Universal Tomonaga-Luttinger liquid phases in one-dimensional strongly attractive fermionic cold atoms
A simple set of algebraic equations is derived for the exact low-temperature thermodynamics of one-dimensional multi-component strongly attractive fermionic atoms with enlarged spin symmetry and Zeeman splitting. Universal multi-component Tomonaga-Luttinger liquid (TLL) phases are thus determined. For linear Zeeman splitting, the physics of the gapless phase at low temperatures belongs to the universality class of a two-component asymmetric TLL corresponding to spin-neutral -atom composites and spin- single atoms. The equation of states is also obtained to open up the study of multi-component TLL phases in 1D systems of -component Fermi gases with population imbalance.
pacs:03.75.Ss, 03.75.Hh, 02.30.Ik, 05.30.Fk
A significant feature of one-dimensional (1D) many-body systems is the universal low energy physics of the gapless phase described by a Tomonaga-Luttinger liquid (TLL) Giamarchi-b (), as recently revealed in experimental measurements on the thermodynamics of spin ladder materials LL-exp (). In such systems magnetic fields drive phase transitions between gapped and gapless phases. In the gapless phase spin excitations (spinons) carrying spin- give rise to universal TLL thermodynamics.
On the other hand, exquisite control over the effective spin-spin interaction between cold atoms Li () has provided a new opportunity to rigorously test spin liquid behaviour via trapped fermionic atoms with higher spin symmetry Ho (); Rapp (); Lecheminant (); quatet (). In particular, fermionic alkaline-earth atoms display an exact spin symmetry with where is the nuclear spin Gorshkov (). For example, for Sr and for Yb. Such fermionic systems with enlarged spin symmetry are expected to display a remarkable diversity of new quantum phases due to the existence of multiple charge bound states.
We derive the universal thermodynamics of the gapless phase in strongly attractive 1D fermions with -component hyperfine states. We thus find that population imbalances controlled by Zeeman splitting can be used to explore and control multi-component TLL physics in 1D interacting fermionic gases with spin symmetry. The universal crossover from the regime of a relativistic multi-component Luttinger liquid to a nonrelativistic quantum critical regime is determined from the specific heat phase diagrams.
In principle, the thermodynamic Bethe ansatz (TBA) can provide the exact thermodynamics of such systems. However, the TBA involves an infinite number of coupled nonlinear integral equations which hinders access to the thermodynamics from both the numerical and analytical points of view kakashvili (); casula (). To overcome this obstacle, a method was proposed to analytically obtain thermodynamic quantities of the two-component attractive Fermi gas ZGLO (). We build on this approach to reduce the multiple charge bound state TBA for attractive 1D fermions with arbitrary spin symmetry to a set of simple algebraic equations to provide universal and exact low-temperature thermodynamics of the model.
The model. We consider a system of interacting fermions of equal mass which may occupy possible hyperfine levels (, ) labeled by isospin states and constrained by periodic boundary conditions to a line of length . The Hamiltonian Sutherland (); Takahashi () is
with Zeeman energy . Here is the number of fermions in state with Zeeman energy (or say, a chemical potential ) determined by the magnetic moments and the magnetic field . The spin-independent contact interaction remains between fermions with different hyperfine states so that the spins are conserved, i.e., with are good quantum numbers. The coupling constant with interaction strength determined by the effective 1D scattering length . For simplicity, we choose the dimensionless units of for numerical calculation and use the dimensionless coupling constant with linear density . The Hamiltonian (1) exhibits symmetry where is associated with the charge degree of freedom and with the hyperfine spin states.
Ground states. The energy eigenspectrum is given in terms of the fermion quasimomenta satisfying the Bethe ansatz equations (BAE) Sutherland (); Takahashi () by . For attractive interaction, the BAE allow charge bound states and spin strings. The symmetry acquires kinds of charge bound states which can be viewed as composites of -atoms with total spin , where and . In order to simplify calculations in the study of magnetism, we rewrite the Zeeman energy as where the independent parameters with characterize the chemical potentials for unpaired fermions and charge bound states of -atoms. Here we set up explicit relations among the ’s and arbitrary (nonlinear) Zeeman splittings via the relations with . We shall show that equally spaced (linear) Zeeman splitting, i.e., for , drives a smooth phase transition from bound states of -atoms into a normal Fermi liquid at zero temperature, see Fig. 1(A). Nonlinear Zeeman splittings may trigger spin- charge bound states as illustrated in Fig. 1(B), where the magnetic moments of paired states is Ho ().
For strong attraction () these charge bound states are stable and the system is strongly correlated. The corresponding binding energies of the charge bound states are given by . Solving the BAE with strongly attractive interaction, the ground state energy per unit length in the thermodynamic limit is given explicitly by (in units of )
Here and we have defined the functions
where is the step function.
The general result (2) for arbitrary unifies and extends known results GB (); Wadati (); Hu () for isospin and . Furthermore, we find that the low-lying excitations are described by the linear dispersion relations , where with are the velocities for unpaired fermions and charge bound states of -atoms. We denote the corresponding Fermi momentum by . These dispersion relations lead naturally to the universal form Cardy (); Affleck () for finite-size corrections to the ground state energy
Quantum phase transitions. At finite temperatures, the equilibrium states are determined by the minimization of the Gibbs free energy Takahashi-B (), which gives rise to a set of coupled nonlinear integral equations – the TBA equations note (). The TBA equations provide not only finite temperature thermodynamics but also the band fillings with respect to Zeeman splittings and chemical potentials which may be conveniently used to analyze quantum phase transitions at zero temperature. From the TBA equations, we find that complete phase diagrams at zero temperature are determined by the independent external field-energy transfer relations
for . The effective chemical potentials with are given by
characterizing Fermi surfaces of stable charge bound states and unpaired fermions. Here and with
For pure Zeeman splitting, if the external field is less than a lower critical field , a molecular superfluid phase of spin-neutral bound states forms the ground state. We find the value
at which the spin gap is diminished. Here . On the other hand, when , where
the system is fully-polarized into a normal Fermi liquid. For the intermediate regime spin-neutral bound states of -atoms and unpaired fermions coexist. The magnetization gradually increases from to its normalized value as the field increases from to (see the solid line Fig. 2). More subtle phases may be explored using the energy-transfer relations (5) by controlling the nonlinear Zeeman splitting parameters, e.g., the BCS pairing phase (see Fig. 1) by setting and for . All phase transitions are of second order with a linear field-dependent magnetization in the vicinity of critical points due to the condition of fixed total particle number.
Universal thermodynamics. At high temperatures (setting the Boltzmann constant to unity), thermal fluctuations can break the charge bound states while spin fluctuations lead to an effective ferromagnetic spin-spin interaction coupled to each Fermi sea of spin- charge bound states and unpaired fermions. The effective ferromagnetic spin-spin coupling constants are given by for note (). In this sense, we may simply view the non-neutral charge bound state as a molecule with spin , which could flip its spin antiparallel to the external fields due to thermal fluctuations. However, such spin fluctuations coupled to the channels of unpaired fermions and the spin- charge bound states are suppressed by large fields at low temperatures. Thus the low energy physics is dominated by charge density fluctuations. Indeed we show that in the physically interesting regime the breaking of charge bound states and spin wave fluctuations is strongly suppressed. Thus each dressed energy can be written in a single particle form , where the marginal scattering energies among composites and unpaired fermions and spin-wave thermal fluctuations are considered in the chemical potentials .
The thermodynamics at finite temperatures can thus be obtained from the set of algebraic equations note ()
for . Here is the standard polylogarithm function and denotes the free energy of an effective ferromagnetic spin-spin interaction with coupling constant in the channel of the spin- charge bound state note (). However, because it is a spin singlet, there is no such effective spin-spin interaction for the spin-neutral bound state of -atoms, i.e., and . is the pressure for charge bound states of -atoms. The pressure for the system is given by . The spin string contributions to thermal fluctuations in the physically interesting regime can be asymptotically calculated from the spin string equations for these non-spin-neutral charge bound states and unpaired fermions, i.e., , where . It is important to observe that becomes exponentially small as .
The suppression of spin fluctuations leads to a universality class of a multi-component TLL in each gapless phase, where the charge bound states of -atoms form hard-core composite particles. In order to see this universal TLL physics, we calculate the leading low temperature corrections to the free energy by using Sommerfeld expansion with the TBA equations to give
This result is consistent with the finite-size correction result (4). In the above equation . This result proves the existence of TLL phases in 1D gapped systems at low temperatures. Although there is no quantum phase transition in 1D many-body systems at finite temperatures due to thermal fluctuations, the existence of the TLL leads to a crossover from relativistic dispersion to nonrelativistic dispersion between different regimes at low temperatures Maeda (). Nevertheless, we find that such a field-induced multi-component TLL only lies in a small portion of Zeeman parameter space. Linear Zeeman splitting may result in a two-component Luttinger liquid in a large portion of Zeeman parameter space at low temperatures.
The thermodynamics of the gapless phase for the model (1) can be analytically calculated with linear Zeeman splitting, i.e. with for . In the regime the TBA equations (10) reduce to two coupled equations for and . The rest of the effective . From these two equations, we can obtain the density , magnetization and free energy by iteration. Fig. 2 shows the magnetization for attractive three-component fermions with a pure Zeeman field at different temperatures. We see that the linear field-dependent phase transitions in the vicinity of the critical points are smeared.
The analytic results (10) and (11) indicate that the magnetization develops minima at the same temperatures as when the two-component TLL is broken. Indeed, a deviation from the linear temperature-dependent specific heat and the magnetization minima occur around the same crossover temperatures – see the white-diamond-lines in Fig. 3 where we show a contour plot of the specific heat for and . The hard-core -atom composite particles and single atoms form an asymmetric two-component TLL lying below the white-diamond-lines. The filled-black-circles which separate the different regimes are determined by the magnetization values and . The peaks at the critical points are expected when the unpaired (bound state) band starts to fill (empty) Vekua ().
To conclude, our analytic results for universal thermodynamics and quantum phase transitions provide a unified description of attractive 1D fermions in the presence of external fields. Our formulae also provide the essential equation of states for studying trapped -component Fermi gases with population imbalance. They thus pave the way for the study of multi-component TLL phases in 1D systems of ultracold fermionic atoms.
Acknowledgment. This work has been supported by the Australian Research Council. S. C. is supported by NSFC and 973 programs (China). We thank M. Oshikawa, W. Vincent Liu, X.-J. Liu, H. Hu and P. Drummond for discussions.
- (1) T. Giamarchi, Quantum Physics in one dimension, Oxford University Press, Oxford (2004)
- (2) M. Klanjsek et al., Phys. Rev. Lett. 101, 137207 (2008); Ch. Rüegg et al., Phys. Rev. Lett. 101, 247202 (2008)
- (3) T. B. Ottenstein, T. Lompe, M. Kohnen, A. N. Wenz and S. Jochim, Phys. Rev. Lett. 101, 203202 (2008); J. H. Huchans et al., Phys. Rev. Lett. 102, 165302 (2009); J. R. Williams et al., Phys. Rev. Lett. 103, 130404 (2009)
- (4) T.-L. Ho and S. Yip, Phys. Rev. Lett. 82, 247 (1999)
- (5) Á. Rapp, G. Zaránd, C. Honerkamp and W. Hofstetter, Phys. Rev. Lett. 98, 160405 (2007); C. Honerkamp and W. Hofstetter, Phys. Rev. Lett. 92, 170403 (2004)
- (6) P. Lecheminant, E. Boulat and P. Azaria, Phys. Rev. Lett. 95, 240402 (2005); S. Capponi et al., Phys. Rev. A, 77, 013624 (2008)
- (7) C. Wu, J.-P. Hu and S.-C. Zhang, Phys. Rev. Lett. 91, 186402 (2003); D. Controzzi and A. M. Tsvelik, Phys. Rev. Lett. 96, 097205 (2006)
- (8) A. V. Gorshkov, et al, arXiv:0905.2610; M. A. Cazalilla, A. F. Ho and M. Ueda, New J. Phys. 11, 103033 (2009)
- (9) P. Kakashvili and C. J. Bolech, Phys. Rev. A 79, 041603(R) (2009)
- (10) M. Casula, D. M. Ceperley and E. J. Mueller, Phys. Rev. A 78, 033607 (2008)
- (11) E. Zhao, X.-W. Guan, W. V. Liu, M. T. Batchelor and M. Oshikawa, Phys. Rev. Lett. 103, 140404 (2009)
- (12) B. Sutherland, Phys. Rev. Lett. 20, 98 (1968)
- (13) M. Takahashi, Prog. Theor. Phys. 44, 899 (1970)
- (14) X.-W. Guan, M. T. Batchelor, C. Lee and M. Bortz, Phys. Rev. B 76, 085120 (2007); X.-W. Guan, M. T. Batchelor, C. Lee and H.Q. Zhou, Phys. Rev. Lett. 100, 200401 (2008); X.-W. Guan, M. T. Batchelor, J. Y. Lee and C. Lee, EPL 86, 50003 (2009)
- (15) T. Iida and M. Wadati, J. Phys. Soc. Jpn, 77, 024006 (2008)
- (16) X.-J. Liu, H. Hu and P. D. Drummond, Phys. Rev. A 77, 013622 (2008)
- (17) H. W. J. Blöte, J. L. Cardy and M. P. Nightingale, Phys. Rev. Lett. 56, 742 (1986)
- (18) I. Affleck, Phys. Rev. Lett. 56, 746 (1986)
- (19) Y. Maeda, C. Hotta and M. Oshikawa, Phys. Rev. Lett. 99, 057205 (2007)
- (20) M. Takahashi, Thermodynamics of One-Dimensional Solvable Models, Cambridge University Press, Cambridge, (1999)
- (21) The details of the derivation of the TBA equations and the result (10) will be presented elsewhere.
- (22) T. Vekua, Phys. Rev. B 80, 201402(R) (2009)