Pis'ma v ZhETF, vol.92, iss. 10, pp.751-756
© 2010 November 25
Fermions with cubic and quartic spectrum
T. T. Heikkilii -1), G.E. Volovik*+11 * Low Temperature Laboratory, Aalto University, School of Science and Technology, FI-00076 AALTO, Finland
+ Landau Institute for Theoretical Physics RAS, 119334 Moscow, Russia
Submitted 14 October 2010
We study exotic fermions with spectrum E2 oc p2N. Such spectrum emerges in the vicinity of the Fermi point with multiple topological charge N, if special symmetry is obeyed. When this symmetry is violated, the multiple Fermi point typically splits into N elementary Fermi points - Dirac points with N = 1 and spectrum E2 oc jr.
1. Introduction. There is a fundamental interplay of symmetry and topology in physics, both in condensed matter and relativistic quantum fields. Traditionally the first role was played by symmetry (symmetry classification of crystals, liquid crystals, magnets, superconductors, superfluids, etc.). The phenomenon of spontaneously broken symmetry remains one of the major tools in physics. The last decades demonstrated the opposite tendency in which topology is primary. Topology in momentum space is becoming the main characteristics of quantum vacua - ground states of the system at T = 0 [1-14]. For example in topological matter, nodes in the spectrum of fermionic quasiparticles are protected by topology, and this property is insensitive to the details of the microscopic many-body Hamiltonian (see e.g. review [8]). The momentum-space topological invariants determine universality classes of the topological matter and the type of the effective theory which emerges at low energy and low temperature. In many cases they also give rise to emergent symmetry. Examples are provided by the nodes of the energy spectrum characterized by the elementary topological charges, N = +1 or N = — 1. Close to such nodes the effective Lorentz invariance emerges: the Fermi point with N = +1 or N = —1 represents the Dirac point and the spectrum forms the relativistic Dirac cone. This is the consequence of the Atiyah-Bott- Shapiro construction [15].
However, in many systems (including condensed matter and relativistic quantum vacua), the Fermi points with iV = +loriV=^l may merge together forming the points either with N = 0 or with multiple N (i.e. \N\ > 1 [16]). In this case topology and symmetry become equally important, because it is the symmetry which may stabilize the degenerate node. Example is provided by the Standard Model of particle physics, where 16 fermions of one generation have degenerate
Dirac point at p = 0 with the trivial total topological charge N = 8^8 = 0. In the symmetric phase of Standard Model the nodes in the spectrum survive due to a discrete symmetry between the fermions and they disappear in the non-symmetric phase forming the fully gapped vacuum [8]. In case of degenerate Fermi point with \N\ > 1, situation is more diverse. Depending on symmetry, interaction between fermionic flavors may lead to splitting of the multiple Fermi point to elementary Dirac points [17]; or gives rise to the essentially non-relativistic energy spectrum E±(p —t 0) ±pN, which corresponds to different scaling for space and time in the infrared: r —t Ar, t ANt. The particular case of anisotropic scaling with N = 3 was suggested by Horava for quantum gravity at short distances [18-20], and anisotropic scaling in the infrared in Ref. [21].
The non-linear spectrum arising near the Fermi point with N = 2 has been discussed for different systems including graphene, double cuprate layer in high-Tc superconductors, surface states of topological insulators and neutrino physics [22 - 29]. The spectrum of (quasi)particles in the vicinity of the doubly degenerate node (say, with topological charge N = N3 = ±2 for Fermi points of co-dimension 3) depends on symmetry. One may obtain: two Weyl fermions, if there is some special symmetry; exotic massless fermions with quadratic dispersion at low energy,
„2
JMp)* Hi-
fi)
or semi-Dirac fermions with linear dispersion in one direction and quadratic dispersion in the other,
E±(P) « ±\
f^W)
(2)
^e-mail: tero.heikkila8tkk.fi, volovikôboojum.hut.f
Here we discuss the class of effective Hamiltonians which has a Fermi point with higher degeneracy, described by the symmetry protected topological invariant N = 3 and N = 4, and quantum phase transitions
which may occur in these systems where topology of the spectrum changes.
2. Cubic spectrum. Let us consider first the case with N = 3. Examples are three families of right-handed Weyl 2-component fermions in particle physics; three cuprate layers in high-Tc superconductors; three graphene layers, etc. If the Fermi point is topologically protected, i.e. there is a conserved topological invariant N, the node in the spectrum cannot disappear even in the presence of interaction, but it can split into N nodes with elementary charge N = 1. The splitting can be prevented if there is a symmetry in play, such as rotational symmetry. Here we provide an example of such symmetry.
For simplicity we study the 2+1 systems. In general the nodal points in 2+1 dimensions (Fermi points of co-dimension 2) obey Z2 topology [15, 23], that is why we also need an additional symmetry K which extends the group Z2 to the full group Z and thus makes the multiple Fermi point possible. The corresponding invariant protected by symmetry K [23, 30] is:
NK =
Am
tr
j> dl KG{uj = 0, p)5iG_1 (w = 0, p),
(3)
where G is Green's function matrix, and C is contour around the Fermi point in 2D momentum space (px,py), or around the Fermi surface if the Fermi point expands to the Fermi surface. The matrix K commutes or anti-commutes with matrix G(w = 0,p). The single particle Green's function at zero energy represents the effective single-particle Hamiltonian %(p) = G^1(w = 0, p).
We consider 3 species (families or flavors) of fermions, each of them being described by the invariant N = NK = +1 and an effective relativistic Hamiltonian emerging in the vicinity of the Fermi point
•Ho(p) = 0" • p = axpx + (Tvpy.
(4)
The matrix K = az anticommutes with the Hamiltonian. This supports the topologically protected node in the spectrum, which is robust to interactions. The position of the node here is chosen at p = 0:
E2=p2
(5)
The total topological charge of three nodes at p = 0 of three fermionic species is NK = +3. Let us now introduce mixing of the fermions, which obeys some symme-
tries which may follow from the underlying microscopic theory. We consider two cases
/
fti(p) =
a p
921CT
\ 931^
and
/
n2( P) =
a p
921CT \ 931<7~
912 a • p
Si 20-T a • p
\
S230-+ 0- p /
S130-+ \
923 a p /
(6)
(7)
where a± = \{ax ± iav) are ladder operators. Both Hamiltonians anti-commute with K = az and thus mixing preserves the topological charge NK in (3). Hamiltonian (6) is symmetric with respect to the Z3 group of permutations if all the couplings gmn are equal. For general couplings, Hamiltonian (6) is symmetric under the group of rotations by 2ir/3 combined with gauge transformations from the SU(3) family group:
or bP) — ^gauge^spm^l (p)f^spmf^gauge • (8)
Here 0orb and C4pm are orbital rotations by 2ir/3 in momentum and spin space respectively, and f/gauge = diag (l,e27ri/3,e^27ri/3) is the element of the SU(3) family group.
The Hamiltonian (7) at px = py = 0 is independent of the spin rotations up to a global phase of the coupling constants. Under spin rotation by angle 9 all elements in the upper triangular matrix are multiplied by e,e, while all elements in the lower triangular matrix are multiplied by . This symmetry of triangular matrices generates a specific property of the spectrum. Mixing between fermions does not split the multiple Fermi point at p = 0, as a result the gapless branch of the spectrum in Fig.l has the cubic form at low energy, E ^ 0, which corresponds to the topological charge NK = +3:
E2
73 =
I012II023I'
(9)
In the low-energy limit the spectrum in the vicinity of the multiple Fermi point (9) is symmetric under rotations. But in general the spectrum is not symmetric as demonstrated in Figs.l and 2. There is only the symmetry with respect to reflection, (px,py) (px,—py)-The rotational symmetry of spectrum (9) is an emergent phenomenon. For trilayer graphene this spectrum has been discussed in Ref.[31].
The cubic spectrum (9) becomes singular when either one of the coupling constants gi2 or g23 nullify. At this
0
Px !g
Fig.l. Spectrum of the Hamiltonian (7) showing cubic dispersion for the lowest two eigenvalues around the point p = 0. Different curves correspond to different eigenvalues, the spectrum is shown as a function of px at py = 0. The spectra have been calculated with equal coupling strengths gi2 = gw = P23 = g
Fig.2. Contour plot showing the equi-energy curves in the vicinity of the degenerate Fermi point with NK = 3 and cubic dispersion. Here 312 =323 = g = 313/2
At 531 = 0 one also has the quantum phase transition, since at 531 = 0 the Hamiltonians (7) and (6) coincide: ■H2(p,S3i = 0) = 'Hi(p,S3i = 0). Then when £31 grows from zero in (6) the multiple Fermi point splits into 3 elementary Dirac points, i.e. Fermi points with NK = +1. For equal coupling strengths 912 = 9i3 = 523 = 9 one has
/ 0 P+ 0 9 0 0 \
p- 0 0 0 9 0
0 0 0 P+ 0 9
9 0 P- 0 0 0
0 9 0 0 0 P+
V 0 0 9 0 P- 0 /
(10)
where p± = px ± ipy. In this case the full permutation symmetry takes place and three Fermi points with N = 1 each form the configuration obeying the three-fold rotational symmetry, see Figs.3 and 4. For general cou-
3
^ 0 -1
-2
—3 h 1 ■ » 1 1 1 » 1 1
-2 -1 0
Px 'g
J_I_I_I_I_I_I_I_I_L
1 2
Fig.3. Spectrum of the permutation symmetric Hamiltonian (6) shows three Dirac points (Fermi points with N = 1 and linear dispersion), located at {px,py}/g = {-1, 0}, {1/2, ^3/2}, {1/2,-V3/2}, first of them is shown in the figure. Different curves correspond to different eigenvalu
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