We consider theoretically the possibility of observing unusual quantum fluid
behavior in liquid $^{3}$He and solutions of $^{3}$He in $^{4}$He systems
confined to nano-channels. In the case of pure ballistic flow at very low
temperature conductance will be quantized in units of $2m^{2}/h$. We show that
these steps should be sensitive to increases in temperature. We also use of a
random scattering matrix simulation to study flow with diffusive wall
scattering. Universal conductance fluctuations analogous to those seen in
electron systems should then be observable. Finally we consider the possibility
of the cross-over to a one-dimensional system at sufficiently low temperature
where the system could form a Luttinger liquid

Beenakker C. W. J.

G. Gervais

G. Lambert

Holt J. K.

Lee P. A.

W. J. Mullin

English

arXiv

This is the pre-published version which is collected from arXiv. The published version is at 
http://ltp.aip.org/resource/1/ltpheg/v34/i4/p249_s1?isAuthorized=noWe consider theoretically the possibility of observing unusual quantum fluid behavior in liquid 3He and solutions of 3He in 4He systems confined to nanochannels. In the case of pure ballistic flow at very low temperature the conductance will be quantized in units of 2m2/h. We show that these steps should be sensitive to increases in temperature. We also use a random scattering matrix simulation to study flow with diffusive wall scattering. Universal conductance fluctuations analogous to those seen in electron systems should then be observable. Finally we consider the possibility of crossover to a one-dimensional system at sufficiently low temperature, where the system could form a Luttinger liquid.249-25

Lambert, G

Gervais, G

Mullin, WJ

ScholarWorks@UMass Amherst

Quantum-limited mass flow of liquid He-3

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Quantum-limited mass flow of liquid He3

We consider theoretically the possibility of observing unusual quantum fluid behavior in liquid 3He and solutions of 3He in 4He systems confined to nanochannels. In the case of pure ballistic flow at very low temperature the conductance will be quantized in units of 2m2/h. We show that these steps should be sensitive to increases in temperature. We also use a random scattering matrix simulation to study flow with diffusive wall scattering. Universal conductance fluctuations analogous to those seen in electron systems should then be observable. Finally we consider the possibility of crossover to a one-dimensional system at sufficiently low temperature, where the system could form a Luttinger liquid

University of Massachusetts AmherstScholarWorks@UMass AmherstPhysics Department Faculty Publication Series Physics2008Quantum-limited mass flow of liquid He-3G LambertG GervaisWJ Mullinmullin@physics.umass.eduFollow this and additional works at: https://scholarworks.umass.edu/physics_faculty_pubsPart of the Physics CommonsThis Article is brought to you for free and open access by the Physics at ScholarWorks@UMass Amherst. It has been accepted for inclusion in PhysicsDepartment Faculty Publication Series by an authorized administrator of ScholarWorks@UMass Amherst. For more information, please contactscholarworks@library.umass.edu.Recommended CitationLambert, G; Gervais, G; and Mullin, WJ, "Quantum-limited mass flow of liquid He-3" (2008). LOW TEMPERATURE PHYSICS. 25.Retrieved from https://scholarworks.umass.edu/physics_faculty_pubs/25arXiv:0801.4006v2  [cond-mat.mes-hall]  4 Feb 2008Quantum-limited mass flow of liquid 3HeG. Lamberta, G.Gervaisa, and W. J. MullinbaDepartment of Physics, McGill University,3600 rue Universite´, Montre´al, Qc, CanadabDepartment of Physics, University of Massachusetts,Amherst, Massachusetts 01003 USAAbstractWe consider theoretically the possibility of observing unusual quantum fluid behavior in liquid3He and solutions of 3He in 4He systems confined to nano-channels. In the case of pure ballisticflow at very low temperature conductance will be quantized in units of 2m2/h. We show thatthese steps should be sensitive to increases in temperature. We also use a random scatteringmatrix simulation to study flow with diffusive wall scattering. Universal conductance fluctuationsanalogous to those seen in electron systems should then be observable. Finally, we consider thepossibility of the cross-over to a one-dimensional system at sufficiently low temperature where thesystem could form a Luttinger liquid.Keywords: liquid 3He, solutions of 3He in liquid4He, nanotubes, quantized conductance, universalconductance fluctuations, Luttinger liquidPACS numbers: 67.30.ej, 67.60.gj, 61.46.Fg, 71.10.Pm1INTRODUCTIONAt very low temperatures, the thermal motion of the 3He atoms in the liquid becomes verysmall as the de Broglie wavelength becomes comparable with the distance between the atomsand so quantum effects become significant enough to dictate the macroscopic properties ofthe liquid. In a recent paper[1], one of us noted that degenerate dilute solutions of 3Hein liquid 4He might be cooled by flow through an array of nano-channels whose diametersare comparable to the de Broglie wave length of the channel. Aligning the Fermi energyof a container of fermions with the lowest allowed band in the channel would allow onlythe hot gases through the channel, thereby cooling the remaining gas in the container. Thepresence of such separated bands in a channel immediately suggests further possible unusualbehavior. For fluids confined over such small scales, one may legitimately ask whether ornot the physics of the system, for example, the flow properties, are still the same as atlarger scale, or if perhaps a breakdown of the fluid mechanics might occur. Indeed recentlySato et al[2] considered the feasibility of seeing quantized conductance in dilute solutionswhen there was pure ballistic flow. Recent advances in materials research have opened upthe possibilities to design and engineer single, or arrays of pores with diameters of tensof nanometer in ceramic membranes[3], and down to only ∼1 nm in tailor-made carbonnanotubes membranes[4]. These materials provide a radial length scale small enough toquantize the transverse motion of the helium atoms so that a realization of quantum-limitedballistic flow of helium, similar to that observed in the electronic transport of quantumwires[5], truly seems within experimental reach.We consider here possible experiments on pure 3He as well as dilute solutions. We firstexamine the temperature dependence of the conductance quantization for pure ballistic flow.We then investigate the effect of disorder and how wall scattering in diffusive flow causes theonset of “universal” conductance fluctuations in the mass flow. In the ultra-low temperaturelimit T → 0, we also argue that one-dimensional quantum fluids should crossover to anentirely new exotic type of quantum matter known theoretically as a Luttinger liquid[6].Such liquid should possess entirely different flow properties and excitation spectra owing toits ‘spin-mass’ separated ground state[7].This study presented here constitutes only a preliminary analysis to illustrate the richnessof flow properties that one might expect in experiments on helium in nanoscaled cylindrical2pores. More thorough investigations at the theoretical and experimental level are in progress.QUANTIZED CONDUCTANCEThe de Broglie wave length λdB gives the scale of the diameters of the nanochannels neces-sary for seeing unusual quantum effects; the Fermi temperature TF tells us what temperaturewe need for degeneracy. For pure 3He λdB ≈ 0.7 nm, TF ≈ 2 K while for a 1% solution of3He in liquid 4He, λdB ≈ 3 nm and TF ≈ 120 mK. Moreover, the interparticle mean freepath is in the micron range because it contains the factor (TF/T )2. In each case a dilutionrefrigerator is able to produce adequate degeneracy. Ref. [2] considered the possibility ofusing such arrays of channels to see the quantization of the conductance. Unfortunatelydispersion in the channel size can smooth out the quantization steps. In the present studywe will ignore this problem and assume that technology will soon provide the feasibility ofdetecting flow from a single channel or a very small number of channels. Initially we willneglect any interparticle scattering and many-body interaction; the only scattering we willconsider is that from the walls.A natural definition of the mass conductance, analogous to that used in the case of anelectric current, is given by[2]G =I∆µ/m(1)where I is the mass current, m the particle mass, and ∆µ the difference in chemical potentialbetween the ends of the channel. The mass current is taken to beI =2mh∑n∫ ∞−∞dpzpzm[n(ǫz + ǫn − µ−∆µ)− n(ǫz + ǫn − µ)] T (ǫz) (2)where pz is the longitudinal momentum, and we have written the energy in the channel as alongitudinal part ǫz = p2z/2m, plus ǫn, the discrete transverse energies in the channel. TheFermi distribution isn(ǫ− µ) =1eβ(ǫ−µ) + 1. (3)The quantity T (ǫz) is the transition probability for the scatterers in the channel, and β =1/kbT . In Eq. (2) we have assumed the temperature in the channel to be uniform; anygradient can easily be incorporated into the formula. Eq. (2) is just the Landauer formula[8]for the conductance. We have assumed T for each band is just a function of ǫz . Expanding3the Fermi distributions in small ∆µ we findG =2m2h∑n∫ ∞0dzez−αn(ez−αn + 1)2T (kbTz) (4)where z = βǫz and αn = β(µ−ǫn). Note that if z < 0, the transmission probability vanishes.For extremely low temperature the derivative of the Fermi distribution is sharply peakedaround z = αn so we can writeG =2m2h∑nT (ǫF − ǫn) (5)where ǫF is the Fermi energy. The states in a cylindrical nano-channel are denoted by(n,m, kz) where n and m are the radial and azimuthal quantum numbers and kz is thelongitudinal wave number. We assume that the states of different m are all degenerate andthat a band is characterized by (n, k(n)z ) and a degeneracy factor. If an atom of energy Eenters a channel from a reservoir whose Fermi energy matches a state in the Nth band, thenit can be in a superposition of bands (1, k(1)z ) to (N, k(N)z ) such that E = ǫn+ ~2k(n)2z /2m. Ifthere is no scattering (that is, there is only specular scattering at the walls and the lattersimply provide boundary conditions for the wave functions), then particles in each kz statestay there and do not exchange from one band to another. The transmission probability forany band is just a step functionT (ǫz) = Θ(ǫz). (6)The conductance is then simplyG =2m2hN. (7)This function is plotted in Fig. 1. The conductance is clearly quantized in units of 2m2/h.Double steps occur because of a two-fold degeneracy of some states.Suppose now we turn up the temperature while still assuming that particles in the channelmaintain their kz values and do not elastically scatter to other bands. Eq. (2) can be writtenasG =2m2h∑n∫ ∞0dzez−αn(ez−αn + 1)2Θ(z)=2m2h∑n1(e−αn + 1). (8)The states at the Fermi energy feeding into the channel are not sharp and the steps arerounded as shown in Figs. 2 and 3. The steps are already reduced substantially at T/TF =0.1. The steps are completely gone by T/TF = 0.5.42 4 6 8 10 12Ν24681012GFIG. 1: Conductance versus ν = ǫF/ǫ0 (where ǫ0 is the lowest band edge) for pure ballistic flow atT = 0.2 4 6 8 10 12Ν24681012GFIG. 2: Conductance versus ν at T/TF = 0.05 for ballistic flow.CONDUCTANCE FLUCTUATIONSThe above analysis considered only ballistic flow. However wall scattering can causediffusive flow. We return here to the T = 0 situation. An atom entering the channel is in asuperposition of bands (1, k(1)z ) to (N, k(N)z ) as above. Backscattering is possible so that thestate at the entrance (left end) to the channel isψleft = φ+φ− , (9)52 4 6 8 10 12Ν24681012GFIG. 3: Conductance versus ν at T/TF = 0.1 for ballistic flow.where φi+ = {(1, k(1)z ), · · · , (N, k(N)z )} and φ− = {(1,−k(1)z ), · · · , (N,−k(N)z )}. That is, ψleftis a 2N dimensional vector. This state has N scattering channels. An elastic scattering ofthe particle from (n, k(n)z ) can be to any of the N bands as long as energy remains the same.If ǫ0 < E < ǫ1, only one channel is available and we have simple 1D scattering.The Landauer formula[8] for multichannel conductance isG =2m2h∑a,bTab (10)where Tab is the transmission coefficient for scattering from channel a to channel b.If we assume that a scattering center on the wall has random position and strength thena method that simply uses a series of completely random scattering matrices gives a suitablesimulation. Such random matrix theory has been used extensively to treat conductance inelectron systems.[9]. If the state after a scattering event isψ′ = φ′+φ′− . (11)Then the transfer matrix M satisfiesψ′ = Mψ (12)or  φ′+φ′− =M++ M+−M−+ M−− φ+φ− (13)6where each Mij is an N ×N matrix. The S matrix, also a 2N ×2N matrix that we write asS = r tt′ r′ , (14)connects outgoing (on the left of the equation) and incoming waves (on the right of theequation) according to  φ−φ′+ = r tt′ r′ φ+φ′− . (15)where t, t′, r, and t′ are N ×N matrices. Suppose we have a two-channel system with justone scattering site so that the final state has φ′− = 0. That is, after the scattering one hasonly the transmitted wave. Then we have φ− = rφ+ and φ′+ = tφ+; r represents reflectionand t transmission. In detail for the two-channel system we have φ′+1φ′+2 = t11 t12t21 t22 φ+1φ+2 . (16)If a particle starts in channel 1 (φ+2 = 0), the probability of scattering to final channel 1 is|t11|2 and the probability of scattering to final channel 2 is |t21|2 . The total probability ofscattering out of channel 1 is T1 = |t11|2 + |t21|2 . The total conductance for an incoherentmixture of all incoming states is thenG =2m2h∑a,b|tab|2 =2m2hTr[t†t]. (17)The same result holds for many channels.The S matrix is unitary and symmetric if we impose current conservation and time-reversal invariance, which also impose conditions on the M matrix. We want a wholeseries of randomly chosen transfer matrices because the transfer matrix for P consecutivescatterings satisfiesMtot = M1M2 · · ·MP . (18)Because S and M both connect φ±a and φ′±a we can relate the elements of one to the otherto find M . We then uset = (M−−)−1 . (19)It is not difficult to construct these random matrices. Some results are shown in thefigures. In Fig. 4, there is only a single scattering. Steps are clearly established depending on71 2 3 4 5 6Ν123456GFIG. 4: Conductance versus ν for one scatterer. The regular stepped line is the conductance withpure ballistic flow.the number of channels. On the other hand, there are fluctuations in the conductance havingamplitude close to one conductance unit; these are the “universal conductance fluctuations”observed in electron systems.[10]In Fig. 5 and Fig. 6 we increase the number of scatterers to three and ten, respectively.The step sizes are reduced in each case, as one might expect from an increase in backscat-tering. All steps maintain the one unit fluctuation value except the single-channel case,which is strongly reduced for three scatterers and has disappeared for ten. One-dimensionalmotion has reached full localization for 10 scatterers. In that case the step structure hasalmost disappeared as well.This random matrix method seems to capture many of the effects we expect experimen-tally. However this method is so generic that it says nothing specific, for example, aboutthe scattering being just at the walls of the channels. Alternative techniques more specificto helium are being investigated.LUTTINGER LIQUIDS?Based on simple estimates of the transverse quantum zero-point motion, we argue that inthe regime of nanometric pore size and at sufficiently low temperatures, the quantum flowproperties may not be described by a fluid with independent particles, as used above, but81 2 3 4 5 6Ν123456GFIG. 5: Conductance versus ν for 3 scatterers.1 2 3 4 5 6Ν123456GFIG. 6: Conductance versus ν for ten scatterers.can enter a new kind of low-dimensional 3He ground state with experimentally unknownexcitation spectra and transport properties. In this regime, the 3He Fermi system shouldreach the strongly interacting one-dimensional limit for which a breakdown of the LandauFermi liquid picture is expected[6] and the system becomes a Luttinger liquid. This expectedcrossover to the one-dimensional regime should be considered in a theory describing thetransport of 3He particles inside very narrow channels. The ideal gas approximation is nolonger valid in this situation and interactions are vital to the existence of this phase.As an idealized case, let’s consider the transport of pure 3He inside a nanochannel witha diameter d ∼ 1 nm and of length L such that L ≫ d. If the fluid is effectively confined9radially in a cylindrical nanochannel, the fluid will be in the one-dimensional limit whenthe occupied states are all in the lowest band. The degree of occupation is measured by theone-dimensional Fermi energy, given byTF1D =~22m3kB(πn1D2)2, (20)where the number density n1D is the reciprocal of the interatomic distance. A characteristictemperature measuring the separation between bands is of orderT ⋆1D =~2m3d2kB. (21)We expect a crossover to the one-dimensional regime in nanochannels to require TF1D < T⋆1D.Assuming an interatomic distance of ∼ 1 nm, we estimate TF1D ∼ 200 mK and T⋆1D ∼ 160mK for d = 1 nm. With TF1D and T⋆1D comparable, reaching the 1D liquid in the pure caseis a possiblity if the temperature is low enough. In addition, a new unidimensional state ofmatter will arise only when the 3He atoms are in the strongly correlated regime attainedat temperature T ≪ TF1D, say, at T . 0.1TF1D ∼ 20 mK. The conditions needed may bewithin experimental reach for a few nanometer hole in the pure case.For a 1% concentration of isotopic solution the one-dimensional Fermi temperature is oforder 10 mK, and T ⋆1D ∼ 17 mK for d = 3 nm. The crossover around ∼ 1 mK is within theregime of the cooling technique by filtering discussed by Ref. [1]. The use of isotopic heliumsolutions may allow one to adjust the parameters into the needed ranges.The excitation spectrum of the one-dimensional fermionic system differs entirely from aFermi liquid, and, for electrons, this state has also been known to give rise to interestingpower law transport behaviour. Depending on whether the interactions are attractive orrepulsive, the 1D-quenched 3He system could lead to interesting new physics analogous tothat of one-dimensional electronic system such as, for example, the highly sought-after spin-charge separation for which there are now important experimental hints[11]. Having bothpure 3He and dilute solutions available allows the possibility of a considerable alteration ofthe interatomic interactions leading to differing physical properties.CONCLUSIONWe have shown that many of the interesting effects known for electron systems canbe studied experimentally and theoretically in 3He in nanochannels. We have provided10examples of quantum-limited mass flow giving rise to a quantization of the conductance.We have also calculated its temperature dependence and have shown the appearance ofconductance fluctuations when diffusive scattering takes place. We have also argued that avery interesting new state of matter, a Luttinger liquid, which may have quite new transportproperties and excitations, could perhaps form in 3He confined to nano-channels. While thistheoretical study has been of a preliminary nature, we present it in the hopes of stimulatingfurther discussion of this interesting system.We would like to acknowledge helpful discussions with I. Affleck. This work has beensupported by the Natural Sciences and Engineering Research Council of Canada (NSERC),the Canadian Institute for Advanced Research (CIFAR), and the Alfred P. Sloan Foundationfor their support under their fellowship program (G.G).[1] W.J. Mullin and N. Kalechofsky, Phys. Rev. Lett 97, 010801 (2006).[2] Y. Sato, B.H. Eom and R. Packard, J. Low Temp. Phys. 141, 99 (2005).[3] Synkera Technologies, Inc. http://www.synkera.com/products/nanomemb.htm.[4] J. K. Holt, H. G. Park, Y. Wang, M. Stadermann, A. B. Artyukhin, C. P. Grigoropoulos, A.Noy, and O. Bakajin. Science 312, 5776 (2006).[5] Wees, J. van et.al. Phys. Rev. Lett. 60, 848 (1988).[6] J.M. Luttinger, J. of Math. Phys., 4, 1154 (1963); F.D.M. Haldane, J. Phys. C- Sol. StatePhys., 14, 2585 (1981).[7] I. Affleck, private communication.[8] For a review see, C. W. J. Beenakker and H. van Houten, Solid State Phys. 44, 1 (1991).[9] C. W. J. Beenakker, Rev. Mod. Phys, 69, 731 (1997).[10] See for example, P. A. Lee, A. D. Stone, and H. Fukuyama, Phys. Rev. B 35,1622 (1985).[11] O.M. Auslaender, H. Steinberg, A. Yacoby, Y. Tserkovnyak, B. I. Halperin, K. W. Baldwin,L. N. Pfeiffer, and K. W. West, Science 308, 88 (2005).11

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Quantum-limited mass flow of liquid $^{3}$ He

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Quantum-limited mass flow of liquid $^{3}$ He