Exotic pairing states in a Fermi gas with three-dimensional spin-orbit coupling

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<ul><li><p>PHYSICAL REVIEW A 87, 063606 (2013)</p><p>Exotic pairing states in a Fermi gas with three-dimensional spin-orbit coupling</p><p>Xiang-Fa Zhou,1 Guang-Can Guo,1 Wei Zhang,2,* and Wei Yi1,1Key Laboratory of Quantum Information, University of Science and Technology of China,</p><p>CAS, Hefei, Anhui 230026, Peoples Republic of China2Department of Physics, Renmin University of China, Beijing 100872, Peoples Republic of China</p><p>(Received 14 February 2013; published 6 June 2013)</p><p>We investigate properties of exotic pairing states in a three-dimensional Fermi gas with three-dimensionalspin-orbit coupling and an effective Zeeman field. The interplay of spin-orbit coupling, effective Zeeman field, andpairing can lead to first-order phase transitions between different phases, and to interesting nodal superfluid stateswith gapless surfaces in the momentum space. We then demonstrate that pairing states with zero center-of-massmomentum are unstable against Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states, with a finite center-of-massmomentum opposite to the direction of the effective Zeeman field. Unlike conventional FFLO states, these FFLOstates are induced by the coexistence of spin-orbit coupling and Fermi surface deformation, and have intriguingfeatures like first-order transitions between different FFLO states, nodal FFLO states with gapless surfacesin momentum space, and exotic fully gapped FFLO states. With the recent theoretical proposals for realizingthree-dimensional spin-orbit coupling in ultracold atomic gases, our work is helpful for future experimentalstudies and provides valuable information for the general understanding of pairing physics in spin-orbit-coupledfermionic systems.</p><p>DOI: 10.1103/PhysRevA.87.063606 PACS number(s): 67.85.Lm, 03.75.Ss, 05.30.Fk</p><p>I. INTRODUCTION</p><p>Ultracold atomic gases have been considered as idealplatforms for the quantum simulation of interesting modelsin a variety of different physical contexts, ranging fromcondensed matter physics to nuclear physics [13]. Withpowerful tools like Feshbach resonance, optical lattices, etc.,strongly correlated models can be studied in this noveltype of experimental system. Of particular importance is thepossibility of stimulating models that have no counterparts inother systems. In the past decade or so, interesting phenomenalike BCS-BEC crossover, the superfluid to Mott insulatortransition in a Bose-Hubbard model etc., which do not occurnaturally in condensed matter systems, have been investigatedexperimentally using ultracold atom gases [1,2].</p><p>Recently, by engineering the atom-laser coupling, syntheticspin-orbit coupling (SOC) has been realized experimentally inultracold atoms [47]. As SOC is considered to play a keyrole in systems with interesting properties like the quantumspin Hall effect or topological phases [8], the experimentalrealization of SOC in ultracold atoms has greatly extendedthe possibility of quantum simulation in these systems. Aconsiderable amount of theoretical and experimental effort hassince been dedicated to the characterization of the rich physicsin spin-orbit-coupled atomic gases [937]. Most of thesestudies have focused on Rashba or Dresselhaus SOC, whichalso exist in solid state systems; or on the NIST SOC, whichcan be readily implemented with present technology [47,38].More exotic forms of SOC, such as a three-dimensional (3D)SOC, which have not been observed in natural condensedmatter systems, have also attracted some attention recentlyin bosonic systems [3945]. With the recent proposals forimplementing 3D SOC in ultracold atomic gases [4650], it</p><p>*wzhangl@ruc.edu.cnwyiz@ustc.edu.cn</p><p>is hoped that interesting physical effects like 3D topologicalinsulators [8], Weyl semimetals [51], or pairing physics in aFermi gas, etc. [37], could be investigated in these systems.</p><p>Indeed, a particularly interesting problem that has been un-der intensive theoretical study lately is the pairing superfluidityin an attractively interacting Fermi gas with SOC and effectiveZeeman fields. In a two-dimensional (2D) Fermi gas withRashba SOC and an axial Zeeman field, it has been shownthat a topological superfluid state can be stabilized, whichsupports topologically protected edge states and Majorana zeromodes at the core of vortex excitations [9,10,25,52], while fora 2D Fermi gas with the experimentally available NIST SOCor for a 3D Fermi gas with Rashba SOC, nodal superfluidstates with gapless excitations exist [15,18,19,29,3235].Furthermore, it has been suggested that for a 2D Fermigas under NIST SOC and crossed Zeeman fields, finitecenter-of-mass momentum Fulde-Ferrell-Larkin-Ovchinnikov(FFLO) states become the ground states of the system [33].These FFLO states are different from the conventional FFLOpairing states in a polarized Fermi gas in that they are drivenby SOC-induced spin mixing and transverse-field-inducedFermi surface deformation [28,33,36,37]. A natural questionthat follows is whether such an exotic pairing state is aunique feature of systems with SOC and Fermi surfaceasymmetry.</p><p>In this work, we study the pairing states in a 3D attractivelyinteracting Fermi gas with 3D SOC and an effective axialZeeman field [53]. As previous studies with 3D SOC havemostly focused on bosonic systems, our work provides astarting point for the understanding of the effects of 3D SOCin a fermionic system. Additionally, the Fermi surfaces of sucha system are asymmetric along the direction of the effectiveZeeman field. Hence it is an ideal system for the study ofpairing states under SOC and Fermi surface asymmetry. Toget a qualitative understanding of the pairing physics, wefocus on the zero-temperature phase diagram on the mean-fieldlevel.</p><p>063606-11050-2947/2013/87(6)/063606(8) 2013 American Physical Society</p><p>http://dx.doi.org/10.1103/PhysRevA.87.063606</p></li><li><p>XIANG-FA ZHOU, GUANG-CAN GUO, WEI ZHANG, AND WEI YI PHYSICAL REVIEW A 87, 063606 (2013)</p><p>For the zero center-of-mass momentum pairing states, wefind that exotic nodal superfluid states exist with either twoor four closed gapless surfaces in momentum space. As theanisotropy of the 3D SOC along the axial direction increases,these gapless surfaces shrink gradually, and will eventuallycollapse into gapless points as the 3D SOC is reduced to aRashba form. Therefore, these nodal superfluid states can beseen as the counterparts of the nodal superfluid states in a 3DFermi gas with Rashba SOC and an effective Zeeman field[19]. We also find that the interplay between SOC, effectiveZeeman field, and pairing in the system can lead to first-orderphase boundaries between different superfluid states, whichimplies the existence of a phase-separated state in a uniformgas and spatial phase separation in a trapped gas. Thesefindings provide a valuable starting point for understandingfinite center-of-mass momentum pairing states.</p><p>When the finite-center-of-mass momentum pairing is takeninto account, the phase diagram is qualitatively modified.Similar to the effects of a transverse field in the NIST scheme,the effective axial Zeeman field in combination with the 3DSOC makes the Fermi surface asymmetric. As a result, pairingstates with zero center-of-mass momentum become unstableagainst FFLO states, with the center-of-mass momentum ofthe pairs determined by the Fermi surface asymmetry [33].These FFLO states are characteristic of pairing states inthe presence of spin-orbit coupling and Fermi surface de-formation, which induce a shift of the local minima in thethermodynamic potential landscape onto the plane of finitecenter-of-mass momentum. Consequently, these FFLO statesretain many features of their zero center-of-mass momentumcounterparts, e.g., the existence of first-order boundariesbetween different FFLO states, the stabilization of nodal FFLOstates with gapless surfaces in momentum space, and thepresence of a continuous transition between the gapless and afully gapped FFLO state. As similar scenarios have also beenreported in Fermi gases with NIST SOC in different dimen-sions [33,36], we expect that these should be unique featuresfor pairing states under SOC and Fermi surface asymmetry,and should be independent of the concrete form of the SOC.</p><p>We stress that the FFLO state considered in this workis in fact the Fulde-Ferrell (FF) state, i.e., a pairing statewith a single center-of-mass momentum Q. More generally,one should also consider the Larkin-Ovchinnikov (LO) state,where the center-of-mass momentum of the pairing state hasboth Q and Q components. The stability of the LO statein a spin-orbit-coupled system is a subtle issue and canbe investigated, for example, by solving the BogoliubovdeGennes equation [54,55].</p><p>The paper is organized as follows: In Sec. I, we presentour mean-field formalism; in Sec II, we first discuss the phasediagram when only zero center-of-mass momentum pairingstates are considered; we take finite center-of-mass pairingstates into account in Sec. III, where we also discuss thegeneral physical picture of pairing under SOC and Fermisurface asymmetry. Finally, we summarize in Sec. IV.</p><p>II. MODEL</p><p>We consider a 3D uniform two component Fermi gas withan axially symmetric 3D spin-orbit coupling k + zkz,</p><p>where i (i = x,y,z) are the Pauli matrices, and the parameter depicts the anisotropy of the 3D SOC. The system is furthersubject to an effective Zeeman field along the z direction. TheHamiltonian of the system can be written as</p><p>H =</p><p>k</p><p>(k )(akak + bkbk) h</p><p>2</p><p>k</p><p>(akak bkbk)</p><p>+</p><p>k</p><p>k sin k(eikakbk + eikbkak)</p><p>+ UV</p><p>k,k,q</p><p>ak+q/2b</p><p>k+q/2bk+q/2ak+q/2</p><p>+ </p><p>k</p><p>k cos k(akak bkbk), (1)</p><p>where k = k(sin k cosk, sin k sink, cos k),V is the quan-tization volume in 3D, k = h2k2/(2m), ak (ak) and bk (bk)represent the annihilation (creation) operators for atoms ofdifferent spin species, is the chemical potential, h isthe effective Zeeman field, and is the spin-orbit-couplingstrength. We have assumed s-wave contact interaction betweenatoms of different spin species, where the bare interaction rateU should be renormalized following the standard relation [56]</p><p>1</p><p>U= 1</p><p>Up 1</p><p>V</p><p>k</p><p>1</p><p>2k. (2)</p><p>The physical interaction rateUp is defined asUp = 4h2as/m,with as the s-wave scattering length between the two spinspecies, which can be tuned via the Feshbach resonancetechnique. We note that while the SOC becomes isotropic whenthe anisotropy parameter = 1, it reduces to the Rashba typeSOC for = 0.</p><p>The Hamiltonian (1) can be diagonalized under themean-field approximation, for which we define the FF pairingorder parameter [57] Q = U/V</p><p>kbk+Q/2ak+Q/2.</p><p>The resulting effective Hamiltonian can be arrangedinto a matrix form under the basis k+Q/2 =(ak+Q/2,bk+Q/2,a</p><p>k+Q/2,b</p><p>k+Q/2)</p><p>T :</p><p>Heff = 12</p><p>k</p><p>k+Q/2M</p><p>Qk k+Q/2 +</p><p>k</p><p>(k+Q/2 )</p><p> VU|Q|2, (3)</p><p>with</p><p>MQk =</p><p>(H0(k + Q/2) iQyiQy H 0 (k + Q/2)</p><p>). (4)</p><p>Here, H0(k) = (k )I ( h2 kz)z + k,where I is the identity matrix. Diagonalizing the effectiveHamiltonian (3), we may then evaluate the zero-temperaturethermodynamic potential</p><p> = 14</p><p>k,,</p><p>(Ek, Ek,) +</p><p>k</p><p>(k+Q/2 + |Q|</p><p>2</p><p>2k</p><p>)</p><p> VUp</p><p>|Q|2. (5)</p><p>063606-2</p></li><li><p>EXOTIC PAIRING STATES IN A FERMI GAS WITH . . . PHYSICAL REVIEW A 87, 063606 (2013)</p><p>Here, the quasiparticle (-hole) dispersion Ek, (, = ) arethe eigenvalues of the matrix MQk in Eq. (3). Without lossof generality, we assume h &gt; 0, 0 , and Q to be realthroughout the work.</p><p>Typically, the ground state of the system can be foundby solving the gap equation /Q = 0 and the numberequation n = (1/V )/ simultaneously, where n isthe total particle number density. However, the competitionbetween pairing and the effective Zeeman field leads toa double-well structure in the thermodynamic potential forboth the Q = 0 and the finite-Q cases. Therefore, to get thecorrect ground state of a uniform gas, one must explicitlytake into account the possibility of phase separation betweenthe phases that correspond to the degenerate local minimaof the thermodynamic potential [58]. Indeed, for calculationsin a canonical ensemble where the total particle number isfixed, a phase-separated state must be taken into considerationwhen minimizing the Helmholtz free energy. As we will showin Sec. IV, this is reflected in the fact that for certain totalnumber densities, the number equation and the gap equationcannot be solved simultaneously, unless a phase-separatedstate is considered. On the other hand, for a fixed chemicalpotential, one does not need to solve the number equation;hence there is no phase separation. In this work, we fix thechemical potential and look for the global minimum of thethermodynamic potential (5) as a function of the pairing orderparameter Q and center-of-mass momentum Q. Consideringthe fact that different spin states are mixed in the presenceof SOC, the phase diagram obtained from this algorithm canbe easily connected to that of a uniform system with a fixedtotal particle number. In addition, the global minimum of thethermodynamic potential for a fixed chemical potential alsocorresponds to the local ground state in a trapped gas under thelocal density approximation (LDA). The total particle numberin the trap can then be evaluated by integrating the local numberdensities from the trap center to its edge, i.e., by integrating thenumber equation with decreasing chemical potentials. Thus,it is straightforward to reveal the phase structure in slowlyvarying potential traps from the resulting phase diagram underthe LDA.</p><p>III. PAIRING STATES WITH ZEROCENTER-OF-MASS MOMENTUM</p><p>We start by analyzing the pairing states with zero center-of-mass momentum, which will provide us with valuableinformation for the FFLO pairing states that we will discuss inthe following section. For Q = 0, an analytical expression forthe quasiparticle (-hole) dispersions is typically not available,except for those along the kz axis, which can be written as</p><p>Ek=0, = [h</p><p>2</p><p>(k + kz)2 +2</p><p>]. (6)</p><p>Apparently, two of the branches Ek, can cross zero, leadingto nodal superfluid states with gapless excitations. As thedispersion spectra have the symmetry E+kz, = Ekz, , thegapless points on the kz axis must be symmetric with respectto the origin. Furthermore, we find that the gapless pointsin momentum space typically form closed surfaces that are</p><p>(a)</p><p>1.5 1.0 0.5 0.0 0.5 1.0 1.51.5</p><p>1.0</p><p>0.5</p><p>0.0</p><p>0.5</p><p>1.0</p><p>1.5</p><p>kx kh</p><p>k zk h</p><p>(b)</p><p>1.5 1.0 0.5 0.0 0.5 1.0 1.51.5</p><p>1.0</p><p>0.5</p><p>0.0</p><p>0.5</p><p>1.0</p><p>1.5</p><p>kx kh</p><p>k zk h</p><p>(c)</p><p>1.5 1.0 0.5 0.0 0.5 1.0 1.51.5</p><p>1.0</p><p>0.5</p><p>0.0</p><p>0.5</p><p>1.0</p><p>1.5</p><p>kx kh</p><p>k zk h</p><p>(d)</p><p>1.5 1.0 0.5 0.0 0.5 1.0 1.51.5</p><p>1.0</p><p>0.5</p><p>0.0</p><p>0.5</p><p>1.0</p><p>1.5</p><p>kx kh</p><p>k zk h</p><p>(e)</p><p>1.5 1.0 0.5 0.0 0.5 1.0 1.51.5</p><p>1.0</p><p>0.5</p><p>0.0</p><p>0.5</p><p>1.0</p><p>1.5</p><p>kx kh</p><p>k zk h(f)</p><p>1.5 1.0 0.5 0.0 0.5 1.0 1.51.5</p><p>1.0</p><p>0.5</p><p>0.0</p><p>0.5</p><p>1.0</p><p>1.5</p><p>kx kh</p><p>k zk h</p><p>FIG. 1. (Color online) Gapless contours of the ground states withQ = 0 in the kx-kz plane (ky = 0) for = 0.5 and (khas)1 = 1,with the parameters (a) kh/h = 0.85, /h = 0.8, /h 0.06;(b) kh/h = 1, /h = 0.8, /h 0.25; (c) kh/h = 1.05, /h =0.8, /h 0.31; (d) kh/h = 1.25, /h = 0.8, /h 0.49; (e)kh/h = 1.26,/h = 0.35,/h 0.22; (f) kh/h = 1.25,/h =0.2,/h 0.01. Note that the gapless contours have axial symmetryaround the z axis. The effective Zeeman f...</p></li></ul>