A Effective Hamiltonian

Long-Distance Entanglement of Spin Qubits via Quantum Hall Edge States

Abstract

The implementation of a functional quantum computer involves entangling and coherent manipulation of a large number of qubits. For qubits based on electron spins confined in quantum dots, which are among the most investigated solid-state qubits at present, architectural challenges are often encountered in the design of quantum circuits attempting to assemble the qubits within the very limited space available. Here, we provide a solution to such challenges based on an approach to realizing entanglement of spin qubits over long distances. We show that long-range Ruderman-Kittel-Kasuya-Yosida interaction of confined electron spins can be established by quantum Hall edge states, leading to an exchange coupling of spin qubits. The coupling is anisotropic and can be either Ising-type or XY-type, depending on the spin polarization of the edge state. Such a property, combined with the dependence of the electron spin susceptibility on the chirality of the edge state, can be utilized to gain valuable insights into the topological nature of various quantum Hall states.

pacs:
73.43.Fj, 73.63.Kv, 03.67.Lx

I Introduction

Quantum computers, exploiting entanglement and superposition of quantum mechanical states, promise much better performance than classical computers tackling a collection of important mathematical problems (1). Over the past few decades, a variety of solid-state systems have been studied for the implementation of qubits, the building blocks of a quantum computer. Among such systems, a very promising candidate (2) makes use of the spin of electrons confined in semiconductor quantum dots (QDs). In that scheme, entanglement of qubits is achieved through the direct exchange interaction between confined electrons and manipulation of individual qubits can be realized by magnetic or electrical means. (3) Recent advances in QD technology have established long coherence times (4) exceeding  ms and fast gate-operation times (3) on the order of tens of nanoseconds for spin qubits in QDs.

With the great progress in the development of quality spin qubits, scalability becomes the next major challenge towards building a functional quantum computer capable of performing fault-tolerant quantum computing (5). The implementation of quantum-error-correction algorithms (6) requires that the system reach a size of several thousands of qubits. In practice, however, one faces tremendous difficulties in assembling so many spin qubits, among which entanglement must be selectively established and maintained. Indeed, the nearest-neighbor nature of the direct exchange interaction, the primary source of entanglement, restricts drastically access of each qubit to the rest of the system and thus the space that can be used for installing the quantum circuits. A natural way to overcome such difficulties is to employ means of entangling spin qubits over long distances, which creates extra space for wiring the quantum circuits. In principle, this may be achieved by coupling the spin qubits to an electromagnetic cavity (7); (8); (9); (10), a floating metallic gate (11), or a dipolar ferromagnet (12). Recently, it was shown that coupling of distant spin qubits can also be realized via photon-assisted cotunneling (13).

In this article, we propose a new mechanism to achieve long-distance entanglement of spin qubits. We make use of the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction (14); (15); (16) between confined electron spins in QDs, mediated by the conducting edge states of quantum Hall (QH) liquids (17), to which the QDs are tunnel coupled (18). The spin qubit coupling obtained in such a way is particularly interesting. Depending on whether the edge state is spin-polarized or not, the induced coupling between the spin qubits can be Ising-type and perpendicular to the plane of the system, or XY-type and in-plane. This offers great versatility in the design of large-scale quantum circuits. The advantage of using QH edge states is twofold. First, the edge states and the QDs can be formed in the same material (by top gates) such as a two-dimensional electron gas (2DEG) in GaAs heterostructures. Second, the QH edge states are topologically stable and thus much more robust against disorder effects compared to one-dimensional (1D) conduction channels in nano- or quantum wires. Moreover, we find that the spin susceptibility of QH edge states manifests the inequivalence between the opposite directions, “clockwise” and “counterclockwise”, along the QH edge. In chiral edge states, conduction electrons propagate in only one direction, leading to a “rectified” spin susceptibility in the propagation direction of electrons. In non-chiral edge states, the spin susceptibility is nonzero in both directions along the QH edge, but with different magnitudes. The spin susceptibility has the same type of anisotropy as the coupling between qubits. Thus, measuring the spatial dependence of the spin susceptibility (19) can serve as a powerful probe of the chirality and spin polarization of the edge state, and thus of the topological order (17) in a QH liquid.

Ii Model

Figure 1: The basic setup consisting of two QDs (yellow disks) tunnel coupled to the edge (white lines and arrows) of a QH liquid (blue sheet) confined in the - plane. In general, the QH edge may support multiple edge modes, propagating in the same or opposite direction(s), which we do not depict explicitly. The QDs are separated by a distance along the QH edge. Each QD contains a single electron (blue spheres), whose spin (red arrows) serves as a qubit. The coupling strength between the QDs and the QH edge is controlled by gates (not shown). We assume no direct interaction between the localized electron spins and in the QDs.

We now discuss the physics of RKKY interaction mediated by QH edge states. The basic setup is shown in Fig. 1. Two QDs are placed adjacent to a QH liquid, separated by a distance and labelled by the site index . Conduction electrons in the QH edge state can tunnel into and out of the QDs (18) and thus can interact with the localized spins in them. This establishes coupling between the QH edge and the QDs. For simplicity, we treat the QDs as two spatial points. The Hamiltonian describing such a system has the form

(1)

where is the Hamiltonian of conduction electrons in the edge state, denotes the localized spin in the th QD, and denotes the spin of conduction electrons coupled to , with coupling strength . Experimentally, can be tuned by gating. We define to be the spin density in the edge state multiplied by the confinement length of the QDs. For the setup, we assume is large so that there is no direct interaction between the spins in the QDs.

In the weak tunnel coupling regime such that , where is the Fermi energy of conduction electrons, the dynamics of the spins in the QDs effectively decouples from that of the conduction electrons. In such a case, one can derive an effective Hamiltonian for the spins in QDs, valid in the adiabatic regime, by performing a Schrieffer-Wolff transformation (20); (21) of Eq. (1) followed by tracing out the degrees of freedom of conduction electrons (see Appendix A for the derivation of effective Hamiltonian and a discussion of adiabaticity),

(2)

where the spin-component indices . The first term is the RKKY interaction, with . Here is the static spin susceptibility of conduction electrons, , where and denotes the average determined by . Physically, conduction electrons in the vicinity of a QD develop a spin-density oscillation due to their interaction with the spin in the QD. This spin-density response, determined by , can be perceived by the spins in other QDs coupled to the QH edge. In this way the RKKY interaction is established. For spin-unpolarized QH states, we assume , such that . On the other hand, the in-plane spin operators , are less relevant (in the renormalization group sense) than the out-of-plane ones in a QH state with full spin polarization, as we discuss below. In this case we set and hence . Thus, in general we have . The RKKY interaction leads to an effective exchange coupling , as a function of the interdot distance , between the localized spins and . The effective onsite Zeeman fields are a direct consequence of time-reversal (TR) symmetry breaking in QH systems. We find . In spin-polarized QH states, (for more details and estimates we refer to Appendix A).

Iii RKKY interaction in various QH states

The RKKY interaction in Eq. (2) is by nature long-ranged and can be used as an approach to entangle spin qubits over long distances. Thus, it is important to understand how the interaction looks like in various QH systems. To this end, it is convenient to adopt a continuum description of the QH edge states that is well approximated by the chiral Luttinger liquid (LL) model at low energy (17). In general, the edge of a QH liquid may support (electron-) density-fluctuation modes as well as Majorana fermions (zero-modes), with the action

(3)

written in the bosonization language (17) (throughout the article we set ). The bosonic fields describe the density modes, and denote the Majorana fermions. The symmetric matrix encodes the topological properties of the QH state, while the positive-definite symmetric matrix specifies the velocities and interactions of . The parameter is the velocity of : () if the is left-moving (right-moving).

Upon passing to the continuum limit, we replace the spin operators with spin density operators , where is the confinement length of the QDs and is the position of the th QD. The nonvanishing components of the spin susceptibility are given by . Assuming translation invariance along the QH edge, which is justified for clean samples, we may further write , (22) where

(4)

with the time-ordering operator. The correlators are evaluated in the zero-temperature limit. We define

(5)

where is the sum of the most-relevant electron operators with spin on the QH edge.

The number of operators is not necessarily equal to that of operators since TR symmetry is broken. For instance, the most-relevant electron operators have the same spin in a spin-polarized QH state, so that . This is in contrast to the situation in 1D systems where TR symmetry is present (23); (24). Using bosonization, we express in terms of the fields and , and compute the spin susceptibility.

We sketch the calculation of the spin susceptibility for a generic QH edge state (for particular examples, see Appendix B). First of all, we assume separation of charged and neutral degrees of freedom in the QH edge state. This phenomenon, as has been demonstrated experimentally in a number of QH systems (26); (25), results from strong Coulomb interaction among the elementary density modes and resembles “charge-spin separation” in a generic TR-invariant 1D system (22). As a result, the physical modes that propagate on the QH edge are the charged and neutral collective modes as well as Majorana fermions. The physical parameters relevant to experiment are the velocities and interactions of these propagating modes, whose magnitudes are set by different energy scales in the QH system. For instance, the charged-mode velocity, determined by the dominant Coulomb energy scale, is much greater than the velocity of neutral mode and other parameters (25). We make use of this fact in our calculation. For a moment, we consider the case of two density modes in the edge theory, see Eq. (III). To compute the correlators in Eq. (4), we define a new set of fields which diagonalize the action of the density modes . The action takes the form

(6)

in the basis of new fields and . Here () if the edge states are chiral (non-chiral) and . New velocities and are well approximated by the velocities of the physical charged mode and neutral mode, respectively, so that . Upon expressing the spin density operators in terms of the free fields , , and , it is straightforward to compute the correlators,

(7)

where is an infinitesimal and is the gauge-invariant momentum difference between the edge modes. The case corresponds to the scattering of an edge mode with itself. Here we have omitted the terms that are less relevant, and assumed as both of the velocities are determined by less dominant energy scales in the system. The exponents , are functions of the matrices and and as we show and (see Appendix B for the expressions in different QH states). Evaluating the time integral in Eq. (4), we obtain , which in general may contain multiple terms for different momentum differences. We keep only the most-relevant terms.

The various QH states can be divided into three types: (i) Those with a chiral edge state containing a single density mode, such as the Laughlin states at filling factors , where is an odd integer. (ii) Those with a chiral edge state containing multiple interacting density modes, such as the QH state at . (iii) Those with a non-chiral edge state, such as the particle-hole dual states (27) of Laughlin states.

For QH states of type (i), we find , taking into account the most-relevant spin operators in the edge state. Thus, to the lowest order the RKKY interaction cannot be established. Physically, the vanishing spin susceptibility reflects the homogeneous electronic structure in an independent QH edge mode, a property originating from the incompressibility of the QH liquid which prevents the formation of electronic spin texture. In reality, however, a small nonzero spin susceptibility may still be measured, due to higher-order processes involving virtual transitions to edge states in higher Landau levels.

In QH edge states of type (ii) and type (iii), the spin susceptibility is nonzero to the lowest order. In these cases, the inter-edge interactions introduce inhomogeneous degrees of freedom (“noise”) to the stream of conduction electrons, allowing for the development of spin-density oscillations. We find

(8)

for left-moving type (ii) edge states, where , is the Heaviside step function, and are functions of and , whose explicit definitions are given in Appendix B. If the edge state is right-moving, one replaces with , and sends in . These findings suggest that the spin susceptibility in type (ii) edge states is “rectified”, i.e., directed in the down-stream direction of the propagation of conduction electrons, see Fig. 2(a), where left- and right-moving directions are defined with respect to the lower edge of the QH liquid (the same in Fig. 2(b)). This result is not surprising and can be understood also intuitively. In a left-moving edge state, conduction electrons move in the direction, leading to the factor in the expression of . Formally, such an interesting form of the spin susceptibility is a manifestation of the causality principle in 1D chiral systems, where information is transported one-way and novel physical rules can emerge, e.g., see Ref. (28) for fluctuation-dissipation relations in chiral QH systems.

Lastly, we find

(9)

for type (iii) edge states, where and are functions of and , defined in Appendix B. The spin susceptibility in this case is “both-way”, as shown in Fig. 2(b), with different magnitudes in the and directions, i.e., . This again reflects the inequivalence between left-moving and right-moving edge modes. Imagining now the chirality of all edge modes are reverted, e.g., by TR operation, the profile of the spin susceptibility should also be reverted. Indeed, we find that are related to by the exchange of arguments and , which technically carries out the chirality-reverting procedure (see Appendix B). In the above discussion, we have assumed that spin excitations do not extend into the part of the QH edge, where is the total edge length. In practice, this is realized by grounding the part or by choosing the sample such that .

(a)
(b)
Figure 2: Spin susceptibility in QH edge states of (a) type (ii) and (b) type (iii). For the type (ii) case, the spin susceptibility is directed in the propagation direction of the edge modes. For the type (iii) case, the spin susceptibility is nonzero in both directions along the QH edge.

The exponents , where , determine how the RKKY interaction scales with distance. In Table 1, we list them in different QH states. In general, depend on both the chirality and spin polarization of the QH edge state. For chiral edge states, i.e., those of type (ii), these exponents are integral invariants depending on the topological order of the bulk liquid, whereas for non-chiral edge states they are nonuniversal and depend on the parameters in the Hamiltonian. In the latter case, we write , where is the integer part of . As shown in Appendix B, for all the non-chiral edge states in the table, assuming “charge-neutral separation” on the edge. Moreover, we find that the in-plane components of the RKKY interaction vanish in a spin-polarized QH state, leading to an Ising-type exchange coupling of spin qubits. On the other hand, the RKKY interaction has zero out-of-plane component and equal in-plane components in a spin-unpolarized QH state, which is XY-type. This suggests that a transformation of the anisotropy type of the RKKY interaction may be observed in the QH liquid at , which was found to be spin-unpolarized at low fields and spin-polarized at high fields (29).

The QH state at is also of special interest. We consider both Abelian and non-Abelian topological orders proposed to describe this state. The former include the Halperin state (30) and 113 state (31), and the latter include the Moore-Read (Pfaffian) state (32), the anti-Pfaffian state (33); (34) and the state (35). The state and state can be both spin-polarized and -unpolarized, just like the QH state. The Pfaffian state, like the Laughlin states, supports a single density mode on the edge and thus has vanishing RKKY interaction. The particle-hole dual state of the Pfaffian state, the anti-Pfaffian state, has a non-chiral edge state and a non-integer scaling exponent. For the state, we assume that the Majorana fermion and the neutral collective mode propagate at different velocities, as they should in reality, which is necessary to obtain a nonvanishing scaling exponent. Such careful treatment is not essential for other states. We have assumed that the RKKY interaction is mediated solely by the fractional edge modes in the second Landau level, while the integer edge modes in the lowest Landau level do not play a role. Experimentally, this can be fulfilled, using the fact that edge modes in different Landau levels are spatially separated (36). For instance, the QDs in Fig. 1 can be moved out of the plane of the QH liquid and formed in a second two- or quasi-one-dimensional electron gas in the vertical direction (37); (38); (39); (40), such that they are in tunnel contact with the fractional edge modes but far away from the integer edge modes. The coupling between the integer edge and the QDs and the interaction between the integer edge and the fractional edge can be neglected to a good approximation.

 QH state               331      113            
RKKY type XY XY Ising XY Ising XY Ising Ising Ising
Table 1: Scaling exponents and anisotropy type of the RKKY interaction in various QH states. An overline is used to indicate a spin-polarized state, e.g., denotes the spin-polarized QH state at . We consider several topological orders at , including both Abelian ones (the 331 state and the 113 state, denoted as 331/ and 113/, respectively) and non-Abelian ones (the Pfaffian state, the anti-Pfaffian state, and the state, denoted as , , and , respectively). The 331 state and the 113 state both have spin-unpolarized and spin-polarized versions. For chiral edge states, the exponents are integers and we add arrows to indicate that the spin susceptibility is non-zero only in the down-stream direction. For non-chiral edge states, the exponents are non-integers and we enter the integer parts of the exponents. We put “” in the entry if the corresponding component of the spin susceptibility (and thus that of the RKKY interaction) vanishes. The RKKY interaction is XY-type in spin-unpolarized states and Ising-type in spin-polarized states.

Iv Discussion

Let us estimate the coupling between the two spin qubits in Fig. 1, given by . In Appendix B, we obtain the dimensional part of the spin susceptibility,

(10)

for both type (ii) and type (iii) edge states, where is the lattice constant of the underlying material hosting the QH system. For example, let us consider the QH state at , realized in GaAs heterostructures. We have  nm for GaAs, , and  m/s (25). Using  meV and  nm (see Appendix C for the estimates), we find eV for m. This is about one order of magnitude smaller than the direct exchange strength eV in typical GaAs double QDs (2) and is experimentally measurable. The RKKY interaction established by QH edge states thus provides a way to realize entangled quantum gates over mesoscopic distances. The implementation of two-qubit gates using Hamiltonians of the form of Eq. (2) is well known: see, e.g., Ref. (2) (footnote 13) for Ising-type coupling and Ref. (7) for XY-type coupling. The eV exchange strength converts to gate-operation times of the order of nanoseconds, which is well below the coherence times (3) of spin qubits.

It is interesting to compare the RKKY interaction in QH edge states with that in semiconductor quantum wires. Assuming spin-rotation symmetry, the dimensional part of the spin susceptibility in quantum wires can be found in Ref. (23). The ratio

(11)

characterizes the relative strength of the RKKY interaction in the two sorts of systems, where is the Fermi velocity in the quantum wire and depends on the interaction of electrons. In non-interacting case, . Consider the QH edge state and GaAs quantum wire. We find for m, using and m/s (23). In principle, quantum wires can also be used to mediated RKKY interaction between spin qubits. However, using QH edge states offers more advantages. From technical aspect, the edge states and the spin qubits can be realized in the same material, for instance, in a 2DEG in GaAs heterostructures, which is more experimentally accessible than a setup with quantum wires. More importantly, the topologically protected QH edge states are more immune to disorder effects and perturbations in the system than quantum wires. This guarantees a better quality of the long-distance quantum gates.

Our discussions so far have focused on the RKKY interaction between spin qubits. Interestingly, the treatments can also be applied to obtain the RKKY interaction between nuclear spins embedded in the 1D QH edge state (see also Ref. (41)). To this end, let and in Eq. (1) and the following equations, where is the hyperfine coupling constant, is the number of nuclear spins in a cross section (labelled by ) of the QH edge, and is the total nuclear spin operator in a given cross section. Given a non-chiral edge state with both spin-up and spin-down electrons, e.g., the spin-unpolarized state at , the nuclear spins may form a helical magnetic order (23) at low temperatures, induced by the RKKY interaction. The nuclear magnetic order acts back on the electronic system by gapping out conducting edge modes. Experimentally, such an order is evidenced by the reduction of the conductance at low temperatures (40).

By measuring the spatial dependence of RKKY interaction (19); (42), one can obtain information about the chirality and spin polarization of the QH edge state, which in turn are related to the topological order of the bulk QH liquid (17). In particular, this technique may be used to detect the nature of the QH liquid at : One can distinguish between a chiral edge state and a non-chiral edge state by confirming whether the spin susceptibility is unidirectional along the edge. One can rule out either a spin-polarized state or a spin-unpolarized state by comparing the in-plane and out-of-plane components of the RKKY interaction, by measuring the spin states in the QDs. For this one can make use of experimental techniques based on spin-to-charge conversion (2) developed for read-out of spin qubits in QDs (43); (44); (45). The numerical values of the scaling exponents also help to identify the true state. The advantages of measuring the spin susceptibility are obvious, compared with other approaches detecting topological orders based on edge-bulk correspondence (17), such as measuring the temperature and voltage dependence of quasiparticle tunneling (46). First, it is easier to vary the sampling point in space than in temperature or voltage, e.g., one may use the setup in Fig. 1 with an array of QDs. Second, information encoded in spin degrees of freedom is more robust than that encoded in charge current, against unfavorable modification due to long-range Coulomb interaction in the device (47). Compared with electronic Fabry-Pérot (48); (49) and Mach-Zehnder (50); (51); (52) interferometries, our setup probes the non-Abelian topological orders at with a much simpler device geometry and more straightforward data.

The scenario becomes more complicated if one replaces the QDs with quantum anti-dots (53). In that case, tunneling of quasiparticles, rather than electrons, defines the coupling between the QH edge and the anti-dots. It is still possible to define an RKKY interaction mediated by quasiparticles in the edge state, whose spatial dependence can be used to distinguish different Abelian QH states. For non-Abelian states, however, there are ambiguities in the scaling behavior of the RKKY interaction, arising from the multiple fusion channels of non-Abelian quasiparticles.

To conclude, we have introduced a novel approach to achieving long-distance entanglement of spin qubits confined in QDs, based on the RKKY interaction mediated by QH edge states. The approach allows for the implementation of quantum gates with long coupling ranges and fast operation times, which would greatly facilitate the development of large-scale quantum computers. From fundamental point of view, the ability to probe the chirality and the spin polarization of a QH edge state via measuring the spatial form of the RKKY interaction opens up a new venue for studying electronic and spin physics in QH systems.

Acknowledgements.
We acknowledge support from the Swiss NSF and NCCR QSIT.

Appendix A Effective Hamiltonian

Our starting point is the Hamiltonian in Eq. (1). For weak tunnel coupling between the QH edge and the QDs, we can treat as a perturbation and make a Schrieffer-Wolff transformation (20); (21) to remove terms linear in from the Hamiltonian. The transformed Hamiltonian reads

(12)

where satisfies . Written in terms of the Liouvillian superoperator , . The leading-order terms in in are given by

(13)

Using , where , we find

(14)

where we have defined for an operator and used , with the Levi-Civita symbol. Summation over repeated spin-component indices (Greek letters) is implied throughout this appendix.

Next, we take the expectation over the electronic degrees of freedom in the QH edge state. This gives an effective Hamiltonian describing the dynamics of localized spins in the adiabatic limit,

(15)

where we have identified the spin susceptibility of conduction electrons,

(16)

and defined effective onsite Zeeman fields for the QDs,

(17)

This is the Hamiltonian in Eq. (2).

In deriving the above effective Hamiltonian, we have neglected the external magnetic field that leads to the formation of the QH liquid. To justify this, let us estimate and (in unit of energy). For spin-unpolarized QH states, let us assume that the two terms in Eq. (A) have the same order of magnitude after taking the expectation and performing the integration, for typical time scales related to the dynamics of conduction electrons. This gives , where denotes the dimensional part. Passing to the continuum limit, let , where is given by Eq. (10) and is the natural short-distance cut-off, taken as the lattice constant of the host material. An estimation similar to that for the effective exchange coupling finds  meV. Meanwhile,  meV for typical field strengths of several Tesla in QH liquids. Thus, is large compared with in spin-unpolarized states.

For spin-polarized QH states, applying the assumption yields . In this case, we consider fluctuations in the next order, associated with the next-most-relevant spin operators in the edge theory. We have . The fluctuations give rise to effective onsite Zeeman fields

(18)

which are fully out-of-plane, . In other words, the spin-polarized edge states tend to polarize the spin qubits. Simple dimensional analysis shows that the order of magnitude of differs from those of the (nonvanishing) most-relevant spin operators by a factor of , where is the mean edge velocity and is a typical time scale for the dynamics of conduction electrons. Accordingly, the factor enters the relative strength of the effective onsite fields () in spin-unpolarized states to that () in spin-polarized states (where ). Our estimation shows that , so that . In the main text we have neglected for simplicity.

In principle, the Zeeman terms (assuming ) should be included in the unperturbed Hamiltonian in the Schrieffer-Wolff procedure, i.e., in Eqs. (12) and (13) and the definition of time evolution. As a consequence, the first localized-spin operator appearing in the two terms in Eq. (A) acquires time dependence, in addition to the time dependence in the first conduction-spin operator . The dynamics of , set by the Zeeman energy , however decouples from that of , set by the Fermi energy , since according to the estimation above. Thus, to a good approximation we may neglect the time dependence in . We do this for spin-unpolarized states. For spin-polarized states, Eq. (A) is exact: Only the terms with survive in the equation and we have since commutes with .

We note moreover that also appears in Eq. (15) for both spin-unpolarized and spin-polarized QH states. In the main text, we have neglected this term for simplicity. However, must be taken into account for the purpose of implementing two-qubit quantum gates.

The effective Hamiltonian in Eq. (15) describes the system in Fig. 1 in equilibrium. Given a change in the spin state of one of the qubits, the entire electronic system readjusts to achieve new equilibrium. The change of the qubit must be adiabatic in order for the other qubit to sense the change and respond. This means that the switching time of the first qubit satisfies . On the other hand, if the qubit state is changed very fast (non-adiabatically), there will be no effect on the second qubit within time . In that case, the process is dynamic and is described by the spin susceptibility at finite frequencies. For m,  ps, which is much shorter than the ideal gate-operation time  ns. Thus, the requirement for adiabaticity does not place much restriction on the operation of spin qubits.

Appendix B Spin Susceptibility

In this appendix, we calculate the spin susceptibility for the QH states listed in Table I. The formula is given by Eq. (4). First, we compute the correlators in the zero-temperature limit

(19)

where . We focus on the scaling behaviors of these correlators and neglect the proportionality constants. Next, we evaluate the time integral

(20)

where . Restoring the proportionality constants, we obtain the full expression of the spin susceptibility.

b.1 Correlators

Laughlin states at

The Lagrangian density that describes the edge state of the ( is an odd integer) Laughlin state is

(21)

where is the velocity of the edge mode described by bosonic field . We assume the edge state is left-moving. Electrons in the edge state are described by the vertex operator , where is the short-distance cut-off and is the Fermi momentum. Here and throughout this appendix we omit the Klein factors in the electron operators, which will drop out when evaluating the average. Since the edge state is spin-polarized, all the electrons have the same spin . Let us assume . Using Eq. (5) and neglecting transitions to higher Landau levels, we find , and

(22)

The correlator of can be read from Eq. (21), , where is defined as a positive infinitesimal throughout the appendix. This gives

(23)

whereas . Substituting Eq. (23) in Eq. (20) we obtain the spin susceptibility in Laughlin states.

The QH state at

The QH state has two bosonic edge modes , propagating in the same direction, where has spin up and has spin down. The Lagrangian density is

(24)

where is the velocity of and is the repulsive Coulomb interaction between and . We assume the edge modes are left-moving. The most-relevant electron operators are , where is the Fermi momentum of . The spin density operators are

(25)

where is the gauge-invariant momentum difference, proportional to the magnetic flux penetrating between the two edge modes.

To compute the correlators, we define eigenmodes

(26)

where , which diagonalize the edge theory,

(27)

where . According to the experiment (25), as a result of the strong Coulomb interaction . Expressing the spin density operators in eigenmodes, it is straightforward to obtain

(28)

where the functions . Notice that , i.e., the scaling exponents of the correlators are integral invariant, independent of the angle which depends on the inter-edge interaction. This is a well-known property of chiral QH edge states (17).

The QH state at

The QH state can be spin-unpolarized at low fields and spin-polarized at high fields (29).

We first consider the spin-unpolarized state. It has two bosonic edge modes and , where has spin up and has spin down. The Lagrangian density is

(29)

where

(30)

with the velocity of and the inter-edge interaction. The eigenvalues of the -matrix have opposite signs, so the edge state is non-chiral.

Experiment (26) revealed that the edge state consists of a charged mode and a neutral mode, moving in opposite directions. To connect the parameters in the edge theory described by Eq. (29) with experiment, we change to the physical basis of charged mode and neutral mode ,

(31)

where , , and . In general, due to finite Zeeman splitting. The charged-mode velocity , determined by the large Coulomb energy scale, is expected to be much greater in order of magnitude than the neutral-mode velocity and the interaction . We therefore assume . In particular, we assume that the scaling dimensions of quasiparticle operators in the real case do not deviate much from those in the case . With this assumption, we can determine the most-relevant electron operators in the edge theory, which are , with spin up, and , with spin down, where and are momentum-like constants related to the spatial locations of the edge modes and . The spin density operators are obtained by computing the operator product expansions (OPEs) of the electron operators and keeping the most-singular terms. We find

(32)

where .

In terms of eigenmodes

(33)

where , the edge theory is diagonalized,

(34)

where and . Since , we have and thus , and . The correlators are evaluated to be