Quantum Hall system in Tao-Thouless limit

Quantum Hall system in Tao-Thouless limit

E.J. Bergholtz    A. Karlhede Department of Physics, Stockholm University
AlbaNova University Center
SE-106 91 Stockholm, Sweden
September 4, 2019
Abstract

We consider spin-polarized electrons in a single Landau level on a torus. The quantum Hall problem is mapped onto a one-dimensional lattice model with lattice constant , where is a circumference of the torus (in units of the magnetic length). In the Tao-Thouless limit, , the interacting many-electron problem is exactly diagonalized at any rational filling factor . For odd , the ground state has the same qualitative properties as a bulk () quantum Hall hierarchy state and the lowest energy quasiparticle exitations have the same fractional charges as in the bulk. These states are the limits of the Laughlin/Jain wave functions for filling fractions where these exist. We argue that the exact solutions generically, for odd , are continuously connected to the two-dimensional bulk quantum Hall hierarchy states, ie that there is no phase transition as for filling factors where such states can be observed. For even denominator fractions, a phase transition occurs as increases. For this leads to the system being mapped onto a Luttinger liquid of neutral particles at small but finite , this then develops continuously into the composite fermion wave function that is believed to describe the bulk system. The analysis generalizes to non-abelian quantum Hall states.

pacs:
73.43.Cd, 71.10.Pm, 75.10.Pq

I Introduction

The two-dimensional electron gas in a perpendicular magnetic field, the quantum Hall (QH) system, is remarkably rich. The integer QH effect shows a conductance precisely quantized in integer units of vonk , and in the fractional QH regime the conductance is quantized in fractions of and there are fractionally charged quasiparticles that obey (abelian) fractional statistics tsui ; stat1 ; stat2 . Whereas the integer effect can be understood in terms of non-interacting electrons it1 ; it2 , the fractional effect is caused by the interaction between the electrons Laughlin83 . Other states observed in this strongly correlated electron system are metallic states that resemble a free two-dimensional Fermi gas jiang ; willett90 ; wang and inhomogenous striped states in higher Landau levels striped1 ; striped2 . A further example of an exotic quantum state that may form is a non-abelian state where the quasiparticles obey non-abelian fractional statistics mr ; nayak ; readrezayi .

The fractional QH effect is understood as an incompressible quantum liquid with fractionally charged quasiparticles; this is based on Laughlin’s wave functions for filling fractions Laughlin83 . For other fractions, a hierarchy construction where quasiparticles condense to form new quantum liquids just as electrons form the Laughlin states was proposed by Haldane hierarchyHaldane , Laughlin hierarchyLaughlin and by Halperin hierarchyHalperin . An alternative view, where electrons supposedly capture magnetic flux to form composite fermions that see a reduced magnetic field was developed by Jain jain89 ; jainrev1 ; jainrev2 ; jainbook . Successful mean field theories that support this idea of flux attachment have been developed zhang ; read89 ; lopez . The composite fermion approach has received strong support by experiments performed in the half-filled Landau level. Near , ballistic transport is consistent with particles moving in a reduced magnetic field in accordance with the composite fermion prediction ballistic1 ; ballistic2 ; ballistic3 .111However, an alternative consistent interpretation is that it is a quasiparticle, with a reduced charge compared to the electron, that moves in the original full magnetic field; what is measured is the product of the charge and the magnetic field. At , the external magnetic field is completely absorbed by the electrons and the composite fermions see no magnetic field—they form a two-dimensional Fermi gas. The mean field theory of this state hlr ; kalmeyer is in excellent agreement with surface acoustic wave experiments willett90 ; Willettreview .

In spite of these impressive results it is our opinion that there are basic questions concerning the QH system that remain to be answered.222A critical discussion of the status of the theory of the QH effect is presented by Dyakonov Dyakonov . Most importantly, a microscopic understanding is lacking. It is true that good many-body wave functions exist, but there is no microscopic understanding or derivation of them. For example, according to the composite fermion picture the fractional QH ground state is formed when composite fermions fill an integer number of effective Landau levels in the reduced magnetic field. No derivation of this scenario from the many body wave function exists. Mean field theory, which is successful, does give support to the idea of electrons binding flux quanta; however, it is not understood why mean field theory works as well as it does. A further question is the relation between the original hierarchy description and composite fermions; are they alternative descriptions of the same thing as argued by Read read90 and by Blok and Wen blokwen , or are they fundamentally different as argued by Jain jain89 ; jainrev1 ; jainrev2 ; jainbook ? Our aim is to contribute to a solution to these problems.

In this article we consider spin-polarized interacting electrons within a single Landau level. It has recently been realized that there is a limit in which this problem can be exactly solved for any rational filling factor —and that the solution is physically relevant bk1 ; bk2 ; we06 ; hierarchy ; conformal . We here expand on, and provide details of, our work presented in Refs. bk1, ; bk2, . We consider the interacting electron gas on a torus where it becomes equivalent to a one-dimensional lattice model with a complicated long-range interaction. The solvable limit is the thin torus: , where is one of the circumferences of the torus (the other circumference being infinite). In this limit the interaction in the one-dimensional lattice problem becomes purely electrostatic and the ground state is a ”crystal” of electrons occupying fixed positions on the lattice as far apart from each other as possible. For every third site is occupied; interestingly enough this is the state introduced by Tao and Thouless in 1983 to explain the fractional QH effect Tao83 and we call these crystal states Tao-Thouless (TT) states. The fractionally charged quasiparticles are domain walls separating the degenerate ground states that exist on the torus.

It should be noted that the mapping of the lowest Landau level, or of any single Landau level for that matter, onto a one dimensional lattice problem is exact and that varying may alternatively be thought of as varying a parameter in the hamiltonian that controls the range of the interaction while keeping the two-dimensional space fixed (and possibly infinite). To stress this we refer to the thin limit as the Tao-Thouless (TT) limit. If the ground state in the TT-limit remains the ground state as we may thus conclude that the experimentally accessible ground state is adiabatically connected to the ground state in the TT-limit.

The simple limit may at first seem to be of little physical interest—it is really an extreme case: the interaction is both very short range and anisotropic and is furthermore purely electrostatic. The surprising fact is that the ground state in this limit has all the qualitative properties of a fractional quantum Hall state, such as a gap, the correct quantum numbers and quasiparticles with the correct fractional charge. We argue that the simple TT-ground states obtained in the TT-limit develop continuously, without a phase transition, into the fractional QH hierarchy states, as , for filling factors where such states are observed. Thus, we argue that the TT-state in general describes the fractional QH phase observed as , in the sense that these states are adiabatically connected. We show that the TT-states are the limits of the Laughlin and Jain wave functions for filling fractions where these exist. In the TT-limit the original hierarchy construction is manifest: the TT-ground states are condensates of quasiparticles.

The hierarchy structure of states in the TT-limit has recently emerged within a conformal field theory (CFT) construction of bulk hierarchy wave functions for all fractions that are obtained by successive condensation of quasielectrons (as opposed to quasiholes) hierarchy ; conformal . These wave functions reduce to the correct TT-states in the TT-limit and are obtained by a natural generalization of the conformal construction of the composite fermion wave functions hans . This supports the adiabatic continuity fro m the TT-limit to the bulk for general hierarchy states.

The TT-states are the ground states also for the even denominator fractions in the TT-limit; however, for these fractions we claim that there is always a phase transition when increases. This is supported by numerical studies and by a detailed analysis of , which we believe is a representative case for the even denominator fractions.

At , there is a phase transition from the gapped TT-state to a one-dimensional gapless state at . (Lengths are measured in units of the magnetic length, .) Also for this gapless phase there is an exact solution: For a hamiltonian that is a good approximation at , the low energy sector consists of non-interacting neutral fermions (dipoles); the ground state is a one-dimensional Fermi sea and there are gapless neutral excitations. This provides an explicit example of interacting electrons in a magnetic field being equivalent to free particles that do not couple to the magnetic field. The ground state in the exact solution is a version of the composite fermion state jainbook given by Rezayi and Read rr . There is strong numerical evidence bk2 that this gapless one-dimensional state develops continuously, without a phase transition, into the two-dimensional bulk version of the Rezayi-Read state that is believed to describe the observed metallic state at .

The Moore-Read pfaffian state mr , believed to describe the half-filled second Landau level, , also exists in the TT-limit. A simple construction gives the quarter-charged quasiholes and quasielectrons as domain walls between the six-fold degenerate ground states and the non-trivial degeneracies for these excitations is obtained haldaneAPS ; we06 . Similar results for other non-abelian states exist seidel06 ; read06 ; weunpublished ; see also Ref. haldanejack, .

Over the years there have been many interesting attempts to improve the understanding of the QH system and it is impossible to here mention them all. In our work what emerges is a one-dimensional theory of the quantum Hall system which depends on a dimensionless parameter , where the Tao-Thouless states Tao83 are the exact solutions as and the fractionally charged quasiparticles are domain walls between the degenerate ground states. The TT-states are adiabatically connected to bulk QH states as —there is no phase transition if a QH state is observed. Searching the literature one finds hints and suggestions for such a scenario. Anderson noted in 1983 that Laughlin’s wave function has a broken discrete symmetry and that the quasiparticles are domain walls between the degenerate ground states anderson . Furthermore he noted that the TT-state is non-orthogonal to the Laughlin state and suggested that it can be thought of as a parent state that develops into the Laughlin wave function, without a phase transition, as the electron-electron interaction is turned on. In 1984, Su concluded, based on exact diagonalization of small systems on the torus, that the QH state at is -fold degenerate and that the lowest energy excitations are quasiparticles with charge that are domain walls between the degenerate ground states su1984 ; su2 . In 1994, Rezayi and Haldane studied the Laughlin wave function on a cylinder as a function of its radius and noted that it approaches the Tao-Thouless state on the thin cylinder. Implicit in their work is the fact that the Laughlin wave function is the exact and unique ground state to a short range pseudo-potential interaction on a cylinder for any circumference Haldane94 . In retrospect, this makes a very strong case for an adiabatic evolution from the TT-state to the Laughlin wave function. A one-dimensional approach to QH states was also considered by Chui in 1985 chui1985 ; chui86 . More recently, this has been explored in connection with Bose-Hubbard models by Heiselberg heiselberg , and it should also be mentioned that Dyakonov presents a one-dimensional toy QH-model Dyakonov .

At , we find that the low energy sector consists of weakly interacting dipoles. This relates to earlier descriptions in terms of dipoles, in particular to the field theory of Murthy and Shankar ms and the work by Read readearly ; read.5 , Pasquier and Haldane pasq , Lee dhlee and Stern et al stern ; for reviews see Ref. rev1, . It is of course also reminiscent of composite fermions in general jainbook in that the particles do not couple to the magnetic field and are weakly interacting.

We would also like to draw the attention to the construction of composite fermion wave functions directly in the lowest Landau level by Ginocchio and Haxton haxton , the series of work by Wojs, Quinn and collaborators, see Ref. wojs, and references therein. Explicit wave functions have been obtained within the original hierarchy construction by Greiter greiter94 .

The content of the article is as follows. The one-dimensional lattice model for interacting electrons in a single Landau level is introduced in Sec. II. In Sec. III we solve this problem exactly in the TT-limit , ie we diagonalize the interacting electron hamiltonian for any rational filling factor in this limit. The ground states and the fractionally charged quasiparticles are identified and it is found that the former are condensates of the latter thus proving the original hierarchy construction in this limit hierarchy . The quasiparticles are domain walls between the degenerate ground states and their charge is determined by the Su-Schrieffer counting argument. The energy of a quasielectron-quasihole pair is determined as and it is found that the gap to creating an infinitely separated such pair at decreases monotonously with increasing but is independent of ; this is in surprisingly good agreement with experiments, which are performed in the two-dimensional bulk system, . The Laughlin and Jain fractions, as well as those observed by Pan et al pan are considered explicitly as examples, and finally the relation to composite fermions and emergent Landau levels is commented on.

The transition to the two-dimensional bulk system as is considered in Sec. IV, first for odd denominator fractions leading to the QH hierarchy states, then for the half-filled Landau level; comments on other even denominator fractions and on non-abelian states are included. The conclusion is that the rich structure and different phases of matter present in the QH system exist also in the TT-limit, where it can be studied in detail starting from a microscopic hamiltonian.

Technical details are burried in a series of appendices. The mathematics that we need for a single Landau level on a torus Haldane85PRL including the construction of the lattice hamiltonian is given in Appendix A. In Appendix B we prove that the relaxation procedure given in Sec. III actually gives the ground state. In Appendix C, we show that the quasiparticle charge at is ; the result is obtained in the limit using the Su-Schireffer counting argument. The energy of a quasielectron-quasihole pair is obtained in Appendix D. In Appendix E we show that the limit of Laughlin’s and Jain’s wave functions are the TT-ground states obtained in Sec. III. Appendix F shows that Laughlin’s wave function at is the exact ground state, and that there is a gap to excitations, for all for a short-range interaction. Appendix G contains details of the exact solution at .

Ii Model

We consider a single Landau level of spin-polarized electrons on a torus with lengths in the and -directions respectively. We use units such that and choose one-particle states , , , that have -momentum and are gaussians centered at , see Appendix A. This provides an exact mapping of the Landau level onto a one-dimensional lattice model with lattice constant , see Fig. 1. Each site can be either empty, 0, or occupied by an electron, 1. Thus a basis of many particle states is provided by , where ; alternatively, states can be characterized by the positions (or, equivalently, the -momenta) of the particles.

Figure 1: Mapping of a single Landau level onto a one-dimensional lattice model. The two-dimensional space is a cylinder with circumference (or a torus if the ends are identified) (top panel). The single particle wave functions are centered along circles and are gaussians of width one along the cylinder (central panel); they are numbered by their momenta which also give the position along the cylinder. This gives a one-dimensional lattice model with lattice constant where each site is either empty (0) or occupied by en electron (1). As an example the Tao-Thouless state 001 at is shown (bottom panel).

A general two-body interaction, , that depends on the distance, , between two electrons only, such as a Coulomb or a short-range delta function interaction, leads to the one-dimensional hamiltonian

(1)

Here creates an electron in state , , and . consists of the two-body terms that preserve the -momentum, ie, the position of the center of mass of the electron pair, see Fig. 2. is the amplitude for two electrons separated lattice constants to hop symmetrically to a separation of lattice constants. is the electrostatic repulsion (including the exchange interaction) between two electrons separated lattice constants.333Note that is multiplied by because of the delta-function—this compensates for the fact that for even every process with appears twice in (1).

Figure 2: A general translationally invariant two-electron interaction consists of the terms where two electrons preserve the position of their center of mass. There are electrostatic terms , where the electrons do not move, and hopping terms , where the electrons hop symmetrically.

For a given real-space interaction , when is small the lattice constant is large and, hence, the dominant are those with small . Furthermore, the wave functions are gaussians in the -direction, with width of the order of the magnetic length, hence their overlap vanishes rapidly as , and the electrostatic terms dominate in this limit. The physics is thus very simple in the TT-limit —it is determined by electrostatic repulsion only. In the two-dimensional limit, , the range of the one-dimensional interaction measured in number of lattice constants goes to infinity for any real space two-dimensional interaction; this is true also for a local interaction such as .

The symmetry analysis of many-electron states on the torus was given by Haldane Haldane85PRL ; a simple version adapted to our needs can be found in Appendix A. We give here the results. There are two translation operators, , , that commute with the hamiltonian ; they obey , see (38). These operators have eigenvalues . corresponds to -translations and is the total -momentum (in units of ). translates the system one lattice constant in the -direction, ie along the one-dimensional lattice and increases by . At filling factor (where and are relatively prime) commutes with : is a maximal set of commuting operators. , generate degenerate orthogonal states, which have different , when acting on any state—this is the -fold center of mass degeneracy. Hence, each energy eigenstate is (at least) -fold degenerate and we choose to characterize it by the smallest . Thus, the energy eigenstates are characterized by a two-dimensional vector , where is the -eigenvalue.

Iii Exact solution

In this section we solve the problem of interacting spin-polarized electrons exactly at any rational filling factor in the Tao-Thouless limit; we diagonalize the hamiltonian (1) and construct explicitly the ground state as well as the low energy charged excitations which turn out to have charge . As noted above, the hopping elements, , vanish rapidly as decreases, whereas the electrostatic elements, , decrease much more slowly. Hence in the limit effectively only the electrostatic interaction survives and the hamiltonian (1) becomes

(2)

where and periodic boundary conditions have been imposed, . in (2) defines the Tao-Thouless limit. At filling fraction , the energy eigenstates are simply the states

(3)

where electrons occupy fixed positions on lattice sites,

(4)

iii.1 Ground states, quasiparticles and the hierarchy

Here we determine the ground state at in the TT-limit, ie for the hamiltonian in (2). This is the classical electrostatics problem of finding the position of electrons, on a circle with sites, that minimizes the energy. Hubbard gave an algorithm for constructing this ground state hubbard ; we will present two simple alternative constructions. The first gives the ground state for given , and provides an intuitive understanding of why the energy is minimized; the second starts from and constructs the ground states at all other filling factors iteratively from this state as repeated condensations of quasiparticles—in addition to the ground states this gives the fractionally charged quasiparticles and makes the Haldane-Halperin hierarchy construction manifest in the TT-limit.

The states obtained here are the ground states for any interaction that obeys the concavity condition

(5)

which implies, by iteration,

(6)

for all . When, , is simply the electrostatic interaction energy between two rings of circumference separated a distance ; hence he concavity condition is fulfilled by a generic electron-electron interaction when . It implies that the interaction energy of one electron with two other electrons that have fixed positions is minimized if the first electron is as close to the midpoint between the fixed electrons as possible, ie if the distances to the two fixed electrons differ by at most one lattice constant, see Fig. 3. 444Hubbard assumes, in addition, that ; this need not be fulfilled when periodic boundary conditions are assumed as is the case here.

The crucial observation in obtaining the ground state is to realize that it is possible to minimize the energies of the :th nearest neighbors separately for each for an interaction that obeys (6) hubbard .

Figure 3: The interaction energy of one movable electron with two fixed electrons is minimized when the distances to the two fixed electrons differ by at most one lattice site.

In our first construction, the ground state is obtained by placing the electrons equidistantly on a circle and then letting them relax to the closest lattice sites. The ground state is periodic with a unit cell of length containing electrons and this cell is obtained as follows. Consider a circle with equidistant electrons and a lattice with (equidistant) lattice sites as in Fig. 4. Move each electron to its closest lattice site; an electron that is equally far from two sites is moved to one of these sites. 555There is at most one such electron, since otherwise the period would be less than . The obtained configuration is the unit cell of the ground state, for the proof we refer to Appendix B. An algebraic expression for the unit cell is obtained by noting that electron is at site

(7)

where and denotes the integer closest to . For example, for , (7) gives and hence the unit cell is in agreement with Fig. 4. We use a chemical notation where the subscript denotes the number of times the quantity is repeated.666Hubbard uses a more compact notation where the distance between consecutive ones is given, eq, . That the unit cell at has length implies that the ground state is -fold degenerate. This is the center of mass degeneracy of any state at discussed in Sec. II Haldane85PRL . For the quantum Hall states this is the topological degeneracy of the ground state identified by Wen and Niu wen ; wenreview .

Figure 4: Ground state unit cell at by relaxation on circle. a) p=5 equidistant electrons on a circle with q=13 lattice sites; b) each electrons moved to its closest site, unit cell is .

Incidentally, the initial configuration with equidistant electrons is clearly the ground state in the continuum problem when there is no lattice—when there is a lattice the ground state is as similar to the continuum one as possible. Because of the periodicity, the picture of course generalizes immediately to the full system consisting of unit cells and periodic boundary conditions: the ground state for electrons on sites is obtained by starting with equidistant electrons and moving each electron to its closest site.

At the procedure gives the unit cell . Thus, the ground state is obtained by placing one electron on every :th site; this obviously minimizes the electrostatic repulsion. For odd , these are the Laughlin fractions and the state with unit cell was in fact proposed by Tao and Thouless in 1983 as an explanation of the fractional quantum Hall effect Tao83 . Although this state has a small overlap with the exact ground state (for large ), and with the Laughlin state, it does in fact play an important role in the quantum Hall effect and we will call the crystal ground states at general Tao-Thouless (TT) states. We claim that the TT-states are QH-states in the sense that they are adiabatically connected to the bulk QH states bk2 ; hierarchy .

Note that the TT-states have a gap to excitations—the lattice sites are fixed in space and there are no phonons in these crystals; the TT-states should not be confused with Wigner crystals, which have gapless excitations due to the broken translational invariance. This is a further reason to call the states considered here TT-states rather than crystal states.

We will now present an alternative, iterative, construction of the TT-states that brings out the connection to the hierarchy of fractional quantum Hall states hierarchyHaldane ; hierarchyHalperin and determines the fractionally charged quasiparticles. Let be the unit cell for a TT-state at level ; this level is defined by the iteration process and is identical to the one in the hierarchy construction. At the first levels we have

(8)

where . The unit cells for the TT-states at level are obtained iteratively as

(9)

where indicates that is repeated times; is the complement of in the unit cell , ie and . At the second level we obtain:

(10)

We can now connect to the original hierarchy construction. As will be further discussed below, is the quasihole, and is the quasielectron, with charges , in the ground state . Thus the unit cells at level two consists of (or ) copies of the unit cell at level one, followed by a quasihole (or quasielectron) in the level one ground state—the new ground state at level two is a condensate of quasiparticles in the level one ground state.

At the next level, level three, we find the unit cells

(11)

and the corresponding filling factors

(12)

Generalizing this, we find at level , the filling factors

(13)

where and the corresponding unit cells are obtained using (III.1). In this construction, if is constructed with () in (III.1). A state at level is uniquely characterized by the parameters . Equation (13) is the continued fraction form of the filling factors as given by the hierarchy scheme, except that now also even denominators are obtained. Restricting to and , for , (13) is identical to Haldane’s formula for the filling factors hierarchyHaldane , which is known to give each odd denominator fraction once.

This construction gives each rational filling factor and by inspecting the unit cells one finds that they are such that the distances between :th nearest neighbors differ by one lattice constant at most, hence they minimize the energy and are identical to the ground states obtained by the relaxation procedure above. To be precise, we have not proven that the unit cells obtained by the two methods are always identical, but we have checked this in many examples and are convinced that this is the case.

The interpretation of (III.1) as a condensate of quasiparticles generalizes to arbitrary level in (III.1):

(14)

are the quasihole and the quasielectron (which is which depends on the state) in the ground state with unit cell , and the ground state at level , , is a condensate of these quasiparticles in accordance with the hierarchy construction. That the proposed quasiparticles, and , have the expected charges at follows from the Su-Schrieffer counting argument Schrieffer , see Appendix C. Furthermore, they are domain walls between the degenerate TT ground states. This was noted by Anderson anderson and stressed by Su su1984 ; su2 based on exact diagonalization studies on the torus. The quasiparticles discussed here are the ones with the elementary charge and the QH states are the simple abelian ones; as pointed out by Wen wenreview other quasiparticles may also in principle condense to form more complicated ground states.

The iteration formula (III.1) gives the one quasiparticle excitations in the state with unit cell and shows that these are the lowest energy excitations at the corresponding filling factors; for example, inserting in the ground state with unit cell gives the quasielectron and it is the lowest energy state at .

The energy of a quasielectron-quasihole pair at can be calculated in the limit . According to the discussion above, the quasiparticles in a TT-state with unit cell are and . Thus, a minimally separated particle-hole pair is obtained by the replacement

(15)

in the ground state. Note that the replacement in (15) amounts to a translation of the unit cell with periodic boundary conditions on the cell itself, thus it creates two domain walls (with the expected charge). It can be shown, see Appendix D, that differ from only in that one electron has been moved one lattice constant.

A separated particle-hole pair is obtained by translating consecutive cells as in (15), ie with periodic boundary conditions on each cell separately, or, equivalently, on all of them together. This is equivalent to inserting a string of unit cells between the particle and hole

(16)

and moves electrons (one in each unit cell) one lattice constant in the same direction.

The replacement (15) or (16) implies an ordering of the particle-hole pair. The opposite ordering is obtained by instead making the reverse replacement

(17)

Again the replacement (16) or (17) amounts to a translation of assuming periodic boundary conditions.

It is shown in Appendix D that the energy of the separated particle-hole pair in (16,17) is

(18)

where

(19)

is the change in energy for an electron with its two neighbors unit cells away when the initial electron is moved one lattice constant. This energy, which is the second derivative of , is positive due to the concavity condition (5), which thus ensures the stability of the TT-ground state to particle-hole formation. Note that is independent of the numerator of the filling factor.

It can be shown that the nearest neighbor pair excitation , which has energy

(20)

according to (18), is the lowest energy excitation at fixed filling fraction, see Appendix D.

An important quantity is the energy of an infinitely separated particle-hole pair; this is what is measured in activated transport and is a measure of the stability of a quantum Hall state. When the separation of the particles goes to infinity, , we find

(21)

Note that the gap depends only on the denominator , ie on the fractional charge , and not on the numerator in .

This is natural in the sense that the denominator determines the charge of the quasiparticles, but the result is non-trivial since the properties of the ground states depend on both and . Furthermore, is a monotonic function that approaches zero from above as the denominator increases.777This statement is true for most imaginable interactions. A sufficient (but not necessary) condition is that (or its derivative) is monotonic in . In this context it should be mentioned that Halperin within the original hierarchy construction of fractional QH states predicted a gap that is predominantly determined by and decreases monotonously with increasing hierarchyHalperin .

Thus we find that the gap to creating a separated quasiparticle pair decreases monotonously with increasing . This motivates Fig. 5 which shows for each , odd hierarchy . Each point corresponds to a TT-state that we claim is adiabatically connected to a bulk QH-state. The higher up a point is, the larger is the gap in the corresponding TT/QH-state and the more stable is the state. The points marked by plusses are filling fractions where a dip in the longitudinal resistance is reported in the experiment by Pan et al pan . At the crosses, we infer a very small dip in the same data. This experiment, which covers the range marked in the figure, is performed on the highest available mobility samples and exhibits the largest number of QH states. We note that to a surprisingly good approximation a dip in is observed at a filling factor if, and only if, . This is in agreement with the gap, , being independent of .

Figure 5: The Tao-Thouless states and their stability. Each point , odd, corresponds to a TT-state that we claim is adiabatically connected to a bulk QH-state. The gap, and hence the stability, increases with decreasing . At the points marked by plusses or crosses indications of QH-states are observed pan . The line is at constant gap and shows the range of the experiment in . It is an approximate lower boundary in for the observed states.

The structure shown in Fig. 5 is a fractal, self-similar, one: enlarging any region of reproduces the original figure mani ; goerbig . This fractal structure is connected to the hierarchy construction of fractional QH states. The TT-states and their quasiparticles obtained when makes the hierarchy construction of fractional QH states manifest as discussed above. According to (III.1), for each state, the parent state, condensation of quasiparticles gives rise to two sequences of daughter states with filling fractions approaching that of the parent state from above and below respectively and with decreasing , see Fig. 6. This holds for each and explains the fractal structure in Fig. 5. For a more complete discussion of the connection to the hierarchy theory, including the connection to the global phase diagram global ; lutken , we refer to Ref. hierarchy, .

Figure 6: The hierarchy state at is the parent state for two sequences of daughter states formed by condensation of quasielectrons and quasiholes in the parent state.These sequences approach , with decreasing gap , from above and below respectively. Repeating this construction ad infinitum gives the fractal structure in Fig. 5.

Using Fig. 5 we can predict what QH-states are next in line to be discovered when higher mobility samples become available. For example, in the region we first expect, in addition to 7/17 and 8/13 included above and new Jain states at , states at 10/17, 11/17, 6/19, 7/19 and 8/19.

iii.2 Examples

We here give explicit examples of TT ground states and quasiparticles for prominent filling fractions.

iii.2.1 Laughlin/Jain fractions

For the Jain sequences that for fixed approach from below as increases, the unit cell of the TT-state is ; in the hierarchy notation this corresponds to . Thus the states in the Jain sequences are those in the hierarchy where all but the first condensate has maximal density. At , the unit cell is . Explicitly for the experimentally most prominent sequence, , the unit cells are

(22)

By taking the particle-hole conjugate, , of the states in (III.2.1) one obtains the Jain series that approaches from above.

We now turn to the fractionally charged quasiparticles. Consider to start with the TT-state at . According to (III.1), its unit cell is and the quasiparticles are and . A quasihole with charge is created by inserting somewhere:

The first line in (III.2.1) is the ground state; the second line has one extra 0 inserted—comparing to the ground state it is clear that this creates a domain wall between two degenerate ground states. In the third line, three well-separated 0’s are inserted, creating three domain walls. Comparing to the ground state in the first line one sees that far away from the three 0’s the state is unchanged and to maintain the size of the system the underlined unit cell has to be removed. In between the inserted 0’s the state is a rigid translation of the original ground state and thus indistinguishable from this state by a measurement that only refers to the translated state. Thus, the net change is that one electron has been removed and this charge is divided on the three well-separated domain walls which thus have charge each. This is of course nothing but the Su-Schrieffer counting argument Schrieffer . The argument is also closely related to Laughlin’s original argument for the fractional charge, where a quasihole was created by adiabatic insertion of a flux quantum Laughlin83 —this corresponds to adding one empty site as there is one site per flux quantum.

The quasielectron is created by inserting : 00100100101001001001001. Again this is a domain wall and the charge is, by the counting argument . Note that particle-hole symmetry is manifest; this is not the case for the bulk wave functions. It is straightforward to show that the excitations just given are the limits of the two-dimensional bulk quasiparticles constructed by Laughlin, see Appendix E.

It follows from Sec. III that inserting or removing 01 create quasiparticles in all the TT ground states in the Jain sequence that approaches as increases. For example, a quasielectron at is obtained as:

(24)

For Jain sequences with general , starting from the Laughlin state , quasiholes (quasielectrons) are created by removing (inserting) . In all cases, the Su-Schrieffer counting argument gives the expected charge, , for these quasiparticles and they are the limits of the Laughlin and Jain quasiparticles in the two-dimensional bulk system, see Appendix E.

iii.2.2 Non-Jain fractions

Until recently, all experimentally well-established fractional quantum Hall states where for Laughlin-Jain filling fractions. However, in 2003 Pan et al reported a new set of states in ultra-high mobility samples pan . Preliminary indications of such a state at 4/11 was reported by Goldman and Shayegan goldman . Fractional quantum Hall states were seen at the following odd denominator filling factors and (the ones in parenthesis are inferred by us from the data in Ref. pan, but were not claimed in this reference). 888The experiment shows similar features, a dip in the longitudinal resistance, at even denominator fractions, eg at —these are presumably not QH states, see Sec. IV.2.2. Using the methods above, we readily find the TT ground states and the quasiparticles with charge at these filling fractions. In Table 1 the unit cells are given in the hierarchy form, ie as in (III.1). The anti-quasiparticle, which is not given in the table, is obtained by taking the complement of the quasiparticle in the given ground state unit cell, according to (14). The unit cell for given is most easily obtained using the relaxation method, (7); it is then easily transformed to the hierarchy form by identifying its parent using Figs. 5 and 6. The parent is one of the two nearest neighbors whose denominator is smaller than . —tt Check this

ground state quasiparticle
4/11
7/11
4/13
5/13
8/13
5/17
6/17
7/17
Table 1: Ground states and quasiparticles in the TT-limit for the odd denominator fractions observed by Pan et al pan that do not belong to the Jain sequence.

iii.3 Emergent Landau levels and composite fermions

We have seen that in the TT-limit the TT ground states give a microscopic realization of the original hierarchy construction of fractional quantum Hall states. Today the alternative construction of quantum Hall states in terms of composite fermions has become the dominant framework and one may ask how the TT-construction is related to this. We believe, as argued by Read read90 and by Blok and Wen blokwen but contrary to the opinion of Jain jainrev1 ; jainrev2 ; jainbook , that the original hierarchy construction and composite fermions are just two different ways to view the same phenomena. In our approach, this is supported by the fact that both the hierarchy construction and the emergent Landau level structure is manifest in the TT-limit and that the TT-ground states and the quasiparticles are the limits of Jain’s composite fermion wave functions.

According to the composite fermion idea, fractional QH states are created when composite fermions fill an integer number of Landau levels in a reduced magnetic field. This implies the existence of an emergent structure of effective Landau levels within the lowest Landau level; such a structure is indeed seen in numerical studies emergent ; jainrev1 . We here describe how this comes about in the TT-limit. To be specific we imagine that we start at filling factor with the state , ie the ground state has unit cells. When the magnetic field is decreased, quasielectrons with charge are created by inserting 01. One such quasielectron can be inserted in different equivalent places; these degenerate states form an effective Landau level for these quasiparticles. When is further decreased more quasielectrons are added to this effective Landau level; the quasielectrons repel eachother and are hence pushed as far apart as possible. Eventually, one 01 has been added per unit cell 001, the effective Landau level is filled and one has reached the state , ie the 2/5 state with unit cell 00101, see Fig. 7.

Figure 7: Quasiparticles and emergent Landau levels in the TT-limit. The TT-ground state 001 at ; in three quasielectrons with charge are created by inserting 01 in three places. When one 01 is inserted for every third unit cell 001, the TT-ground state at is obtained . In the 2/5 ground state is obtained by inserting one quasielectron for every unit cell 001. The ground states in can be interpreted as filling emergent Landau levels.

Of course, when decreasing the magnetic field and adding quasielectrons 01 to the 1/3 TT-state many other, infinitely many in fact, TT-ground states are obtained before the 2/5 state is reached, see Fig. 5. Among them are the states with unit cells , see Fig. 6; these are just as 2/5 obtained by a condensation of quasielectrons in the 1/3 state albeit with a lower density of quasielectrons. However, these states have a filling factor with larger denominator and hence a smaller gap and are thus less stable to disorder and may not form in a given sample.

Continuing decreasing the magnetic field creates quasielectrons in the -state by adding 01 which have charge (adding 01 just before reaching 2/5 removes quasiholes of charge ). After having added 01 an additional times one reaches the 3/7 ground state with unit cell 0010101. Continuing the process gives all the TT ground states in the Jain sequence approaching .999As well as states at all fractions in between just as was the case when going from 1/3 to 2/5. Note that whereas the operation of inserting 01 is one and the same throughout this process, its interpretation in terms of quasiparticles varies: Near , adding 01 creates quasielectrons with charge and when approaching , adding 01 removes quasiholes of charge .

In the discussion above we started from the 1/3 state. However, due to the self-similar fractal structure established for the TT-ground states, it is clear that the procedure is general—it applies mutatis mutandis to any fraction .

Iv Transition to bulk

We have seen in the previous section that the TT-limit is remarkably simple but still rich and non-trivial. However, this is a limit that is most likely impossible to realize experimentally. Its raison d’tre lies in what it says about the experimentally realizable two-dimensional bulk limit, ,—this is the topic of this section.

We propose that the TT-ground state is the limit of a two-dimensional bulk QH state for any , odd. If this QH state is the ground state (in the bulk) then it is adiabatically connected to the TT-state—there is in general no phase transition from the TT-state as goes from 0 to .101010Of course more complicated scenarios are possible where there are intermediate phases and the QH phase is reentrant. This statement may need clarification. When changes, the size of the physical system changes; normally when one talks about adiabatic continuity one has in mind that a parameter in the hamiltonian changes. However, all that is happening when changes is that the matrix elements change. Thus for any , we can think of the problem as the infinite two-dimensional QH system with an interaction that depends on . This interaction is unusual: for finite it is anisotropic, it becomes purely electrostatic in the TT-limit, and it cannot, presumably, be written as a real space interaction . However, in the limit it approaches the chosen isotropic electron-electron interaction, whereas it is exactly solvable in the TT-limit. Thus, we have the standard situation when adiabatic continuity can be discussed. In passing, we note that this point of view shows that the TT-states give an exact solution to the two-dimensional QH-problem albeit with a peculiar interaction.

We will present arguments for our proposal about adiabatic continuity below; the strength of these arguments varies with . For the Laughlin states adiabatic continuity holds for a short-range pseudo-potential interaction, for the Jain states it can be strongly argued and for more general hierarchy states a case for it is emerging. In our opinion, the overall evidence for the correctness of the claim is convincing.

Apart from fractional QH states, there may of course be other ground states in the bulk. An example of this are Wigner crystals that are expected at low filling factors, in particular at , for small enough. In these cases, there is a phase transition from the TT-state as increases.

The even denominator filling factors are special. For them we propose that there is always a phase transition from the gapped TT-state as increases. The suggestion is based on: 1) a detailed analysis of the case and 2) exact diagonalization of small systems. For , we find a phase transition to a gapless Luttinger liquid that can be identified with a version of the Rezayi-Read state rr ; when increases further the state develops smoothly into the Rezayi-Read state that is believed to describe the system in the limit . Exact diagonalization of all with shows that for each filling factor with even denominator there is an abrupt change in the ground state, where its quantum numbers change, as increases, whereas there is no such change for any odd denominator filling factor bk2 .

Below we first discuss the odd denominator hierarchy states, concentrating on the Laughlin and Jain fractions; results for more general hierarchy fractions are summarized. We then turn to even denominator fractions; we discuss the half-filled Landau level in detail and comment on general such fractions and on the difference between and . The TT-limit is useful also to analyze non-abelian states; we end by discussing the Moore-Read pfaffian state at as an example of this.

iv.1 Hierarchy states

We begin by comparing the qualitative properties of the TT-states and the bulk hierarchy states. First of all, both have a gap to excitations. Note that the crystalline TT-states have no phonons: once the boundary conditions, ie the flux through the torus, are fixed the lattice sites are fixed in space. Thus the TT-states are not the Wigner crystals expected in the quantum Hall system at low filling factor; these would have gapless phonons. Second, the TT-state and the bulk hierarchy state at both have quasiparticle excitations with charge . A further similarity is that the ground state degeneracy on the torus in both cases is , but this is rather trivial as it is true for any state at .

On the torus, the symmetries are the magnetic translations and the TT and Laughlin-Jain states have the same quantum numbers, , with respect to these. This follows since the former are the limit of the latter, see Appendix E.

The TT-state is inhomogeneous, whereas the bulk hierarchy state should be homogeneous—a property that is normally considered fundamental for the quantum Hall fluids. However, it should be noted that imposing periodic boundary conditions on the problem of electrons moving in a perpendicular magnetic field, ie considering the problem on a cylinder or a torus, implies that the continuous translational invariance present on the infinite plane, is broken, in one direction, to a discrete invariance, translation by one lattice constant , along the cylinder, see Appendix A. As a consequence, for any state on a finite cylinder or torus there will be ripples in the density that disappear completely only as the size of the torus goes to infinity haldanebook .

Furthermore, it should be noted that if one takes the alternative ”adiabatic” point of view where the TT-limit is just a change in the interaction in the infinite two-dimensional system (where the lattice constant vanishes), then the TT-state is homogeneous.

iv.1.1 Laughlin-Jain fractions

For the Laughlin and Jain filling fractions, , explicit wave functions are known for the ground states as well as for the quasiparticle excitations—these allow us to further investigate the connection to the TT-states.

Rezayi and Haldane noted in 1994 that the limit of Laughlin’s wave function (at ) is the TT-state with unit cell 001 Haldane94 . This analysis can be generalized to general Jain states and also to quasiparticle excitations thereof. Taking the limit of these wave functions, one finds the TT-states and the quasiparticles of Sec. III bk2 . One direct consequence of this is that the TT-state has the same quantum numbers as the Laughlin/Jain state. The details of this analysis are given in Appendix E.

On the plane, the Laughlin state at is the unique ground state to a certain short range interaction and there is a gap to all excitations haldanebook ; trugman ; pokrovsky . Since this result on the plane is a consequence of the behavior of the wave function when two electrons approach each other it is natural that it holds in other geometries as well. This is indeed the case: For the short range interaction on the torus the Laughlin state is the unique ground state (up to the center of mass degeneracy) and there is a gap to all excitations for any circumference . An explicit statement or proof of this has not, to the best of our knowledge, appeared, but it is implicit in Ref. Haldane94, .111111However, there is a confusing comment in Ref. Haldane94, on observed gapless modes which seems to say that such bulk modes are expected. For a proof see Appendix F.

The transition from an inhomogeneous charge density wave state to a homogeneous-looking state as grows was studied in Haldane94 for the Laughlin state and a rapid crossover was found at . This crossover was recently studied numerically in detail by Seidel et al Lee05 . They confirm that there is no phase transition but an exponentially fast crossover to a virtually homogeneous state. That the Laughlin state spontaneously breaks the discrete translational symmetry on the thin cylinder has been rigorously proven by Jansen et al Jansen .

To conclude, we consider it firmly established that the TT ground states are adiabatically connected to the bulk QH states for the Laughlin fractions and that a strong case has been made for this being true also for the Jain fractions.

iv.1.2 Non-Jain hierarchy fractions

For the fractional QH states observed by Pan et al pan that are not of the Laughlin-Jain type discussed above no agreed upon wave functions exist. However, recently wave functions were constructed for all filling fractions in the hierarchy scheme that are obtained as repeated condensates of quasielectrons (as opposed to quasiholes) hierarchy ; conformal . These wave functions are obtained by a natural generalization of the conformal field theory construction of Jain’s composite fermion wave functions hans . The limit of the proposed wave functions are the TT ground states and the wave function at 4/11 has a large overlap with the exact ground state for small systems. We take this as a strong indication that the non-Jain hierarchy states in the bulk are also adiabatically connected to the TT ground states. Details of this conformal field theory construction of hierarchy wave functions will be presented elsewhere conformal . Wave functions at these fractions can also be constructed within the hierarchy scheme as condensates of quasiparticles greiter94 and as fractional QH states of composite fermions JainGoldman ; Quinn ; jain4over11 . For further discussion of these states, see Ref. lopez2, ; wojs, ; scomp2, ; pashitskii, .

iv.2 Even denominator fractions

So far we have discussed odd denominator filling factors and argued that the TT-states that are the ground states in the TT-limit are adiabatically connected to the two-dimensional bulk QH hierarchy states. In this section we discuss the even denominator filling factors. We begin by considering the half-filled Landau level; this case has been analyzed in detail and we believe it is representative for other gapless even denominators. (The gapped state at is discussed in Sec. IV.3.)

iv.2.1 Half-filled Landau level

In the half-filled lowest Landau level, , a metallic, ie, gapless state is observed jiang . The composite fermion explanation for this behaviour is that each electron binds two flux quanta, thereby removing all the external magnetic flux. The state is then supposedly described by non-interacting, or weakly interacting, composite fermions in zero magnetic field, ie, by a free two-dimensional Fermi gas. This is also the picture that emerges from the mean field theory where the statistical magnetic field completely cancels the external magnetic field hlr . These descriptions successfully explain experimental data and constitute the main evidence for the correctness of the composite fermion idea. The microscopic wave function of composite fermion type describing the state was provided by Rezayi and Read rr —on the torus it reads rrhaldane

(25)

where are the guiding center coordinates and is the bosonic Laughlin state at , see Appendix A. The wave function is not unique but depends on a set of parameters, ”momenta”, . In the two-dimensional bulk limit these are assumed to form a circular Fermi sea.

The TT-state at has unit cell 01, ie it is a state with one electron on every other site. This clearly minimizes the electrostatic repulsion, and hence is the ground state, in the TT-limit . As any TT-state it has a gap to excitations and it has quasiparticles with charge . Clearly, this is very different from the gapless state that is observed in the two-dimensional bulk limit and is believed to be described by (25).

A gapless state does in fact exist also for small but finite and there is strong numerical evidence that this state develops without a phase transition into the gapless two-dimensional state described by (25). The first evidence for this was obtained in a numerical study of the QH system on a thin cylinder bkdmrg , using the density matrix renormalization group method white . At , a sharp transition from the TT-state with a finite amplitude of the density oscillations to a virtually homogeneous state was observed at (for a short-range interaction ) and there were indications of gapless excitations. This behaviour was dramatically different from other fractions such as where no transition was observed as was varied—the amplitude of the density wave decreased continuously as increased. The transition at takes place in a region where only the first few shortest-range terms are important. This made it possible to find an exact solution—ground state and low energy excitations—for a hamiltonian that is a very good approximation to the short-range interaction for bk1 .

We now describe the exact solution, for details see Appendix G. Consider to start with a hamiltonian (1) that consists only of the shortest range hopping term, . Define a restricted Hilbert space by requiring that each pair of nearby sites contains exactly one electron, ie include the states and , but not 00 or 11.121212If one instead groups sites , one gets a restricted Hilbert space that is related to by a symmetry operation. The hamiltonian in (1) have vanishing matrix elements between the states in and since is conserved and these spaces have no common states except for the two TT-states with unit cell 01. Note that the interaction preserves the restricted Hilbert space . In spin notation, becomes and hence the model with the shortest range hopping term is, within , equivalent to a nearest neighbour spin-1/2 -chain,

(26)

where , , and . Note that creates a neutral particle, a dipole. The hamiltonian (26) is diagonalized by mapping the spins to fermions, , via a Jordan-Wigner transformation—the hamiltonian then becomes that of free one-dimensional fermions with nearest neighbor hopping.

Thus, within , the shortest range hopping term gives a one-dimensional free Fermi gas of neutral particles. But to make this a reasonable approximation to (1) for small but finite we must consider also the short-range electrostatic terms. Including the two shortest range such terms, and , with relative strength one finds that all states in have the lowest possible electrostatic energy. There are states not in (or ) with the same electrostatic energy but a strong case can be made that the hopping term creates a gap to these ( contains the most hoppable states) bk1 ; this is also supported by the numerics discussed below. Thus, the low energy sector for the short-range hamiltonian

(27)

where , is contained in and is that of a free one-dimensional Fermi gas of neutral particles. The ground state is a filled Fermi sea

(28)

and excitations out of this sea gives gapless neutral excitations. This provides an exact and explicit mapping of the low-energy sector of a system of interacting electrons in a magnetic field onto a system of non-interacting neutral particles, ie onto free particles that do not interact with the magnetic field.

We note that dipoles have many of the properties expected of composite fermions, most notably that they do not couple to the magnetic field. Earlier approaches to a microscopic theory in terms of dipoles were made by Read readearly ; read.5 , Murthy and Shankar ms , Pasquier and Haldane pasq , Lee dhlee and by Stern et al stern .

An important feature of the exact solution is that the number of dipoles is not conserved. These particles are neutral and their number is not tied to the number of electrons in the system as is the case for the composite fermions. For example, there are excitations where a single dipole is added to, or removed from, the Fermi sea. Moreover, the number of dipoles making up the Fermi sea in (28) is determined dynamically, ie by filling the negative energy dipole states; this leads to there being one dipole for every second electron. As this is a consequence of energetics, the number of dipoles presumably changes as increases. Assuming the ground state develops into a two-dimensional Fermi gas of dipoles as , this suggests that the number of dipoles, and hence the Fermi momentum, is determined dynamically rather than being simply given by the number of electrons. In this context we note that the Fermi momentum measured in surface acoustic wave experiments Willettreview disagrees with the mean field theory result which follows from that the number of composite fermions in the Fermi sea is equal to the number of electrons hlr .

The solvable case is a good approximation for a small finite . For the real-space short-range interaction , corresponds to . The short-range hopping term is then , whereas the leading ignored terms are small: and . This is close to the solvable point.

Longer range hopping terms in (1) will in general not preserve the subspace . However, since there is a gap to states not in , the effect of small such terms can be included perturbatively in an effective spin-1/2 hamiltonian that acts within . To zeroth order the effective hamiltonian is

(29)

where and is given in (19)131313Note that need not be positive for finite . In particular, this effective hamiltonian, , contains an Ising term that changes the non-interacting Fermi gas to an interacting Luttinger liquid with interaction parameter . If such terms come with large coefficients, they may cause a phase transition that opens up a gap; the results below strongly indicate that this does not happen in this system. In higher order perturbation theory, one gets increasingly more complicated spin-1/2 models containing renormalized quadratic as well as higher order terms. However, our numerical calculations suggest that these higher order terms may be very small for a range of —we find that the projection of the exact ground state of the Coulomb interaction on is virtually unity in the ”solvable” region as discussed below.

We have numerically identified the state obtained in exact diagonalization in the solvable region with the exact solution (28). We continuously varied from the values that give the exact solution to values that correspond to a in this region and verified that the ground state develops continuously without any level crossing. Whereas gives a qualitative understanding, we suggest that gives a quantitatively very accurate description of the low energy sector within this region. Thus we conclude that the exact solution is stable in a finite neighborhood of the solvable point where it develops into an interacting gapless one-dimensional model, a Luttinger liquid.

However, the perturbative argument for this is of course restricted to hamiltonians close to the solvable one—when increases, hopping terms of increasing range and with large coefficients will appear—leading to a hamiltonian that is very different from (27) and that definitely does not preserve the restricted hilbert space . To investigate this region we compared the ground states obtained in exact diagonalization with the Rezayi-Read states (25) bk2 .

The plane waves in (25) are periodic on the torus, with our normalization this means that are integers. A straightforward calculation shows that these momenta determine the conserved quantum numbers of the state:

(30)

where is the momentum of particle