Pis'ma v ZhETF, vol. 91, iss. 2, pp. 61-67 © 2010 January 25
Topological invariants for Standard Model: from semi-metal to
topological insulator
G. E. Volovik1^
Low Temperature Laboratory, Helsinki University of Technology, FIN-02015 HUT, Finland Landau Institute for Theoretical Physics RAS, 119334 Moscow, Russia Submitted 14 December 2009
We consider topological invariants describing semimetal (gapless) and insulating (gapped) states of the quantum vacuum of Standard Model and possible quantum phase transitions between these states.
1. Introduction. Recently topological insulators, semimetals, superconductors, superfluids and other topologically nontrivial gapless and gapped phases of matter have attracted a lot of attention. Probably the first discussion of the 3D topological insulators in crystals can be found in Refs. [1, 2]; the two-dimensional massless edge states of electrons at the interface between topologically different bulk states have been discussed in [2]. The fully gapped 3D superfluid with nontrivial topology is represented by the phase B of superfluid 3He; the corresponding 2D gapless quasiparticles living at interfaces have been discussed in [3]. Examples of the 2D topological fully gapped systems are provided by the films of superfluid 3He in the phase A and in the planar phase; the topological invariants give rise to quantization of the Hall and spin-Hall currents in these films in the absence of external magnetic field [4]. The three-dimensional 3He-B and the two-dimensional planar phase of triplet superfluid/superconductor belong to the time-reversal invariant topological states of matter.
Different aspects of physics of topological matter have been discussed, including topological stability of gap nodes; classification of fully gapped vacua; edge states; Majorana fermions; influence of disorder and interaction; topological quantum phase transitions; intrinsic Hall and spin-Hall effects; quantization of physical parameters; experimental realization; connections with relativistic quantum fields; chiral anomaly; etc. [5]-[47]
The vacuum of Standard Model (SM) is also a topological substance: both known states of the SM vacuum - gapless semimetal state and fully gapped insulating state - possess the non-trivial topological invariants. Here we present the explicit expression for the relevant topological invariants of SM, and discuss possible topological quantum phase transitions occurring between the vacuum states.
^e-mail: volovik0boojum.hut.fi
2. Green's function as an object. The object for the topological classification must be the Green's function rather than Hamiltonian. Then it is applicable even in cases when one cannot introduce the effective low energy Hamiltonian, for example when Green's function does not have poles, see [31, 48] in condensed matter, unparticles in relativistic quantum fields [49] and phenomenon of quark confinement in QCD with suggested anomalous infrared behavior of the quark and gluon Green's functions [50, 51, 52].
Green's function topology has been used in particular for classification of topologically protected nodes in the quasiparticle energy spectrum of systems of different dimensions including the vacuum of Standard Model in its gapless state [53, 23, 26, 31]; for the classification of the topological ground states in the fully gapped 2+1 systems, which experience intrinsic quantum Hall and spin-Hall effects [4, 14, 20, 21, 23, 31]; in relativistic quantum field theory of 2 + 1 massive Dirac fermions [54, 55, 56]; etc.
The quantum phase transition occurs when some parameter of the system crosses the critical value at which the momentum-space topology of the Green's function changes. In SM the role of such parameter may be played by the high-energy cut-off scales, such as the ultraviolet E\jy and compositeness Ec energy scales introduced in Ref. [57]. In the limit E\jy/Ec —t oo, all three running coupling constants vanish, («i,«2,0:3) —^ 0 [57]. In this zero-charge limit fermions become uncoupled from the gauge fields, the gauge invariance becomes irrelevant, and the gauge groups of SM may be considered as the global groups which connect the fermionic species. That is why in this limit the Green's function is well defined.
In general case when the gauge invariance is important there are two different approaches to treat the Green' function, (i) One may use the gauge-fixing conditions. Though in this approach the Green's function depends on the choice of the gauge, the topological in-
variants are robust to continuous deformations and thus should not depend on the choice of the gauge (only large gauge transformations are prohibited since they may change the value of topological invariant), (ii) One may use the two-point Green's function with points connected by a special path-ordered exponential, which produces the parallel transport of color from one point to the other and thus makes the propagator gauge invariant. In this case the Green's function is path-dependent, and we assume the straight contours to ensure translational invariance, which allows us to consider the Green's function in momentum space.
Simplest examples of the Green's function are 2x2 matrix Green's function for chiral Weyl fermions
S =
Z(p2
iu) ± a ■ p
(1)
with + sign for right and — sign for left fermions, and 4x4 matrix Green's function for Dirac fermions:
S =
Z(p2
-»7"^ + M (p2)'
(2)
Here a are spin Pauli matrices; Dirac matrices will be chosen according to Sec. 5.4 in Ref. [58]:
7° = -m, /y = T2er, 75 = ^¿7°717273 = r3.
(3)
For the topological classification of the gapless vacua, the Green's function is considered at imaginary frequency Po = «•<■>, i.e. the Euclidean propagators are used and
2 2 P = P
2 2 Po = P
• U)
(4)
This allows us to consider only the relevant singularities in the Green's function and to avoid the singularities on the mass shell, which exist in any vacuum, gapless or fully gapped.
There is the following topological invariant expressed via integer valued integral over the S3 surface a around the point p2 = 0 in momentum space [23]:
N= tr J dSa S8PßS-^S-^S-K (5)
Here trace is over all the fermionic indices of the Green's function matrix including spinor indices. This invariant (5) equals the difference between numbers of right and left fermionic species,
N = tir — n /..
(6)
One has N = +1 for a single right fermionic species in (1); N = —1 correspondingly for a single left fermion. For massless Dirac fermions one has N = +1 — 1 = 0,
which demonstrates that there is no topological protection, and interaction may produce mass term in (2). Eq.(6) is applicable to the general case when interaction between the fermions is added, or Lorentz invariance is violated. Nonzero value N of the integral around some point in momentum space tells us that at least N fermionic species are gapless and have nodes in the spectrum at this point.
3. Topological invariant protected by symmetry in semi-metal state. We assume that SM contains equal number of right and left Weyl fermions, nn = til = 8ng , where ng is the number of generations (we do not consider SM with Majorana fermions, and assume that in the insulating state of SM neutrinos are Dirac fermions). For such Standard Model the topological charge in (5) vanishes, N = 0. Thus the masslessness of the Weyl fermions is not protected by the invariant (5), and arbitrary weak interaction may result in massive particles.
However, there is another topological invariant, which takes into account the symmetry of the vacuum. The gapless state of the vacuum with N = 0 can be protected by the following integral [23]:
N' =
&Qtß\±V
24tt2
tr
L
K / dSc
8dpß 8
lSdp„S-
lsd0,.s-
(7)
where liy is the matrix of some symmetry transformation. In SM there are two relevant symmetries, both are the Z2 groups, K2 = 1. One of them is the center subgroup of SU(2)l gauge group of weak rotations of left fermions, where the element K is the gauge rotation by angle 27r, K = enTZL. The other one is the group of the hypercharge rotation be angle 67r, K = et6irY. In the G(224) Pati-Salam extension of the G(213) group of SM, this symmetry comes as combination of the Z2 center group of the SU(2)r gauge group for right fermions, et7rT3R, and the element e37rt(B-L) 0f the Z, center group of the SU(4) color group - the Pm parity (on the importance of the discrete groups in particle physics see [59, 60] and references therein). Each of these two Z2 symmetry operations changes sign of left spinor, but does not influence the right particles. Thus these matrices are diagonal, K^ = diag(l, 1,... , —1, —1,...), with eigen values 1 for right fermions and — 1 for left fermions.
In the symmetric phase of Standard Model, both matrices commute with the Green's function matrix Sy, as a result N1 is topological invariant: it is robust to deformations of Green's function which preserve the symmetry. Simple explanation is the following. The Z2 symmetries K completely forbid the mixing, M(p2) = 0, and
one has two independent sectors in the Green's function matrix with two independent topological invariants N = N1 = ur for right fermions and N = —N' = —m for left fermions. Thus the symmetric phase of Standard Model contains N' = 16n9 massless fermions, which remain massless even if the interaction is introduced. Since the mixing between leptons and quarks is negligibly small at low energies, the invariant (7) splits into two separate invariants for leptonic and baryonic sectors
N1 = N1
leptons
•^baryons
12 na.
This is one of numerous examples of integer valued topological invariants which are supported by discrete or continuous symmetry. Other examples in condensed matter systems and in quantum field theory can be found inRefs/[4, 17, 20, 23, 31].
The integral (7) can be applied also to vacua with massive Dirac fermions. In the massive case this integral is no more the topological invariant: it depends in particular on th
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