Parity non-conservation in a condensed matter system
aa r X i v : . [ c ond - m a t . o t h e r] F e b Parity non-conservation in a condensed mattersystem
Veit ElserLaboratory of Atomic and Solid State PhysicsCornell UniversityIthaca, NY 14853-2501USAFebruary 19, 2020
Abstract
The nuclear spin of a He quasiparticle dissolved in superfluid He sees an apparent magnetic field proportional to the Fermi coupling con-stant, the superfluid condensate density, and the electron current atthe He nucleus. Whereas the direction of the current must be paral-lel to the quasiparticle momentum, calculating its magnitude presentsan interesting theoretical challenge because it vanishes in the Born-Oppenheimer approximation. We find the effect is too small to beobserved and present our results in the hope others will be inspired tolook for similar effects in other systems. As one of the cleanest condensed matter systems, superfluid He is a goodcandidate for precision experiments. With but one exception — the isotopeHe — nothing dissolves in superfluid He . And unlike trapped cold atomsystems, superfluid He samples are truly macroscopic and can be observedover long times. At low concentration and low temperature, a dissolved He atom behaves as a simple quasiparticle whose only degree of freedom is itsmomentum with respect to the superfluid condensate. The physics of diluteHe in superfluid He is that of an ideal, weakly interacting fermi gas.The He quasiparticles also have a nuclear spin. In the infinite dilu-tion limit and at low temperatures, the nuclear spin would appear to be a1 bscure Physics Quarterly P : H = P M ∗ − v P · σ . (1)Here M ∗ is the quasiparticle mass (enhanced over the nuclear mass by inertiain the superfluid flow), σ are the Pauli operators of the He nuclear spin,and v is a parameter. Galilean invariance rules out the second term forparticles in vacuum, but this is suspended for the He quasiparticle becausethe superfluid condensate defines a preferred rest frame. This term is oddunder parity and v could only be nonzero if the weak interaction played apart in its origin.We will argue that the parity non-conserving coupling v is indeed nonzero.Although its magnitude is far too small to be observed, even in this cleanestof condensed matter systems, it is interesting that the reach of the weakinteraction extends even to the low energy properties of a condensed mattersystem. In particular, from Hamiltonian (1) we know that the ground stateof a He quasiparticle is a definite helicity state with nonzero momentum ofmagnitude P = M ∗ | v | .The estimation of v is interesting theoretically because it brings to-gether particle, atom/molecule, and condensed matter physics. Of these themolecular physics turns out to be the most challenging because one must gobeyond the Born-Oppenheimer approximation to obtain a nonzero v . The effective Hamiltonian for the coupling of a nuclear spin- σ to the atomicelectrons by the weak interaction is derived in Commins and Bucksbaum [2]: H weak = λ G F c ( j e · σ + i j e × σ e · σ ) . (2)Here G F is the Fermi coupling constant, j e and j e × σ e are respectivelythe electron and electron spin current densities at the nucleus, and λ is adimensionless constant given by the Weinberg angle θ w and the nucleon axialcharge g A : λ = 1 − θ w √ g A ≈ . . (3) bscure Physics Quarterly H weak : j e = 12 m X i (cid:0) p i δ ( r i ) + δ ( r i ) p i (cid:1) . (4)Here m is the electron mass, the sum is over all electrons, and the electronpositions are relative to the nucleus. In a superfluid a nonzero fraction f of the atoms can be treated as occupyinga zero momentum condensate [6]. In this picture the constituent pairs ofelectrons on the He atoms of the condensate form a uniform electron density2 f n , where n is the density of He atoms and f ≈
9% has been measuredby elastic neutron scattering and Green’s function Monte Carlo (summarizedin [7]). In the rest frame of a He quasiparticle of momentum P , where thesuperfluid condensate has velocity − P /M ∗ , the electron current density atthe He nucleus would be h j e i = − γ (2 f n )( P /M ∗ ) . (5)Here γ is a numerical factor meant to correct important correlation effectsthat were left out in this analysis. To appreciate these, consider the pair ofelectrons on the He atom itself. These have zero current at the nucleus andwould seem to shield the nucleus from any electron current in the superfluidenvironment. It seems clear that γ is less than one and probably is quitesmall. The primary motivation for this work was to make a convincing casethat γ is nonzero.Taking γ = 1 as a generous upper bound on the true electron currentdensity, combining (2) and (5) we arrive at the following upper bound onthe parameter v in (1): v = (cid:18) λ G F c (cid:19) (cid:18) f nM ∗ (cid:19) = 8 × − m/s . (6)To put this number in perspective, we calculate the magnetic field B eq thatwould produce the same NMR resonance as that produced by the weakinteraction on a He quasiparticle with a root-mean-square momentum inthe superfluid corresponding to thermal equilibrium at T = 1 K: B eq = 2 v √ M ∗ k B T ~ γ = 5 × − T . (7) bscure Physics Quarterly γ is the He gyromagnetic ratio. For the nuclear dipole magnetic fieldsproduced by other quasiparticles to be below this value the He concentra-tion would have to be less than 10 − . And even if such concentrationswere not prohibitive for collecting signal, it seems unlikely that a resonancefrequency of order ω = γ B eq = 10 − Hz (8)could ever be detected.When the restriction of solubility is relaxed, in experiments with heliumnanodroplets, a coupling of angular and linear momentum closely analogousto the P · σ term in (1) is realized by chiral molecules immersed in thedroplets [5]. While the strength of the coupling is no longer dependent onthe weak interaction, its detection is complicated by other factors. Much of He superfluid phenomenology is qualitatively reproduced when theinteractions between atoms is treated in the pairwise approximation [1]. Wewill show that this extends to the formation of an electron current at thenucleus of an impurity He atom.In the pair approximation, the interaction of the He atom with thecondensate is approximated as interactions with individual He atoms. Suchan interacting pair has Hamiltonian H pair = − ~ M ∇ R cm − ~ µ ∇ R + H e ( R ) , (9)where M and µ are the total and reduced masses of the two nuclei, M and M , R cm is the center of mass of the two nuclei, R = R − R is theposition of the condensate atom nucleus relative to the He nucleus, and H e is the electron Hamiltonian for fixed, specified nuclear positions.We will need at least two eigenstates of H e ( R ): the ground state andone (or more) excited states. Their wave functions only depend on R andthe electron positions relative to R : φ ( r , r , r , r ; R ) φ ′ ( r , r , r , r ; R ) . (10)The ground state φ is spin singlet and has zero angular momentum about R , or of type Σ. We assume the same for φ ′ and most of our derivationof the electron current only depends on these properties. After we see howthe current depends on φ ′ we will know which excited dimer wave functions bscure Physics Quarterly Δ EU'U Figure 1: Schematic rendering, based on reference [4], of the electronic en-ergies U ( R ) and U ′ ( R ), of respectively the ground state and one excitedstate of the helium dimer, as a function of nuclear distance. The two curvesare highly offset for clarity: for large R , U ′ ( R ) − U ( R ) ∼
20 eV, while thedepth of the U ′ ( R ) minimum is typically only about 1 eV. The excitationenergy ∆ E , to an unbound excited-state dimer, arises in the calculation ofsection 4.2.to focus on. We do not need to keep track of the electron spins if we agreethat electrons 1 and 2 are spin-up while 3 and 4 are spin-down and the wavefunctions are antisymmetric with respect to those electron pairs. Both wavefunctions are real and have the following normalization:1 = Z d r · · · d r φ . (11)Plots of the corresponding electron energies (eigenvalues of H e ), denoted U ( R ) and U ′ ( R ) as they only depend on the internuclear distance, areshown in Figure 1. The 2-body potential for nuclear motion in the electronicground state is essentially repulsive: the very weak minimum in U ( R ) (toosmall for the resolution in Figure 1) barely binds two He atoms but nota He -He pair. Whereas the potentials of the excited states usually haveminima that support bound states, they share the property with U ( R ) of bscure Physics Quarterly U ′ ( R ) willplay a role in the electron current. In the Born-Oppenheimer approximation the He -He pair is described astwo nuclei moving in the lowest energy potential, U ( R ). The lowest energywave function (of the six particle system) has zero center-of-mass momentumand zero angular momentum:Ψ = r n V χ ( R ) φ ( r , r , r , r ; R ) . (12)Here χ ( R ) is the nuclear wave function in the potential U ( R ) with zeroasymptotic nuclear kinetic energy, n = f n is the density of condensateatoms and V is the system volume. The normalization factors are explainedby our convention: χ ( R ) ∼ , R → ∞ . (13)From this we see that Z V d R d r · · · d r | Ψ | = n (14)gives the correct density of condensate atoms in the impurity atom’s envi-ronment. We note that f ≈ n < n is missed in our pairwise scattering approximation.When the He quasiparticle has a small momentum P , the many-bodywave function can be argued to be the following modification of the groundstate wave function [3]: Ψ (0) P = e i P · R / ~ Ψ . (15)Being still just a multiple of a real wave function for the electrons, this Born-Oppenheimer wave function has h j e i = 0. We will have to apply pertur-bation terms from the nuclear kinetic energy that admix excited electronicwave functions φ ′ in order to get a nonzero current at the He nucleus.The excited state wave functions associated with the electronic wavefunction φ ′ have the formΨ P klm = e i P · R cm / ~ √ V (cid:18)r R V χ kl ( R ) Y lm ( θ, ϕ ) (cid:19) φ ′ ( r , r , r , r ; R ) . (16)Here ( R, θ, ϕ ) are the usual spherical coordinates of the nuclear separation R and χ kl is a real nuclear radial wave function in the potential U ′ ( R ) aug-mented by the centrifugal potential for angular momentum l . The quantum bscure Physics Quarterly k is the wave number associated with the asymptotic nuclear kineticenergy of the excited state with total energy E k :( ~ k ) µ = E k − U ′ ( ∞ ) . (17)For large R , χ kl ( R ) ∼ R cos( kR − ϕ kl ) , (18)where the phase ϕ kl is determined by the vanishing of χ kl for R →
0. Inorder for Ψ P klm to have proper unit normalization, the radial wave functionshave normalization 2 R V Z R V χ kl R dR = 1 , (19)where R V is the radius of a large bounding sphere. The wave numbers k arediscrete because of the boundary condition on the sphere and have density R V /π . We see that, in the excited state wave function (16), the momentum P is shared by the scattering He -He pair. The first correction to the Born-Oppenheimer approximation is generatedby terms in (9) where the nuclear kinetic energy operator acts both on thenuclear wavefunction as well as the parametric dependence of the electronicwavefunction on the nuclear positions. Since the latter only depends on R ,we write the perturbation term as H ′ = − ~ µ ∇ R n · ∇ R e , (20)where ∇ R e acts on φ and ∇ R n acts on the nuclear wavefunction that mul-tiplies φ . The corrected (six particle) wave function has the formΨ (1) P = Ψ (0) P − X klm h Ψ P klm | H ′ | Ψ (0) P i E k − E (0) Ψ P klm , (21)where E (0) = U ( ∞ ) is the energy of two separated ground-state heliumatoms. We will be interested in corrections only up to linear order in themomentum P . At this order we will find that the sum over the angularquantum numbers only includes the term ( l = 1 , m = 0), with the conventionthat the momentum direction ˆ P defines the positive z -axis of our sphericalcoordinate system. bscure Physics Quarterly We now evaluate the matrix element in (21): M P klm = h Ψ P klm | H ′ | Ψ (0) P i (22)= − ~ µ √ n V r R V Z d R d R d r · · · d r (23) (cid:16) e − i P · R cm / ~ χ klm Y lm ( θ, ϕ ) φ ′ (cid:17) (cid:16) ∇ R e i P · R / ~ χ (cid:17) · ( ∇ R φ ) . Using R = R cm − ( M /M ) R , the gradient acting on the nuclear wavefunc-tion generates two terms and the integrand becomes independent of R . Theintegrals over the electron positions, Z d r · · · d r φ ′ ∇ R φ = u ( R ) ˆ R , (24)produces a purely radial function of R by symmetry and defines a scalarfunction u ( R ). Expanding (23) in powers of P and keeping only terms upto first order, we obtain, M P klm = − ~ µ √ n r R V Z d R Y lm ( θ, ϕ ) (cid:18) − i M M ( P · R / ~ ) + · · · (cid:19) (25) χ klm (cid:18) − i M M (cid:16) P · ˆ R / ~ (cid:17) χ + dχ dR (cid:19) u = i ~ PM √ n r R V Z d R Y lm ( θ, ϕ ) cos θ χ klm (cid:18) χ + R dχ dR (cid:19) u, (26)where we have retained only terms proportional to P . For the only nonzerocase, ( l = 1 , m = 0), we obtain M P k = i ~ PM √ n r π r R V A k , (27)where A k = Z χ k (cid:18) χ + R dχ dR (cid:19) u R dR (28)and we have dropped the l and m indices on χ klm since the only remainingsum is over k . bscure Physics Quarterly P cos θ = P · ˆ R in our coordinate system, the six-particlewave function with Born-Oppenheimer correction to first order in P isΨ (1) P = r n V (cid:16) e i P · R / ~ χ φ (29) − i ~ M R V X k A k ∆ E k e i P · R cm / ~ ( P · ˆ R ) χ k φ ′ ! , (30)where ∆ E k = E k − U ( ∞ ). When evaluating the electron current density h j e i = Z d R d R d r · · · d r Ψ (1) P ∗ j e Ψ (1) P (31)the term of order P has a real electron wavefunction and therefore vanishingcurrent. The cross-terms, which are of order P , also have the factors e ± i ( M /M ) P · R / ~ , (32)which when expanded only produce higher orders in P . Integration of theelectron positions in the cross terms produce another radially symmetricfunction: Z d r · · · d r X i =1 δ ( r i ) (cid:0) φ ∇ r i φ ′ − φ ′ ∇ r i φ (cid:1) = v ( R ) ˆ R . (33)This follows from the cylindrical symmetry of both φ and φ ′ about theinter-nuclear axis and because the current density is evaluated at the He nucleus on this axis. The result of the current density calculation is h j e i = − (cid:18) ~ m Z A k B k ∆ E k dk (cid:19) n ( P /M ) , (34)where B k = Z χ k χ v R dR (35)and we used the density of wave numbers to convert the sum over k into anintegral. bscure Physics Quarterly The integrals (28) and (35) involving the nuclear wave functions that define A k and B k can be simplified in the semiclassical limit, when χ k is the onlyrapidly oscillating function. This is the case for the problem at hand, sincethe repulsive barrier in U ′ ( R ) ensures the integral in (34) includes wavenumbers that satisfy ka B ≫
1. Here the Bohr radius a B represents thelength scale of slow variation in the functions u and v . The case k → R where the electronic functions u and v become very small.If maximizing the orbital mixing ( u ) or the current integral ( v ) werethe only considerations, the focus would be on small R and the sum overexcitations would include nuclear bound states. However, the contributionsto the current density from such excitations is strongly suppressed by therapid exponential decay of χ , for small R , in the repulsive potential U .The assumption of slowly varying functions in the semiclassical evaluationof nuclear integrals will also apply to χ .Let R k be the turning point for wave number k , the nuclear separationwhere the classical velocity is instantaneously zero when scattering withasymptotic relative momentum ~ k . The semiclassical limit of integrals of χ k with slow functions is given by just the contribution at the turning point.In the appendix we show this corresponds to the replacement χ k ( R ) → s πk µg k ~ R k δ ( R − R k ) , (36)where g k = − dU ′ dR (cid:12)(cid:12)(cid:12)(cid:12) R k (37)is the gradient at the turning point. Using this the integrals A k and B k reduce to the values of χ , dχ /dR , u and v at the turning point. Droppingthe index k on R with the understanding that this is the turning point, wecan also transform the integral over k in (34) to an integral over R with theJacobian dkdR = µg ~ k . (38)The result of taking these steps is h j e i = − γ (2 n )( P /M ) , (39)where γ = π ~ m Z (cid:18) χ + R dχ dR (cid:19) χ (cid:16) uv ∆ E (cid:17) R dR (40) bscure Physics Quarterly g at the turning point and ∆ E = U ′ ( R ) − U ( ∞ ) is the energy of the excited state when its nuclear wavefunction has turning point R . To properly evaluate the dimensionless constant γ in the current density(39), the integral over turning points (40) would have to be computed for the( u, v, ∆ E ) of each Σ excited state of the helium dimer — their contributionsadd. An extensive study of the excited dimer states by Guberman andGoddard [4] is helpful in identifying the states and integration range wherewe can expect the largest contributions. Since ∆ E is very similar for thelow lying excitations and is essentially flat (in absolute terms) as a functionof R , the functions u ( R ) and v ( R ) should be our focus.When the separation R of the He -He pair is large it is easy to see thatthe product uv is small. In this limit, neglecting antisymmetry betweenelectrons on different atoms, the wave functions are approximately productsof 2-electron wave functions: φ ≈ φ ( r , r ; 0) φ ( r , r ; R ) (41)Here φ and φ are ground state helium wave functions centered, respec-tively, on the He nucleus and the He nucleus (at R ). The Born-Oppenheimerperturbation generates the wave function ∇ R φ ≈ φ ( r , r ; 0) φ ∗ ( r , r ; R ) (42)where φ ∗ = ∇ R φ is a combination of helium excited states with p -symmetry.The p -states on the He , when combined with the ground s -state (and a rel-ative phase), generates a current — but at the wrong nucleus. In fact, by(24) the only excited state — again in the product approximation — thatcan give a nonzero u has the form φ ′ ≈ φ ( r , r ; 0) φ ′ ( r , r ; R ) , (43)where φ ′ is a particular helium excitation with p -symmetry. But when (43)and (41) are used in (33) for the current at the He nucleus, the resulting v is zero.In order for the perturbation on the He nucleus to produce a currentat the He nucleus, the electrons on the two atoms must interact. Themost direct manifestation of an interaction is the repulsive barrier in the 2-body potential U ′ ( R ). Another consideration, for current at the nucleus, is bscure Physics Quarterly p -type atomic character. The two lowestexcited states, called A ( Σ u ) and C ( Σ g ), would appear to be ruled out bythis because they correspond to a (resonating) 2 s -atomic excitation at large R . However, Guberman and Goddard [4] find that the 2 s -like “Rydberg”orbital develops p -like character for R < . Σ u ), which is 2 p -like already at large R . Moreover, the D state has a moresharply rising barrier, and therefore the promise of a coupling between theHe position and the He current at larger R . In fact, the barrier for theD state rises in a range where U is essentially flat and the nuclear wavefunction χ has not yet started to decay significantly. The effect considered in this paper does not open a new low energy windowto the weak interaction, nor promise a novel technique for measuring theelusive condensate fraction of a superfluid. As the estimate of section 3showed, the magnitude of the spin-momentum coupling is many orders ofmagnitude too small to be detected. The corresponding NMR frequency isso small the He nuclear spins would only have precessed a small fraction ofa period before they are randomized when the quasiparticles on which theyreside scatter from the walls containing the superfluid.What really motivated this paper was a theoretical question about thenature of superfluids. As a low energy phenomenon one automatically treatsthe helium atoms as single entities: the particles of the superfluid. Animpurity He is also a single entity, albeit one that is distinguishable from theother helium atoms. Lost in this abstraction is the fact that, although thetwo kinds of nuclei are clearly distinguishable, the electrons that surroundthem are not. One could describe the magnitude of the numerical factor γ we have calculated in (40) as quantifying the degree to which the He atom accepts the “condensate electrons” of all the other helium atoms as itsown. Only with respect to these shared zero-momentum electrons can theHe nucleus experience an electron wind. It is unfortunate that the weakinteraction appears to be the only mechanism for detecting this wind. Acknowledgements
We thank Gordon Baym, Peter Lepage, Quentin Quakenbush and CyrusUmrigar for helpful discussions. bscure Physics Quarterly Here we derive the strength of the turning point contribution (36) to radialnuclear wave function integrals in the semiclassical limit. Let R k be theturning point of the nuclear motion in the potential U ( R ). Near R k thenuclear wave function χ satisfies the Schr¨odinger equation − ~ µ d χdx − g k x χ = 0 , (44)where x = R − R k and g k > U at R k . Thelength scale a = (cid:18) ~ µg k (cid:19) / (45)satisfies a ≪ a B for m ≪ µ and lets us express χ near the turning point asthe Airy function: χ ∼ ( Z/a )Ai( − x/a ) . (46)Here Z is a normalization constant to be determined and represents thestrength of the turning point contribution since Z ∞−∞ Ai( y ) dy = 1 . (47)The constant Z can be determined by matching limits of the semiclassicalapproximation of χ : χ ( R ) = A ( R ) cos ϕ ( R ) . (48)Here A is the slowly varying amplitude and and A / ± v ( R ) where v ( R ) = r µ ( U ( R k ) − U ( R )) , (49)and we assume R > R k in the following. The conserved radial flux ofprobability j , of one wave packet, j = 2 πR A v, (50)gives us an explicit expression for the amplitude in terms of the velocity: A ( R ) = j πR v ( R ) . (51) bscure Physics Quarterly v ( ∞ ) = ~ k/µ , we infer j = 2 π ~ kµ . (52)To make contact with the Airy function at the turning point, we also considerthe limit of (51) for R − R k = x →
0. Expanding (49) for small x we thenfind A ( x ) ∼ s ~ kµR k (cid:18) µ g k (cid:19) / x / . (53)Using this amplitude in (48) near the turning point, comparing with (46)and the asymptotic behavior of the Airy function,Ai( − y ) ∼ √ π y / cos ϕ ( y ) , y → ∞ , (54)we obtain Z = s πk µg k ~ R k . (55) References [1] N. Bogoliubov. On the theory of superfluidity.
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