Density matrix in a rotating box

InSubsection3.1.3,we determined the densitymatrix of a freequantum particle in a box with periodic boundaryconditions. We nowdiscuss the physicalcontentof boundaryconditions and the related counter-intuitive behavior of a quantum system under slow rotations. Our discussion is

a prerequisite for the treatmentof superfluidityinquantumliquids(see Subsection 4.2.6), but also for superconductivity in electronic systems, a subject beyond the scopeof this book.

...

0 L

0 = L

Fig. 3.7A one-dimensional quantum system on a line of lengthL(left), and rolled up into a slowly rotating ring (right).

Let us imagine for a moment a complicatedquantum system, for ex-ample a thin wire rolled up into a closed circleof circumferenceL, or a circular tube filledwith a quantum liquid, ideally ⁴He (see Fig. 3.7).

Both systems are described by wave functions with periodic boundary conditionsψ_n(0) =ψ_n(L). More precisely, thewave functions in thelab system forN conduction electronsorN helium atoms satisfy

ψ_n^lab(x1, . . . , xk+L, . . . , xN) =ψ^lab_n (x1, . . . , xk, . . . , xN). (3.24) These boundary conditions for the wave functions in the laboratory frame continue to hold if the ring rotates with velocity v, because a quantum system—even if partsof it are moving—must be described ev-erywhere by a single-valued wave function. The rotating system can be described in the laboratory reference frame using wave functions ψ^lab_n (x, t)and the time-dependentHamiltonianH^lab(t),which represents the rotating crystal latticeor the rough surfaceof the tube enclosing the liquid.

The rotating system is more convenientlydescribed in the corotating reference frame,using coordinatesx^rot rather than thelaboratory coor-dinatesx^lab (see Table3.1). In this reference frame, the crystal lattice or the container walls are stationary, so that the Hamiltonian H^rot is time independent. For veryslow rotations, we can furthermore neglect centripetalforces and also magnetic fields generated by slowly moving charges. This implies that theHamiltonianH^rot(with coordinatesx^rot) is the same as thelaboratory Hamiltonian H^lab atv = 0. However,we shallsee that the boundaryconditionson the corotatingwave functions ψ^rot are nontrivial.

Table 3.1Galilei transformation for a quantum system. E^rot, p^rot_n , and x^rot are defined in the moving reference frame.

Reference frame

Lab. Rot.

x^lab=x^rot+vt x^rot p^lab_n =p^rot_n +mv p^rot_n E_n^lab=₂¹

m

`p^lab_n ´2

E_n^rot=₂¹

m

`p^rot_n ´2

Todiscuss this point,we nowswitch back from the complicated quan-tum systems toa free particleon a rotating ring, described bya Hamil-tonian

H^rot=−1 2

∂² (∂x^rot)².

We shallgothrough the analysisofH^rot, but keep in mind that thevery distinction between rotating andlaboratoryframes is problematic for the

noninteracting system, becauseof the missing contactwith the crystal latticeor the container boundaries. Strictlyspeaking, the distinction is meaningfulfor a noninteracting systemonlybecause thewave functions of the interacting system can be expanded in a plane-wave basis. In the rotating system, plane waves can bewritten as

ψ^rot_n (x^rot) =exp

ip^rotx^rot .

This same planewave can alsobewritten, at alltimes, in thelaboratory frame,where it must be periodic(see eqn(3.24)). The momentumof the plane wave in the laboratory system is related to that of the rotating system bythe Galilei transformofTable3.1.

p^lab_n =2

Ln =⇒ p^rot_n = 2

Ln−mv (n=−∞, . . . ,∞). (3.25) It follows from the Galilei transform of momenta and energies that the partition function Z^rot(β)in the rotating reference frame is

Z^rot(β) = In the rotating reference frame, each planewavewith momentump^rot_n contributesvelocityp^rot_n /m. The meanvelocitymeasured in the rotating reference frame is with energies and momenta given by eqn (3.25) that satisfy E_n^rot = (p^rot_n )²/(2m). In the limit of zero temperature, only the lowest-lying state contributes to this rotating reference-frame particlevelocity. The momentumof this state is generallynonzero, because thelowest energy state, then= 0state atverysmall rotation,E₀^rot, has nonzero momen-tum(it corresponds toparticlevelocity−v). The particlevelocityin the laboratoryframe is )

v^lab*

=) v^rot*

+v. (3.27)

Atlowtemperature, this particlevelocitydiffers from thevelocity of the ring(see Fig. 3.8).

The particle velocity in a ring rotating with velocity v will now be obtainedwithin the frameworkof densitymatrices, similarlytotheway weobtained the densitymatrixfor a stationaryring, inSubsection3.1.3.

Writingxforx^rot(theyare the same at timet= 0, andwe donot have tocomparewave functions at different times),

0

Fig. 3.8Mean lab-frame velocity of a particle in a one-dimensional ring (from eqns (3.26) and (3.27) withL=m=

The densitymatrixat positionsxandx in the rotating system is:

g(n), compare with eqn (3.16)

= wherewe have againused the Poisson sum formula(eqn(3.17)). Setting φ=2φ/L−v and changingwinto −w, we reach Integrating the diagonaldensitymatrix over all the positionsx, from 0 toL,yields the partition function in the rotating system:

Z^rot(β) = Trρ^rot(x, x, β) (see eqn (3.6)). For small velocities, we expand the exponential in the above equation. The zeroth-order term gives the partition function in the stationary system, and the first-order term vanishes because of a symmetryfor the winding numberw→ −w. The term proportionalto w² is nontrivial. We multiply and divide by the laboratory partition function at rest,Z_v=0^lab , andobtain

Z^rot(β) = Noting that the free energyisF =−log(Z)/β,weobtain the following fundamentalformula:

F^rot=F_v=0^lab +v²L²) w²*

v=0

2β +· · ·. (3.29)

This is counter-intuitive because we would naively expect the slowly rotating system torotate alongwith the reference system and have the same free energyas the laboratorysystem at rest, in the same way as a bucket of water, under very slow rotation, simply rotates with the walls and satisfiesF^rot=F_v=0^lab. However,we sawthe same effect in the previous calculation, using wave functions and eigenvalues.Part of the system simplydoes not rotatewith the bucket. It remains stationaryin thelaboratorysystem.

The relation between the mean squared winding number and the change of free energies upon rotation is due to Pollock and Ceperley (1986). Equation (3.29) contains no more quantities specific to nonin-teracting particles; it therefore holds also in an interacting system, at least for small velocities. We shall understand this more clearlyby rep-resenting the density matrix as a sum of paths, which do not change with interactions, andonlyget reweighted.Equation(3.29)isvery con-venient for evaluating the superfluid fractionof aliquid, that partof a veryslowlyrotating systemwhich remains stationaryin thelaboratory frame.

In conclusion, we have computed in this subsection the density ma-trixfor a free particle in a rotating box. As mentioned severaltimes, this calculation has far-reaching consequences for quantum liquids and su-perconductors. Hess and Fairbank(1967)demonstrated experimentally that aquantumliquid in a slowlyrotating container rotates more slowly than the container,or even stops to rest. The reader isurged to study the experiment, and Leggett’s classic discussion (Leggett1973).