Pis'ma v ZhETF, vol.90, iss.8, pp.639-643
© 2009 October 25
Topological invariant for superfluid 3He-B and quantum phase
transitions
G. E. Volovik1^
Low Temperature Laboratory, Helsinki University of Technology, FIN-02015 HUT, Finland Landau Institute for Theoretical Physics RAS, 119334 Moscow, Russia Submitted 22 September 2009
We consider topological invariant describing the vacuum states of superfluid 3He-B, which belongs to the special class of time-reversal invariant topological insulators and superfluids. Discrete symmetries important for classification of the topologically distinct vacuum states are discussed. One of them leads to the additional subclasses of 3He-B states and is responsible for the finite density of states of Majorana fermions living at some interfaces between the bulk states. Integer valued topological invariant is expressed in terms of the Green's function, which allows us to consider systems with interaction.
PACS: 71.23.An
1. Introduction. General classification schemes based on topology [1 - 7] suggest existence of the topological insulators and fully gapped topological superflu-ids/superconductors which have the gapless edge states on the boundary or at the interface. Superfluid 3He-B belongs to the special class of three-dimensional topological superfluids with time-reversal symmetry. Topological invariant which describes the ground states (vacua) of 3He-B has been discussed in Refs. [1, 8]. Here we present the explicit expression for the relevant topological invariant, and discuss topological quantum phase transitions occurring between the vacuum states and fermion zero modes living at the interfaces between the bulk states.
2. Topological invariant protected by symmetry. Usually in the topological classification of ground states people use the Hamiltonian of free particles or the corresponding effective Hamiltonian such as Dirac and Bogoliubov-de Gennes Hamiltonians [1-4]. However, in this classification the natural problem arises, what is the effect of interaction between particles. Moreover, the original first-principle many-body Hamiltonian of, say, liquid 3He
V2
U^iiM =
2 to
•M
+ i I dxdyt/(x - y)tl>Hx)tl>Hy)tl>(y)tl>(x), (1)
has no information on the topological structure of the ground state of the system - superfluid 3He-B. The accurate procedure to reduce such a strongly interacting many-body system to the effective coarse-grained Hamil-
tonian is absent. However, the microscopic Hamiltonian allows us at list in principle to calculate the Green's function G(w,p) - the quantity which determines the main properties of the translational invariant or periodic ground states of the system. That is why 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 [6, 9].
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; for the classification of the topological ground states in the fully gapped 2+1 systems, which experience intrinsic quantum Hall and spin-Hall effects [10-12,5,6,13], including multi-band topological insulators [14]; in relativistic quantum field theory of 2 + 1 massive Dirac fermions [16-18; etc.
The integer-valued topological invariants were expressed via the Green's function, which was considered at imaginary frequency to avoid the zeroes, poles or other possible singularities on the mass shell. In the fully gapped systems one may consider the Green's function not only at imaginary frequency, but also at real frequency if it is below the gap. For the classification of the 3He-B states we shall use the Green's function at zero frequency: £?(p) = G(w = 0,p). The typical example of the integer valued topological invariant is the following 3-form:
^e-mail: volovik0boojum.hut.fi
N =
24^2
tr
I
,fPgdVig 'GdV]G 'go,,, g
(2)
x
640
G.E. Volovîk
where integration is over the whole momentum space for translational invariant systems, or over the Brillouin zone in crystals. For 3He-B obeying time-reversal symmetry this invariant is identically zero, N = 0. However, the discrete symmetries of 3He-B give rise to the other invariants. Examples of additional integer valued topological invariants which appear due to symmetry in different condensed matter systems and in quantum field theory can be found in Refs. [10, 19, 12, 5, 6].
Due to symmetry, one or several Pauli matrices 7 of spin, pseudospin or Bogoliubov-Nambu spin may either commute or anti-commute with the Green's function matrix:
1/m*
or
lQ( P) = -É?(P)7,
lG{ p) = G(p)l-
(3)
(4)
This leads to the following integer valued topological invariants:
N - tr
7" 24tt2
I
7 / dZpgd^g-'gdp.g-'gd^g
(5)
with i/(p) and 7 obeying either (3) or (4).
The topological classes of the 3He-B states can be represented by the following Green's function, which plays the role of effective Hamiltonian:
g-1(p) = M(p)T3
Ti (axc.xpx „2
M(p) =
P
2m*
VyCyPy
Mi
VzCzPz), (6)
(7)
where r,- are Pauli matrices of Bogolyubov-Nambu spin. The overall 'conformal' factor, which may depend on p, is omitted since it does not influence the topological invariant. In the isotropic 3He-B all 'speeds of light' are equal, [c^,.[ = \cy\ = |c-| = c. The topological invariant relevant for 3He-B is iV7 in (5) with 7 = 72:
N _ eijh 7 " 24tr2
tr
/
r2 / <rP gd^gdp.g^gdp.g
(8)
The T2 matrix plays the role of the 7-matrix, which anti-commutes with the Green's function in (3). The T2 symmetry is combination of time reversal and particle-hole symmetries in 3He-B.
3. Phase diagram in mass plane. Fig.l shows the phase diagram of topological states of 3He-B in the plane (/t, 1/m*). On the line 1/m* = 0 one obtains Dirac fermions with mass parameter M = ^/t:
g^1(p) = Mt3 + n (axc.xpx + (jycypy + azc.zpz). (9)
Strong coupling
3
3He-B
N = 0
= -1 0 N=+1
Dirac
N=-2 N = 0
Week coupling
3
3He-B
Week coupling
3
3He-B
N=+2
Strong coupling
3
3He-B
Fig.l. Phase diagram of topological states of 3He-B in equation (6) in the plane {¡i, 1/m*) for the speeds of light cx > 0, cy >0 and c- >0. States on the line 1/m* = 0 correspond to the Dirac vacua, which Hamiltonian is non-compact. Topological charge of the Dirac fermions is intermediate between charges of compact 3He-B states. The line 1/m* = 0 separates the states with different asymptotic behavior of the Hamiltonian at infinity: —► ±T3p2/2m*. The line ¡j, = 0 marks topological quantum phase transition, which occurs between the weak coupling 3He-B (with ¡j, > 0, m* > 0 and topological charge iV7 = 2) and the strong coupling 3He-B (with ¡i < 0, m* > 0 and iV7 =0). This transition is topologically equivalent to quantum phase transition between Dirac vacua with opposite mass parameter M = ±|/a|, which occurs when ¡1 crosses zero along the line 1/m* = 0. The interface which separates two states contains single Majorana fermion in case of 3He-B, and single chiral fermion in case of relativistic quantum fields. Difference in the nature of the fermions is that in Bogoliubov-de Gennes system the components of spinor are related by complex conjugation. This reduces the number of degrees of freedom compared to Dirac case
Topological invariant (8) for Dirac fermions is
iV7 = ^Sign (McxC.yCz) = Sign (l-lCxCyCz).
(10)
In relativistic quantum field theory, 72 matrix in (8) is the operator of CT symmetry [20]. Hamiltonian for Dirac fermions is non-compact, with different asymptotes at p 00 for different directions of momentum p. As a result, the topological charge of the Dirac fermions is intermediate between charges of the compact states of 3He-B below and above the horizontal axis (see Refs. [21, 1, 5] on the marginal behavior of fermions with relativistic spectrum; also note that the topological invariant iV7 in (8) has values twice larger than the invariants introduced in Refs. [3, 8]). When the line
Topological invariant for superiluid 3He-B and quantum phase transitions
641
1/m* = 0 is crossed, the asymptotic behavior of the 3He-B Green's function changes from +T$p2/2m*
to g-hp) ^t3p2/2m*.
The real superfluid 3He-B lives on the weak coupling side of the phase diagram: at /t > 0, m* > 0, /t m*c2. However, in the ultracold Fermi gases with triplet pairing the strong coupling limit is possible near the Fesh-bach resonance [22]. When /t crosses zero the topological quantum phase transition occurs, at which the topological charge iV7 changes from iV7 = 2 to iV7 = 0. Previously the topological quantum phase transition occurring at /t = 0 has been considered for the chiral p-wave states of the 3He-A type: the transition from the gapless state at /t > 0 to the fully gapped state at /t < 0 [19, 22]. In the 3He-B case, both states are fully gapped, while the intermediate state at /t = 0 is gapless: it has two point nodes at p = 0 with opposite chiralities. There is a general relation between topological invariants of the two states and the number of point nodes in the intermediate gapless state [5], which for a given case reads:
•^point nodes = |iV7(/t > 0) -JV7(/i < 0)1 = 2. (11)
This relation also determines the number of 2+1 fermion zero modes living at the interface between the two states:
iVpzM = 7T |iV7(/t > 0) - iV7(/t < 0)| = 1.
(12)
This analog of the index theorem [23] implies that such interface contains single Majorana fermion.
The same quantum phase transition occurs when // crosses zero along the line 1/m* = 0, i.e. the line of the relativistic Dirac fermions. At this transition the mass M = —¡j, of Dirac fermions changes sign. The rules (11) and (12) are also applicable for the Dirac fermions. However, in relativistic quantum field theory the domain wall separating vacua with opposite M contains chiral fermion rather than
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