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Topological invariant for the orbital spin of the electrons in the quantum Hall regime and Hall viscosity
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2015 J. Phys.: Condens. Matter 27 075601 (http://iopscience.iop.org/0953-8984/27/7/075601) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 132.239.1.231 This content was downloaded on 09/06/2017 at 16:01 Please note that terms and conditions apply.
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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 27 (2015) 075601 (6pp)
doi:10.1088/0953-8984/27/7/075601
Topological invariant for the orbital spin of the electrons in the quantum Hall regime and Hall viscosity V Gurarie Physics Department, University of Colorado, Boulder, CO 80309, USA E-mail: [email protected] Received 4 September 2014, revised 17 November 2014 Accepted for publication 15 December 2014 Published 28 January 2015 Abstract
Hall conductance of noninteracting fermions filling a certain number of Landau levels can be written as a topological invariant. A particular version of this invariant when expressed in terms of the single particle Green’s functions directly generalizes to cases when interactions are present including those of fractional Hall states, although in those cases this invariant no longer corresponds to Hall conductance. We argue that when evaluated for both integer and fractional Hall states this invariant gives twice the total orbital spin of fermions which in turn is closely related to the Hall viscosity, a quantity characterizing the integer and fractional Hall states which recently received substantial attention in the literature. Keywords: fractional quantum Hall effect, Hall viscosity, topological invariants (Some figures may appear in colour only in the online journal)
Energy levels of a particle moving in a 2D plane in a magnetic field perpendicular to that plane form bands called Landau levels. If all the levels lying below a certain fixed energy located in a band gap (Fermi energy) are filled by noninteracting fermions, we obtain what can be called the integer quantum Hall system. The conductance of this system when written in units of e2 / h is integer valued and can be thought of as a topological invariant, the sum of Chern numbers of the filled bands [1]. It is possible to re-express this conductance as a certain topological winding of the fermionic Green’s function. Indeed, the Green’s function of this system of fermions is Gab (ω, k), where ω is frequency conjugate to imaginary time, k is the 2D (quasi-)momentum, and indices a, b refer to the Landau levels. It can be mathematically thought of as a matrix valued function of frequency ω and quasimomentum k. Such matrix valued functions are known to fall into equivalence classes such that a Green’s function belonging to one class cannot be smoothly deformed into a Green’s function belonging to a different class (in mathematics those are usually called the homotopy classes). These classes can be labelled by integers. It has been known for quite some time that these integers coincide with the Hall conductance expressed in units of e2 / h, in other words with combined Chern numbers of filled bands [2]. The fact that 0953-8984/15/075601+06$33.00
the integer Hall system has an excitation gap implies that the Green’s function is never infinity and is a smooth function of frequency and momenta, which is necessary for these classes to be well-defined. Consider now a system of interacting fermions moving in the same 2D plane in a perpendicular magnetic field. Depending on their density, they can still be in an integer quantum Hall phase, but they can also be in a fractional Hall state, or in some cases in a gapless state. As long as the system has an excitation gap (that is, the Green’s function is not singular), and as long as the Green’s function never vanishes for any frequency and momenta, it is still a smooth function of its arguments and is also labelled by integers representing the homotopy classes discussed earlier [3]. But these integers no longer necessarily coincide with the Hall conductance. First of all, the expression for the Hall conductance in terms of the single-particle fermionic Green’s function is derived in the absence of any interactions, and the derivation clearly breaks down once the interactions among fermions are turned on. Moreover, the Hall conductance in fractional Hall states when expressed in terms of e2 / h is a non-integer fraction, and cannot be directly related to the integers labeling homotopy classes of Green’s functions. Establishing the meaning of those integers 1
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J. Phys.: Condens. Matter 27 (2015) 075601
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in a fractional Hall regime is therefore not entirely trivial. In [4] these integers were calculated for certain fractional Hall states, but no conclusions were drawn as to their meaning. We would like to argue that these integers, when properly defined as explained below, give twice the total orbital spin of the fermion [5]. In turn, this spin is known to coincide with the shift S of the fractional quantum Hall state [5] in some although not all cases [6]. Additionally, the shift is known to represent the Hall viscosity [7], a certain non-dissipative transport coefficient η which received significant attention in the literature recently, via the relation η=
h ¯ nS , 4
Quantum Hall 0 Fluid
perturbation G → G + δG N does not change (furthermore, it is possible to show that N reflects the homotopy class of the map from the ω, k space into the space of complex matrices represented by G). However this assumes that G is nonsingular, that is its eigenvalues are neither zero nor infinity. The eigenvalues of G cannot be infinity in a system with a gap. Furthermore, they cannot be zero in the absence of interactions [3]. This results in the invariant N taking a specific value within each noninteracting state of electrons. When interactions are present the eigenvalues of G can be zero, and this raises a possibility of several values of the invariant N within the same gapful phase of the system of electrons under study. This in fact occurs in a 1D analog of equation (2) for certain 1D systems as discussed in [16] (in 1D space this has a natural explanation in view of the well known collapse of the Z classification of some 1D topological insulators down to Zn with a certain n in the presence of interactions [17, 18]). However, when equation (2) is applied to a fractional quantum Hall state in 2D space it takes a a specific single value within each state as will follow from the analysis in this paper. In the absence of interactions this invariant can be shown to be equal to the Hall conductance up to a factor of e2 / h via the application of the Kubo formula, thus not only it takes a specific value within each phase, but also different phases correspond to different values of the invariant. Once interactions are turned on, it loses its interpretation as the Hall conductance. The invariant remains an integer, although two distinct fractional quantum Hall states may now end up with the same value of N . Therefore one should not expect that N labels distinct fractional Hall states uniquely like it does in the integer Hall regime. Nevertheless it may still present a useful characterization of fractional Hall states. The general physical meaning of N in the fractional Hall regime can now be established in the following way. Consider a half-plane geometry where a 2D system extends to x < 0 and terminates at x = 0 as depicted in figure 1. In the presence of the boundary, the Green’s function of this system is given by GBab (ω, p; x, x ) = −i dτ eiωτ T aˆ a (τ, p; x) aˆ b† (0, p; x ) ,
(1)
Here aˆ and aˆ † are annihilation and creation operators of fermions in bands labelled by the subscript of these operators at an appropriate imaginary time and at appropriate momenta and T is the usual imaginary time ordering operation [15]. In terms of this Green’s function the integer-valued topological invariant can be written in the following form [2] α,β,γ
αβγ tr
x
Figure 1. A 2D half-plane geometry with a boundary perpendicular to the x-axis. Quantum Hall fluid is confined to x < 0. The axis perpendicular to x is labeled by p, the momentum along the y-axis.
where n is the fermion number density [6, 8–11]. Taken together, this means that the Hall viscosity is often, although not always, given by a topological invariant, which is the main result of this paper. More precisely, simple arguments [6] can be given that the total orbital spin of the fermion (equal in particular for Abelian quantum Hall states to I tI sI , where t and s and charge and spin vectors respectfully [12]) coincides with the shift for all the states whose wave functions are given by the so-called conformal blocks of relevant conformal field theories [13], including the Laughlin states, the Read–Moore (Pfaffian) state, as well as Read–Rezayi (parafermion) states [14]. In other cases, including the Abelian hierarchy states and some of the states obtained from the Read–Rezayi states by a particlehole transformation the shift might not be equal to the total spin of the fermion (and might not even be an integer), instead being represented by an ‘average’ spin as opposed to the total spin [5, 6]. In those cases, the direct relationship between the invariant and the Hall viscosity breaks down, although a slightly more indirect relationship remains. Let us now present the derivation of this result. A Green’s function of a system of elections moving in a 2D lattice without boundaries (for example, with periodic boundary conditions) can be defined as Gab (ω, k) = −i dτ eiωτ T aˆ a (τ, k) aˆ b† (0, k) . (2)
N=
p
dωd2 k −1 G ∂α GG−1 ∂β GG−1 ∂γ G, (3) 24π 2
where G−1 ab (ω, k) is a matrix inverse to Gab (ω, k), the variables α, β and γ take values 0–2, and ∂0 = ∂/∂ω, ∂1 = ∂/∂kx , ∂2 = ∂/∂ky . It is straightforward to verify that this expression is a topological invariant for example by showing tha under a small
(4) where a(τ, ˆ p; x) annihilates a fermion with momentum p along the boundary at the position x perpendicular to the boundery and at imaginary time τ . Consider 2
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that is, gn are indeed the Green’s functions of modes labelled by their momentum along the boundary. Now equation (5) can be rewritten in the following way dk µ Nedge = (10) ∂µ ln gn (ω, p). 2π i n C
This is nothing but a sum of phases accumulated by gn over a circular contour in the ω − p plane, normalized by 2π i. This will necessarily be an integer, justifying the identification of Nedge with a topological invariant. Now among gn most describe fermions moving far away from the boundary. However some of those gn correspond to the gapless edge modes. To elucidate this further, let us examine the contribution to Nedge coming from typical boundary modes encountered in a quantum Hall fluid [21]. A simple integer Hall boundary mode with the dispersion vp where v is the velocity of the mode has a Green’s function g = 1/(iω − vp). The phase it accumulates over a circular contour in the ω − p plane when normalized by 2π i is simply equal to 1. A more complicated fractional Hall mode, for example in a Laughlin state with a filling fraction ν = 1/(2m+1) where m is some positive integer has a boundary fermion Green’s function which is easiest to find in the coordinate-imaginary time domain g(τ, x) ∼ 1/(x − ivτ )2m+1 . However for its use in equation (10) we need to know it in the frequency-momentum domain. The appropriate Fourier transform is not straightforward as the Green’s function has a strong UV singularity which needs to be regularized. A correct regularization was discussed in the literature on Luttinger liquids ( [22]; see also [23]) and involves integrating over x first and then over τ dx eipx p 2m iωτ g(ω, p) = dτ e ∼ . (11) (x − ivτ )2m+1 iω − vp
Figure 2. A contour of integration C in equation (5) where v is some arbitrarily chosen velocity.
the following expression dk µ −1 Nedge = dxdx GBab (ω, p; x, x ) 2π i a,b,µ C
×∂µ GBba (ω, p; x , x).
(5)
Here k 0 = ω, k 1 = p, and the integral proceeds over a contour C which is a circle in the ω − p plane as shown in figure 2. We will see below that Nedge is also an integer-valued topological invariant. The following crucial for our purposes result has been established in [19, 20]. Define the bulk Green’s function by the following Wigner transform r r G(ω, k) = lim dr e−irkx GB ω, ky ; R + , R − . R→−∞ 2 2 (6) The limit R → −∞ is needed to move far away from the boundary. It is natural to assume G(ω, k) defined in this way coincides with the Green’s function of the system without boundaries. Using G(ω, k) found in this way we can construct the invariant N as in equation (3). It can be shown that N from equation (3) and Nedge from equation (5) are always equal to each other, as long as equation (6) is used to relate GB to G. This allows us to calculate N by calculating Nedge instead. Even though Nedge appears to be more complicated than N , in fact it will be straightforward to evaluate in a general fractional Hall state. To do that, we first introduce the eigenvectors and eigenvalues of the Green’s function in the presence of the boundary GB , dx GBab (ω, p; x, x ) ψb(n) (x ) = gn (ω, p) ψa(n) (x). (7)
This Green’s function does not have a well defined phase as p is taken to zero. Equation (10) with it is not well defined. Instead, we could define a Green’s function with a rotationally invariant cutoff [21] f (x 2 + v 2 τ 2 ) ei(ωτ +px) (iω + vp)2m gr (ω, p) = dxdτ ∼ . 2m+1 (x − ivτ ) iω − vp (12) Here f is some function whose precise form is unimportant, but which quickly goes to 1 if its argument exceeds some arbitrarily chosen small cutoff a 2 and goes to zero sufficiently quickly as its argument goes to zero to make the integral convergent, and the right hand side of this equation is valid for ω/v and p much smaller than 1/a. It is possible to chose a specific form of the function f for practical calculations. For example, the choice f = θ(x 2 + v 2 τ 2 − a 2 ) where θ is the usual step function θ(x) = (sign x + 1) /2 allows to compute the Fourier transform in equation (12) exactly. However, the right hand side of equation (12) could be easily seen to be independent of the choice of f . Unlike g from equation (11), gr from equation (12) has a well defined phase. When substituted for gn in equation (10), gr produces the contribution of 2m + 1. Note however that unlike the more physical function g(ω, p),
b
gn (ω, p) can be thought of as Green’s functions of modes propagating along the boundary. Indeed, it is straightforward to see that we can define an annihilation operator of the boundary mode aˆ n (τ, p) = dx ψa∗ (n) (x) aˆ a (τ, p; x). (8) a
With its help, equation (7) implies that
gn (ω, p) = −i dτ eiωτ T aˆ n (τ, p) aˆ n† (0, p) ,
(9) 3
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gr (ω, p) can not be a conventional Green’s function at all frequencies. For example, it grows at large frequency while a conventional Green’s function must always decay as 1/ω [15]. Yet gr (ω, p) must be the correct low energy Green’s function of a rotationally invariant theory, as g(ω, p) is not rotationally invariant while g(τ, x) is. More generally, borrowing the notations from conformal field theory applicable at the boundary [13], we can imagine a boundary Green’s function defined by a boundary fermion operator with holomorphic and antiholomorphic dimensions ¯ , , 1 . (13) g∼ (x − ivτ )2 (x + ivτ )2¯
where K is the Fourier transform of f . We can then do it systematically for every n and construct GBr (ω, p; x, x ) from GB (ω, p; x, x ) according to (n) GBr (ω, p; x, x ) = ψa∗ (x)gr n (ω, p)ψb(n) (x ), (16) n
where ψ (n) are the eigenvectors defined in equation (7). This GBr satisfies the bulk boundary correspondence in the same way as GB . This implies that Nedge calculated with gr n substituted for gn in equation (10) is equal to N calculated with Gr substituted for G in equation (22). In turn, Gr can directly be constructed from the bulk Green’s function according to Gr (ω, kx , ky ) = dω dky K(ω − ω , ky − ky )G(ω , kx , ky ).
While the physical Fourier transform along the lines of equation (11) produces the Green’s function in frequencymomentum space without a well defined phase, just as in the simpler example of equation (11), the phase if rotationally invariant cutoff is employed gr =
dxdτ
f (x 2 + v 2 τ 2 ) ei(ωτ +px) (x − ivτ )2 (x + ivτ )2¯
(17) Defined in this way, N computed in the bulk thus gives twice the total conformal spin of the fermion boundary operator, as promised earlier. In turn, the total conformal spin coincides with the sum of all the components of the spin vector as defined in [5]. This observation also matches the explicit evaluation of the invariant in [4]. The values of the invariant with G defined via the rotationally invariant cutoff have been explicitly calculated in [4] for the Abelian hierarchy states [12], for the Read–Rezayi states [14], and for the anti-Pfaffian [24, 25], the particle-hole transformed Moore–Read state at half filling, and in all those cases they indeed coincide with the total conformal spin of the fermion boundary operator. Finally, a conformal spin of the electron boundary operator coincides with the conformal spin of the electron bulk operator and is known to be equal to half the shift in many although not all quantum Hall states [6]. A straightforward argument supporting this proceeds as follows. We use the expected bulkboundary correspondence of the fermion creation operator to argue that the total flux given by the phase accumulated by the electron operator in the bulk taken around all other fermions is nothing but twice the conformal spin of the electron operator (up to a term proportional to the number of fermions produced by the Jastrow factor of the wave function). The shift is the relationship between the flux Nφ and the number of fermions Nf as in Nφ = Nf /ν − S , and its equality to twice the conformal spin can thus be etablished. Since the shift is related to bulk viscosity via equation (1), we find that the Hall viscosity is directly related to the topological invariant N from equation (3) computed with Gr from equation (17) according to η = h ¯ nN/4, as was promised at the beginning of this paper. A more detailed discussion of this argument relating spin and shift can be found in [10]. However, the simple relationship between the spin and the shift indeed breaks down with more complex quantum Hall states. The most dramatic manifestation of this breakdown occurs in the quantum Hall states which are particle-hole transformed versions of the quantum Hall states produced by conformal blocks of conformal field theories [13]. In some cases (although not in the case of the Pfaffian case at half filling) the particle-hole transformation produces Hall viscosity which is not integer in units of h ¯ n/4 [10], explicitly
(14)
is clearly well defined. Its accumulation over a circular contour, which can be read directly off equation (14) without having to do the integral in it explicitly thanks to its rotational ¯ which is twice the so-called conformal invariance, is 2(− ), spin of this fermion boundary operator. The message coming from these considerations indicate that we need to consider Green’s function with a rotationally invariant cutoff. Note that an even more general Green’s function would include several branches of excitations at different velocities. Arguments can be given however that the Fourier transform with a rotationally invariant cutoff with an arbitrary velocity v which does not have to match any of the actual physical velocities of the edge excitations still produces the Green’s function with the phase accumulating according to the conformal spin of the boundary operator [4]. Let us also note that bulk modes (the ones which propagate far from the boundary) do not contribute at all to Nedge . The Green’s functions for these modes have a form g ∼ 1/(iω − (p)) where (p) never changes sign as a function of p to maintain the bulk gap. Such function does not wind at all along a circular contour in the frequency-momentum domain (it phase accumulation along this contour is zero). The end result of these arguments is that Nedge calculated with gr n when summed over n gives twice the total conformal spin of the fermion boundary operator summed over all the boundary modes. To be able to relate Nedge to N computed in the bulk, we imagine applying a rotationally invariant cutoff to the functions gn (ω, p). This can be accomplished by Fourier transforming them into the euclidean time-space domain and then transforming them back into the frequency-momentum domain while employing the rotationally invariant cutoff. Together the relevant transformation looks like gr n (ω, p) = dω dp K(ω − ω , p − p )gn (ω, p), (15) 4
J. Phys.: Condens. Matter 27 (2015) 075601
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C = 1. Note that in case of integer Hall effect in the absence of interactions GLLL = 1/(iω + |0 |) where 0 < 0 is the energy of the lowest Landau level and W = 1, giving N = C as expected (it has to coincide with the Chern number of the lowest Landau level). Therefore, a practical numerical evaluation of N may proceed with the following steps. First the Green’s function is computed numerically by exact diagonalization and spectral decomposition [15] entirely within the lowest Landau level
contradicting the concept of the Hall viscosity being an integervalued topological invariant in these units. As discussed earlier, similar breakdown between the spin and shift occurs in Abelian fractional quantum Hall states with more than one branch of the electron boundary excitation [5]. A practical application of this result is that in the exact diagonalization studies it is now possible to compute this invariant numerically to help identifying the approximately found ground states with possible candidate states. One difficulty one needs to overcome is that its direct evaluation according to the equation (3) may not be easy. Indeed, it involves the knowledge of the entire matrix-valued G(ω, k) including G computed in higher Landau levels. In a typical fractional Hall study, we examine fermions confined to a lowest Landau level, and evaluating the entire Green’s function in all Landau levels for subsequent substitution into equation (3) may be problematic. Fortunately, just like in case of integer Hall effect when the evaluation of equation (3) can be replaced by the evaluation of the Chern number of the filled bands, in case of fractional Hall effect in the lowest Landau level it is possible to reduce equation (3) to the study of the phase winding of the Green’s function in the lowest Landau level in the following way. We imagine studying a fractional Hall state employing basis wave functions of Landau levels with periodic boundary conditions, satisfying
GLLL =
m
(18)
where is the magnetic length and n labels Landau levels. These functions can be easily found explicitly [26]. Similar functions can also be found in case if the motion occurs on a lattice. In the basis of these functions the Green’s function of a fractional Hall state confined to the lowest Landau level (that is, neglecting Landau level mixing justified at sufficiently weak interactions) is essentially diagonal. Its lowest diagonal entry GLLL (ω, k) = G00 (ω, k) is computed entirely in the lowest Landau level according to equation (2) with a = b = 0. Its higher diagonal entries look essentially like non-interacting Green’s function Gnn = 1/(iω − n ) for n > 0, where n > 0 are the energy of the corresponding Landau levels. We then evaluate (3) by calculating the trace in the basis of functions n . This calculation is rather technical, and so we leave it until the end of the paper, with the result that one needs to evaluate the phase winding of the Green’s function ∞ W = −∞
dω ∂ω ln GLLL (ω, k) πi
+
m
iω + Em − E0
.
(21)
Here ∂1 = ∂kx , ∂2 = ∂ky . We emphasize that what we use here is not the Green’s function itself, but the Green’s function Gr transformed with the operation equation (17), but to avoid cluttering equations we drop the subscript r everywhere. We study a fractional Hall state in the lowest Landau level, in the regime where Landau level mixing can be neglected. In this regime the Green’s function is diagonal in the space of Landau levels. This means its eigenfunctions coincide with n defined in equation (18). As explained in the main body of the paper, in the basis of these eigenfunctions the Green’s function in the excited Landau levels can be thought of as simply the Green’s functions of noninteracting electrons,
(19)
gn = Gnn =
at arbitrary k. In terms of this number, as long as it is odd, the invariant N is given by W +1 C, N= 2
iω − Em + E0
m| aˆ 0 (k) |0 2
Here aˆ 0† (k) creates a particle in the lowest Landau level with momentum k (in the state 0 (k)), |0 represents an exact fractional Hall state, and Em are the energies of exact eigenstates of the interaction Hamiltonian containing one more or one less particle then the fractional Hall state under study. Given GLLL the operation equation (17) has to be applied to it. Finally, its phase winding W has to be determined by employing equation (19) or by a direct examination of its phase. Note that the expression equation (21) by itself cannot give winding different from W = 0 or W = ±1. This is easiest to see if one notes that Im G = 0 implies ω = 0, thus as ω is taken from −∞ to +∞ the total phase accumulation of G cannot exceed π . Therefore, the operation equation (17) is a crucial step to convert G into Gr whose phase can wind multiple number of times. The winding W gives the invariant N according to equation (20). Let us now go over the derivation of equation (20). We begin with the expression equation (3) for the topological invariant, which we rewrite for convenience as dωd2 k −1 N= αβ tr G ∂ω GG−1 ∂α GG−1 ∂β G. (22) 2 8π α,β=1,2
n (x + , y; kx , ky ) = e2π iy/+ikx n (x, y; kx , ky ), n (x, y + ; kx , ky ) = eiky n (x, y; kx , ky ),
2 † m| aˆ 0 (k) |0
1 , n = 1, 2, . . . iω − n
(23)
where n > 0 are the single-particle energy of the excited Landau levels (note that the operation equation (17) does not change gn because its Fourier transform is UV convergent). The lowest Landau level Green’s function is a nontrivial function of frequency and momenta, g0 = G00 (ω, k), transformed with equation (17), which encodes in itself the information about the fractional Hall state.
(20)
where C is the Chern number of the lowest Landau level. For a particle moving in continuous 2D space in a magnetic field (as opposed to moving in a lattice), this Chern number is simply 5
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We explicitly evaluate the trace in equation (3) to find
dωd2 k ∂ ω gn n| |m m| |n ∂ G ∂ G (24) αβ α β 8π 2 n,m αβ gn2 gm
Hall states. The invariant equation (2) is also known to give Hall conductance in integer, although not in fractional, quantum Hall states. An interesting consequence of this is that the Hall conductance of integer quantum Hall states and their Hall viscosity are proportional to each other, something which is already known indepedently [5, 10]. This work gives further support to the arguments that Hall viscosity is robust against perturbations, and gives another way of identifying quantum Hall states numerically, via the evaluation of equation (2) or equivalently via equation (20). Finally in recent work it has been argued that the Hall viscosity is closely related to Hall conductivity at finite momentum [11, 27]. It would be interesting to see whether this sheds further light on the meaning of the topological invariant equation (3).
Here the summations over n and m go over 0, 1, 2, . . .. The Dirac notations here imply computing the expectation of the matrix ∂G with respect to the basis defined in equation (18). Note that while G itself is diagonal in this basis, its derivatives are not. In turn, this can be recast in the following form ∂ω ln ggmn dωd2 k n| ∂α G |m m| ∂β G |n αβ 16π 2 n =m αβ gn gm (25) Now we can take advantage of the standard relation
Acknowledgments
n| ∂α G |m = δnm ∂α g + (gm − gn ) n| ∂α |m . Substituting it gives (gn − gm )2 ∂ω ln dωd2 k αβ 16π 2 n =m αβ g n gm × n| ∂α |m m| ∂β |n .
(26) This work has been supported by the NSF grants DMR1205303 and PHY-1211914. The author is grateful to A Essin, P Ostrovsky, T Giamarchi, N Cooper and N Read for useful conversations at various stages of this project.
gm gn
(27)
References
Introduce a variable zmn = gm /gn . In terms of this variable we find 2 d2 k 1 1 − αβ dzmn + 2 16π 2 zmn zmn n =m αβ × m| ∂α |n n| ∂β |m .
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(28)
As ω is taken from −∞ to +∞, the variable znm goes over a certain contour in the complex plane which is being integrated over in equation (28). If both n and m are larger than 0, thanks to the explicit form equation (23), zmn does not circle about zero and the integral over it is zero (since the only singularities in equation (28) are at zmn = 0). Therefore, either n or m are zero. It is sufficient to restrict just one of them to zero, to find 2 2 d k 1 1 − αβ dzn + 8π 2 zn zn2 n>0 αβ × n| ∂α |0 0| ∂β |n ,
(29)
where zn = gn /g0 . Now zn winds about zero (W + 1)/2 times, where W was defined in equation (19). Thus the integral over zn is π i(W + 1) times the residue of the pole at zn = 0. This gives d2 k 0| ∂α |n n| ∂β |0 . (W + 1) (30) 4π i n0 αβ Here the sum could be extended over all n 0 as the expression above at n = 0 is zero. Finally, this expression up to a factor of (W + 1)/2 is the Chern number C of the lowest Landau level, equation (5) from [1]. Therefore, the answer is given by equation (20). In conclusion, we have shown that the invariant equation (2) produces twice the conformal spin of the electron in fractional Hall states, which coincides with Hall viscosity measured in units of h ¯ n/4 in many, although not all, quantum 6
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Topological invariant for the orbital spin of the electrons in the quantum Hall regime and Hall viscosity
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2015 J. Phys.: Condens. Matter 27 075601 (http://iopscience.iop.org/0953-8984/27/7/075601) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 132.239.1.231 This content was downloaded on 09/06/2017 at 16:01 Please note that terms and conditions apply.
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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 27 (2015) 075601 (6pp)
doi:10.1088/0953-8984/27/7/075601
Topological invariant for the orbital spin of the electrons in the quantum Hall regime and Hall viscosity V Gurarie Physics Department, University of Colorado, Boulder, CO 80309, USA E-mail: [email protected] Received 4 September 2014, revised 17 November 2014 Accepted for publication 15 December 2014 Published 28 January 2015 Abstract
Hall conductance of noninteracting fermions filling a certain number of Landau levels can be written as a topological invariant. A particular version of this invariant when expressed in terms of the single particle Green’s functions directly generalizes to cases when interactions are present including those of fractional Hall states, although in those cases this invariant no longer corresponds to Hall conductance. We argue that when evaluated for both integer and fractional Hall states this invariant gives twice the total orbital spin of fermions which in turn is closely related to the Hall viscosity, a quantity characterizing the integer and fractional Hall states which recently received substantial attention in the literature. Keywords: fractional quantum Hall effect, Hall viscosity, topological invariants (Some figures may appear in colour only in the online journal)
Energy levels of a particle moving in a 2D plane in a magnetic field perpendicular to that plane form bands called Landau levels. If all the levels lying below a certain fixed energy located in a band gap (Fermi energy) are filled by noninteracting fermions, we obtain what can be called the integer quantum Hall system. The conductance of this system when written in units of e2 / h is integer valued and can be thought of as a topological invariant, the sum of Chern numbers of the filled bands [1]. It is possible to re-express this conductance as a certain topological winding of the fermionic Green’s function. Indeed, the Green’s function of this system of fermions is Gab (ω, k), where ω is frequency conjugate to imaginary time, k is the 2D (quasi-)momentum, and indices a, b refer to the Landau levels. It can be mathematically thought of as a matrix valued function of frequency ω and quasimomentum k. Such matrix valued functions are known to fall into equivalence classes such that a Green’s function belonging to one class cannot be smoothly deformed into a Green’s function belonging to a different class (in mathematics those are usually called the homotopy classes). These classes can be labelled by integers. It has been known for quite some time that these integers coincide with the Hall conductance expressed in units of e2 / h, in other words with combined Chern numbers of filled bands [2]. The fact that 0953-8984/15/075601+06$33.00
the integer Hall system has an excitation gap implies that the Green’s function is never infinity and is a smooth function of frequency and momenta, which is necessary for these classes to be well-defined. Consider now a system of interacting fermions moving in the same 2D plane in a perpendicular magnetic field. Depending on their density, they can still be in an integer quantum Hall phase, but they can also be in a fractional Hall state, or in some cases in a gapless state. As long as the system has an excitation gap (that is, the Green’s function is not singular), and as long as the Green’s function never vanishes for any frequency and momenta, it is still a smooth function of its arguments and is also labelled by integers representing the homotopy classes discussed earlier [3]. But these integers no longer necessarily coincide with the Hall conductance. First of all, the expression for the Hall conductance in terms of the single-particle fermionic Green’s function is derived in the absence of any interactions, and the derivation clearly breaks down once the interactions among fermions are turned on. Moreover, the Hall conductance in fractional Hall states when expressed in terms of e2 / h is a non-integer fraction, and cannot be directly related to the integers labeling homotopy classes of Green’s functions. Establishing the meaning of those integers 1
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in a fractional Hall regime is therefore not entirely trivial. In [4] these integers were calculated for certain fractional Hall states, but no conclusions were drawn as to their meaning. We would like to argue that these integers, when properly defined as explained below, give twice the total orbital spin of the fermion [5]. In turn, this spin is known to coincide with the shift S of the fractional quantum Hall state [5] in some although not all cases [6]. Additionally, the shift is known to represent the Hall viscosity [7], a certain non-dissipative transport coefficient η which received significant attention in the literature recently, via the relation η=
h ¯ nS , 4
Quantum Hall 0 Fluid
perturbation G → G + δG N does not change (furthermore, it is possible to show that N reflects the homotopy class of the map from the ω, k space into the space of complex matrices represented by G). However this assumes that G is nonsingular, that is its eigenvalues are neither zero nor infinity. The eigenvalues of G cannot be infinity in a system with a gap. Furthermore, they cannot be zero in the absence of interactions [3]. This results in the invariant N taking a specific value within each noninteracting state of electrons. When interactions are present the eigenvalues of G can be zero, and this raises a possibility of several values of the invariant N within the same gapful phase of the system of electrons under study. This in fact occurs in a 1D analog of equation (2) for certain 1D systems as discussed in [16] (in 1D space this has a natural explanation in view of the well known collapse of the Z classification of some 1D topological insulators down to Zn with a certain n in the presence of interactions [17, 18]). However, when equation (2) is applied to a fractional quantum Hall state in 2D space it takes a a specific single value within each state as will follow from the analysis in this paper. In the absence of interactions this invariant can be shown to be equal to the Hall conductance up to a factor of e2 / h via the application of the Kubo formula, thus not only it takes a specific value within each phase, but also different phases correspond to different values of the invariant. Once interactions are turned on, it loses its interpretation as the Hall conductance. The invariant remains an integer, although two distinct fractional quantum Hall states may now end up with the same value of N . Therefore one should not expect that N labels distinct fractional Hall states uniquely like it does in the integer Hall regime. Nevertheless it may still present a useful characterization of fractional Hall states. The general physical meaning of N in the fractional Hall regime can now be established in the following way. Consider a half-plane geometry where a 2D system extends to x < 0 and terminates at x = 0 as depicted in figure 1. In the presence of the boundary, the Green’s function of this system is given by GBab (ω, p; x, x ) = −i dτ eiωτ T aˆ a (τ, p; x) aˆ b† (0, p; x ) ,
(1)
Here aˆ and aˆ † are annihilation and creation operators of fermions in bands labelled by the subscript of these operators at an appropriate imaginary time and at appropriate momenta and T is the usual imaginary time ordering operation [15]. In terms of this Green’s function the integer-valued topological invariant can be written in the following form [2] α,β,γ
αβγ tr
x
Figure 1. A 2D half-plane geometry with a boundary perpendicular to the x-axis. Quantum Hall fluid is confined to x < 0. The axis perpendicular to x is labeled by p, the momentum along the y-axis.
where n is the fermion number density [6, 8–11]. Taken together, this means that the Hall viscosity is often, although not always, given by a topological invariant, which is the main result of this paper. More precisely, simple arguments [6] can be given that the total orbital spin of the fermion (equal in particular for Abelian quantum Hall states to I tI sI , where t and s and charge and spin vectors respectfully [12]) coincides with the shift for all the states whose wave functions are given by the so-called conformal blocks of relevant conformal field theories [13], including the Laughlin states, the Read–Moore (Pfaffian) state, as well as Read–Rezayi (parafermion) states [14]. In other cases, including the Abelian hierarchy states and some of the states obtained from the Read–Rezayi states by a particlehole transformation the shift might not be equal to the total spin of the fermion (and might not even be an integer), instead being represented by an ‘average’ spin as opposed to the total spin [5, 6]. In those cases, the direct relationship between the invariant and the Hall viscosity breaks down, although a slightly more indirect relationship remains. Let us now present the derivation of this result. A Green’s function of a system of elections moving in a 2D lattice without boundaries (for example, with periodic boundary conditions) can be defined as Gab (ω, k) = −i dτ eiωτ T aˆ a (τ, k) aˆ b† (0, k) . (2)
N=
p
dωd2 k −1 G ∂α GG−1 ∂β GG−1 ∂γ G, (3) 24π 2
where G−1 ab (ω, k) is a matrix inverse to Gab (ω, k), the variables α, β and γ take values 0–2, and ∂0 = ∂/∂ω, ∂1 = ∂/∂kx , ∂2 = ∂/∂ky . It is straightforward to verify that this expression is a topological invariant for example by showing tha under a small
(4) where a(τ, ˆ p; x) annihilates a fermion with momentum p along the boundary at the position x perpendicular to the boundery and at imaginary time τ . Consider 2
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that is, gn are indeed the Green’s functions of modes labelled by their momentum along the boundary. Now equation (5) can be rewritten in the following way dk µ Nedge = (10) ∂µ ln gn (ω, p). 2π i n C
This is nothing but a sum of phases accumulated by gn over a circular contour in the ω − p plane, normalized by 2π i. This will necessarily be an integer, justifying the identification of Nedge with a topological invariant. Now among gn most describe fermions moving far away from the boundary. However some of those gn correspond to the gapless edge modes. To elucidate this further, let us examine the contribution to Nedge coming from typical boundary modes encountered in a quantum Hall fluid [21]. A simple integer Hall boundary mode with the dispersion vp where v is the velocity of the mode has a Green’s function g = 1/(iω − vp). The phase it accumulates over a circular contour in the ω − p plane when normalized by 2π i is simply equal to 1. A more complicated fractional Hall mode, for example in a Laughlin state with a filling fraction ν = 1/(2m+1) where m is some positive integer has a boundary fermion Green’s function which is easiest to find in the coordinate-imaginary time domain g(τ, x) ∼ 1/(x − ivτ )2m+1 . However for its use in equation (10) we need to know it in the frequency-momentum domain. The appropriate Fourier transform is not straightforward as the Green’s function has a strong UV singularity which needs to be regularized. A correct regularization was discussed in the literature on Luttinger liquids ( [22]; see also [23]) and involves integrating over x first and then over τ dx eipx p 2m iωτ g(ω, p) = dτ e ∼ . (11) (x − ivτ )2m+1 iω − vp
Figure 2. A contour of integration C in equation (5) where v is some arbitrarily chosen velocity.
the following expression dk µ −1 Nedge = dxdx GBab (ω, p; x, x ) 2π i a,b,µ C
×∂µ GBba (ω, p; x , x).
(5)
Here k 0 = ω, k 1 = p, and the integral proceeds over a contour C which is a circle in the ω − p plane as shown in figure 2. We will see below that Nedge is also an integer-valued topological invariant. The following crucial for our purposes result has been established in [19, 20]. Define the bulk Green’s function by the following Wigner transform r r G(ω, k) = lim dr e−irkx GB ω, ky ; R + , R − . R→−∞ 2 2 (6) The limit R → −∞ is needed to move far away from the boundary. It is natural to assume G(ω, k) defined in this way coincides with the Green’s function of the system without boundaries. Using G(ω, k) found in this way we can construct the invariant N as in equation (3). It can be shown that N from equation (3) and Nedge from equation (5) are always equal to each other, as long as equation (6) is used to relate GB to G. This allows us to calculate N by calculating Nedge instead. Even though Nedge appears to be more complicated than N , in fact it will be straightforward to evaluate in a general fractional Hall state. To do that, we first introduce the eigenvectors and eigenvalues of the Green’s function in the presence of the boundary GB , dx GBab (ω, p; x, x ) ψb(n) (x ) = gn (ω, p) ψa(n) (x). (7)
This Green’s function does not have a well defined phase as p is taken to zero. Equation (10) with it is not well defined. Instead, we could define a Green’s function with a rotationally invariant cutoff [21] f (x 2 + v 2 τ 2 ) ei(ωτ +px) (iω + vp)2m gr (ω, p) = dxdτ ∼ . 2m+1 (x − ivτ ) iω − vp (12) Here f is some function whose precise form is unimportant, but which quickly goes to 1 if its argument exceeds some arbitrarily chosen small cutoff a 2 and goes to zero sufficiently quickly as its argument goes to zero to make the integral convergent, and the right hand side of this equation is valid for ω/v and p much smaller than 1/a. It is possible to chose a specific form of the function f for practical calculations. For example, the choice f = θ(x 2 + v 2 τ 2 − a 2 ) where θ is the usual step function θ(x) = (sign x + 1) /2 allows to compute the Fourier transform in equation (12) exactly. However, the right hand side of equation (12) could be easily seen to be independent of the choice of f . Unlike g from equation (11), gr from equation (12) has a well defined phase. When substituted for gn in equation (10), gr produces the contribution of 2m + 1. Note however that unlike the more physical function g(ω, p),
b
gn (ω, p) can be thought of as Green’s functions of modes propagating along the boundary. Indeed, it is straightforward to see that we can define an annihilation operator of the boundary mode aˆ n (τ, p) = dx ψa∗ (n) (x) aˆ a (τ, p; x). (8) a
With its help, equation (7) implies that
gn (ω, p) = −i dτ eiωτ T aˆ n (τ, p) aˆ n† (0, p) ,
(9) 3
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gr (ω, p) can not be a conventional Green’s function at all frequencies. For example, it grows at large frequency while a conventional Green’s function must always decay as 1/ω [15]. Yet gr (ω, p) must be the correct low energy Green’s function of a rotationally invariant theory, as g(ω, p) is not rotationally invariant while g(τ, x) is. More generally, borrowing the notations from conformal field theory applicable at the boundary [13], we can imagine a boundary Green’s function defined by a boundary fermion operator with holomorphic and antiholomorphic dimensions ¯ , , 1 . (13) g∼ (x − ivτ )2 (x + ivτ )2¯
where K is the Fourier transform of f . We can then do it systematically for every n and construct GBr (ω, p; x, x ) from GB (ω, p; x, x ) according to (n) GBr (ω, p; x, x ) = ψa∗ (x)gr n (ω, p)ψb(n) (x ), (16) n
where ψ (n) are the eigenvectors defined in equation (7). This GBr satisfies the bulk boundary correspondence in the same way as GB . This implies that Nedge calculated with gr n substituted for gn in equation (10) is equal to N calculated with Gr substituted for G in equation (22). In turn, Gr can directly be constructed from the bulk Green’s function according to Gr (ω, kx , ky ) = dω dky K(ω − ω , ky − ky )G(ω , kx , ky ).
While the physical Fourier transform along the lines of equation (11) produces the Green’s function in frequencymomentum space without a well defined phase, just as in the simpler example of equation (11), the phase if rotationally invariant cutoff is employed gr =
dxdτ
f (x 2 + v 2 τ 2 ) ei(ωτ +px) (x − ivτ )2 (x + ivτ )2¯
(17) Defined in this way, N computed in the bulk thus gives twice the total conformal spin of the fermion boundary operator, as promised earlier. In turn, the total conformal spin coincides with the sum of all the components of the spin vector as defined in [5]. This observation also matches the explicit evaluation of the invariant in [4]. The values of the invariant with G defined via the rotationally invariant cutoff have been explicitly calculated in [4] for the Abelian hierarchy states [12], for the Read–Rezayi states [14], and for the anti-Pfaffian [24, 25], the particle-hole transformed Moore–Read state at half filling, and in all those cases they indeed coincide with the total conformal spin of the fermion boundary operator. Finally, a conformal spin of the electron boundary operator coincides with the conformal spin of the electron bulk operator and is known to be equal to half the shift in many although not all quantum Hall states [6]. A straightforward argument supporting this proceeds as follows. We use the expected bulkboundary correspondence of the fermion creation operator to argue that the total flux given by the phase accumulated by the electron operator in the bulk taken around all other fermions is nothing but twice the conformal spin of the electron operator (up to a term proportional to the number of fermions produced by the Jastrow factor of the wave function). The shift is the relationship between the flux Nφ and the number of fermions Nf as in Nφ = Nf /ν − S , and its equality to twice the conformal spin can thus be etablished. Since the shift is related to bulk viscosity via equation (1), we find that the Hall viscosity is directly related to the topological invariant N from equation (3) computed with Gr from equation (17) according to η = h ¯ nN/4, as was promised at the beginning of this paper. A more detailed discussion of this argument relating spin and shift can be found in [10]. However, the simple relationship between the spin and the shift indeed breaks down with more complex quantum Hall states. The most dramatic manifestation of this breakdown occurs in the quantum Hall states which are particle-hole transformed versions of the quantum Hall states produced by conformal blocks of conformal field theories [13]. In some cases (although not in the case of the Pfaffian case at half filling) the particle-hole transformation produces Hall viscosity which is not integer in units of h ¯ n/4 [10], explicitly
(14)
is clearly well defined. Its accumulation over a circular contour, which can be read directly off equation (14) without having to do the integral in it explicitly thanks to its rotational ¯ which is twice the so-called conformal invariance, is 2(− ), spin of this fermion boundary operator. The message coming from these considerations indicate that we need to consider Green’s function with a rotationally invariant cutoff. Note that an even more general Green’s function would include several branches of excitations at different velocities. Arguments can be given however that the Fourier transform with a rotationally invariant cutoff with an arbitrary velocity v which does not have to match any of the actual physical velocities of the edge excitations still produces the Green’s function with the phase accumulating according to the conformal spin of the boundary operator [4]. Let us also note that bulk modes (the ones which propagate far from the boundary) do not contribute at all to Nedge . The Green’s functions for these modes have a form g ∼ 1/(iω − (p)) where (p) never changes sign as a function of p to maintain the bulk gap. Such function does not wind at all along a circular contour in the frequency-momentum domain (it phase accumulation along this contour is zero). The end result of these arguments is that Nedge calculated with gr n when summed over n gives twice the total conformal spin of the fermion boundary operator summed over all the boundary modes. To be able to relate Nedge to N computed in the bulk, we imagine applying a rotationally invariant cutoff to the functions gn (ω, p). This can be accomplished by Fourier transforming them into the euclidean time-space domain and then transforming them back into the frequency-momentum domain while employing the rotationally invariant cutoff. Together the relevant transformation looks like gr n (ω, p) = dω dp K(ω − ω , p − p )gn (ω, p), (15) 4
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C = 1. Note that in case of integer Hall effect in the absence of interactions GLLL = 1/(iω + |0 |) where 0 < 0 is the energy of the lowest Landau level and W = 1, giving N = C as expected (it has to coincide with the Chern number of the lowest Landau level). Therefore, a practical numerical evaluation of N may proceed with the following steps. First the Green’s function is computed numerically by exact diagonalization and spectral decomposition [15] entirely within the lowest Landau level
contradicting the concept of the Hall viscosity being an integervalued topological invariant in these units. As discussed earlier, similar breakdown between the spin and shift occurs in Abelian fractional quantum Hall states with more than one branch of the electron boundary excitation [5]. A practical application of this result is that in the exact diagonalization studies it is now possible to compute this invariant numerically to help identifying the approximately found ground states with possible candidate states. One difficulty one needs to overcome is that its direct evaluation according to the equation (3) may not be easy. Indeed, it involves the knowledge of the entire matrix-valued G(ω, k) including G computed in higher Landau levels. In a typical fractional Hall study, we examine fermions confined to a lowest Landau level, and evaluating the entire Green’s function in all Landau levels for subsequent substitution into equation (3) may be problematic. Fortunately, just like in case of integer Hall effect when the evaluation of equation (3) can be replaced by the evaluation of the Chern number of the filled bands, in case of fractional Hall effect in the lowest Landau level it is possible to reduce equation (3) to the study of the phase winding of the Green’s function in the lowest Landau level in the following way. We imagine studying a fractional Hall state employing basis wave functions of Landau levels with periodic boundary conditions, satisfying
GLLL =
m
(18)
where is the magnetic length and n labels Landau levels. These functions can be easily found explicitly [26]. Similar functions can also be found in case if the motion occurs on a lattice. In the basis of these functions the Green’s function of a fractional Hall state confined to the lowest Landau level (that is, neglecting Landau level mixing justified at sufficiently weak interactions) is essentially diagonal. Its lowest diagonal entry GLLL (ω, k) = G00 (ω, k) is computed entirely in the lowest Landau level according to equation (2) with a = b = 0. Its higher diagonal entries look essentially like non-interacting Green’s function Gnn = 1/(iω − n ) for n > 0, where n > 0 are the energy of the corresponding Landau levels. We then evaluate (3) by calculating the trace in the basis of functions n . This calculation is rather technical, and so we leave it until the end of the paper, with the result that one needs to evaluate the phase winding of the Green’s function ∞ W = −∞
dω ∂ω ln GLLL (ω, k) πi
+
m
iω + Em − E0
.
(21)
Here ∂1 = ∂kx , ∂2 = ∂ky . We emphasize that what we use here is not the Green’s function itself, but the Green’s function Gr transformed with the operation equation (17), but to avoid cluttering equations we drop the subscript r everywhere. We study a fractional Hall state in the lowest Landau level, in the regime where Landau level mixing can be neglected. In this regime the Green’s function is diagonal in the space of Landau levels. This means its eigenfunctions coincide with n defined in equation (18). As explained in the main body of the paper, in the basis of these eigenfunctions the Green’s function in the excited Landau levels can be thought of as simply the Green’s functions of noninteracting electrons,
(19)
gn = Gnn =
at arbitrary k. In terms of this number, as long as it is odd, the invariant N is given by W +1 C, N= 2
iω − Em + E0
m| aˆ 0 (k) |0 2
Here aˆ 0† (k) creates a particle in the lowest Landau level with momentum k (in the state 0 (k)), |0 represents an exact fractional Hall state, and Em are the energies of exact eigenstates of the interaction Hamiltonian containing one more or one less particle then the fractional Hall state under study. Given GLLL the operation equation (17) has to be applied to it. Finally, its phase winding W has to be determined by employing equation (19) or by a direct examination of its phase. Note that the expression equation (21) by itself cannot give winding different from W = 0 or W = ±1. This is easiest to see if one notes that Im G = 0 implies ω = 0, thus as ω is taken from −∞ to +∞ the total phase accumulation of G cannot exceed π . Therefore, the operation equation (17) is a crucial step to convert G into Gr whose phase can wind multiple number of times. The winding W gives the invariant N according to equation (20). Let us now go over the derivation of equation (20). We begin with the expression equation (3) for the topological invariant, which we rewrite for convenience as dωd2 k −1 N= αβ tr G ∂ω GG−1 ∂α GG−1 ∂β G. (22) 2 8π α,β=1,2
n (x + , y; kx , ky ) = e2π iy/+ikx n (x, y; kx , ky ), n (x, y + ; kx , ky ) = eiky n (x, y; kx , ky ),
2 † m| aˆ 0 (k) |0
1 , n = 1, 2, . . . iω − n
(23)
where n > 0 are the single-particle energy of the excited Landau levels (note that the operation equation (17) does not change gn because its Fourier transform is UV convergent). The lowest Landau level Green’s function is a nontrivial function of frequency and momenta, g0 = G00 (ω, k), transformed with equation (17), which encodes in itself the information about the fractional Hall state.
(20)
where C is the Chern number of the lowest Landau level. For a particle moving in continuous 2D space in a magnetic field (as opposed to moving in a lattice), this Chern number is simply 5
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We explicitly evaluate the trace in equation (3) to find
dωd2 k ∂ ω gn n| |m m| |n ∂ G ∂ G (24) αβ α β 8π 2 n,m αβ gn2 gm
Hall states. The invariant equation (2) is also known to give Hall conductance in integer, although not in fractional, quantum Hall states. An interesting consequence of this is that the Hall conductance of integer quantum Hall states and their Hall viscosity are proportional to each other, something which is already known indepedently [5, 10]. This work gives further support to the arguments that Hall viscosity is robust against perturbations, and gives another way of identifying quantum Hall states numerically, via the evaluation of equation (2) or equivalently via equation (20). Finally in recent work it has been argued that the Hall viscosity is closely related to Hall conductivity at finite momentum [11, 27]. It would be interesting to see whether this sheds further light on the meaning of the topological invariant equation (3).
Here the summations over n and m go over 0, 1, 2, . . .. The Dirac notations here imply computing the expectation of the matrix ∂G with respect to the basis defined in equation (18). Note that while G itself is diagonal in this basis, its derivatives are not. In turn, this can be recast in the following form ∂ω ln ggmn dωd2 k n| ∂α G |m m| ∂β G |n αβ 16π 2 n =m αβ gn gm (25) Now we can take advantage of the standard relation
Acknowledgments
n| ∂α G |m = δnm ∂α g + (gm − gn ) n| ∂α |m . Substituting it gives (gn − gm )2 ∂ω ln dωd2 k αβ 16π 2 n =m αβ g n gm × n| ∂α |m m| ∂β |n .
(26) This work has been supported by the NSF grants DMR1205303 and PHY-1211914. The author is grateful to A Essin, P Ostrovsky, T Giamarchi, N Cooper and N Read for useful conversations at various stages of this project.
gm gn
(27)
References
Introduce a variable zmn = gm /gn . In terms of this variable we find 2 d2 k 1 1 − αβ dzmn + 2 16π 2 zmn zmn n =m αβ × m| ∂α |n n| ∂β |m .
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(28)
As ω is taken from −∞ to +∞, the variable znm goes over a certain contour in the complex plane which is being integrated over in equation (28). If both n and m are larger than 0, thanks to the explicit form equation (23), zmn does not circle about zero and the integral over it is zero (since the only singularities in equation (28) are at zmn = 0). Therefore, either n or m are zero. It is sufficient to restrict just one of them to zero, to find 2 2 d k 1 1 − αβ dzn + 8π 2 zn zn2 n>0 αβ × n| ∂α |0 0| ∂β |n ,
(29)
where zn = gn /g0 . Now zn winds about zero (W + 1)/2 times, where W was defined in equation (19). Thus the integral over zn is π i(W + 1) times the residue of the pole at zn = 0. This gives d2 k 0| ∂α |n n| ∂β |0 . (W + 1) (30) 4π i n0 αβ Here the sum could be extended over all n 0 as the expression above at n = 0 is zero. Finally, this expression up to a factor of (W + 1)/2 is the Chern number C of the lowest Landau level, equation (5) from [1]. Therefore, the answer is given by equation (20). In conclusion, we have shown that the invariant equation (2) produces twice the conformal spin of the electron in fractional Hall states, which coincides with Hall viscosity measured in units of h ¯ n/4 in many, although not all, quantum 6
