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the unitary fermi gas  ( ufg ) is expected to be one of the most strongly correlated systems in nature as it saturates the unitarity bound for the @xmath1-wave cross section @xmath2 , where @xmath3 is the relative wave vector of scattering particles . the strong correlations are responsible for a multitude of interesting phenomena . the most surprising ones include the existence of the pseudogap regime between the superfluid state and the normal state  @xcite and the nearly ideal hydrodynamic behavior  @xcite . therefore the properties of the ufg attract enormous attention from several communities including atomic physics , nuclear physics , relativistic heavy - ion collisions , and high-@xmath4 superconductivity ( see review papers @xcite ) . experimentally the ufg has been realized with trapped fermionic atoms by means of feshbach resonances  @xcite and currently represents one of the most controllable quantum systems . however , a theoretical description poses a real challenge due to the lack of any small , dimensionless expansion parameter . fortunately , for unpolarized ufg the genuine nonperturbative quantum monte carlo ( qmc ) techniques are free from the infamous fermionic sign problem , and can be used to produce reliable predictions , providing a solid benchmark for experimental as well as other theoretical studies . in particular , the energy of the ground state agrees with experimental results within @xmath5@xcite and the same accuracy was obtained for the equation of state for the system being in the normal state  @xcite . for temperatures around the critical temperature the discrepancy between experiment and the simulations does not exceed @xmath6  @xcite . other properties well established by qmc include the evolution of the pairing gap  @xcite , the critical temperature for the superfluid - to - normal phase transition  @xcite , and the calculation of transport coefficients of the ufg , namely , the shear viscosity  @xcite and the spin - diffusion coefficient  @xcite . determination of transport coefficients is currently of great interest as these provide a verification of various hypothetical bounds having their roots directly in quantum mechanics . the best known example is a conjecture formulated by kovtun , son , and starinets ( kss ) of the existence of a lower bound @xmath7 on the ratio of the shear viscosity @xmath8 to the entropy density @xmath1 , for all fluids  @xcite . due to strong correlations which imply short mean free path , the ufg appears as one of the best candidates for being a `` perfect fluid '' ( besides the quark - gluon plasma created in heavy ion collisions ) , defined as the one with the lowest ratio @xmath9 . experimental measurements for trapped systems yield @xmath10  @xcite , while the recent qmc simulations set the minimum value to be @xmath11  @xcite . still there are contradicting scenarios of how the shear viscosity behaves in the low temperature regime . whereas some authors expect the pairing correlations to reduce the viscosity significantly and even to decrease it to zero at @xmath12  @xcite , others favor the upturn originating from the phonon contribution  @xcite . recently it was shown by chafin and schfer that classical hydrodynamic fluctuations in a non - relativistic fluid lead to the appearance of the minimum in the shear viscosity to density ratio @xmath13 as a function of the temperature  @xcite . in the vicinity of the critical temperature the bound was established to be @xmath14 . this result was confirmed by romatschke and young  @xcite . while the experimental results for trapped systems are consistent with the hydrodynamic bound , the qmc results for uniform systems violate this bound as they predict values @xmath15 in the superfluid phase  @xcite . however , to extract the shear viscosity within qmc approach one has to perform an analytic continuation of the imaginary time correlator to real frequencies , which is a highly nontrivial procedure . it is known that this step represents an ill - posed numerical problem , where statistical fluctuations which are tied up with the qmc are greatly enhanced during the continuation process . indeed , it was pointed out in the supplement of  @xcite , and also in  @xcite , that the statistical accuracy of the quantum monte carlo signal allows to produce the ratio @xmath13 consistent with the hydrodynamic bound .    here we present qmc results for a temperature evolution of the shear viscosity of higher accuracy than in our previous studies  @xcite and discuss the errors associated with the analytic continuation . under the assumption of the smoothness of frequency dependent shear viscosity @xmath0 , we show that the new results violate the classical hydrodynamic bound @xcite for @xmath13 in a low temperature regime , while the ratio @xmath9 stays above the kss bound . moreover the hydrodynamic bound is violated for the temperatures at which the cooper pairs are formed in the system . we show that the agreement with hydrodynamic bound can be achieved only if there exists a low frequency sharp peak in the @xmath0 which is overlooked by analytic continuation procedure .    in order to determine the shear viscosity of the ufg we employ the qmc technique on the lattice , which provides numerical results with controllable accuracy , up to quantifiable systematic uncertainties ( for details see ref . these simulations are very similar to those of ref . @xcite and we therefore only briefly describe the main stages of the computational process , focusing mainly on improvements . henceforth we define units : @xmath16 . we performed simulations using three lattice sizes @xmath17 with corresponding average densities @xmath18 and @xmath19 , respectively . for these lattices the systematic errors were estimated to be less than @xmath20 and are the most severe in the superfluid phase ( for detailed discussion in the context of the transport coefficients see the supplemental material of refs . these errors are related mainly to corrections coming from the nonzero effective range @xmath21 and exclude the universal high momenta tail in the occupation probability by finite momentum cut - off @xmath22 . we obtain rather moderate values of @xmath23 and @xmath24 , respectively for lattices @xmath25 and @xmath26 [ @xmath27 stands for fermi momentum ] . the statistical uncertainties are typically of the order of @xmath5 as an ensemble of @xmath28 uncorrelated samples were collected at each temperature . the fact that we have used three different lattices allows us to check the stability of the results as we approach the thermodynamic @xmath29 and continuum @xmath30 limits , where @xmath31 ( the lattice constant is set to unity ) . these new results were obtained with a new ( parallelized ) code and for densities significantly lower than those reported in ref . @xcite , and thus are closer to the desired limit @xmath32 . the frequency dependent shear viscosity @xmath0 is obtained from the imaginary - time ( euclidean ) stress tensor - stress tensor correlator @xmath33 evaluated at zero momentum @xmath34 , by inversion of the relation @xmath35 } { \sinh\left ( \omega\beta/2\right ) }    d\omega ,    \label{eqn : inversionproblem}\ ] ] where @xmath36 is the inverse of the temperature . taking the limit of zero frequency we obtain the static shear viscosity @xmath37 . the average in eq . ( [ eqn : gpi ] ) is performed over the grand canonical ensemble , @xmath38 , @xmath39 is the hamiltonian of the system , @xmath40 is the chemical potential , and @xmath41 is the particle number operator . for evaluation of the shear viscosity it is sufficient to use the kinetic part of the stress tensor only ( see , e.g. , ref . @xcite ) : @xmath42 the correlator @xmath43 was evaluated for at least 61 points in imaginary time @xmath44 , uniformly distributed in the interval @xmath45 $ ] . since the correlator is symmetric with respect to @xmath46 , we restrict the inversion procedure to the interval @xmath47 $ ] , and apply the symmetrization : @xmath48/2 $ ] in order to decrease statistical fluctuations .    the inversion of eq . ( [ eqn : inversionproblem ] ) represents numerically an ill - posed problem . however , by using known theoretical constraints the solution of this problem becomes easier to determine . besides the non - negativity of the shear viscosity @xmath49 , the sum rule @xmath50 = \dfrac{\varepsilon}{3 } ,    \label{eqn : sumrule}\ ] ] and the asymptotic tail behavior @xmath51 have been used as _ a  priori _ information . the tan contact density @xmath52 and the energy density @xmath53 are obtained consistently within the same simulation , and are in agreement with results reported in previous studies  @xcite . to perform the inversion we applied the methodology which combines two complementary methods : singular value decomposition ( svd ) and maximum entropy method ( mem ) , both described in ref . the inversion procedure consists of two steps : ( i ) the svd method provides us with the projection of the solution onto the subspace , where the inverse problem is well - posed @xmath54 , ( ii ) the inversion is performed using the self - consistent mem and we impose the svd solution as additional external constraint , i.e. , @xmath55=\tilde{\eta}$ ] , where @xmath56 stands for the projection operator onto svd subspace . in order to estimate the stability of the method with respect to the algorithm parameters , the  bootstrap " strategy was applied . for more details see the supplemental material of  @xcite . an important ingredient of the self - consistent mem is an appropriately chosen class of _ a  priori _ models for the solution . we assume that the frequency dependent shear viscosity @xmath0 has lorentzian - like structure at low frequencies , smoothly evolving into the asymptotic tail behavior : @xmath57\ , \mathcal{l}(\omega,\{\mu,\gamma , c\}),\end{aligned}\ ] ] where @xmath58 and @xmath59 the parameters @xmath60 describe admissible degrees of freedom of the model and are adjusted automatically in the self - consistent manner . this choice of model space is reasonable as it is compatible with the low frequency behavior of the @xmath0 predicted by hydrodynamic fluctuations formalism  @xcite and with results obtained within luttinger - ward theory  @xcite . finally , we assume that there is no sharp structure in the shear viscosity @xmath0 in the low frequency limit . this assumption is of crucial importance , as it is well known , that sharp structures can be overlooked by numerical analytic continuation @xcite . clearly , note that adding @xmath61 to the shear viscosity @xmath0 corresponds to increasing the imaginary time correlator by a constant value @xmath62 . lattice for two different temperatures : ( upper panel ) @xmath63 ( close to the critical temperature ) and ( lower panel ) @xmath64 ( normal phase ) . the gray bands show uncertainty of the reconstruction . in the insets , the corresponding qmc correlators ( blue points ) are presented . the lines show the quality of the reproduction of the qmc data . [ fig : eta_dynamic ] ]    in fig . [ fig : eta_dynamic ] , the dimensionless frequency dependent shear viscosity @xmath65 is shown for two selected temperatures obtained for a @xmath66 lattice : close to the critical temperature of superfluid - to - normal phase transition @xmath67 and in the normal phase @xmath64 , where @xmath68 is the fermi energy of the non - interacting gas . in the normal phase @xmath69 , one may observe that @xmath0 possesses a lorentzian - like structure at low frequencies with a maximum in the vicinity of zero frequency . for temperatures below @xmath70 the suppression of @xmath0 develops for the zero frequency . we associate this depletion with the onset of pairing correlations , which has been identified to exist in ufg above the critical temperature up to temperatures @xmath71 , commonly referred to as pseudogap regime  @xcite .     as a function of temperature for an @xmath72 lattice [ solid ( red ) circles ] , a @xmath66 lattice [ ( blue ) squares ] , and a @xmath73 lattice [ ( green ) diamonds ] . vertical black dotted lines indicate the critical temperature of superfluid - to - normal phase transition for @xmath67 and the onset of cooper pair formation , @xmath74 , respectively . the black line indicates smooth approximation of the qmc results and the ( yellow ) band reflects uncertainty of the viscosity computation . the ( purple ) asterisk shows the shear viscosity from the qmc calculations of ref . the phonon contribution to the viscosity , evaluated according to ref . @xcite is shown as a dot - dashed ( brown ) line and it is accounted for in the ( yellow ) band . the hydrodynamic bound of ref . @xcite is plotted by the black dashed line . the inset shows the ratio of the shear viscosity to entropy density ( @xmath9 ) as a function of @xmath75 , obtained from the smooth approximation of the qmc results . the black dashed line indicates kss bound : @xmath76 . [ fig : viscosity ] ]    in fig . [ fig : viscosity ] , the dimensionless static shear viscosity @xmath13 is shown as a function of dimensionless temperature @xmath77 . results obtained for different lattices exhibit reasonable agreement , even though they are affected by significant uncertainty of the order of @xmath78 . for the @xmath73 lattice we provide results only for temperatures greater than @xmath79 as we were not able to produce the correlator of required accuracy to perform reliable inversion for lower temperatures . the temperature dependence of the shear viscosity can be resolved , which is clearly visible when a smooth approximation of qmc results is performed , while the amount of scattering of the qmc points can be used to estimate final uncertainty of the viscosity computation ( yellow band in  fig . [ fig : viscosity ] ) . moreover , the new results are consistent with previous estimate of the viscosity coefficient of ref . we have checked the stability of our results with respect to the variation of the density , e.g. , for the lattice size @xmath66 and @xmath80 and obtained @xmath81 and @xmath82 for densities @xmath83 and @xmath84 , respectively . in the low temperature regime no upturn originating from phonon contribution is observed , however , this does not imply that there is no upturn at all . precisely , the boxes used in the simulations are not capable of properly capturing the phonon - related physics . it becomes clear if we note that the phonon occupation probability @xmath85 $ ] ( where @xmath86 is the sound velocity ) has a peak at momenta of the order of @xmath87 , which for @xmath88 is smaller than the smallest lattice momentum @xmath89 . thus infrared momentum cutoff @xmath89 removes almost entirely the pure phonon contribution . to take into account this source of the systematic error we modified the estimated error of the shear viscosity computation in such a way as to encapsulate the pure phonon contribution of ref . phonons appear to contribute significantly only well below the critical temperature @xmath4 . above the critical temperature , when the system is in a normal state and the superfluid order parameter is vanishing the calculation approach adopted in ref . @xcite is not valid anymore . this approach is based on an effective theory for the phonon field alone , valid at very low temperatures only , where cooper pair breaking and pseudogap physics can be ignored . comparing our _ ab initio _ results with the hydrodynamic bound we find that as long as the system is in the normal phase ( for @xmath90 ) the static shear viscosity is located above the bound . the results obtained for all three lattices indicate that the hydrodynamic bound is violated when the system enters the pseudogap regime , which occurs above the critical temperature , at @xmath91 . interestingly , the corresponding ratio @xmath9 stays above the kss bound @xmath92 for all considered temperatures , with the minimal value @xmath93 , located in the vicinity of the critical temperature . for the temperature evolution of the entropy we have used the mit results of ref . @xcite , as our results reproduce experimental data with reasonable accuracy ( see  @xcite ) . the presented methodology , which produces the most probable results , together with the smoothness assumption of @xmath0 clearly allows for the hydrodynamic bound to be violated in the low temperature regime for @xmath94 . the hydrodynamics bound was obtained by adding the contribution due to thermal fluctuations to an undefined classical value of the viscosity @xcite and with an amplitude depending on a cutoff parameter , which can be estimated only by order of magnitude , and thus a factor of order unity can not be refuted . these estimates of a possible lower bound for viscosity are purely classical in character and they show that the role of classical thermal fluctuations could become important if the bare viscosity , evaluated in the absence of such thermal fluctuations , is small . the bare viscosity can be evaluated in a classical kinetic approach , but to what extent such an estimate is valid in the case of a unitary fermi gas remains an open question . even though a unitary fermi gas is dilute in the sense that @xmath95 , two - body collisions do not dominate because @xmath96 , where @xmath97 is the interaction cross section between fermions at the fermi surface . except the role of very low momentum phonons at very low temperatures which can not be properly included in qmc within the presently tractable lattice sizes , all other effects ( pairing , pseudogap , fermion collisions ) are included . as seen from fig . [ fig : viscosity ] phonons show a significant contribution only for temperatures below the critical temperature @xmath4 . it implies that even if the phonons were included , the viscosity would have remained below the classical hydrodynamic bound @xcite up to temperatures smaller than @xmath98 . as another possibility , one can assume that hydrodynamic bound originates from long - wavelength fluctuations , which are not included properly into our simulations due to finite size of the box . however , in this case we would expect to see a systematic trend in the results as we increase the box size , while we do not detect it .    even though we have no reason to expect that any alleged unaccounted for thermal fluctuations of the kind advocated in refs . @xcite might appear in the pseudogap regime below the temperature @xmath99 , we included such a contribution into the spectral function to investigate whether our qmc data are consistent with this classical hydrodynamical bound . three tests were performed using data obtained for a @xmath66 lattice at the critical temperature . hydrodynamic fluctuations  @xcite suggest nonanalytic contributions for small frequencies of the form @xmath100 where the momentum diffusion constant @xmath101 is positive . thus , @xmath0 is predicted to be a monotonically decreasing function for small frequencies . in the first test , the mem solution is forced to be consistent with this assumption by replacing the lorentzian - like structure at low frequencies  ( [ eqn : lorentzian ] ) by the expression @xmath102 where @xmath103 . this modification implies that the space for default models contains only solutions which are monotonically decreasing and explicitly consistent with predictions taking into account hydrodynamic fluctuations . in addition the constraint provided by svd is removed . the resulting frequency dependent viscosity is shown with a ( red ) solid line in fig . [ fig : tests ] . in the second test , the modified model is fit to the imaginary time correlator , in a similar way as has been done in ref . this method typically violates the sum rule  ( [ eqn : sumrule ] ) and the resulting @xmath0 is shown with a ( blue ) dot - dashed line in  fig .  [ fig : tests ] . it is noted that our fit reveals a significantly different result than reported in ref . @xcite , which may be related to the fact that we use different models for the fitting procedure . finally , the qmc correlator was fit to the model proposed by enss _ _ in the paper  @xcite @xmath104 where the only free parameter is the viscous transport scattering time @xmath105 , as the total drude weight @xmath106 is fixed by the sum rule  ( [ eqn : sumrule ] ) [ see ( red ) dotted line in  fig . [ fig : tests ] ] . all three approaches are able to reproduce the imaginary time correlator within its error bars . these simple tests result in a static shear viscosity below the hydrodynamic bound . for mem with the model ( [ eqn : lorentzianm ] ) , we find that @xmath13 is typically located below the bound . however , we also find instances of the mem algorithm that produce results above the bound as reflected in the error band depicted in  fig . [ fig : tests ] . similar results were determined for other temperatures . obtained by three different methodologies of the reconstruction as described in the text . the ( purple ) asterisk indicates the hydrodynamic bound obtained in refs . [ fig : tests ] ]    based on these tests we conclude that the qmc results are compatible with the hydrodynamic bound only if we assume the existence of a very narrow peak located around @xmath107 . the classical hydrodynamic fluctuation formalism predicts this type of peak , which has its origin in nonanalytic contributions to the shear viscosity . in ref .  @xcite it was estimated that the width of the nonanalytic structure can be of the order of @xmath108 and at the critical temperature it gives @xmath109 , which is in agreement with our first test . however , we have to point out several important aspects of our results . ( i ) we do not observe any abrupt change in the frequency dependent shear viscosity profile as we decrease the temperature , which is a typical signal of losing the resolution . instead , the profile we construct evolves smoothly between the two cases presented in  fig . [ fig : eta_dynamic ] . ( ii ) our qmc simulations admit the existence of the pseudogap regime for the ufg  @xcite . other theoretical approaches that admit the presence of the pseudogap predict the shear viscosity to be significantly suppressed , as in systems where a fermionic gap opens at @xmath99 , and exhibit a clear violation of the hydrodynamic bound  @xcite . similar behavior was predicted for the spin - diffusion coefficient  @xcite . our _ ab initio _ results are qualitatively consistent with these observations and support scenarios provided by these theories . ( iii ) we find it difficult to rationalize the existence of a nonanalytical classical hydrodynamical correction to the spectral function deep in the quantum regime . in summary , we have presented the temperature evolution of a shear viscosity obtained within fully _ ab initio _ simulations . assuming the smoothness of the @xmath0 the results violate the hydrodynamic bound for @xmath13 in a low temperature regime , while the ratio @xmath9 stays above the kss bound for all studied temperatures . violation of the hydrodynamic bound occurs simultaneously with the occurrence of the cooper pairs in the system . the accuracy of the qmc signal allows a result with the static viscosity being above the hydrodynamic bound only if we assume an existence of a sharp structure in the @xmath0 located in the vicinity of zero frequency . we thank p. romatschke for making the results of ref . @xcite available to us . we acknowledge support under u.s . doe grants no . de - fg02 - 97er41014 and no . de - fc02 - 07er41457 , and contract no . n n202 128439 of the polish ministry of science . one of the authors ( g.w . ) acknowledges the polish ministry of science for support within the program `` mobility plus - i edition '' under contract no . calculations reported here have been performed at the interdisciplinary centre for mathematical and computational modelling ( icm ) at warsaw university and on the university of washington hyak cluster funded by the nsf mri grant no . this research also used resources of the national center for computational sciences at oak ridge national laboratory , which is supported by the office of science of the department of energy under contract no . de - ac05 - 00or22725 . stewart , j.p . gaebler , and d.s . jin , nature * 454 * , 744 ( 2010 ) . j. p. gaebler , j. t. stewart , t. e. drake , d. s. jin , a. perali , p. pieri , and g. c. strinati , nature phys . * 6 * , 569 ( 2010 ) . a. perali , _ et al . 106 * , 060402 ( 2011 ) . a. turlapov , j. kinast , b. clancy , l. luo , j. joseph , j.e . thomas , j. low temp . * 150 * , 567 ( 2008 ) . c. cao , e. elliott , j. joseph , h. wu , j. petricka , t. schaefer , j. e. thomas , science * 331 * , 58 ( 2011 ) . c. cao , e. elliott , h. wu and j.e . thomas , new j. phys . * 13 * , 075007 ( 2011 ) . s giorgini , l.p . pitaevskii , s. stringari , rev . mod . phys . * 80 * , 1215 ( 2008 ) . i. bloch , j. dalibard , w. zwerger , rev . phys . * 80 * , 885 ( 2008 ) . t. schfer and d. teaney , rep . . phys . * 72 * , 126001 ( 2009 ) . _ the bcs - bec crossover and the unitary fermi gas _ lecture notes in physics , edited by w. zwerger ( springer - verlag , berlin , 2012 ) , vol . 836 . _ ultracold fermi gases _ , proceedings of the international school of physics `` enrico fermi , '' courseclxiv , varenna , june 20 - 30,2006 , edited by m. inguscio , w. ketterle , and c. salomon ( ios press , amsterdam , 2008 ) . j. carlson _ et al . _ , a * 84 * , 061602(r ) ( 2011 ) . k. van houcke , f. werner , e. kozik , n. prokofev , b. svistunov , m. j. h. ku , a. t. sommer , l. w. cheuk , a. schirotzek , and m. w. zwierlein , nature phys . * 8 * , 366 ( 2012 ) . drut , t.a . lhde , g. wlazowski , p. magierski , phys . a * 85 * , 051601(r ) ( 2012 ) . j. carlson and s. reddy , phys . * 95 * , 060401 ( 2005 ) . a. gezerlis and j. carlson , phys . c * 77 * , 032801(r ) ( 2008 ) . p. magierski , g. wlazowski , a. bulgac , and j.e . drut , phys . rev . lett . * 103 * , 210403 ( 2009 ) . p. magierski , g. wlazowski , and a. bulgac , phys . lett . * 107 * , 145304 ( 2011 ) . e. burovski , n. prokofev , b. svistunov , and m. troyer , phys . rev * 96 * , 160402 ( 2006 ) . e. burovski , e. kozik , n. prokofev , b. svistunov , and m. troyer , phys . * 101 * , 090402 ( 2008 ) . a. bulgac , j.e . drut , and p. magierski , phys . a * 78 * 023625 ( 2008 ) . g. wlazowski , p. magierski , and j.e . drut , phys . lett . * 109 * , 020406 ( 2012 ) . g. wlazowski , p. magierski , j.e . drut , a. bulgac , and k.j . roche , phys . * 110 * , 090401 ( 2013 ) . kovtun , d.t . son , and a.o . starinets , phys . lett . * 94 * , 111601 ( 2005 ) . bruun and h. smith , phys . a * 75 * , 043612 ( 2007 ) . h. guo , d. wulin , c .- c . chien , and k. levin , phys . * 107 * , 020403 ( 2011 ) ; new j. phys . * 13 * , 075011 ( 2011 ) . g. rupak and t. schfer , phys . a * 76 * , 053607 ( 2007 ) . c. chafin , t. schfer , phys . a * 87 * , 023629 ( 2013 ) . p. romatschke , r.e . young , phys . a * 87 * , 053606 ( 2013 ) . t. enss , r. haussmann , w. zwerger , ann . phys . * 326 * , 770 ( 2011 ) . p. magierski and g. wlazowski , comput . phys . commun . * 183 * , 2264 ( 2012 ) . ku , a.t . sommer , l.w . cheuk , m.w . zwierlein , science * 335 * , 563 ( 2012 ) . d. wulin , h. guo , c .- c . chien , and k. levin , phys . a * 83 * , 061601(r ) ( 2011 ) .
we present an _ ab initio _ determination of the shear viscosity for the unitary fermi gas based on finite temperature quantum monte carlo ( qmc ) calculations and the kubo linear - response formalism . 
 the results are confronted with the bound for the shear viscosity originating from hydrodynamic fluctuations . 
 assuming smoothness of the frequency dependent shear viscosity @xmath0 , we show that the bound is violated in the low temperature regime and the violation occurs simultaneously with the onset of the cooper paring in the system . in order to preserve the hydrodynamic bound in qmc @xmath0 
 has to possess a sharp structure located in the vicinity of zero frequency which is not resolved by an analytic continuation procedure .