Wess-Zumino-Witten terms in graphene Landau levels
Junhyun Lee1 and Subir Sachdev1, 2 1Department of Physics, Harvard University, Cambridge MA 02138, USA 2Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
(Dated: November 24, 2014)
We consider the interplay between the antiferromagnetic and Kekul´e valence bond solid orderings in the zero energy Landau levels of neutral monolayer and bilayer graphene. We establish the presence of Wess-Zumino-Witten terms between these orders: this implies that their quantum ﬂuctuations are described by the deconﬁned critical theories of quantum spin systems. We present implications for experiments, including the possible presence of excitonic superﬂuidity in bilayer graphene.
PACS numbers:

arXiv:1411.5684v1 [cond-mat.str-el] 20 Nov 2014

Introduction

A number of recent experimental [1–7] and theoretical [8–18] works have focused on the presence of antiferromagnetism (AF) in neutral monolayer and bilayer graphene in an applied magnetic ﬁeld. It has also been argued that a non-magnetic state with lattice symmetry breaking in the Kekul´e valence bond solid (VBS) pattern (see Fig. 1) is proximate to the AF state [12, 13, 16, 17]. Bilayer graphene o↵ers a particularly attractive area for studying the interplay between the AF and VBS order because it may be possible to tune between them by applying a transverse electric ﬁeld [1, 13, 17].
The presence of the competing AF and VBS orders sets up the possibility [17] of novel quantum criticality between these orders, similar to that found in insulating quantum spin models [19–29]. However these quantum spin models apply in the limit of very large on-site Coulomb repulsion between the electrons, and this is not the appropriate parameter regime for graphene. Here we examine a complementary limit of large magnetic ﬁeld and moderate interactions, so that it is permissible to project onto an e↵ective Hamiltonian acting only on the zero energy Landau levels. Such a limit has been widely used with considerable success in describing the properties of graphene. Our main new result is that the Landau level projected e↵ective action for the AF and VBS orders has a topological Wess-Zumino-Witten (WZW) term [30–32] for both the monolayer and bilayer cases. The presence of this term implies [33–35] that the ﬁeld theories of deconﬁned criticality [20, 21] apply to graphene, and we will discuss experimental implications.
Model and results
We begin by directly stating the Hamiltonian of the low energy graphene bands (see e.g. Refs. [15, 17] for

FIG. 1: AFM (left) and Kekul´e VBS states of bilayer graphene. The blue (red) lines indicate the honeycomb lattice of the top (bottom) layer. The ellipses in the VBS state denote the links between the top and bottom layers which are equivalently distorted with respect to the parent lattice.

details)

✓

H =v

0 a†q

aq ◆ 0

(1)

where v is a Fermi velocity, a = px ipy (e/c)(Ax Ay) with (px, py) the electron momentum, (Ax, Ay) is the vector potential of the applied magnetic ﬁeld, the matrix acts on the graphene sublattice index, and q = 1 for monolayer graphene, while q = 2 for the bilayer case. For bilayers, the sublattice index co-incides with the layer index. For both monolayers and bilayers, there is an additional 2-fold valley degeneracy, along with the usual 2-fold spin degeneracy (in the absence of a Zeeman coupling).
The a, a† obey commutation relations proportional to those of the ladder operators of a harmonic oscillator, and so it is easy to diagonalize H. In this manner we obtain q zero energy Landau levels, which are spanned by the orthonormal eigenfunctions `(r), where ` = 1 . . . qN , with N the number of ﬂux quanta. So we write the electron annihilation ﬁeld operator projected to the zero

2

energy Landau levels as

qXN

(r) =

`(r)c`.

`=1

(2)

where c` is canonical fermion annihilation operator. In the zero energy Landau levels, the valley, sublattice, and
layer indices all co-incide; henceforth we will refer to this
as a valley index, and it can take 2 values. The fermion
operators also carry a spin index with 2 possible values,
and we do not explicitly display the spin or valley indices. We now introduce Pauli matrices x,y,z which act on
the spin space, and a second set ⇢x,y,z which act on the
valley space (here we follow the conventions of Ref. [17]).
In terms of these matrices, the 3-component AF order is measured by (⇢z x, ⇢z y, ⇢z z) while the 2-component VBS order parameter is (⇢x, ⇢y).
It is convenient to write the above matrices as

1 = ⇢z x , 2 = ⇢z y , 3 = ⇢z z , 4 = ⇢x , 5 = ⇢y

and to notice that the 5 a matrices anti-commute and square to unity; indeed these are the 5 Dirac gamma matrices. Their 10 products i a b (a 6= b) realize the Lie algebra of SO(5), and the 15 matrices a and i a b realize the Lie algebra of SU(4).
Next, we introduce a 5-component unit vector na(r, ⌧ ), where r = (x, y) are the spatial co-ordinates and ⌧ is imaginary time, representing the combined spacetime ﬂuctuations of the AF and VBS orders. Then the imaginary time Lagrangian of the electrons projected to the zero energy Landau levels is

L

=

qXN

c†`

@c` @⌧

`=1

Z d2r na(r, ⌧ ) †(r, ⌧ ) a (r, ⌧ ) ,
(3)

where is the coupling of the electrons to the AF and

VBS orders, and there is an implicit sum of a over 5

values, and also over the spin and valley indices.

Now we can state our primary result. We integrate

over the the c` electrons in L and obtain an e↵ective action for unit vector na(r, ⌧ ). Apart from the usual terms of the O(5) non-linear sigma model considered in

Ref. [16] (and anisotropies due to the Zeeman coupling

and electron-electron interactions [16]), the e↵ective ac-

tion has a topological WZW term at level q

SWZW W [na]

= =

82⇡3⇡2iqZW01[dnuaZ]

(4) d2rd⌧ ✏abcdena@xnb@ync@⌧ nd@une

Here we have introduced the extra co-ordinate u, and
na(r, ⌧, u) is any function which smoothly extrapolates from the physical na(r, ⌧ ) at u = 1 to a ﬁxed value (say) na = (1, 0, 0, 0, 0) at u = 0. The choice of the extrapolation can only change W [na] by integers, and so e2⇡iqW is well deﬁned.

In the case of graphene in zero magnetic ﬁeld and weak interactions, the same WZW term between the N´eel and VBS orders is also present [36]. However, for the experimentally important case of bilayer graphene, there is no such WZW term for the AF and VBS orders at zero ﬁeld and weak interactions [17, 37] (although, E.-G. Moon has noted such a term for the quantum spin Hall order [37]); so, in this case the zero energy Landau level projection is crucial for obtaining the topological coupling.
Such a WZW term has a strong impact in the interplay between the order parameters. As we will review below, it topologically links AF order to defects of the VBS order, and vice versa.

Derivation

We provide two derivations of Eq. (4).
First, pick any 3 of the 5 na components, say a = u, v, w, and set the other 2 to zero. Then we have unit 3-vector ﬁeld N~ = (nu, nv, nw). Now consider a static Skyrmion texture in N~ (r). Then by a computation parallel to that in Section III.B of K. Moon et al. [38] (and its generalization to q = 2 [39]), the Skyrmion acquires a “charge”. In the present situation the charge is measured by i u v w and its spatial density is

⌦

†(r) i u

v

w

↵ (r)

=

q 2⇡

N~

·

(@xN~

⇥

@yN~ )

(5)

where the angular brackets represent the expectation
value over the occupied states in the zero energy Landau level perturbed by the texture in N~ as in L. Now
consider a VBS vortex i.e. a 2⇡ vortex in (n4, n5) applied to L. For a two-component order, the core of the vortex
has a singularity, but this can be relieved by orienting na in a third direction, say (±1, 0, 0, 0, 0). Now the VBS vortex is equivalent to a half-Skyrmion in N~ = (n1, n4, n5), and after integrating Eq. (5) over all space, this vortex has h xi = ±q. Similarly, vortex cores in the directions (0, ±1, 0, 0, 0) and (0, 0, ±1, 0, 0) yield h yi = ±q and h zi = ±q. So we reach the important conclusion
that the VBS vortex has total spin S = q/2, and has an
associated (q + 1)-fold degeneracy. For q = 1, note that
this is precisely the situation considered in Ref. [40] for
quantum spin models (see also [41]). Alternatively, we can examine the fate of SWZW in the presence of such VBS vortices: following a computation by Grover and
Senthil [35], we ﬁnd that the WZW term reduces to the
quantum Berry phase of a single spin with S = q/2. From
this we conclude that Eq. (5) implies Eq. (4).
For a second derivation of the WZW term from Eq. (3), we examine a diagrammatic expansion of L. Consider
a situation where na is polarized near, say, (0, 0, 0, 0, 1). Then, we can write na = (⇡1, ⇡2, ⇡3, ⇡4, 1) where |⇡i| ⌧ 1 for i = 1 . . . 4. Then to zeroth order in the ⇡i, the c`

3

FIG. 2: Box diagram leading to S⇡. The full lines are the Green’s function in Eq. (6) at the labeled frequencies, and the vertices are the term in Eq. (3).

operators in L have the Green’s function

G = (i! + 5) 1

(6)

where ! is the frequency of the electron propagator. We

now proceed to integrate out the electrons, and derive

an e↵ective action for the ⇡i. At fourth order in the ⇡i, we consider the box diagram in Fig. 2; this can be evalu-

ated by methods similar to those in [17], but with the G

above, and the vertices contributing the factors implied

by Eq. (3). A computation described in the supplement

yields the contribution

S⇡

=

i3q 16⇡

Z

d2rd⌧ ✏ijkm⇡i@x⇡j @y⇡k@⌧ ⇡m .

(7)

It can be checked that Eq. (4) reduces to S⇡ for na = (⇡1, ⇡2, ⇡3, ⇡4, 1), and so SWZW is the explicitly SO(5) invariant form of S⇡.

Theoretical consequences

We now turn to a discussion of the theoretical conse-

quences of the WZW term for the vicinity of the AF-

VBS transition. For q = 1, it has been demonstrated in

Refs. [34, 35, 40] that the O(5) non-linear sigma model

with O(3)⇥O(2) anisotropy and a level 1 WZW term is

equivalent to the CP1 model in 2+1 dimensions. This is

the same model appearing in the AF-VBS transition of

SU(2) quantum spin models [20, 21], and is a relativistic

ﬁeld theory with a U(1) gauge ﬁeld and a 2-component

complex scalar z↵. In terms of these ﬁelds, the AF or-

der

is

z↵⇤

s ↵

z

,

with

s

=

x, y, z;

so

the vector AF

or-

der has been ‘fractionalized’ into spinons z↵. Alterna-

tively, we can also view the z↵ quanta as representing the vortices/anti-vortices in the VBS order [40] which,

as we have just seen, carry spin S = 1/2.

Presently, the experimentally accessible case of the AFVBS transition is in bilayer graphene, so we focus now on the q = 2 case. With a level 2 WZW term, the VBS vortices carry spin S = 1, and therefore we need complex scalar ﬁelds with 3 components: we write these as Zs, with s = x, y, z. The ﬁeld theory of the Zs quanta is now the CP2 model with anisotropic quartic terms; such a ﬁeld theory was considered in Ref. [42] in a di↵erent context:
Lcp = |(@µ iAµ)Zs|2 +g|Zs|2 +u1(|Zs|2)2 +u2(Zs2)(Zt⇤2).
Here µ is a spacetime index, Aµ is the emergent U(1) gauge ﬁeld, g is the coupling which tunes the AF to VBS transition, and u1,2 are quartic couplings. In terms of the degrees of freedom in Lcp, the 3-component AF order parameter is now i✏stuZt⇤Zu, while the complex VBS order h⇢x + i⇢yi ⇠ ei✓ is the monopole operator in the U(1) gauge ﬁeld [19, 21].
For both the CP1 and CP2 models mentioned above, both ﬁrst and second order transitions are possible between the AF and VBS states. A recent numerical study [16] on a single layer model indicates a ﬁrst order transition for the parameters studied.
Experimental implications
Finally, we turn to experimental consequences for bilayer graphene. The deﬁning characteristic of deconﬁned criticality is the presence of a gapless ‘photon’ excitation of an emergent U(1) gauge ﬁeld [20]. This is associated with the Aµ above, and can also be interpreted as a ‘spin-wave’ excitation involving ﬂuctuations of the angle ✓. Our deﬁnition of ✓ shows that it is the angular phase associated with o↵-diagonal-long-range order (ODLRO) in valley space. The valley anisotropy terms in graphene is very small [12, 16], because it is suppressed by powers of the lattice spacing to the magnetic length; so we expect a nearly gapless ✓ spin-wave mode to be present (and most of the remarks below to also apply) even in the case of a ﬁrst-order transition.
Now recall the fact, noted earlier, that in the zero energy Landau levels the valley index coincides with the layer index of bilayer graphene (and also the sublattice index). So ODLRO in valley space is accompanied by ODLRO in the layer space i.e. ✓ is also the angular phase of interlayer excitonic superﬂuidity. Signatures of excitonic superﬂuidity have been observed in quantum Hall states in GaAs bilayers [43]. However, in the GaAs bilayers there is negligible tunneling of electron between the layers, and this crucial to the emergence of a U(1) symmetry which is broken by the excitonic condensate. So it might seem surprising that a similar superﬂuidity can be present in graphene bilayers, in the presence of very strong tunneling between the layers. The resolution is the identiﬁcation of the layer and valley indices

4

in the zero energy Landau levels of bilayer graphene: in the absence of intervalley scattering, and the irrelevancy of valley anisotropy terms to be presented below, there is also an emergent interlayer U(1) symmetry in bilayer graphene.
The counterﬂow electrical current can be written in terms of the gauge ﬁeld

Jtµ

Jbµ

=

4e 2⇡

✏µ⌫

@⌫ A

(8)

where Jt and Jb are the currents in the top and bottom layers, and e is the charge of the electron. The factor of 4
is deduced from Eq. (5), which shows that a AF Skyrmion in (n1, n2, n3) has excitonic charge density h⇢zi = 4 for q = 2. The counterﬂow conductivity can be computed
from Eq. (8) using Lcp: at the g = gc deconﬁned quantum critical point this implies a universal value of order the quantum unit of conductance e2/h.
For g > gc, in the VBS state, the conductivity should be computed using an e↵ective action for ✓ which includes
the inﬂuence of monopoles. Now the current is

Jtµ Jbµ = 4e⇢s@µ✓

(9)

where ⇢s is the sti↵ness of the excitonic superﬂuidity appearing the e↵ective Lagrangian density

L✓

=

⇢s 2

(@µ✓)2

y3 cos(3✓),

(10)

with y3 the fugacity of tripled monopoles which are allowed by the 3-fold rotational symmetry of the under-
lying honeycomb lattice [19]. The sti↵ness vanishes as ⇢s ⇠ (g gc)⌫ by the Josephson relation, where ⌫ is the correlation length exponent. Away from the critical point, the bare value y30 is proportional to the very small 3-fold valley anisotropy term [12, 16]; y3 has a further suppression [21, 44] from the critical ﬂuctuations of Lcp leading to y3 = y30(g gc)⌫ , where is the scaling dimension of the tripled monopole operator. So the e↵ec-
tive “interlayer tunneling” term, y3, is highly suppressed near the deconﬁned quantum critical point. We also note
that numerical studies on square lattice antiferromagnets
have provided striking evidence for the emergent U(1)
symmetry due to the suppression of monopoles [45], and
there is direct evidence for the suppression of monopoles
on the honeycomb lattice in the work of Block et al. [26].
In GaAs bilayers [43], the excitonic superﬂuidity
is most directly observed in counterﬂow experiments,
where electric currents ﬂow in opposite direction in the
two layers. This would be technically more di cult in
bilayer graphene, given the close spacing of the layers,
but experiments of this type would be ideal. In the
bilayer graphene experiments of Weitz et al. [1], there is
a Zeeman coupling to the magnetic ﬁeld (whose conse-
quences have been studied earlier [21]), and an electric
ﬁeld is applied transverse to the layers. The electric

ﬁeld provides a small breaking of the layer-exchange symmetry. In the presence of such a symmetry breaking, there is a coupling between the counterﬂow and parallel current modes, and a vestige of the counterﬂow superﬂuidity would also be present in a measurement of the total current in both layers. Weitz et al. observe a phase transition out of the (presumed) AF state, signaled by the enhancement of the conductivity. We propose that this enhancement is due to the coupling to counterﬂow superﬂuidity.
Acknowledgements: We thank D. Abanin, Eun-Gook Moon, E. Shimshoni, T. Senthil, I. Sodemann, and A. Yacoby for useful discussions. This research was supported by the NSF under Grant DMR-1360789, the Templeton foundation, and MURI grant W911NF-14-1-0003 from ARO. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation. J. L. is also supported by the STX Foundation.
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Supplement: Wess-Zumino-Witten terms in graphene Landau levels
Junhyun Lee and Subir Sachdev

DIAGRAMMATIC DERIVATION OF WZW TERM

We give a detailed derivation of Eq. (7). We work in Landau gauge and separate the quantum number l in Eq.

(2) into n and X, where n is the Landau level index and X is the center of the magnetic oscillator. For monolayer

graphene (q = 1) n is 0, and for bilayer graphene (q = 2) n is over 0 and 1. There are N of X labels, so the total number of states in the zero energy Landau level is qN .

The spatial wavefunction n,X (r) in Eq. (2) is:

n,X (r) = eiXy'n,X (x),

(S1)

where 'n,X (x) are the n-th harmonic oscillator eigenfunction:

'0,X (x)

=

1 ⇡p1/4

e

(x

X )2 /2 ,

'1,X (x) =

2(x ⇡1/4

X

)

e

(x

X )2 /2 .

(S2)

Note that we work in the unit where the magnetic length lB = 1. We also set the volume of the system as 1 to avoid volume factors in Fourier transform.

Now we can write the Hamiltonian in cn,X basis. Z

H=

d2r na(r) †(r) a (r)

=

Z d2r d2q e iq·rna( q)

X

c†n,X 'n,X (x)e iXy a eiX0y'n0,X0 (x)cn0,X0

=

Z d2q na(

n,X,n0 ,X 0
q) X e iqxX c†n,X qy/2

a cn0,X+qy/2Fn,n0 (q)

n,n0 ,X

(S3)

6

The form factors Fn,n0 (q) are calculated as,

F00(q) = e q2/4

F01(q)

=

(

iqpx qy) e 2

q 2 /4

F10(q)

=

(

iqpx + qy) e 2

q 2 /4

F11(q) = (1 q2/2)e q2/4.

(S4)

The di↵erence in Eq. (S3) between q = 1 and q = 2 case comes from the summation limit of n and n0.
As in the main text, consider the situation na(r) = (⇡1, ⇡2, ⇡3, ⇡4, 1) where na is polarized near (0, 0, 0, 0, 1). Substituting this to Eq. (S3) we get the Hamiltonian,

H = H0 + H⇡
= X c†n,X 5cn,X
n,X

X4 Z d2q ⇡i(
i=1

X q) e
n,n0 ,X

iqxX c†n,X

qy /2

i cn0,X+qy/2Fn,n0 (q).

(S5)

From the above equation, we integrate out the fermions to get an e↵ective theory for the ﬂuctuating order parameters. The coupling between the order parameters appear at fourth order of one-loop expansion. In momentum space, the four point coupling between the order parameter ﬁelds are,

Z Y3

S1 =

dp↵Kpjk1pm2;pi 3 ⇡j (p1)⇡k(p2)⇡m(p3)⇡i( p1 p2 p3).

↵=1

(S6)

Among these terms, we are most interested in the topological term,
Z S⇡ = iK d2rd⌧ ✏ijkm⇡i@x⇡j @y⇡k@⌧ ⇡m.

(S7)

To extract the coe cient K from S1 we expand Kpjk1pm2;pi 3 in powers of momenta and frequency, and consider the terms linear in p1p2p3:

Kpjk1pm2;pi 3 = · · · + K↵jkm;ip↵1 p2 p3 + · · · .

(S8)

Here, ↵, , are spacetime indices ⌧, x, y. In real space, these terms correspond to the derivative expansion.

S1 = · · · + i

X4

Z K↵jkm;i d2rd⌧ ⇡i@↵⇡j @ ⇡k@ ⇡m + · · ·

i,j,k,m=1

(S9)

Comparing Eq. (S7) with Eq. (S9), we obtain the expression for K in terms of K↵jkm;i’s.

24K = ✏↵ ✏ijkmK↵jkm;i

(S10)

Note the summation of repeated indices are implicit, and thus the right-hand side of the above equation consists of 144 terms.
Now we only need to calculate K↵jkm;i to obtain K. This can be done by calculating the box diagrams in Fig. 2. The kinetic energy of fermions are quenched, and the propagators are momentum independent. Therefore, the momentum dependence comes from the vertices and frequency dependence comes from the propagator.
Recalling Eq. (S5), the inverse Green’s function is written as G 1 = i! + H0, where ! is the Matsubara frequency of the fermions. The e↵ective action of order ⇡4 in perturbation theory, which is of our interest according to Eq. (S9), is:

S [⇡4 ]

=

1 4

tr

(GH⇡

GH⇡

GH⇡

GH⇡

)

.

(S11)

7

Considering the momentum ﬂow in Fig. 2, we obtain the expression for K↵jkm;i.

K↵jkm;i

=

1 4

⇣ @p↵1 @p2 @p3 tr

G(k)H⇡j (p1)G(k + p1)H⇡k(p2)G(k + p1 + p2)

⇥H⇡m(p3)G(k + p1 + p2 + p3)H⇡i ( p1 p2 p3)

p1 =p2 =p3 =0

(S12)

We used a shorthand notation H⇡a for H⇡a = H⇡(⇡i=a = 1, ⇡i6=a = 0). We get the value of K by calculating Eq. (S12). In zero temperature, the frequency integral can be done analytically,

K

=

1 2⇡

Z

d!

X

2q (!2 +

5
2)3

X

=

X

3q 8

X

=

3q 16⇡

.

(S13)

The individual diagram is independent of X, merely reﬂecting momentum conservation. Therefore the X summation is just multiplying N , which equals (sample area)/(2⇡). This gives the result in Eq. (7).