9 shows the development spin). Increasing U separates the spectrum into dif- of the low energy states as a function of the magnetic ferent groups. The lowest group corresponds to states ÿeld and U inthe case of four electronsineight sites. where all the electrons are mainly at di7erent lattice When U is reduced from inÿnity, the di7erent spin sites while the two higher bands correspond to states states separate, causing the periodicity of the yrast line where one or two sites have double occupancy, re- to change from 0=N at U = ∞ to 0 at U = 0. This spectively. This canbe seenby lookingat the struc- change of the periodicity is addressed in more detail ture of the many-body states or simply by noticing inSection 12. S. Viefers et al. / Physica E 21 (2004) 1–35 19

-4

-5 energy

-6

-7 U=0 U=7 U=15 U=50 U=1000

ΦΦΦΦΦUUUU

Fig. 9. Flux and U dependence of the many-body states of the Hubbard model with N = 4 and L = 8. For each ÿxed U the 8ux goes from 0 to 0; inbetween, = 0 and U increases linearly to the next ÿxed value. The lowest overall energy and the lowest energy state corresponding to the maximum spin are shown as thick lines. Note that the maximum spin state is independent of U and that the periodicity of the yrast state changes from 0 to 0=4 when U increases from 0 to ∞.

The e7ect of ÿnite, but large U, is to split the de- 1 generacy of the di7erent spin states of the t-model. If there are no empty sites, the Hamiltonian approaches 0 that of the tJ-model, Eq. (63). We have solved the spectrum of four electrons as a function of the 8ux, in- -1 creasing the number of sites from 4 to 12. The results show that when the number of empty sites increases, energy the spectra are still infair agreementwith those of the -2 tJ -Hamiltonian, but the e7ective coupling between the spins, J, is not any more 4t2=U, but decreases rapidly -3 when the number of empty sites increases. Fig. 10 shows the di7erence of the energy spectra -4 0 1 2 3 derived from the exact Hubbard spectrum and from angular momentum M the t-model, for di7erent values of empty sites. The results are scaled so that the energy di7erence be- Fig. 10. The Heisenberg model energy spectrum for four electrons tweenthe two spinstates correspondingto M = 0 are determined from the Hubbard model with empty sites. The black the same. We cansee that whenthe numberof empty dots are the results for four electrons with U = 100 and di7erent number of sites: For each state the dots from left to right correspond sites increases, the di7erence spectrum approaches to L = 5, 6, 8, 10, and 12, respectively. For comparison, the solid that of the Heisenberg Hamiltonian. These results lines denote the half-ÿlling case (no empty sites), i.e. L =4.In suggest that the separationof the Hubbard Hamilto- each case the energy di7erence of the two M =0 states is adjusted nian (for large U) into the t-model and Heisenberg to be 1. Opencircles show the same for U = 10. Hamiltonians is accurate in the two limits, N=L → 1 and ∞. Moreover, the agreement with all numbers of Yu and Fowler [54] have used the Bethe ansatz empty sites is surprisingly good, even if U is as small to study the large U limit for any number of empty as 10. sites and shown that the e7ective Je7 is related to the 20 S. Viefers et al. / Physica E 21 (2004) 1–35

1.2

1 N=4

0.8

0.6 JU/4

0.4

0.2

0 0 2 4 6 8 10 12 14 L

Fig. 11. The dependence of the e7ective Heisenberg coupling J onthe numberof sites ( L). The crosses show the asymptotic large U limit of Eq. (64), the black dots (opencircles) the result determinedfrom the exact solutionfor U = 100 (U = 10).

quantum numbers kj as they were a natural continuation of earlier calculations done for electrons in harmonic two-dimensional quan- 4 N J = sin2k : (64) tum dots [36]. Usually the quantum ring is described e7 LU j j=1 with a displaced harmonic conÿnement (although several other models have beenused [ 69,70]) Fig. 11 shows the e7ective coupling constant deter- V (r)= 1 m !2(r − r )2; (65) mined by this equation as a function of the number of 2 e 0 0 sites for a four electronring,compared to those ob- where r0 is the radius of the ring and !0 the per- tained from the direct diagonalization of the Hubbard pendicular frequency of the 1D wire. Note that we Hamiltonian for U =10andU = 100. The agreement still assume the ring to be strictly two-dimensional, is fairly good for any number of empty sites even for i.e. it is inÿnitelythininthe directionperpen- the relatively small U = 10. The decrease of Je7 as a dicular to the plane of the ring. The parameters function of L canbe explainedwith electron“localiza- describing the ring, r0 and !0 canbe related to tion”: When the number of empty sites increases, the the density parameter rs =1=(2n0) of the 1D sys- localized electrons move further apart reducing the ex- tem (n0 is the 1D density) and to a parameter change interaction. We ÿnd, once again, that in the 1D describing the degree of one-dimensionality of system the electrons with repulsive (even -function) the wire. For the latter, Reimann et al. [45,71] interactions behave as localized particles. used a parameter CF deÿned with the relation 2 2 2 ˝!0 = CF˝ =(32mrs ). The physical meaning of CF is that it is the ratio of the ÿrst radial excitation 10. Quasi-1D-continuum rings: Exact CI method in the ring to the Fermi energy of the 1D electron and e&ective Hamiltonian gas. There are several approaches to solve the many-body Inthis sectionwe review electronicstructure calcu- problem of interacting electrons in the above po- lations for quasi-1D rings, published in Refs. [45,46]; tential. In studying the spectral properties the most S. Viefers et al. / Physica E 21 (2004) 1–35 21 useful method is the brute force diagonalization of it can be determined by solving the antiferromagnetic the many-body Hamiltonian in a proper basis. This Heisenberg model for a ring of six electrons. The ra- conÿguration-interaction (CI) method gives the whole tios of the energies of di7erent spin states are quanti- many-body energy spectrum as well as the corre- tatively the same as inthe Heisenbergmodel. sponding many-body states. Naturally, the solution The lower panel of Fig. 12 shows that this is true can be numerically accurate only for small numbers also for a wider ring. In this case only the vibrational of electrons, typically less than 10. The matrix di- ground state is clearly separated from the rest of the mension can be reduced by ÿxing the orbital angular spectrum. However, the internal structure of this yrast momentum to the desired value. For example, Kosk- band is still very close to that of the Heisenberg model: inen et al. [45] ÿrst expanded the solutions of the Qualitatively the agreement is exact, i.e. each angular single particle part of the Hamiltonian in a harmonic momentumhas the right spinstates inthe right or- oscillator basis and then used these functions as a der. Only the energy ratios are not any more exactly single particle basis for the Fock space in doing the the same as in the Heisenberg model. Koskinen et al. CI calculations. According to their eigenenergies, [46] have studied inmore detail how well the model up to 50 single-particle states were selected to span Hamiltonian (66) describes the exact many-body re- the Fock space and the number of the many-body sults for a ring of six electrons. Fock-states was restricted to about 105 using another Exact CI calculations for a ring of 4 [72], and 5 energy cuto7 (for a given total M). For aneven and7[45] electrons show similar agreement. Fig. 13 number of electrons all spin states can be obtained shows the energy spectra for a four electron ring with with ÿxing N↑ = N↓ = N=2. The total spinof each a comparisonto a four electrondot. With such a small state can afterwards be determined by calculating the number of electrons even the dot shows nearly the expectationvalue of the Sˆ2 operator. same yrast spectrum as the ring. This indicates that Fig. 12 shows the calculated energy spectra ob- also inthe dot the electronswill localize ina square tained with such a calculation for two di7erent rings Wigner molecule (see also Ref. [73]). Similar local- with six electrons in each. It is instructive to introduce ization of electrons and their rotational and vibrational a simple model Hamiltonian [45] spectra are observed also intwo-electronquantumdots ˝2 [74] (where the energy spectrum can be solved exactly H = M2 + ˝! n + J S · S ; (66) e7 2I i j [75]). When the number of electrons in a quantum dot i;j is 6 or larger the classical conÿguration of electrons where ! are the frequencies and the integers n the inthe Wignermolecule [ 76] is not any more a single number of excitation quanta of the various vibrational ring and the spectral properties become more com- normal modes, and I is the moment of inertia of the plicated. Nevertheless, eventhere the polarized case “molecule”. This Hamiltonian is thus simply a com- shows rotational bands consistent with the localization binationofrigid rotationof the whole system, inter- of electrons [77]. We should point out that the idea of nal vibration, and a Heisenberg term to capture the describingfew-electronsystems interms of rigid ro- spin dynamics, and one may examine how well it de- tation and internal vibrations, is not new, but was ÿrst scribes the exact results. To this end, note the follow- applied to quantum dots by Maksym [41]. ing interesting features in Fig. 12: The narrower ring shows clear rotation–vibration bands, very similar to those obtained for electrons in a continuum ring with 11. Exact diagonalization: ÿnite magnetic ÿeld -function interaction, cf. Fig. 5. The only di7erence is that now the ratios of the vibrational states corre- 11.1. Flux inside the ring spond to those determined for classical electrons interacting with 1=r interaction. Let us ÿrst consider a magnetic ÿeld concentrating Each spectral line consists of several nearly degen- in the center of the ring, as described with the vector erate spinstates. The insetshows as anexample the potential (14). Since the r-component of the vector detailed structure of the M =0 state. The spinstructure potential is zero and the ’-component is proportional coincides with that determined from the tJ-model, i.e. to 1=r, the only e7ect of the 8ux is to change the 22 S. Viefers et al. / Physica E 21 (2004) 1–35

Fig. 12. Energy spectra for two quasi-one-dimensional continuum rings with six electrons (in zero magnetic ÿeld). The upper panel is for a narrow ring and it shows several vibrational bands. The lower panel is for a wider ring which shows stronger separation of energy levels corresponding to di7erent spin states (shown as numbers next to the energy levels). Note that also the narrow ring has the same spin-ordering of the nearly degenerate state as expanded for the lowest M = 0 state. From Ref. [45].

angular momentum term of the single particle It is easy to see that for integer =0 the energy lev- SchrTodinger equation as els and single particle states are equivalent to those ˝2m2 ˝2(m − = )2 without the 8ux, they are just shifted to another angu- → 0 : (67) 2 2 lar momentum value. Since each Slater determinant of 2mer 2mer S. Viefers et al. / Physica E 21 (2004) 1–35 23

Dot with 4 electrons The observed periodicity of the ground state energy, or persistent current, as a function of 8ux is determined by the variation of the yrast-line as a function of M, for example as showninFig. 12. If the situa- 1 tionis like inthe upper panelof Fig. 12, i.e. whenthe 0 2 1 ring is very narrow, the energy increases accurately as 2 0 1 0 M . Inthis case Eq. ( 68) gives the periodicity 0=N. 1 Onthe other handinthe case of the lower panelof Fig. 12 the minimum energy jumps from M = 0 to 3 and then to M = 6, whenthe 8ux is increased.This means a periodicity of 0=2. If the ring is made even Ring with 4 electrons wider, eventually the minimum at M = 3 will not be reached and the periodicity changes to 0. Finally, we should notice that if the electron gas were polarized (spinless electrons), the periodicity 0 would always be . This canbe seenfrom Fig. 12 1 0 1 which shows that the maximum spinstate occurs in 2 the lowest vibrational band only at the angular mo- 0 1 0 mentum M = 3 (more generally, at M = N=2 for even 1 number of particles and at M = 0 for odd number of Angular Momentum M M=0 M=4 particles). Fig. 13. Low energy spectra of a four electron quantum dot (upper panel) and a four electron quantum ring (lower panel) as a function 11.2. Homogeneous magnetic ÿeld, no Zeeman of the angular momentum. The numbers next to the energy levels splitting give the total spin. CI calculation from [72]. Experimentally it would be easier to measure the quantum rings in the presence of a homogeneous mag- the many-body state has a good angular momentum netic ÿeld. This case was ÿrst treated by ÿrst princi- quantum number, the same is true for the many-body ple calculationmethods inthe pioneeringpapers by state: The angular momentum M is shifted to that of Chakraborty and PietilTainen [78] and by NiemelTaet M −N when a integer number ==0 of 8ux quanta al. [70], and the present section is mainly based on is penetrating the ring (as in Eq. (17) for a strictly these papers. Inthis case the vector potentialcanbe 1D ring). This means that we get the same picture as expressed, for example, interms of a symmetric gauge discussed before: The e7ect of the increasing 8ux is 1 A = 2 (−By; Bx; 0). This vector potential e7ectively just to tilt the spectrum so that successively higher and adds an additional harmonic conÿnement centered at higher angular momentum values become the ground the origin. The r-dependent single particle potential state. The ground state energy will be periodic with changes as 8ux: EM ()=EM+N( + 0). Note that this is true 1 2 2 1 2 2 evenif the M-dependence of the many-body spectrum 2 me!0(r − r0) → 2 me!B(r − rB) is not exactly proportional to M 2; the only require- +constant; (69) ment is that the 8ux is restricted in the central region of the ring and the magnetic ÿeld does not overlap 2 2 2 2 2 2 2 where !B = !0 + e B =(4me)andrB = r0!0=!B. The with the singleparticle states. Inthis case we canwrite e7ect of the ÿeld is thenjust to changethe parameters the model Hamiltonian describing the yrast spectrum of the conÿning ring. This will change the energy dif- of the exact CI calculationas ferences between the single particle states, but as long as the ring is narrow enough to have only one radial ˝2 N 2 H = J S · S + M − : (68) mode, it cannot change their order. Consequently, in i j 2 2NmeR 0 i;j narrow rings the e7ect of the ÿeld on the many-body 24 S. Viefers et al. / Physica E 21 (2004) 1–35

The right hand side of Fig. 15 shows the energy levels for a quasi-1D-ring with 1=r-interaction. The spectra are essentially the same, the only di7erence being a slight upward shift of the energy when the 8ux changes from 0 to 1. This is due to the harmonic repulsion of the 8ux-dependent e7ective potential, Eq. (69), caused by the homogeneous magnetic ÿeld. Note the similarities of the spectra showninFigs. 15b and d.

12. Periodicity of persistent current

12.1. Strictly 1D rings: Spectrum of rigid rotation

Fig. 14. Energy levels for four noninteracting (dashed lines) and The many-electronexcitationspectrumfor elec- interacting (solid lines) spinless electrons in a ring as a function trons interacting with the inÿnitely strong -function of the magnetic 8ux. The magnetic ÿeld is homogeneous. From interaction was studied using the Hubbard model in Ref. [78]. Section 9, see Fig. 7. As mentioned, this spectrum can be constructed from the Bethe ansatz. In the low-energy part of the spectrum (close to the yrast state is small and only quantitative. Nevertheless, the line) the levels consist only of the compact states of change of the potential shape and the constant term the quantum numbers Ij, i.e. there is no vibrational means that the lowest single particle state increases energy. It is instructive to use the continuum limit with the 8ux [78], as showninFig. 14 for spinless (L=N →∞)) of the Bethe ansatz solution for the electrons. The ÿgure also indicates that the e7ect of t-model to compare these (yrast) energy levels as the electron–electron interactions in the spinless case a function of the 8ux to those of single electron is mainly to shift the spectrum upwards by a constant. states ina ringof radius R. One ÿnds: NiemelTa et al. [70] have performed anextensive ˝2 2 E(M; )= M − study of quasi-1D-rings with 2 to 4 interacting elec- 2Nm R2 h=Ne trons. They used a homogeneous magnetic ÿeld and e neglected the Zeeman splitting. The results for two and (+constant) many electrons; (70) three interacting electrons show periodicities 0=2and ˝2 2 0=3, respectively, i.e. consistent with the 0=N peri- (m; )= m − 2 odicity for a narrow ring. In the case of four electrons 2meR h=e NiemelTa et al. studied inadditionto the 1 =r Coulomb single electron; (71) interaction also a -function interaction (V0(r)) in the case of an inÿnitely narrow ring. It is then inter- where we have now written the 8ux quantum as 0 = esting to compare the results of these two models, as h=e. We notice that the many-electron states are identi- showninFig. 15. For the -function interaction the cal to the singleelectronlevels with the electronmass results are expected to be the same as for the Hubbard and charge, me and e, replaced by the total mass and model with an inÿnite number of lattice sites. Indeed, total charge of all the electrons, Nme and Ne. Indeed the results show the change of periodicity from 0 the strongly correlated electron system with inÿnitely ÿrst to 0=2 and then to 0=4, whenthe strengthof strong delta function interaction behaves as a rigidly the -function interaction (V0) is increased. The re- rotating single particle. We should note that in the sults of Fig. 15 compare well with the results of the spinless case the number p of Eq. (40) is always zero Hubbard model with only 8 sites, i.e. those shown in and consequently one recovers the noninteracting case Fig. 9 for U =0,15and50. (for spinless electrons the -function interaction does S. Viefers et al. / Physica E 21 (2004) 1–35 25

Fig. 15. Energy levels as a function of the magnetic 8ux in a ring of four electrons. The left-hand side shows the results for delta-function interaction in a strictly 1D ring, (a) for noninteracting, (b) and (c) for interacting so that in (c) the interaction is twice as strong as in (b). The right hand panel shows the result for a quasi-1D ring with 1=r-interaction, (c) noninteracting and (d) interacting electrons. Note the similarity of the results in(b) onthe left-handside and(d) onthe right handside. From Ref. [ 71].

not have any meaning due to the Pauli exclusion prin- vibrational level is the ÿrst noncompact state, hav- 2 2 ciple). The rigid rotationof the electronsystem then inganexcitationenergyof about N˝ =2meR . This leads always to the 0 periodicity as discussed already equals the E(M = N; = 0) energy of the yrast band, inSection 3. Eq. (70). For N particles there are thus about N purely We have learned in Sections 10 and 11 onthe basis rotational states below the ÿrst vibrational band. of numerical solutions for narrow quasi-1D rings, that Fig. 7 demonstrates that this is true for N =4. electrons with normal 1=r interactions also produce similar rotational spectrum. Moreover, the solution of 12.2. Periodicity change in quasi-1D rings the Calogero–Sutherland model shows rigorously that a similar spectrum is observed for 1=r2-interaction. Inthis sectionwe will concentrateonthe lowest Both these interactions have the property that they are energy state (the yrast state) and study in more detail inÿnitely strong at the contact, preventing the electron its periodicity in quasi-1D rings, where the electrons to pass each other. It seems obvious that any repulsive are allowed to pass each other. (The periodicity of the interaction which is inÿnitely strong (in such a way persistent current at zero temperature is the same as that electrons with opposite spin are not allowed at that of the ground state energy). As discussed in Sec- the same point) produces for the strictly 1D ring the tion 9.6, the (strictly 1D) Hubbard model suggests that same spectrum of rigid rotation. (i) for spinless electrons the periodicity is always the The yrast spectrum is qualitatively similar for all 8ux quantum 0 and (ii) for electrons with spin the electronnumbers.Inparticular, inexperimentallyde- periodicity changes from 0 ÿrst to 0=2 and then to termined spectra, the number of electrons in the ring 0=N whenthe interaction U increases from zero to canbe seenas a qualitative changeof the spectrum inÿnity, as illustrated in Fig. 9. This change of peri- only by observing the vibrational bands. (The yrast odicity inthe Hubbard model was ÿrst studied by Yu energy alone determines the number of electrons in and Fowler [54] and Kusmartsev et al. [79]. a narrow ring only if the 8ux is quantitatively deter- The changeinperiodicity canbe traced back to the mined.) A way of estimating the number of particles notion that the Hubbard model with empty sites can is to count the number of (purely rotational) states be quite accurately described by the tJe7 -model with below the onset of the ÿrst vibrational band, as can an e7ective exchange coupling Je7 which depends not be seen from the following argument: By considering only on U but also onthe numberof empty sites, as noninteracting spinless electrons (which have exactly demonstrated in Figs. 10 and 11. When Je7 is small, the same energy levels, though not all of them, as the 0=N periodicity is observed, while for large Je7 spinful electrons) we know from Eq. (5) that the ÿrst the 0 periodicity is found (for even numbers of 26 S. Viefers et al. / Physica E 21 (2004) 1–35

narrow ring

wide ring energy

0 1 2 3 4 5 6 7 8 M

Fig. 16. Schematic illustrationof the 0=N and 0=2 periodicities of a ring with eight electrons. The black points show the energy spectra with two di7erent values of J . The parabolas drawn at each point indicate the change of the rotational energy level as a function of the magnetic 8ux. The lowest envelope of the overlapping parabolas gives the ground state energy as a function of the 8ux. electrons). The periodicity of 0=2 results from the question, i.e. all interparticle correlations remain the fact that the solution of the Heisenberg Hamiltonian same. has two close lying states corresponding to angular We canlook at the periodicity changeinmore de- momenta M =N=2 and 0, while the states correspond- tail by making a Fourier analysis of the lowest energy ing to other angular momenta are clearly higher in state. We have done this using the model Hamiltonian energy. The situation is demonstrated in Fig. 16 where (68). A parameter determining the relative weights we show the energy spectrum for the Heisenberg of the energy states of the two separable parts of the model for aneight electronring.The energyspectrum Hamiltonian is then JI, i.e. the product of the moment 2 for the total Hamiltonian as a function of the 8ux can of inertia (I ≈ NmeR ) and the Heisenberg coupling be estimated simply by drawing a parabola at each parameter (in the Hubbard model the corresponding energy level as shown in the ÿgure. Now depending parameter would be Je7 ). Fig. 17 shows the three most on the energy splitting of the Heisenberg model, i.e. important Fourier components as a function of JI for on Je7 , the resulting lowest energy state as a function rings with 6, 7 and 8 electrons. For 6 and 8 electrons of 8ux canhave periodicities, 0=N, 0=2, or 0. we see clearly the period changes. In the case of an Deo et al. [80] have studied the periodicity change odd number of electrons, the solution of the Heisen- using the model Hamiltonian Eq. (68) which is known berg model is qualitatively di7erent: There are two to give good agreement with exact diagonalization degenerate minima corresponding to di7erent angular results of the quasi-1D continuum Hamiltonian. This momenta. Consequently, the period 0=2 stays always model Hamiltonian has the same 8ux dependence more important than the period 0 [80]. as Eq. (70), which was derived from the Hubbard We have learned that the periodicity change as a model. It is thus natural that the same periodicity function of the e7ective width of the quasi-one-dimen- change is observed. In both cases the 8ux only a7ects sional continuum ring is similar to that in a strictly the rotational state of the system by changing its en- 1D Hubbard ring as a function of U, suggesting that 2 2 ergy as M → (M − N=0) , but does not change the ÿnite U inthe Hubbard model mimics the ÿ- the internal structure of the many-body state in nite width of a continuum ring. This similarity can be S. Viefers et al. / Physica E 21 (2004) 1–35 27

N=8

N=7 Fourier component

N=6

0.01 0.1 1 10 JI

Fig. 17. Three largest Fourier components of the periodic ground state energy. JI is the product of the e7ective Heisenberg coupling and the moment of inertia. The dotted, dashed and solid lines are the Fourier components corresponding to the periodicities 0, 0=2and 0=N, respectively. understood as follows. With inÿnite U the electrons the Monte Carlo method [81]. These methods have are forbiddento pass each other. This situationis sim- been extensively applied also for quantum dots [82– ilar to an inÿnitely narrow continuum ring with 1=r in- 86] and quantum rings [87,88] with a few electrons. teractionbetweenthe electrons.When U gets smaller, Monte Carlo approaches are most suitable for studying the electrons (with opposite spin) can hop over each either the ground state properties (variational Monte other, the better the smaller U is. Naturally, inthe Carlo and di7usion Monte Carlo) or average ÿnite continuum ring the electrons are allowed to pass each temperature properties (path integral Monte Carlo), other if the ring has a ÿnite width. Decreasing U in and have thus not been able to produce the detailed the Hubbard model thus corresponds to making the spectral properties with the accuracy of the CI method. continuum ring wider. It also seems that even ÿnding the correct ground state Note that inorder to see the periodicity inthe Hub- is not straightforward by using Quantum Monte Carlo bard model, one has to consider systems where the [89]. 2 number of electrons N is smaller thanthe numberof Pederiva et al. [88] have used so-called ÿxed-node lattice sites L, i.e. empty sites are needed for “free ro- di7usion Monte Carlo for studying six electron quan- tation” of the ring. The resulting periodicity depends tum rings. They calculated also the lowest excited both on the number of empty sites and on the on-site states for M = 0. They found the ground state to have energy U. S = 0 and the ÿrst excited state S = 2, inagreement with the CI calculations and the Hubbard model, while for the second and third excited states they obtained 13. Other many-body approaches for quasi-1D the S =1 and 0 states in opposite order as compared to continuum rings

13.1. Quantum Monte Carlo

2 Still quite recently there have been published several quantum A class of powerful tools to study the low energy Monte Carlo results which give the wrong total spin for as small states of many-body quantum systems are based on system as the four electronquantumdot (see Ref. [ 89]). 28 S. Viefers et al. / Physica E 21 (2004) 1–35 the CI calculations. Nevertheless, as the authors note, = 0=2 the ground state had S = 0. These results the energy di7erences in the narrow rings are so small are in agreement with the CI calculations and the re- that their di7erence starts to be within the statistical sults of the Hubbard model for four electrons. Simi- accuracy of the Monte Carlo method. lar agreement was found for a six electron ring. The The Monte Carlo studies for small quantum dots and spin-densities showed a clear localization of electrons rings show that while these methods can predict accu- in an antiferromagnetic ring. rately the ground state energy, they are not yet capable Density functional theory has the same problem to give reliably the salient features of the many-body as quantum Monte Carlo in that the determination spectrum. of excited states is not straightforward (although time-dependent current-spin-density-functional the- 13.2. Local density approximation ory canprovide some informationonexcitations [92,93,95]). Consequently, it is not possible to con- The density functional Kohn–Sham method is struct the complete excitationspectrum as by using another approach mainly suitable for the determi- the brute force CI method. nation of the ground state structure. In applications to quantum dots and rings (for a review see [36]), the local spin-density approximation (LSDA) is usu- 14. Relation to Luttinger liquid ally made and the system is assumed to be strictly two-dimensional. Generally, the Kohn–Sham method Inÿnitely long one-dimensional systems are often is a “meanÿeld” method, where the electron–electron studied as Luttinger liquids [97,98] (for reviews see correlationis hiddeninane7ective singleparticle Refs. [5–7]). The speciality of the strictly 1D systems potential. This causes the interesting feature that the arises from the fact that the Fermi surface consists of meanÿeld canexhibit symmetry breakingandthe to- only two points (±kF). This leads to a Peierls insta- tal electronandspindensitiescanreveal the internal bility [99] and a breakdown of the Fermi liquid theory symmetry of the ground state, for example the internal in a strictly 1D system. Important low energy excita- shape of nucleus or atomic cluster [90] or the static tions will then be collective, of bosonic nature, and spin-density wave in a quantum dot [91]. Indeed, in have a linear dispersion relation. The Luttinger liq- applications to quantum rings, the LSDA indicated the uid also exhibits the so-called charge-spinseparation: localization of electrons in an antiferromagnetic ring The spin and charge excitations can move with di7er- [71]. Nevertheless, we should add that although the ent velocities. In addition to studying the low energy LSDA canoftenelucidate the internalstructure of a excitations, the Luttinger model has been extensively rotating system, the method is not foolproof: In some used for studying correlation functions [6,8]. cases, for example in rings or dots with high enough It has beenshownthat the tJ -model is a Luttinger electron density, it will not break the symmetry. liquid [6]. In the limit of a narrow ring with many elec- Systematic studies of quantum rings in terms of trons the spectral properties of the quantum rings must density functional methods have been performed in then approach those of a Luttinger liquid. We will now several papers [92–95], and comparisons with “ex- demonstrate that the many-body spectra of quantum act” many-body methods show that the LSDA gives rings are consistent with the properties of the Luttinger accurately the ground state energy [88]. The LSDA liquid. We do this only by qualitative considerations. has been extended to the so-called current-spin-density In an inÿnitely long 1D system, the low energy single functional theory (CSDFT) which can take into ac- particle excitations (of free fermions) are restricted to count the gauge ÿeld [96]. Viefers et al. [94] have ap- have a momentum change of q ≈ 0 due to the fact that plied the CSDFT for studying the persistent current, the Fermi surface is a point. In the Luttinger model it i.e. the ground state energy as a function of the mag- is precisely these excitations that lead to the bosoniza- netic ÿeld, in small quasi-1D quantum rings. For a four tionandcollective plasmonexcitations[ 6] with a lin- electronring(with rs =2:5, CF = 10) they found the ear dispersion relation. In the case of a ÿnite ring these yrast line consisting of two states: At zero 8ux (and single particle excitations are just those described in at = 0) the ground state had S = 1 while around Fig. 1, where one of the last electrons is excited from S. Viefers et al. / Physica E 21 (2004) 1–35 29 the compact state (or similarly inthe Bethe ansatzso- pair correlation functions are quite insensitive on the lutionof the t-model). Now, it is exactly these sin- electron–electron interaction. This is demonstrated in gle particle excitations which lead to the vibrational Fig. 18 where we show the calculated pair correlation model, i.e. longitudinal acoustic phonons in the limit function for two di7erent continuum rings [100]and of a long ring (see Fig. 4), which have a linear dis- compare them to the pair correlationof the Heisen- persionrelation(for small q). The excitationspectrum berg model. We have also studied the pair correlation is inqualitative agreementwith the predictionof the for four electrons using the t-model and found that Luttinger model already in the smallest rings. the correlation is independent of the number of empty Casting the Hamiltonian explicitly into charge sites, as expected from the notion (Section 9.6) that the dependent parts (rotations and vibrations) and a spin–spin correlation is determined from the Heisen- spin-dependent part (Heisenberg Hamiltonian), as in berg model, whatever the number of empty sites. Eqs. (13)and(66) is equivalent to the charge-spin The e7ect of temperature onthe pair correlation separationinthe Luttingermodel. Inaninÿnite function has been studied by Borrmann and Harting system this canbe explicitly donefor the half-full [87] using quantum Monte Carlo. Koskinen et al. [46] Hubbard model (L = N). We have demonstrated in used the model Hamiltonian (66) to determine the Section 9.5 that the spindegrees of freedom canbe temperature dependence by calculating separately the described with a good accuracy with the Heisenberg pair correlationfor each quantumstate. Both meth- model ina much larger variety of quasi-1D rings. ods agree inthe fact that the correlationsbetweenthe electrons vanish as soon as the temperature exceeds the ÿrst excited state of the system. 15. Pair correlation

The internal structure of a many-body electron 16. Interaction of the spin with the magnetic ÿeld: state, especially the possible localizationto a Wigner the Zeeman e&ect molecule, can be studied by examining the correlation functions. We have done this already for noninteract- Throughout most of this paper we have assumed ing electrons in Fig. 3 where we used the N-particle that the magnetic 8ux is conÿned inside the ring so correlation function for identifying the vibrational that the electrons move in a ÿeld-free region. Exper- states. The N-particle correlationfunctionisjust the imentally, however, it might be diWcult to produce a square of the normalized many-body wave function. situation where the magnetic ÿeld is zero at the ring A related analysis of the maximum N-particle correla- site (or the e7ective LandÃe factor is zero). It is then tionfor the Hubbard model, inTable 1, also revealed important to consider also the Zeeman e7ect when the internal structure of the vibrational states. comparing theory with experiments. The interaction The pair correlation function is frequently used to betweenthe electronspinandthe magneticÿeld adds study the internal structure of a many-body state. In to the Hamiltonian a term ;Bg0SzB. Inthe case of a one-dimensional systems the pair-correlations have narrow ring it is beneÿcial to write this as [46] / the property that they decay with distance as 1=r [8]. ˝2 Consequently, the 1D electron system does not have a H = gS ; (72) Z 2m R2 z Wigner crystal with true long-range order. The same e 0 is true for the spin-density oscillations: For example where we have writtenthe ÿeld with help of the 8ux, in the antiferromagnetic Heisenberg model in 1D the ring radius, and an e7ective LandÃe factor g. For ex- spin–spincorrelationdecaysas 1 =r, r being the dis- ample, in the case of a homogeneous magnetic ÿeld 2 tance between the electrons. and an ideal 1D ring = R B,andg = g0. The ad- In ÿnite 1D rings, with only a few electrons, the vantage of writing the Zeeman part of the Hamilto- pair correlationfunctionisevenless informative.The nian as above is that we can study continuously the reasonis againthe Pauli exclusionprinciple,which changeinthe spectra whenwe move from the case prevents electrons with the same spin to be at the (14), no Zeeman e7ect, to a homogeneous magnetic same site. This means that within a short distance, the ÿeld simply by changing g. Moreover, the fact that the 30 S. Viefers et al. / Physica E 21 (2004) 1–35

0.8

0.6

0.4 pair correlation

0.2

0 0 3.14 6.28 angle

Fig. 18. Pair correlation function for six electrons in a quasi-one-dimensional continuum ring, calculated with the CI method. Solid and long-dashed lines: ↑↓-correlationand ↑↑-correlation, respectively, for a narrow ring with rs = 2 and CF = 25. Dashed and dotted lines: ↑↓-correlationand ↑↑-correlation, respectively, for a wider ring with rs = 2 and CF = 4. The ÿlled and open circles show the correlations for a six electron antiferromagnetic Heisenberg ring. The functions are normalized so that the integral of the ↑↓-correlationis 3 andthat of the ↑↑-correlationis 2.

e7ective LandÃe factor insemiconductorsdi7ers from maximum spinstate Sz = N=2 the lowest energy state. that of the free electroncanalso be takenintoaccount This state has always the periodicity 0 as shownin by just changing g. Fig. 9. Ina wide ring(a) where the periodicity is The periodicity of the persistent current, or lowest 0 for g = 0 it changes ÿrst to 0=2 and then again energy state, as a function of 8ux is caused by the to 0 when g increases. This is due to the fact that ground state energy jumping from one angular mo- (in a ring of six electrons) angular momenta M = mentum state to another. If all possible angular mo- 0; 6;:::;are the ground states for the nonpolarized case menta are visited (Je7 is small inthe models discussed while for the polarized case the ground state angular inSection 9.6), the periodicity is 0=N; if angular momenta are M =3; 9;::: . Inthe transitionregiona momenta M =0;N=2;N, etc. (or N=2; 3N=2, etc.) are periodicity 0=2 is observed. In the narrow ring (c) visited the periodicity is 0=2, and if only angular mo- the periodicity changes smoothly from 0=N to 0 menta M =0;N;2N, etc. are visited the periodicity is when g is increased from zero. 0. Inorder to determinethe overall periodicity it is thus suWcient to examine the angular momentum values of the lowest energy state as a function of the 8ux. 17. E&ect of an impurity Fig. 19 shows the angular momentum of the ground state as a function of the 8ux and the e7ective LandÃe There has beenextensiveresearch onthe e7ects of factor discussed above. The results are shownfor three impurities on the persistent current. Already in 1988, di7erent values of the parameter Je7 . Whenthe LandÃe Cheung et al. [101] used a simple tight binding model factor approaches the free electronvalue g = 2 and for studying the e7ect of disorder in 1D rings and the 8ux is large, the periodicity always becomes 0. found the decrease of the persistent current with in- The reasonis the large Zeemane7ect which makes the creasing disorder. Chakraborty and PietilTainen [102] S. Viefers et al. / Physica E 21 (2004) 1–35 31

2.0 interactions and ÿnite width of the ring. Eckle et al. [58,104] have used the Hubbard model to study the 1.6 e7ect of coupling the quantum ring to a quantum dot 0 3 at Kondo resonance. They have shown that at certain 1.2 6 9 values of the 8ux the problem canbe solved exactly g 12 0.8 15 using the Bethe ansatz. The e7ect of animpurity is easy to study inthe 0.4 Hubbard model, where inthe simplest case we can just add a repulsive or attractive potential in one of the (a) lattice sites. Fig. 20 shows how the 8ux dependence 2.0 of the energy levels changes when a repulsive external potential is introduced in one of the lattice sites. The 1.6 e7ect is to opengaps inthe spectrum. The impurity 3 1.2 9 15 potential splits the degeneracy of angular momenta g (modulo 4) and makes all the energy levels oscillate 0.8 with a period of 0 instead of the period of NL0 0.4 seeninthe impurity-free case. Note, however, that the levels corresponding to di7erent angular momenta still (b) cross and the lowest energy state has the same period 2.0 as without the impurity. Inthe above example, the energyof the lattice 1.6 site, say site number 1, was changed by introducing a term (ˆn +ˆn ) in the normal Hubbard Hamiltonian, 1.2 3 9 1↑ 1↓ 15 Eq. (33). Fig. 21 shows how the lowest energy state as g 0.8 function of 8ux changes when is varied from −20 to 20 insteps of 2. For a relatively small value of U 0.4 (=20) andanattractive impurity, the variationof the ground state energy as a function of the 8ux is nearly 0.0 0.5 1.0 1.5 2.0 2.5 3.0 independent of the impurity potential, while a positive (c) / 0 (repulsive) impurity potential reduces the amplitude of the oscillation, as expected from all earlier stud- Fig. 19. Angular momentum of the ground state of a six electron ring as a function of the 8ux and the e7ective Lande factor g. The ies [70,94,102,103]. Since the ground state persistent angular momentum is shown as numbers and it increases with the current is essentially the derivative of the energy wrt. darkness of the grayscale. The uppermost panel is for JI =6 in 8ux, see Eq. (24), this directly re8ects the suppression the model Hamiltonian (66), the center panel for JI =1:8 and the of the current. In the case of a relatively large U (in lowest panel for JI =0. the present example U = 200), we see that the impurity potential changes the original periodicity 0=N to 0=2. used the CI method for studying the e7ect of an im- Inthe case of small systems the Hubbard model purity ina polarized electronring.The basic results caneasily be to used instudyingalso other kindsof of the included Gaussian impurity potential were to impurities. For example, two impurities or evenran- lift some of the degeneracies of the energy levels and dom potentials at lattice sites have qualitatively the decrease the persistent current. Similar ÿndings were same e7ect on a Hubbard ring as a single impurity, as observed by Halonen et al. [103] inthe case where showninFig. 20. The Hubbard model may further- the spinwas included.The suppressionof the persis- more be used to study the coupling of the ring to a tent current due to a Gaussian impurity was also com- quantum dot [58], as mentioned earlier. In Fig. 22 we puted by Viefers et al. [94] using density functional present another example demonstrating how the Hub- theory, thus including the e7ects of both spin, realistic bard model can be extended to quasi-1D rings with 32 S. Viefers et al. / Physica E 21 (2004) 1–35

-1

-2 energy

-3

∆=0 ∆ =1 -4 0 ΦΦ1 ∆ 0 1

Fig. 20. E7ect of an impurity on the energy levels of a six-site Hubbard ring with four electrons. The left panel shows the energy levels as a functionofthe 8ux (inunitsof 0) for an impurity free ring. The middle panel shows the evolution of the energy levels at zero 8ux as a functionofanimpurity potential at one of the lattice sites. The right panel shows the energy levels as a function of 8ux in the case where =1. U = 1000 inall cases.

impurities. Inthe uppermost panelof the ÿgure we WeidenmullerT have shown that interactions e7ectively show the energy spectrum of a 1D Hubbard ring with counteract the impurity suppression of the persistent four electrons and a ÿnite U = 7. The middle panel current [109], inqualitative agreementwith our Hub- represents a quasi-1D ring with inÿnite U (t-model). bard model results. Comparisonof these two models shows that a ÿnite U ina strictly 1D Hubbard model actually mimics well the quasi-one-dimensionality of a more realis- 18. Summary tic ring, as noticed already in comparing the Hubbard model with the continuum models. The lowest panel Inthis paper we have attempted to presenta com- of Fig. 22 shows an “impurity” or a narrow neck in prehensive review of the physics of few-electron quan- the ring.Its e7ect is to opengaps inthe excitation tum rings, with particular focus on their energy spec- spectrum. Qualitatively, the e7ect is the same as ob- tra and the periodicity of the persistent current. We served above for a 1D Hubbard ring with an impurity compared various analytical and numerical theoretical potential. approaches which fall into two main classes—lattice In many experiments, the persistent current is mea- models on the one hand and continuum models on the sured for a collection of rings, possibly consisting of other—and tried to clarify the connections between several propagating channels (radial modes). The per- them. The mainmessage is that all the di7erentap- sistent currents in such rings are not determined by proaches give essentially the same results for the spec- quantum mechanical eigenenergies of a single ring, tra and persistent currents (provided one takes the con- but are a7ected by several complications like disor- tinuum limit of the discrete models to compare them to der and ensemble averaging [105–107]. The electron– the continuum ones). The essential physics is captured electron interactions seem to play a crucial role also by a simple model Hamiltonian which describes the inthe disordered rings[ 108]. Muller-GroelingT and many-body energy as a combination of rigid rotation S. Viefers et al. / Physica E 21 (2004) 1–35 33

-4

-5

-6

0 0.2 0.4 0.6 0.8 1 -7

-8 energy energy

-9 energy -10

-11 0 0.2 0.4 0.6 0.8 1 -7

-8

0 0.5 1 0 0.5 1 -9 flux flux -10 Fig. 21. E7ect of animpurity onthe lowest energystate ina -11 Hubbard ring with L = 6 and N = 4. The left panel is for U =20 0 0.2 0.4 0.6 0.8 1 and the right panel for U =200. The thick line is the result without Φ the impurity and the other lines correspond to rings where the energy of one lattice site is increased (see text) by 2, 4, 6, etc. Fig. 22. Energy spectra as a function of 8ux for four electron (dotted lines above the thick line) or decreased by 2, 4, 6, etc. clusters. Upper panel: Hubbard ring with ÿve sites and ÿnite (dotted lines below the thick line). All lines have been shifted U =7. Middle panel: “Quasi-one-dimensional” t-model ring. Low- vertically so that they appear to be equally spaced. The energy est panel: Quasi-one-dimensional t-model ring with a defect. The scale ineach lineis the same, but it is twice as large for U = 200 geometries ineach case are shownat the left. as compared to U = 20. The 8ux is inunitsof 0 = h=e. Acknowledgements

We would like to thank M. Koskinen for enlight- and internal vibrations of the ring “molecule”, plus a ening discussions. This work has been supported by Heisenberg term which determines the spin dynamics. Nordita and by the Academy of Finland under the In the spectra, the vibrational excitations can be seen Finnish Centre of Excellence Programme 2000–2005 as higher bands, while the lowest (yrast) band is purely (Project No. 44875, Nuclear and Condensed Matter rotational. Its periodicity with respect to the 8ux is al- Programme at JYFL). ways 0 in the case of spinless (polarized) electrons, while for a clean ring of nonpolarized electrons, the periodicity changes from 0 via 0=2to0=N as the References ring get narrower or, in the language of the Hubbard model, as the interaction strength U increases (the ef- [1] Y. Aharonov, D. Bohm, Phys. Rev. 115 (1959) 485. fective Heisenberg coupling J decreases). Impurities [2] M. Buttiker,T Y. Imry, R. Landauer, Phys. Lett. A 96 (1983) in the ring may change the periodicity (to ) even 365. 0 [3] S. Tarucha, D.G. Austing, T. Honda, R.J. van der Haage, in the nonpolarized case, and moreover suppress the L. Kouwenhoven, Phys. Rev. Lett. 77 (1996) 3613. persistent current. [4] A. Lorke, R.J. Luyken, Physica B 256 (1998) 424. 34 S. Viefers et al. / Physica E 21 (2004) 1–35

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