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in ultracold atoms , the typical diluteness condition and short - range interactions can give rise to interesting universal relations among thermodynamic quantities of a many - body system . these universal relations are independent of the microscopic details of two - body interactions , and have stimulated much research interest . in a unitary fermi gas close to an @xmath0-wave scattering , for instance , the central quantity of these universal relations , tan s contact , has been extensively studied both theoretically and experimentally . furthermore , the recent radio - frequency ( r.f . ) spectroscopic measurement near a @xmath1-wave feshbach resonance in @xmath2k atoms opens up new possibilities of studying universal relations and thermodynamics in higher partial - wave scattering channels  @xcite . theoretically , it has also been shown recently that a whole family of contacts and universal relations exist in systems with higher partial - wave scattering  @xcite . motivated by this progress , we investigate universal relations and thermodynamics of a two - component degenerate fermi gas with coexisting @xmath0- and @xmath1-wave interactions . such a system can be experimentally prepared and probed with @xmath2k atoms near the magnetic field @xmath3 g . as the @xmath1-wave feshbach resonance here is also close to an @xmath0-wave feshbach resonance , the @xmath0- and @xmath1-wave interactions can be comparable in strength . it is then expected that the co - existence of both @xmath0- and @xmath1-wave scattering channels can lead to interesting many - body properties . indeed , it has been shown very recently that , at zero temperature , an interesting hybridization of @xmath0- and @xmath1-wave superfluid can be stabilized , which exhibit nontrivial pairing correlations  @xcite . in this work , we focus on thermodynamic properties of the normal state above the superfluid transition temperature . we first derive the general universal relations of a two - component ultracold fermi gas with coexisting @xmath0- and @xmath1-wave interactions . due to the orthogonality of two - body wave functions of different scattering channels , the universal thermodynamic relations of the system , such as the adiabatic relations , pressure , and the energy functional , appear to be direct summations of contributions from @xmath0- and @xmath1-wave interactions . these universal relations are governed by a set of contacts , which can be associated with scatterings within different partial - wave channels . we then numerically evaluate the interaction energy as well as contacts in the high - temperature normal phase using second - order virial expansions . interestingly , due to the interplay of @xmath0- and @xmath1-wave interactions on the many - body level , we find that the contacts , and hence the relevant thermodynamic properties behave differently from those with only @xmath0- or @xmath1-wave interaction potentials . the interplay of multiple partial - wave interactions also leaves experimentally detectable signatures in asymptotic behavior of momentum distribution and r.f . spectrum in the high - momentum , high - frequency regime . another interesting finding is that the interaction energy of the repulsive branch features abrupt changes across the @xmath1-wave resonances , which may serve as a unique signature for the coexistence of @xmath0- and @xmath1-wave interactions in the system . our results can be readily checked experimentally for @xmath2k atoms near the @xmath3 g @xmath1-wave feshbach resonance , where the two scattering channels naturally coexist . the paper is organized as follows : in sec .  [ 2 ] , starting from the two - body density matrix , we derive the general universal relations such as interaction energy , adiabatic relations , pressure relations , and the energy functional . in sec . [ 3 ] , we present the formalism of the quantum virial theorem , and express thermodynamic quantities such as the thermodynamic potential and interaction energy in the normal state using second virial coefficients . in sec . [ 4 ] , we numerically evaluate interaction energy and contacts using virial expansion calculations , adopting typical experimental parameters of @xmath2k atoms near the @xmath3 g @xmath1-wave feshbach resonance . we also calculate the asymptotic behavior of the momentum distribution and the r.f . spectrum in the large - momentum and large - frequency limit , respectively . finally , we summarize in sec .  [ 5 ] . in this section , we study the universal relations for a two - component , partially polarized fermi gas with both @xmath0- and @xmath1-wave interactions . following the standard treatment , we start from the two - body density matrix of the many - body system , decompose it into pair wave functions , and expand the pair wave functions in terms of the two - body wave functions . employing the asymptotic forms of the two - body wave functions in the regime @xmath4 , where @xmath5 is the range of interaction potentials , we derive universal thermodynamic relations based on the two - body density matrices . importantly , these universal relations are governed by a set of contacts , which are associated , respectively , with @xmath0- and @xmath1-wave scattering channels . for the derivations in this section , we consider the general case where @xmath0-wave interactions between the two spin components ( @xmath6 ) and @xmath1-wave interactions between the same spins coexist . following the standard treatment , we consider the two - body density matrix for our many - body system @xmath7 , where @xmath8 are spin indices , and @xmath9 creates a fermion with spin @xmath10 at position @xmath11 . the two - body density matrix can be decomposed as  @xcite @xmath12 where @xmath13 , @xmath14 is the center - of - mass momentum of a pair , @xmath15 are the eigenvalues , and @xmath16 label the quantum numbers of the relative radial direction , the angular momentum , and the @xmath17-axis projection of angular momentum , respectively . moreover , the associated pair wave functions @xmath18 form an orthonormal set .    for the system under consideration , the only nonvanishing pair wave functions are @xmath19 with @xmath20 . the spin wave functions are defined as @xmath21 and the spatial wave functions @xmath22 here , @xmath23 , @xmath24 , @xmath25 is the quantization volume , and @xmath26 is the relevant spherical harmonic function . correspondingly , we denote the relevant eigenvalues as @xmath27 and @xmath28 , for the @xmath0- and @xmath1-wave pair wave functions , respectively . we may then write down the interaction energy of the many - body system in terms of the two - body density matrix , and expand the pair wave functions therein using two - body radial wave functions . the resulting interaction energy reads @xmath29.\nonumber\\\end{aligned}\ ] ] here , we have defined a set of contacts @xmath30 and @xmath31 , which are related to the scattering length @xmath32 , the scattering volume @xmath33 , and effective range @xmath34 , respectively @xmath35 note that the integration over @xmath36 here includes the contributions of the bound states . note that the orthogonality of the two - body wave functions in different scattering channels renders the interaction energy in eq .   a direct summation of contributions from the three scattering channels : one with the @xmath0-wave interaction between @xmath37 and @xmath38 , and two with the @xmath1-wave interaction between two @xmath37s or two @xmath38s . as a result , different scattering channels are decoupled in the definition of contacts as well . as we will see , this effectively allows us to write down the universal relations based on an intuitive extension of the previous works on single - partial - wave interactions . the importance of contacts is that they appear in various universal relations and provide a link between few - body parameters and many - body quantities . this link is most apparent in the so - called adiabatic relations . following the treatment in refs . @xcite , we have @xmath39 where @xmath40 is the total energy , @xmath41 is the reduced planck constant , and @xmath42 is the atomic mass . again , in the universal relations , we see that contacts associated with different scattering channels are decoupled . based on the adiabatic relations , it is straightforward to derive other related universal relations . as two illustrating examples , we show below the pressure and the energy functional .    the pressure for a uniform gas is given by  @xcite @xmath43.\end{aligned}\ ] ] we note that although eq . ( [ ph ] ) is only exact in the absence of external potential , it is approximately valid for each local part of a trapped fermi gas within the local density approximation . in an external harmonic trap @xmath44 , where @xmath45 is the frequency of the harmonic trapping potential , the average energy density is replaced by the local internal energy density , and the contacts are replaced by the local contacts . furthermore , the total energy in the presence of a trapping potential is given by  @xcite @xmath46,\end{aligned}\ ] ] where @xmath47 is the energy of the trapping potential . the relation above is essentially the virial theorem in refs . to further study the thermodynamic properties of the system , we evaluate the thermodynamic potential using virial expansions . for a spin - polarized fermi gas , it is necessary to introduce two fugacities @xmath48 and @xmath49 to distinguish different spin configurations , where @xmath50 , @xmath51 is the chemical potential for spin @xmath10 , @xmath52 is the temperature , and @xmath53 is the boltzmann constant . generally , the thermodynamic potential can be written as  @xcite @xmath54 where @xmath55 is the @xmath56th virial coefficient associated with a cluster containing @xmath57 spin-@xmath58 fermions and @xmath59 spin-@xmath60 fermions . here , @xmath61 and the thermal de broglie wavelength is @xmath62 .    following the common practice , we define @xmath63 , where the superscript  ( 1 ) " denotes an ideal noninteracting system with the same fugacity . for a spin - polarized fermi gas with coexisting @xmath0- and @xmath1-wave interactions , we may rewrite the thermodynamic potential as ( up to the second order ) @xmath64,\end{aligned}\ ] ] where @xmath65 is the thermodynamic potential for a noninteracting ideal fermi gas with the same fugacities as @xmath66 . @xmath67 and @xmath68 are , respectively , the second virial coefficients for the @xmath0- and @xmath1-wave scattering channels .      in an ideal spin - imbalanced fermi gas , the thermodynamic potential for each spin species takes the form @xmath69 where the standard fermi - dirac integral is @xmath70 here , @xmath71 is the gamma function . the total thermodynamical potential of an ideal spin - imbalanced fermi gas is a direct sum of the thermodynamic potentials of each spin component @xmath72.\end{aligned}\ ] ]    accordingly , we can rewrite the thermodynamic potential of a strongly interacting polarized fermi gas as ( up to the second order ) @xmath73.\end{aligned}\ ] ]    therefore , the particle number densities of spin - up and spin - down atoms are given by @xmath74,\\ \label{nh2}n_{\downarrow}&&=-\frac{1}{v}\frac{\partial\omega}{\partial\mu_{\downarrow}}\nonumber\\ & & = \frac{1}{\lambda^{3}}\left[f_{3/2}(z_{\downarrow})+2 z_{\uparrow}z_{\downarrow}\delta b_{2,s}+4z_{\downarrow}^{2}\delta b^{(\downarrow\downarrow)}_{2,p}\right].\end{aligned}\ ] ]      the total energy density can be written as @xmath75 where the logarithm of the grand canonical partition function @xmath76 is given by @xmath77.\end{aligned}\ ] ]    we can further separate the total energy density into the kinetic and the interaction part : @xmath78 . the kinetic energy density is @xcite @xmath79\nonumber\\ & & ~~+\frac{3 n_{\downarrow}k_{b}t}{2}\left[1 + 2^{-5/2}(n_{\downarrow}\lambda^{3})\right].\end{aligned}\ ] ] and the interaction energy density is given by @xmath80,\end{aligned}\ ] ] where @xmath81 and @xmath82 .    for a spin - polarized fermi gas with a fixed polarization @xmath83 , and a fixed total number density @xmath56 , we can obtain two different particle number densities @xmath84 . we then substitute the number densities into eq . ( [ eint ] ) to get the corresponding interaction energy density with the second virial coefficients given in sec . [ 3.4 ] . the contacts can also be expressed in term of the grand thermodynamic potential @xmath85  @xcite @xmath86_{t , v,\mu_{\uparrow},\mu_{\downarrow}},\\ \label{cpr2 } & & c^{(\sigma\sigma)}_{p , r_{m } } = -\frac{8\pi m}{\hbar^{2 } } \left[\frac{\partial \omega}{\partial ( r_{m}^{(\sigma\sigma)})^{-1}}\right]_{t , v,\mu_{\uparrow},\mu_{\downarrow}}.\end{aligned}\ ] ] with these expressions , it is straightforward to numerically evaluate contacts and the associated universal thermodynamic properties once the second virial coefficients are known .      for a uniform system , the second virial coefficients can be expressed in terms of the phase shifts of the corresponding two - body scattering problem . for a spin-1/2 fermi gas , it takes the form  @xcite @xmath87 where the summation is over all the two - body bound states ( with the bounding energy @xmath88 ) and @xmath89 is the phase shift of the @xmath90th partial wave .    in the case of @xmath0-wave scattering  @xcite @xmath91 where @xmath92 is the effective range of interactions . close to resonance , @xmath93 , the second virial coefficient for @xmath0-wave scattering can then be written as  @xcite , @xmath94 where @xmath95 , @xmath96 is the fermi temperature , @xmath97 is the fermi wave vector , @xmath98 is the complementary error function , the heaviside step function @xmath99 in this paper is defined as @xmath100 for @xmath101 and @xmath102 for @xmath103 , the sign function @xmath104 satisfies @xmath105 with @xmath103 , @xmath106 with @xmath107 , and @xmath108 with @xmath101 , and @xmath109 is the @xmath0-wave two - body bound - state energy .    in the case of @xmath1-wave scattering  @xcite @xmath110 where @xmath111 and @xmath112 are , respectively , the scattering volume and the effective range of the @xmath1-wave interaction with a magnetic number @xmath113 . the second virial coefficient in this case can be analytically given by  @xcite @xmath114,\end{aligned}\ ] ] where @xmath115 is the @xmath1-wave two - body bounding energy . in this section , we present our numerical results for the thermodynamic quantities based on the second virial coefficient calculations . to make connections with experiments , we consider @xmath2k atoms close to the @xmath3 g @xmath1-wave feshbach resonance between the same hyperfine states @xmath116 . as this @xmath1-wave resonance is close to a wide @xmath0-wave feshbach resonance at @xmath117 g between the hyperfine states @xmath116 and @xmath118 , both @xmath0- and @xmath1-wave interactions exist in the system  @xcite . here , both the @xmath0- and @xmath1-wave scattering parameters are functions of the magnetic field . for the @xmath0-wave scattering , the scattering length @xmath32 is given by  @xcite @xmath119 where @xmath120 g , @xmath121 , @xmath122 is the bohr radius , and @xmath123 g .    for the @xmath1-wave scatterings , the scattering volume @xmath111 near the @xmath1-wave feshbach resonance can be conveniently calculated using  @xcite @xmath124 here , there are two @xmath1-wave feshbach resonances @xmath125 g and @xmath126 g . the interacting ranges for the @xmath1-wave interactions are  @xcite @xmath127    in the following , we will define @xmath128 and @xmath129 . therefore , the only @xmath1-wave interactions in the system are between two @xmath37 states . for the numerical calculations , we take @xmath130 nm , where the fermi wave vector is @xmath131 with the total particle number density @xmath56 .      with @xmath132 , the interaction energy density becomes @xmath133.\end{aligned}\ ] ]     of @xmath2k atoms as a function of the magnetic field @xmath134 ( in units of g ) near @xmath0- and @xmath1-wave resonances for @xmath135  @xcite with fixed spin polarization @xmath136 . the blue solid line is for the @xmath0-wave feshbach resonance at @xmath137 g , the black solid line denotes the @xmath1-wave feshbach resonance with @xmath138 at @xmath125 g , and the green solid line denotes the @xmath1-wave feshbach resonance with @xmath139 at @xmath126g.,width=302 ]    we show in fig . [ interactionenergy ] the interaction energy density as a function of the magnetic field @xmath134 ( in units of g ) for @xmath135  @xcite with a fixed spin polarization @xmath136 . for comparison , we show the numerical results for a fermi gas at the same temperature with only @xmath0-wave interactions ( dashed black curve ) , with only @xmath1-wave interactions ( dashed - dotted blue curve ) , and with both @xmath0- and @xmath1-wave interactions ( solid red curve ) . we also show the interaction energy densities of the repulsive branch ( thin curves ) , for which the bound - state contributions to the second virial coefficients are excluded in eqs . ( [ b2s ] ) and ( [ b2p ] ) .    an interesting feature of fig . [ interactionenergy ] is the behavior of the interaction energy of the repulsive branch across the @xmath1-wave feshbach resonances . we consider anisotropic @xmath1-wave interactions , in which the @xmath1-wave resonance for angular momentum @xmath140 occurs @xmath141 g , and the @xmath1-wave resonance for angular momentum @xmath142 occurs at @xmath143 g . in the case when only @xmath1-wave interactions are present , the interaction energy of the repulsive branch features an abrupt jump when the magnetic field is swept across the @xmath144 g resonance . physically , this is due to the coexistence of @xmath1-wave interactions with different magnetic angular momentum . for the case with coexisting @xmath0- and @xmath1-wave interactions , similar sudden changes of the repulsive - branch interaction energy show up at both @xmath1-wave resonances at @xmath145 g and at @xmath146 g , which originate from the coexistence of @xmath0- and @xmath1-wave interactions . note that while the interaction energy in the repulsive branch approaches the pure @xmath0-wave value as the magnetic field decreases away from the @xmath1-wave resonance , throughout the @xmath1-wave resonance region , the interaction energy is quite different from that of the pure @xmath0-wave case . this behavior would provide a strong experimental signal for the existence of both scattering channels in the system .          in order to calculate the contacts , we first expand the thermodynamic potential eq . ( [ omegah ] ) with respect to a small fugacity @xmath147 : @xmath148,\end{aligned}\ ] ] where we have taken @xmath132 . the particle number densities eqs . ( [ nh1 ] ) and ( [ nh2 ] ) can also be expanded with respect to a small fugacity @xmath147 : @xmath149,\\ \label{nh22}n_{\downarrow}&&\simeq \frac{1}{\lambda^{3}}\left(z_{\downarrow}-\frac{z_{\downarrow}^{2}}{2^{5/2}}+2 z_{\uparrow}z_{\downarrow}\delta b_{2,s}\right).\end{aligned}\ ] ]    with eqs . ( [ cs2 ] ) , ( [ cpv2 ] ) , ( [ cpr2 ] ) , and eq .  ( [ omegahcontact ] ) the analytical expressions for the contacts become @xmath150_{t , v,\mu_{\uparrow},\mu_{\downarrow}}\nonumber\\ & & = \frac{16\pi m}{\hbar^{2 } } k_{b}t\frac{v}{\lambda^{3}}\sum_{m}\left[\frac{\partial\delta b^{(\uparrow\uparrow)}_{2,p}}{\partial(\nu_{m}^{(\uparrow\uparrow)})^{-1}}\right]_{t}z_{\uparrow}^{2},\\ \label{cprup } c^{(\uparrow\uparrow)}_{p , r}&&=\sum_{m}c^{(\uparrow\uparrow)}_{p , r_{m } } = -\frac{8\pi m}{\hbar^{2 } } \sum_{m}\left[\frac{\partial \omega}{\partial ( r_{m}^{(\uparrow\uparrow)})^{-1}}\right]_{t , v,\mu_{\uparrow},\mu_{\downarrow}}\nonumber\\ & & = \frac{16\pi m}{\hbar^{2 } } k_{b}t\frac{v}{\lambda^{3}}\sum_{m}\left[\frac{\partial\delta b^{(\uparrow\uparrow)}_{2,p}}{\partial ( r_{m}^{(\uparrow\uparrow)})^{-1}}\right]_{t}z_{\uparrow}^{2}.\end{aligned}\ ] ]    numerically , we calculate the two fugacities by solving the coupled number equations eqs . ( [ nh12 ] ) and ( [ nh22 ] ) , and then substitute the fugacities into eqs . ( [ csup ] ) , ( [ cpvup ] ) , and ( [ cprup ] ) to get the corresponding contacts . for convenience , in the following discussions , we only calculate the attractive branch of the contacts . figure  [ contactp ] shows the contacts as functions of spin polarization @xmath151 for a homogeneous fermi gas up to the second - order virial expansion with @xmath152 and @xmath153 g ( solid red line ) . here , the fermi temperature is given by @xmath154 . for comparison , we also show the numerical results for system with only @xmath0-wave interactions ( dashed black line ) , and with only @xmath1-wave interactions ( dashed - dotted blue line ) . note that @xmath155 ( @xmath156 ) for systems with only @xmath1-wave ( @xmath0-wave ) interactions .        in fig . [ contactt ] , we show the temperature dependence of various contacts with a polarization @xmath157 , and a magnetic field @xmath153 g . evidently , at lower temperatures , @xmath158 and @xmath159 for the case with coexisting @xmath0- and @xmath1-wave interactions can be significantly different from those with purely @xmath0- or @xmath1-wave interactions . at high temperatures , as thermal effects dominate over interaction effects , all the contacts approach zero . finally , in fig .  [ contactb ] , we show the dependence of contacts on the magnetic field with a polarization @xmath160 at a given temperature @xmath152 . apparently , the interplay of @xmath0- and @xmath1-wave interactions is most significant near the @xmath1-wave resonances . we also note an interesting sign change of @xmath161 as the magnetic field is swept between two @xmath1-wave resonances .    from figs . [ contactp],[contactt ] , and [ contactb ] , it appears that , in the presence of mixed @xmath0- and @xmath1-wave interactions , despite the apparent decoupling of different scattering channels in the universal relations , both the @xmath0- and the @xmath1-wave contacts are different from those with a single scattering channel . this clearly demonstrates the interesting interplay between @xmath0- and @xmath1-wave interactions on the many - body level . physically , as the spin - up atoms are involved simultaneously in both scattering channels , the inclusion of any one of the two scattering channels on top the other would lead to a spin - dependent shift of the chemical potentials for the many - body system . this amounts to a change of the many - body environment , although the microscopic two - body physics in each channel remains decoupled and unchanged . in terms of numerical calculation , this interplay is reflected in the solutions of the coupled number equations eqs . ( [ nh12 ] ) and ( [ nh22 ] ) . experimentally , contacts can be determined by measuring the asymptotic behavior of the momentum distribution at large momenta . in a system with coexisting @xmath0- and @xmath1-wave interactions , this high - momentum tail would also behave differently .    to see this , we define the single - body density matrix in the asymptotic regime @xmath162 with the interaction potential range @xmath5 and @xmath163 : @xmath164 similarly to the approach in sec . [ 2 ] , it is straightforward to derive    @xmath165\nonumber\\ & & ~~ + \frac{1}{v}\int d^3 \vec{r}_3 \sum_{\vec{p}_{c},j}n^{(\uparrow\downarrow)}_{\vec{p}_{c},j,0,0}e^{i\vec{p}_{c}\cdot(\vec{r}_2-\vec{r}_1)/2}\int dk~a_{j , k}\int dk'~a_{j , k'}\frac{y_{0,0}^{*}(\theta,\varphi)}{|\vec{r}_1-\vec{r}_3|}\frac{y_{0,0}(\theta',\varphi')}{|\vec{r}_3-\vec{r}_2|},\\ \label{rho1down } \rho_1^{(\downarrow)}(\vec{r}_1,\vec{r}_2)&&= \frac{1}{v}\int d^3 \vec{r}_3 \sum_{\vec{p}_{c},j}n^{(\uparrow\downarrow)}_{\vec{p}_{c},j,0,0}e^{i\vec{p}_{c}\cdot(\vec{r}_2-\vec{r}_1)/2}\int dk~a_{j , k}\int dk'~a_{j , k'}\frac{y_{0,0}^{*}(\theta,\varphi)}{|\vec{r}_1-\vec{r}_3|}\frac{y_{0,0}(\theta',\varphi')}{|\vec{r}_3-\vec{r}_2|}.\end{aligned}\ ] ]    note that we have already assumed that no @xmath1-wave interactions exist between spin-@xmath38 atoms . fourier - transforming eqs . ( [ rho1up ] ) and ( [ rho1down ] ) , we obtain the following asymptotic behavior of @xmath166 for @xmath167 : @xmath168\nonumber\\ & & ~~ + \pi\frac{\sum_{\vec{p}_{c},j , m}n^{(\uparrow)}_{\vec{p}_{c},j,1,m}(\int dk'~b_{j , m , k'}^{(\uparrow\uparrow)})^{2}p^{2}_{c}}{vk^{4}}\nonumber\\ & & ~~ + \frac{c_{s}}{vk^{4}},\\ \label{nhdown } n_{\downarrow}(\vec{k } ) & & = \frac{c_{s}}{vk^{4}}.\end{aligned}\ ] ] here , we assume that the distribution of @xmath169 is isotropic , so we have @xmath170 and @xmath171 . equations . ( [ nhup ] ) and ( [ nhdown ] ) show that the momentum distributions are highly asymmetric in spin . especially , the momentum distribution of the spin-@xmath58 component exhibits a quite nontrivial high - momentum tail , which is due to the coexistence of @xmath0- and @xmath1-wave interactions . another experimental tool for the detection of contacts is the r.f . spectroscopy , in which universal scalings in the high frequency tail exist and are governed by contacts  @xcite . coupling hamiltonian @xmath172 transfers fermions into a third spin state @xmath173 , where @xmath174 is the radio - frequency rabi frequency determined by the strength of the radio - frequency signal with frequency @xmath175 . furthermore , the transfer rate can be evaluated as @xmath176,\end{aligned}\ ] ] where @xmath177 , @xmath178 label the initial and final states , @xmath179 denotes the initial state distribution , and @xmath180 , @xmath181 denote the initial and final energies . in the region @xmath182 ( @xmath183 is the fermi energy ) , the asymptotic radio - frequency transition rates for the two spin species become @xmath184\nonumber\\ & & =   \frac{m\omega_{rf}^{2}}{2\pi\hbar}\sum_{m=-1}^{1}\left[\frac{c^{(\uparrow\uparrow)}_{p,\nu_{m}}}{(m\omega/\hbar)^{1/2 } } + \frac{3c^{(\uparrow\uparrow)}_{p , r_{m}}}{2(m\omega/\hbar)^{3/2}}\right]\nonumber\\ & & ~~ + \frac{m\omega_{rf}^{2}}{4\pi\hbar}\frac{c_{s}}{(m\omega/\hbar)^{3/2}},\\ \label{rfdown } \gamma_{rf,\downarrow}(\omega)&&=\frac{m\omega_{rf}^{2}}{4\pi\hbar}\frac{c_{s}}{(m\omega/\hbar)^{3/2}}.\end{aligned}\ ] ]     with spin-@xmath58 as a function of the reduced frequency @xmath185 at magnetic field @xmath153 g for high frequency at temperature @xmath152 with spin polarization @xmath186 . the line styles are the same as in fig . [ contactp].,width=302 ]    again , eqs . ( [ rfup ] ) and ( [ rfdown ] ) show that the radio - frequency transition rates are highly asymmetric in spin . in fig . [ rf ] , we show the reduced radio - frequency transition rate @xmath187 with spin-@xmath58 as a function of the reduced frequency @xmath185 at magnetic field @xmath153 g for the temperature @xmath152 with spin polarization @xmath186 . like the high - momentum tail of the momentum distribution , the high - frequency tail of the r.f . spectroscopy demonstrates the effects of coexistence of @xmath0- and @xmath1-wave interactions . in this work , we study the universal relations and normal - phase thermodynamic properties of a two - component fermi gas with coexisting @xmath0- and @xmath1-wave interactions . similarly to the purely @xmath0-wave or @xmath1-wave case , we start from the two - body density matrix and derive the adiabatic relations , through which a set of contacts can be defined . due to the orthogonality of the two - body wave functions in different scattering channels , contacts from different scattering channels are decoupled . we may therefore identify @xmath0-wave contact and @xmath1-wave contacts based on the adiabatic relations . we then numerically evaluate the interaction energy and contacts using second - order virial expansions . for typical experimental parameters corresponding to @xmath2k atoms , we find that when the interplay of @xmath0- and @xmath1-wave interactions is properly taken into account , all contacts behave differently from those with a single scattering channel . we characterize in detail the dependence of contacts on quantities such as spin polarization , temperature , and magnetic field . an interesting finding is that the interaction energy of the repulsive branch features abrupt changes as the @xmath1-wave resonances are crossed . our results are closely related to the ongoing efforts of realizing and probing strongly interacting quantum gases with high partial - wave interactions . as the interplay of different partial - wave interactions can have important effects on the many - body properties , our work has interesting implications for experiments exploring the @xmath3 g @xmath1-wave resonance in @xmath2k atoms , where interactions in different partial - wave scattering channels naturally coexist . we thank zhenhua yu , shizhong zhang , shao - liang zhang , su wang , and jian - song pan for helpful discussions . this work is supported by the national key r&d program ( grant no . 2016yfa0301700 ) , the national natural science foundation of china ( grants no . 11374177 , no . 11374283 , no . 11404106 , no . 11522545 , no . 11534014 , no . 11622436 ) , and the programs of the chinese academy of sciences . f.q . acknowledges support from the project funded by the china postdoctoral science foundation ( grant no . 2016m602011 ) . w.y . acknowledges support from the `` strategic priority research program ( b ) '' of the chinese academy of sciences , grant no . xdb01030200 .
we study the universal relations and normal - phase thermodynamics of a two - component ultracold fermi gas with coexisting @xmath0- and @xmath1-wave interactions . due to the orthogonality of two - body wave functions of different scattering channels , 
 the universal thermodynamic relations of the system appear to be direct summations of contributions from each partial - wave scattering channels . 
 these universal relations are dictated by a set of contacts , which can be associated with either @xmath0- or @xmath1-wave interactions . 
 interestingly , due to the interplay of @xmath0- and @xmath1-wave interactions on the many - body level , the contacts , and hence all the relevant thermodynamic quantities , behave differently from those with only @xmath0- or @xmath1-wave interactions . 
 these are manifest in our numerical calculations based on second - order virial expansions for @xmath2k atoms under typical experimental parameters . 
 a particularly interesting finding is that , due to the coexistence of @xmath0- and @xmath1-wave scatterings , the interaction energy of the repulsive branch features abrupt changes across the @xmath1-wave resonances . 
 our results can be readily checked experimentally for @xmath2k atoms near the @xmath3 g @xmath1-wave feshbach resonance , where multiple partial - wave scatterings naturally coexist .