Physica 107B (1981) 53-54 BE 6 North-Holland Publishing Company

BROKEN SYMMETRIES AND HYDRODYNAMICS OF SUPERFLUID3p2-NEUTRON STAR MATTER

Helmut Brand and Harald Pleiner

Fachbereich Physik, Universit~t Essen, 4300 Essen, W-Germany

3 We investigate the spontaneously broken symmetries of P2 neutron star matter and we present the nonlinear hydrodynamic equations of this superfluid. Apart from broken gauge symmetry, a fact which is common to all superfluids, we find that the total rotational symmetry of spin and

orbit space is spontaneously broken, a unique feature of P2 neutron star matter.

As ha~ ~ecome clear during the last ~ears~-- there exists very probably a P9 superfluid phase of neutrons in the

in£erior of neutron stars and it is the purpose of the present contribution to clarify the nature of the spontaneously broken continuous symmetries of that phase and to derive the corresponding nonlinear hydrodynamic equations. Our considerations are relevant to neutron stars because

i) the hydrodynamic equations give a macroscopic description (including magnetic fields) of ~P2 neutron star matter and

2) we find a coupling term between the density of linear momentum and the vorticity of the magnetization den- sity which will probably have an im- portant influence on the rotational dynamics of a neutron star.

The or~e{ parameter of 3p~ neutron star matter-'~ is a complex, t~aceless, symmetric 3x3-matrix, denoted by A ~. Under rotations of the orbital coordi- nates A . transforms like a vector with respectU~o the index i and under ro- tations in spin space A transforms

, , 1 ,

llke a vector wlth respect to the index

The operator for the order parameter is defined in ref 4 (for a corresponding ~efinition in the superfluid phases of He we refer to ref 5). For the normali-

zed, unitary equilibrium order parameter we have

A M A A

A°~I = N ei~[u ui + r~ ~i-(l+r)w wi]

where N is a normalization constant and where the parameter r is chosen in such a way that the3Ginzburg Landau free energy is minimized . 6, %% and ~ from a triad of unit vectors in spin or real space, respectively. It seems important to notice that A . is

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symmetric with respect to an interchange of real and spin space indices

A° = A °

and that the only imaginary contribution to the equilibrium structure is the phase factor e ±~. The former property has drastic consequences on the hydrodynamic variables and on the hydrodynamic equations: Every vector in real space can be transformed into a vector in spin space by multiplication with one of the factors 6 6. ~ ~. or %% %% . Of course

' 1 . p~s~ible for the the same ~r~ceddre is transformation of vectors (or tensors) in spin space into corresponding quanti- ties in real space.3This is a feature which is unique to P9 neutron star matter and which has ~ot been found for any hydrodynamic system studied so far and it simply means that spin and orbit indices can be interchanged at will!

As candidates for variables character- izing broken symmetries we have to con- sider the phase deviation 6~ and the de- viations from the unit vectors of the triad (6, ~, ~) in spin or orbit space or equivalently the rotation angles 60. about these axes in th~ corresponding space. As it turns out the correspon- ding o~erators Rave the same structure as in -He-A or~-He-B without external magnetic field- and only the contents of the structure matrix is different. For the commutation relations for the total particle number N and for the total angu- lar momentum J.=L. + S (where L is the 1 1 1 angular momentum opera'or and S i the ope- rator for the spin) we find

<[6~, N]> = - 2i

<[68i,N]> = O

<[6~,Ji]> = O

<[~Sf~]> = i 6ik

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It should be stressed that commutators with L. or S do hot,give rise to a ca- nonica~ relation in 5p~ neutron star matter.' Thus we have i~entified the hy- drodynamic variables characterising spon- taneously broken continuous symmetries of -P~ neutron star matter: One variable, the phase deviation 6~ or the superfluid velocity v~ characterises the broken gauge invarzance and the three variables 68 are characterising the spontaneously

l . . . . broken total rotatlonal lnvarzance, z.e. total angular momentum J =L + S. serves

1 1 1 as the generator of a broken symmetry! The latter property is unique to -P? neutron star matter and has not beeH found for any hydrodynamic system studied previously. It has to be con- trasted e.g. to the spontaneously broken relative spin orbit ~ymmetry of the B- phase of superfluid -He. In this phase one has

<[S i, ~0j]> = - <[L i, ~0j]> or

<[Ji' 6@j]> = O

whereas in all other hydrodynamic sys- tems w~th broken rotational symmetries (e.g. He-Al, nematics, antiferromagnets) these symme£ries are spontaneously bro- ken in real and spin space separately•

To set up the hydrodynamic equations the procedure is as follows. We start with the Gibbs relation which defines the thermodynamic conjugate quantities and then we write down the conservation and quasi-conservation laws for the hydrody- namic variables thus defining the rever- sible and irreversible currents. As con- served quantities we have the density p, the energy density e (or entropy density 0), the density of linear momentum g and the density of total angular momentum. The four variables characterising spon- taneously broken symmetries have already been discussed above• To close the sys- tem of the hydrodynamic equations we pro- ceed in three steps. First we establish the free energy from which the thermody- namic conjugate quantities can be ob- tained by differentiation. In the second step we expand the irreversible currents into the thermodynamic conjugates. In the last step we derive the reversible currents. To achieve this, it is necessary, however, to take into account the anholonomity relation for the hydro- dynamic variables ~0~ (cf ref 4 for the details) ± • Quite analogous relations have been given previously for biaxial nema- tics by the authors. Furthermore we de- monstrate a novel procedure to guarantee conservation of total angular momentum locally. This procedure involves the thermodynamic conjugate force of the to- tal angular momentum density, a quantity

which is local in space!

After the derivation of the compl~te non- linear hydrodynamic equations of -P? neu- tron star matter it is possible to ~heck that e.g. the nonlinear hydrodynamic equations for biaxial nematics are con- tained in the present ones as a special case.

A very interesting static coupling which will probably influence the rotational dynamics of a neutron star is found in the equation for the density of linear momen- tum

gi = "'" + ~ij ejkl ?k hl

where h I is the thermodynamic conjugate to the magnetization density• The three phenomenological coefficients which are involved in ~.. lead to a coupling bet- ween the vort~ity of the magnetization and the density of linear momentum. As hydrodynamic excitations of the lineari- zed equations we find three pairs of pro- pagating spin-orbit-waves which are coupled in an intricate manner to first- and second sound. These complicated couplings can be traced back to the bi- axiality in real and spin space of the present system.

To sum up we have discussed the broken symmetries and the hydrodynamic equations of P2 neutron star matter and we have found several properties which are unique toa-P? superfluid. The most important one among-these seems to be the conclusion that total angular momentum can serve as the generator of a spontaneously broken symmetry!

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[4] Brand, H. and Pleiner, H. to be published

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