Collective excitations across the BCS-BEC crossover induced by a synthetic Rashba spin-orbit coupling
Synthetic non-Abelian gauge fields in cold atom systems produce a generalized Rashba spin-orbit interaction described by a vector that influences the motion of spin- fermions. It was recently shown [Phys. Rev. B 84, 014512 (2011)] that on increasing the strength of the spin-orbit coupling , a system of fermions at a finite density evolves to a BEC like state even in the presence of a weak attractive interaction (described by a scattering length ). The BEC obtained at large spin-orbit coupling () is a condensate of rashbons – novel bosonic bound pairs of fermions whose properties are determined solely by the gauge field. In this paper, we investigate the collective excitations of such superfluids by constructing a Gaussian theory using functional integral methods. We derive explicit expressions for superfluid phase stiffness, sound speed and mass of the Anderson-Higgs boson that are valid for any and scattering length. We find that at finite , the phase stiffness is always lower than that set by the density of particles, consistent with earlier work[arXiv:1110.3565] which attributed this to the lack of Galilean invariance of the system at finite . We show that there is an emergent Galilean invariance at large , and the phase stiffness is determined by the rashbon density and mass, consistent with Leggett’s theorem. We further demonstrate that the rashbon BEC state is a superfluid of anisotropic rashbons interacting via a contact interaction characterized by a rashbon-rashbon scattering length . We show that goes as and is essentially independent of the scattering length between the fermions as long as it is nonzero. Analytical results are presented for a rashbon BEC obtained in a spherical gauge field with .
pacs:03.75.Ss, 05.30.Fk, 67.85.-d, 67.85.Lm, 71.70.Ej
The simulation of quantum condensed matter systemsKetterle and Zwierlein (2008); Bloch et al. (2008); Giorgini et al. (2008) with cold atoms has captivated the imagination and efforts of many. Some of the most recent new developments include the generationJaksch and Zoller (2003); Osterloh et al. (2005); Ruseckas et al. (2005); Gerbier and Dalibard (2010); Dalibard et al. (2011) of synthetic gauge fields in bosonsLin et al. (2009a, b, 2011) and realization of fermionic degeneracy in their presence.Zhai (2012)
Uniform non-Abelian gauge fields produce spin-orbit interactions. The physics of bosons in spin-orbit coupled system has been investigated by many authors.Stanescu et al. (2008); Wang et al. (2010); Ho and Zhang (2010) The rich physics hidden in the fermion problem was revealed by the solution of the two-body problem given in ref. [Vyasanakere and Shenoy, 2011], where it was shown that for certain high symmetry gauge fields, a bound state appears even for an infinitesimal attraction in the singlet channel. The key outcome of this is that a BCS-BEC crossover is induced by increasing the strength of the gauge field even with a weak attractive interaction.Vyasanakere et al. (2011) The BEC that is realized was shown to be a condensate of a new type of boson – the rashbon – whose properties are determined solely by the gauge field and not by the scattering length characterizing the interaction between the fermions. This BEC realized at large gauge coupling is called the rashbon-BEC (RBEC). Concurrently, anisotropic superfluidity of rashbonsHu et al. (2011), zero-temperature BCS-BEC crossover in the presence of Zeeman fieldsGong et al. (2011); Iskin and Subaş ı (2011) (imbalance) was studied, and transition temperatures were estimatedYu and Zhai (2011); Vyasanakere and Shenoy (2011). Dresselhaus like spin-orbit interactionHan and de Melo (2011); Takei et al. (2011) has also been examined. Non-Abelian gauge fields in lower dimensions and lattices have also been investigated.Goldman et al. (2009); He and Huang (2011); Chen et al. (2012) A review of these fast paced recent developments may be found in ref. [Zhai, 2011]. Several aspects of the physics of spin-orbit coupled fermions were reported earlierChaplik and Magarill (2006); Cappelluti et al. (2007) and were independently discovered in the cold atoms context.Vyasanakere and Shenoy (2011); Vyasanakere et al. (2011)
The motivating questions for this work pertain to the properties of the RBEC that is obtained at large gauge coupling at a fixed scattering length . In the usual BCS-BEC crossoverEagles (1969); Leggett (1980, 2006); Noziéres and Schmitt-Rink (1985); Randeria (1995) in the absence of spin-orbit interaction, the BEC state for small positive scattering length is a condensate of bosons (fermionic dimer molecules). This BEC state can be described by the Bogoliubov theory of interacting bosonsAbrikosov et al. (1965), where the boson mass is twice the fermion mass and the effective boson-boson scattering length is proportional to .Randeria (1995); Pethick and Smith (2004) Does a similar description hold for the RBEC obtained by tuning the magnitude of the gauge coupling? How does rashbon-rashbon scattering enter the description, i. e., what is the effective rashbon-rashbon scattering length?
That collective excitations have interesting and unusual features was pointed out in ref. [Zhou and Zhang, 2011] which studied phase stiffness (superfluid density) for an extreme-oblate gauge field (see below for a definition). In the regime , the decreases with increasing gauge coupling. However, for , increases and saturates as attains large values. For all , is less than , the value of phase stiffness for a superfluid without the spin-orbit interaction, where the density and is the mass of the fermions. This is attributedZhou and Zhang (2011) to the lack of Galilean invariance in systems with synthetic non-Abelian gauge fields (see also, ref. [Williams et al., 2012]). While this is true, we conjecture that Galilean will be approximately restored in the system for when an attractive interaction, however weak, is present. The basis of this conjecture stems from the fact that at large the system with even a weak attraction can be thought of as a collection of rashbons which disperse quadraticallyVyasanakere and Shenoy (2011), , albeit with an anisotropic dispersion defined by the direction dependent rashbon mass and is the rashbon binding energy, a result that is valid for . This dispersion is Galilean invariant, and therefore we expect to obtain a phase stiffness tensor (no sum on ), where is the rashbon density, consistent with Leggett’s resultLeggett (1998, 2006). Testing this conjecture regarding emergent Galilean invariance and answering the questions raised in the previous paragraph are the aims of this paper.
To this end, we investigate the collective excitations of superfluids induced by non-Abelian gauge fields using a Gaussian fluctuations theory with a functional integral framework. Our main result is that the rashbon BEC can be described as a collection of weakly interacting rashbons. We obtain an effective rashbon-rashbon scattering length which we show is generically proportional to , and is independent of the scattering length between the fermions to leading order. In addition, we show that the phase stiffness has precisely the form as conjectured above. The RBEC state is a remarkable state where the effective interaction between the emergent bosons (rashbons) is determined by the kinetic energy (spin-orbit coupling ) of the constituent fermions, and not the attraction between the fermions as long as it is non-vanishing. Our theory also provides the phase stiffness, speed of sound and the mass of the Anderson-Higgs boson for any gauge coupling.
Sec. II outlines the functional integral framework used in the analysis of the collective excitations and obtains general formulae for the phase stiffness, sound speed and Anderson-Higgs mass for a generic Rashba like spin-orbit coupled system. Results for a spherical gauge field are discussed in sec. III, and sec. IV contains a discussion of the properties of rashbon BECs. The paper is summarized in sec. V.
The kinetic energy of the spin- fermions is
where, s and s are fermion operators,
is the helicity, . The “vector” describes the configuration of the gauge field that induces a generalized Rashba spin-orbit interaction, where is the magnitude of the gauge coupling and is a unit vector. High symmetry gauge field configurations of interest include the extreme oblate (EO) gauge field with and the spherical (S) gauge field which has . We use units where the fermion mass and are unity. We consider a finite density of fermions which defines a momentum scale such that , and an energy scale .
The interaction piece describes an attraction in the singlet channel as
where is the volume of the system, is the bare interaction parameter. The theory requires an ultraviolet cutoff which can be eliminated by using . Using mean-field theory, it was shown in ref. [Vyasanakere et al., 2011] that increasing induces a BCS to BEC crossover even for a weak attractive interaction (). We aim to study the collective excitations of such superfluids across this crossover.
To this end we use a functional integral framework which has been extensively used in the study of BCS-BEC crossover.Sá de Melo et al. (1993); Randeria (1995); Engelbrecht et al. (1997); De Palo et al. (1999); Dupuis (2004); Diener et al. (2008) Denoting inverse temperature as and chemical potential as , we write the action
is a Nambu vector consisting of Grassmann variables describing the fermions, where is a fermionic Matsubara frequency,
is the Fourier transform of the singlet density with , is a bosonic Matsubara frequency. is the singlet amplitude in a two particle state of and helicities, with centre of mass momentum and relative momentum . It must be noted that satisfy many symmetry properties which are used extensively in the work that follows. Moreover, care must be exercised in the definition of due to the non-zero Chern flux originating from the origin of the momentum space (see ref. [Ghosh et al., 2011]).
We now introduce a Hubbard-Stratanovich pair field to decouple the interaction term to obtain
We integrate out the fermions to obtain the action only in terms of the pairing field
We now perform a saddle point analysis of the action and look for static and homogeneous solutions via the ansatz
where the factor of is introduced for convenience. With this ansatz for the saddle point, the Green’s function is
with . The saddle point condition, after appropriate frequency sums, is
Collective excitations of the system are described by fluctuations about the saddle point state. We treat them at Gaussian level by introducing “small oscillations” about the saddle point value of the pairing field,
After some straightforward, if lengthy, algebra, the action to quadratic order in is
Collective excitations of a superfluid can be conveniently described in terms of spatio-temporally dependent phase and amplitude oscillations. We, therefore, express in terms of two other real fields (amplitude fluctuation) and (phase fluctuation) as
with and . The action in terms of these two fields is
where, using eqn. (24), we find
We now preform the necessary frequency sums to obtain expressions for the s. Here and henceforth in this paper, we focus at zero temperature () and “small” , and do not show the lengthy expressions valid for any temperature and . For small at , we have,
where the quantities depend on the saddle point values of and . is the phase stiffness given by
where , and is a tensor that defines the singlet amplitude for small as
It must be noted that extensive use of the properties of is made in arriving at this expression for the phase stiffness tensor that is valid for any gauge field. The other quantities in eqn. (31),
We have not shown the expression for since it will not be used in the discussion below.
The dispersion of the excitations can be obtained by first analytically continuing to real frequencies and solving . We obtain two modes for a given , one is a gapless sound mode and other is the gapped Anderson-Higgs mode. The speed of sound along direction is given by
and the mass of the Anderson-Higgs mode is obtained as
It must be noted that the amplitude and phase modes are coupledEngelbrecht et al. (1997); their coupling is determined by the quantity .
Equations 34, 37 and 38 are the key results of this paper for the collective excitations of spin-orbit coupled superfluids that are applicable to any Rashba gauge field and scattering length at zero temperature. We have not shown the finite temperature results here to avoid lengthy expressions. In the remainder of the paper, we illustrate the physics of these formulae using the spherical gauge field (next section) and explore the consequences of our results particularly for the rashbon-BEC (sec. IV).
Iii Collective excitations for the spherical gauge field
In this section we discuss collective excitations of superfluids realized in a spherical gauge field with as noted earlier. The two body problem for this gauge field was exhaustively investigated in ref. [Vyasanakere and Shenoy, 2011] where an analytical expression for the binding energy valid for any scattering length is derived along with an analytical expression for the bound state wave function. The binding energy of the rashbonVyasanakere and Shenoy (2011) is
and the rashbon mass (in units of fermion mass) isVyasanakere and Shenoy (2011)
A route to experimental realization of this gauge field has recently been suggested.Anderson et al. (2011) A detailed study of two-body scattering from a finite range box potential is carried out in ref. [Cui, 2011].
iii.1 Analytical Results
Analytical results can be obtained in two regimes of . These correspond to , and the other to .
Two regimes of are tractable analytically for this regime of , both of which are well known; we state them here for the sake of completion.
I. : This regime is studied in detail in ref. [Vyasanakere et al., 2011]. The chemical potential in this regime is set by the value of the noninteracting system (which falls by an amount proportional to ). The gap is essentially unaltered from the well known BCS value. Under these conditions, we obtain the phase stiffness to be with a fall of order . The leading term in the speed of sound is as shown by AndersonAnderson (1958) (with a fall proportional to ) and the Anderson-Higgs mass is exponentially small. This limit corresponds essentially to the BCS limit studied in ref. [Engelbrecht et al., 1997].
This is the regime of interest and corresponds to the rashbon BEC. In this regime, we report new results for the gap
and the chemical potential
By an analysis of the expression for the phase stiffness (eqn. (34)) which is isotropic for this gauge field, we find that
precisely as conjectured in the introductory section (see below for further discussion). Additional analysis provides
As expected, the leading terms for all the quantities of interest are independent of the scattering length between the fermions; scattering length corrections (which we do not show) appear as powers of , which in this regime are small. We emphasize that in this RBEC regime the amplitude and the phase mode are strongly coupled, just like in the usual BEC regime.
iii.2 Numerical Results
In this section we show the results of numerical calculations of evolution of , and with increasing for several scattering lengths.
iii.2.1 Superfluid Phase Stiffness
Fig. 1 shows a plot of the phase stiffness as a function of for various scattering lengths. We see that for small negative scattering lengths, the behaviour of is non-monotonic; it decreases with increasing and attains a minimum near . This is fully consistent with the finding of ref. [Zhou and Zhang, 2011] for the EO gauge field. The new aspect uncovered in our work is that for , the phase stiffness tends to that of a collection of interacting rashbons in exactly same way as the motivating conjecture of this paper. In other words, where is the rashbon number density. The physics behind this is that the rashbon dispersion is Galilean invariant, and hence the phase stiffness as found at is consistent with Leggett’s resultLeggett (1998, 2006). This is a remarkable feature, and corresponds to an emergent infrared symmetry, i. e., in the presence of interactions however small, the system organizes itself to posses a larger symmetry at low energies! A important point that can be inferred is that the nonzero phase stiffness implies that rashbons are interacting bosons. The nature of the interaction is uncovered in the next section.
iii.2.2 Sound Speed
The variation of the sound speed with increasing is shown in fig. 2. We see that there is a monotonic decrease in the sound speed with increasing for all scattering lengths. At large , the sound speed is inversely proportional to as obtained analytically (see eqn. (44)). Again, that there is sound propagation in the medium suggests the presence of interactions between the rashbons.
iii.2.3 Mass of the Anderson-Higgs boson
For small gauge coupling ( ) corresponds to the gap of the amplitude mode for small negative scattering lengths. This mass grows with increasing albeit with some features near for small negative scattering lengths. At large we find the expected behaviour.
The key result of this section is that at large , the system behaves like a Galilean invariant collection of interacting rashbons. Since this regime is the raison d’etre of this paper, we do not pause to consider the interesting regime of which no doubt contains rich physics.
Iv Properties of Rashbon Bose-Einstien Condensates (RBEC)
That the system evolves to a collection of interacting rashbons with increasing is conclusively demonstrated in the previous section. The rashbon dispersion derived in ref. [Vyasanakere and Shenoy, 2011] provides the kinetic energy of the rashbons. What about their interactions? Interestingly, the results of the previous section allow us to answer this question.
Recall from the Bogoliubov theoryAbrikosov et al. (1965) that a collection of bosons of mass with number density and a contact interaction described by a scattering length has a superfluid ground state at zero temperature. The chemical potential of this system is
and the speed of sound is
From eqn. (42), the rashbon chemical potential (measured from the bottom of the rashbon band at ) is
This clearly demonstrates that the rashbon BEC is a condensate of rashbons interacting with a contact interaction. Writing
allows us to calculate the rashbon-rashbon scattering length as
which is approximately equal to . This result is remarkable in the following sense that the effective interaction between rashbons is determined by a scale that enters the kinetic energy of the constituent fermions, and not by the interaction between the constituent fermions (scattering length )!
We emphasize that although our arguments used the spherical gauge fields, the results obtained are applicable to other gauge field configurations described by a general vector (except the extreme prolate gauge field which has only one nonvanishing component, see ref. [Vyasanakere et al., 2011]). For a generic gauge field, the rashbon chemical potential will be
where is a dimensionless number that depends on , and the anisotropic speed of sound in the -direction will be
where is the anisotropic rashbon massVyasanakere and Shenoy (2011) that depends, again, on . The rashbon-rashbon scattering length will be
where is dimensionless number determined by The low energy properties of the rashbon BEC are similar to those of the usual Bogoliubov Bose fluid; in fact, generically, RBEC is a superfluid of anisotropically dispersing rashbons interacting with a contact potential described by a scattering that depends inversely on the spin orbit coupling strength of the fermions. It must be noted that accurate determination of may require further self consistent treatment of the theory.H. Hu et al. (2006); Diener et al. (2008)
In this paper, we explore the properties of the superfluids induced by non-Abelian gauge fields focusing on their collective excitations. We present results for superfluid phase stiffness, sound speed and Anderson-Higgs mass valid for any Rashba gauge field and scattering length. Our main results are
Superfluid phase stiffness has non-monotonic behaviour with increasing , the scale of the spin-orbit interaction. This is in agreement with an earlier reportZhou and Zhang (2011) of superfluid density for the EO gauge field.
A new result is that for large gauge coupling, i e., in the rashbon BEC, the superfluid phase stiffness is determined by the rashbon massVyasanakere and Shenoy (2011). This arises from an emergent Galilean invariance at infrared energies for large gauge couplings, and the phase stiffness is consistent with Leggett’s result.
The sound speed decreases monotonically with increasing gauge coupling. At large gauge coupling it goes as . The Anderson-Higgs mass increases with increasing and goes as in the rashbon-BEC.
A key outcome of this work is that we show that the rashbon-BEC can be described as a collection of anisotropically dispersing rashbons interacting via a contact interaction. We obtain an analytical expression for the rashbon-rashbon interaction for the spherical gauge field showing that it goes as . We argue that this result is true for a generic gauge field (spin-orbit interaction).
We conclude the paper by revisiting the RG flow diagram of the two body problem introduced in ref. [Vyasanakere and Shenoy, 2011]. Fig. 4 is a schematic RG flow diagram in the - plane for the two-particle problem. The key point is that flow from any point with and reaches which is the stable rashbon fixed point corresponding to and . Indeed, the properties of the state attained by a finite density of fermions at large is controlled by the rashbon fixed point; it is therefore a weakly interacting gas of rashbons – the rashbon BEC.
JV acknowledges support from CSIR, India via a JRF grant. VBS is grateful to DST, India (Ramanujan grant), DAE, India (SRC grant) and IUSSTF for generous support.
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