Thermodynamics of the polarized unitary Fermi gas from complex Langevin. Joaquín E. Drut University of North Carolina at Chapel Hill

Size: px

Start display at page:

Download "Thermodynamics of the polarized unitary Fermi gas from complex Langevin. Joaquín E. Drut University of North Carolina at Chapel Hill"

Quentin Harrington
5 years ago
Views:

1 Thermodynamics of the polarized unitary Fermi gas from complex Langevin Joaquín E. Drut University of North Carolina at Chapel Hill INT, July 2018

2 Acknowledgements Organizers Group at UNC-CH (esp. Andrew Loheac) UW & INT J. Braun, L. Rammelmueller + Disclaimer

3 A sequence of papers Thermal equation of state of polarized fermions in one dimension via complex chemical potentials A. C. Loheac, J. Braun, J. E. Drut, D. Roscher Phys. Rev. A 92, (2015) Third-order perturbative lattice and complex Langevin analyses of the finite-temperature equation of state of non-relativistic fermions in one dimension A. C. Loheac, J. E. Drut Phys. Rev. D 95, (2017) Surmounting the sign problem in non-relativistic calculations: a case study with mass-imbalanced fermions L. Rammelmueller, W. J. Porter, J. E. Drut, J. Braun Phys. Rev. D 96, (2017) Polarized fermions in one dimension: density and polarization from complex Langevin calculations, perturbation theory, and the virial expansion A. C. Loheac, J. Braun, J. E. Drut arxiv: Finite-temperature equation of state of polarized fermions at unitarity L. Rammelmueller, A. C. Loheac, J. E. Drut, J. Braun arxiv: Come to Andrew s talk! Tomorrow at 3:15pm

4 Outline Definition, motivation, and some history of the UFG Formalism and technique: Path integrals and CL Results: Equations of state Summary & Conclusions

5 <latexit sha1_base64="tmskt0sdqqdxl86s6zlg0gaj5qq=">aaacjxicjvdns8mwhe3n15xfvy9egkpw4uimob4gqxe8ejhg3wctjc3sltrns5iko/sv8ek/4sxdfmgt/4rzb36h4ipay3u/xy95fskovjb1ahrmzufmf4qlpaxlldu1c33jwsapwmtgmytf20esmmqjrahipj0igikfkzyfno781i0rksb8sg0s4kaox2lamvja8sy6ewies2qe9xjyhx+3i5v93jfp5gvh/qnwhjjiynqwhhvwlpfmcrvijqh/jmuwrdmzh043xmleumimsdmpwolymyquxyzkjsevjee4rd3s0zsjieg3g38zhzta6cigfvpwbcfq10sgiikhka8ni6t68qc3en/zoqkkjtym8irvhopjoiblumvw1bnsukgwygnnebzuvxxiptkdkn1s6x8l2lxkccw6pcg3tqztfmew2aa7oaooqqocgyawaqz34aemwznxbzwaz8blzlrgtdob4bumt3e/iqwg</latexit> <latexit sha1_base64="61vffyq+seupjccljctrvgirr6o=">aaaclhicjvdlsgmxfm3uv62vuzdugkvqxditgroqirpwqwc10kklk95pqzmpkjtigeah3pgririo4tbvmk1fvbq8edg55x5ucrxyco2om7bye5nt0zp52clc/mlikr28cqmjrhgo8uhgqu4xdvkeueobeuqxahz4eq683vhqv7obpuuuxma/hmbaoqhwbwdopjz9eloxr0jdnkhkm/ewpaq7rq8yt8tzyjk6tt9vluy1xirbtmh3s951zwi2i9qtu1guospqv0mrjhhwsh9njicbhmgl07prdmjspkyh4bkygptoibnvsq40da1zalqzjn6b0q2jtkkfkxncpcp1ayjlgdb9wdotacou/uknxd+8rol+fjmvyzwghpxjkz9iihedvkfbqgfh2teecsxmwynvmlmomoil/yuhvikdljzz3wl1anxgnqyrdbjjymspvmkposm1wskdesad8mzdw0/wi/x6mzqzxplv8g3w2zsqsqct</latexit> Definition: Hamiltonian & scales Two species of fermions with a contact two-body force 2 Z Ĥ = d 3 x 4 X s=",# ˆ s (x) ~ 2 r 2 ˆs(x) 2m 3 gˆn " (x)ˆn # (x) 5 Coupling is dimensionful [g] =L Renormalize by solving the two-body problem and relating bare coupling to scattering length 1 g = 1 X L 3 k 2 k 1 E p cot (p) = 1 a r e k

6 <latexit sha1_base64="5j+o7vzwwniredyvegmm/5+ctq4=">aaab/nicjvdlssnafl2pr1pfucgnm8ei1e1nwkfdcevbxfywwmjtmjlo2qgtszizccv24a+4cahi1u9w59+yphyqch64cdjnxu7h+dfnslvwh5gbmz2bx8gvfpawv1bxzpwnaxulklchrdysdr8rypmgjmaa00yskq59tm/8/tniv7mlurfixolbtn0qdwulgme6kzxzq++doxnuqrzi1q4gsddo7f3q0doldtkaa/1nijbf3tpfw52ijcevmncsvno2yu2mwgpgob0wwomimsz93kxnjaocuuwm4/xdtjsphrremhuh0vj9ephiuklb6gebidy99dmbib95zuqhr27krjxoksjkuzbwpcm0kgn1mkre80fgmjesy4pid0tmdfzz4x8lojxycdm6pcjwtqdt5gebdqaenhxcds6gdg4quimheijn4954nf6m18lqzpjebmi3gg+f66otqg==</latexit> Two-body problem Bound state appears at a critical attractive coupling gc Unitarity g EGS Scattering length and density determine the physical dimensionless coupling for the many-body problem 1/(k F a) k F =(3 2 n) 1/3

7 <latexit sha1_base64="3wwr1iil60+nrj6zqpeosciswqm=">aaacfnicjvbns8naen3ur1q/oh69lbbbiyerqb0vbffywdpcu8nmu2mxbjzhd6ku0h/hxb/ixyokv/hmv3ht9qci4iofn+/nmdsvtaxx4lofvmlmdm5+obxywvpewv2z1zeudjipyho0eylqhuqzwsvraafbwqlija4fa4ad08jv3jclesivyziytkx6kkecejbsydsu9gwlgciv3givfkxag+dsot/zruocyb8s7hmzwtcwq57jjoh/jlu0rt2w3/1uqroysaccan323bq6ovhaqwcjip9plhi6id3wnlssmolopr5rhhem0svrosytgmfq14mcxfop49b0xgt6+qdxil957qyio07ozzobk3sykmoenncwieeuv4ycgbpcqolmr5j2isiutjsv/4xq2heohffiofo7mazrrltog+0idx2igjphddrafn2hb/senq1769f6sv4nrsvrorojvsf6+ws/dz4c</latexit> <latexit sha1_base64="d1uist+rjpchzz/82dtcymhmepc=">aaab+3icjvdlssnafj3uv62vajdubovgqqqiqauhkfwxfywvmham00k7ddijmzfseuqvuhgh4tyfceffoh0svbq8cofwzr3cwwltwtu4zodvmjtfwfwqlpdwvtfwn+znrrudzioylyyiubch0uxwyvzginhtqhijq8fayf9s7lfumni8kdcwtjkfk67keacejbty5qy+wd6a40bgartafn4xcuxkrepmgp8mftrdm7dfvu5cs5hjoijo3a45kfg5uccpykosl2mwetonxdy2vjkyat+fhb/hxan0cjqomxlwrp16kzny62ecms2yqe//9mbib147g+jiz7lmm2csth9fmccq4hetummvoycghhcqummkay8oqsh0vfpfce5+9bjqxb1u6qeznopog+2gpvrdh6iolletuyiiixpat+jzurcerrfrdbpasgy3zfqn1tsnmowtqa==</latexit> <latexit sha1_base64="mg+lwlwnarglt00wmp16x6e/oos=">aaab6nicjvbnswmxej2tx7v+vt16crbbu9mkon6kgnis4fqhxuo2nw1dk+ysziwy9c948adi1v/kzx9jtu1brceha4/3zpizf6wcg+v7h15pyxfpeaw8wllb39jcqm7v3jok0wwdlohe30xuooaka8utwltui5wrwhy0uij89j1qwxn1y8cphpiofi85o7aqrr1l0qvwgnv/cvi3qcecrv71vdtpwczrwsaomz2gn9owp9pyjnbs6wygu8pgdiadrxwvamj8euuehdilt+jeu1kwtnwvezmvxoxl5doltupz0yve37xozuptmocqzswqnlsuz4lyhbspkz7xykwyo0kz5u5wwozuu2zdpjx/hrac1c/q/vvxrxk+t6mme7aph9cae2jcfbqgaazdeianepak9+i9ek+z1pi3n9mfb/depghf2o2o</latexit> <latexit sha1_base64="mg+lwlwnarglt00wmp16x6e/oos=">aaab6nicjvbnswmxej2tx7v+vt16crbbu9mkon6kgnis4fqhxuo2nw1dk+ysziwy9c948adi1v/kzx9jtu1brceha4/3zpizf6wcg+v7h15pyxfpeaw8wllb39jcqm7v3jok0wwdlohe30xuooaka8utwltui5wrwhy0uij89j1qwxn1y8cphpiofi85o7aqrr1l0qvwgnv/cvi3qcecrv71vdtpwczrwsaomz2gn9owp9pyjnbs6wygu8pgdiadrxwvamj8euuehdilt+jeu1kwtnwvezmvxoxl5doltupz0yve37xozuptmocqzswqnlsuz4lyhbspkz7xykwyo0kz5u5wwozuu2zdpjx/hrac1c/q/vvxrxk+t6mme7aph9cae2jcfbqgaazdeianepak9+i9ek+z1pi3n9mfb/depghf2o2o</latexit> <latexit sha1_base64="mg+lwlwnarglt00wmp16x6e/oos=">aaab6nicjvbnswmxej2tx7v+vt16crbbu9mkon6kgnis4fqhxuo2nw1dk+ysziwy9c948adi1v/kzx9jtu1brceha4/3zpizf6wcg+v7h15pyxfpeaw8wllb39jcqm7v3jok0wwdlohe30xuooaka8utwltui5wrwhy0uij89j1qwxn1y8cphpiofi85o7aqrr1l0qvwgnv/cvi3qcecrv71vdtpwczrwsaomz2gn9owp9pyjnbs6wygu8pgdiadrxwvamj8euuehdilt+jeu1kwtnwvezmvxoxl5doltupz0yve37xozuptmocqzswqnlsuz4lyhbspkz7xykwyo0kz5u5wwozuu2zdpjx/hrac1c/q/vvxrxk+t6mme7aph9cae2jcfbqgaazdeianepak9+i9ek+z1pi3n9mfb/depghf2o2o</latexit> <latexit sha1_base64="whzveqddgxlcwqtbwor25ptw0bi=">aaacdhicjvdjsgnbfoxxjxglevtsgavpyekcehcc4nisbuefzbb6om+sjj0l3w/emmwpepfxvhhq8eohepnv7mqcvbqsefbu1ap7lz9iodg2362h4zhrsfhcrhfyanpmtjq3f6hjvhfwecxjdeuzdvje4kbacvejahb6ei79zn7pv7wgpuucnwm3as9krugegjm0uqo0fnbwew4wozzk6s51a8v4tp5nm/mjc4kw0oqog6vytwl3qf8mztlaaap05jzjnoyqizdm63rvttdlmelbjerfn9wqmn5hlagbgreqtjf1r8npilganiivmqhpx/26kbfq627om2tisk1/ej3xn6+eyrdtzsjkuosifz4upjjithvv0kzqwff2dwfccfnxytvmfikmwol/sndwkjsv+2yjxnsbtfegi2sjrjiq2si1ckxoium4usx35je8wxfwg/vsvxxgh6zbzgl5buv1a0yem0k=</latexit> <latexit sha1_base64="e2ha2qh9u/hllfstonrszsrtnxy=">aaacbhicjzblswmxfiuz9vxra9sllojfcfwmrvaxqleolis4ttazsya904zmmkosecowgzf+ftcuvnz6i9z5b0wfcxufdwqo59xlki9iofpact6swtz8wujscbm0srq2vmfvbl2rojuuxbrzwlydooazaa5mmkm7kucigemrgj6p+9ytsmvicavhcfgr6qswmkq0ibr2rgejytzybj7fxigjzybdxk0tz2p51y5xk85e+g9trjm1u/a714tpgohqlbololun0x5gpgauq17yugujoupsh46xgksg/gzyixzvm6shw1iaizsepf83mhipnyocmxkrpva/u3h4w9djdxjsz0wkqqzbpxefkcc6xmmkumckum1hxhaqmxkrpgnisggdrvq/cg6tcljxlg/l9bmzjslaqxvoafxreaqjc9relqlodj2gj/rs3vup1ov1oh0twlodbfrn1tsn+g+xyw==</latexit> Scale invariance & universality Lack of scales other than the density 0 r 0 k 1 F a!1 Dimensionful observables must get their units from powers of k F E.g. the ground-state energy E = E FG Bertsch parameter E FG = 3 5 N F F = k2 F 2 Beyond: Tan s contact, epsilon expansion, NR conformal invariance, sum rules, viscosities, Efimov effect,

8 <latexit sha1_base64="v7txgurvvocus82gw6yg/khkry4=">aaab7xicjvbns8naej3urxq/qh69lbbbu0lfug9flx4rgfnoqtlsn+3szsbstoqs+io8efdx6v/x5r9x+3fqufdbwoo9gwbmxbkubj3vw6ksla+srlxx3y3nre2d2u7enckkzbjpmpnptkwnl0jxhwvk3sk1p2ksercprqz+cm+1ezm6xxhoo5qolegeo2ilqpe8mhrpr1zvnrwzyn+kdgu0e7x3sj+xiuukmatgdjtejlfjnqom+cqnc8nzykz0wluwkppye5wzcyfkycp9kmtalkiyu79oldq1zpzgtjolodq/van4m9ctmdmpsqhyarli80vjiqlmzpo76qvngcqxjzrpyw8lbeg1zwgtcv8xgn/sugh4n6f11uuijsocwcecqxpooaxx0ayfgizgaz7g2cmdr+ffez23vpzfzd58g/p2ccrtjro=</latexit> Motivation: Nuclear physics Dilute neutron matter 6Li Neutrons r 0 a 20 Bohr 3 fm 10 4 Bohr -19 fm Bertsch Many-body X Challenge Is the unitary limit stable? If so, what is the ground-state energy?

9 Motivation: Ultracold atoms Bose-Einstein condensates (1995) Fermionic condensates (2004)

10 Motivation: Ultracold atoms Astonishing degree of control - Temperature (Superfluid transitions) - Polarization (LOFF-type phases, polarons) - Coupling (BEC-BCS crossover) - Shape of external trapping potential - Mass imbalance (different isotopes) - Dimension (highly anisotropic traps & lattices) - Bosons, fermions, mixtures: Li, K, Sr, Yb, Dy, Er, and astonishing degree of measurement/detection - Thermodynamics - Phase transitions - Collective modes - Spin response - Hydrodynamic response - Entanglement - Time-dependent dynamics -

11 A bit of history First thermodynamic measurements L. Luo et al PRL 98, (2007)

12 A bit of history Best experiments M. Ku et al Science 335, 563 (2012) Density EoS Pressure EoS

13 A bit of history Best experiments M. Ku et al Science 335, 563 (2012) Pressure-Compressibility plot

14 A bit of history On the theory side: ground-state energy The Bertsch parameter as a function of time Endres et al PRA 87, (2013)

15 A bit of history More on the theory side: finite temperature EoS Bulgac, Drut, Magierski PRL 96, (2006)

16 A bit of history and an update Drut, Lähde, Wlazłowski, Magierski PRA 85, (R) (2012)

17 The 3D BCS-BEC crossover T Normal T c Normal BCS Superfluid?? 1 k F a 1 k F a Unitarity BEC Superfluid T c 1/k F a

18 The 3D BCS-BEC crossover T Normal T c Normal BCS Superfluid?? 1 k F a Unitarity BEC Superfluid T c 1/k F a

19 The 3D BCS-BEC crossover T Normal T c Normal BCS Superfluid?? 1 k F a Unitarity BEC Superfluid T c 1/k F a

20 The 3D BCS-BEC crossover T Normal T c Normal BCS Superfluid?? 1 k F a Unitarity BEC Superfluid T c 1/k F a

21 <latexit sha1_base64="umgqeqpn2kzum1t1ojthvs3q5qg=">aaacdxicjvblswmxgmzwv62vvy9egqvqd9ztedsduptisyjrhxzzsmm2dc0msx6wuvolvphxvhhq8erdm//gdnudioidgcnmn3zjrcmjsnveh5obm19yxmovf1zw19y33m2taywmxmthggl5eyffgoxe11qzcpnkgpkikwbup5/4zvsifrx8sg9teisoy2lmmdjwct1sd57ccjsxydukseoxgpswu3beggfc3ketdivvipcb/k2kyizg6l7bpdyj4rozpfsr6qu6gcgpkwzkxggbrvke+6hlwpzylbavjllvjghjkh0yc2kp1zbtvyzgkffqmer2mkg6p356e/e3r2v0fbymke+njhxpf8wgqs3gpbvyozjgzyawicypfsvepsqr1rbbwv9k8guvk4p3evisn83ayimdsavkoaqoqb1cgabwaqz34ae8gwfn3nl0xpzx6wjomww2wtc4b5+6r5rc</latexit> Objective Explore the thermodynamics and phase transitions of the unitary Fermi gas at finite polarization. T Normal T c Normal BCS Superfluid?? 1 k F a Unitarity BEC Superfluid T c 1/k F a h <latexit sha1_base64="382/gfggvbazzhfv+bj+uvubcne=">aaab7xicjvdlsgnbeoynrxhfuy9ebopgkwxeug9blx4jucaqlgf20kmgzm4um71cwpirxjyoepv/vpk3th4hfqulgoqqbrq7olrjs77/4rwwlldw14rrpy3nre2d8u7enu0yizaqiupmk+iwldqykcsfrdqgjyofzwh0nfwb92istpqtjvmmyz7qsi8fjyc1oxesz8nuuvkr+jowv0kffmh0y++dxikygdujxa1t1/yuwpwbkklhpntjlkzcjpga245qhqmn89m5e3bklb7rj8avjjztv07kplz2heeum+y0td+9qfib186ofx7muqczorbzrf1mmury9hfwkwyfqbejxbjpbmviya0x5biq/s+e4kr6ufvvtiv1y0uartiaqziggpxbha6haqeigmedpmgzl3qp3ov3om8teiuzffgg7+0tufepfg==</latexit> h =(µ " µ # )/2

22 Btw: also interesting for QCD

23 <latexit sha1_base64="gkmgaynvtrdje9wli0zzltnlook=">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</latexit> Many-body formalism Path integral formulation Z =Tr h e (Ĥ µ " ˆN " µ # ˆN# ) i

24 <latexit sha1_base64="gkmgaynvtrdje9wli0zzltnlook=">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</latexit> Many-body formalism Path integral formulation Z =Tr h e (Ĥ µ " ˆN " µ # ˆN# ) i Suzuki-Trotter Hubbard-Stratonovich Z = Z D detm " [ ]detm # [ ]exp( S[ ])

25 <latexit sha1_base64="gkmgaynvtrdje9wli0zzltnlook=">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</latexit> Many-body formalism Path integral formulation Z =Tr h e (Ĥ µ " ˆN " µ # ˆN# ) i Suzuki-Trotter Hubbard-Stratonovich Z = Z D detm " [ ]detm # [ ]exp( S[ ]) Big problem: Conventional QMC approaches require a constant-sign integrand

26 Methods: Stochastic quantization Langevin approach: use random force field to explore configuration space = S + S[ ]= ln P [ ] P [ ]=detm[ ]e S g[ ] Computational sampling 3 4 Langevin algorithm: Sample configuration space using random noise (closely related to hybrid Monte Carlo) 1 2

27 Methods: Complex Langevin Stochastic quantization gone complex R = Re apple S + = S + I = Im apple S Computational sampling 3 4 Complex Langevin algorithm: Make your field complex Sample configuration space using random noise 1 2

28 Methods: Complex Langevin Problems Numerical instabilities Uncontrolled excursions into the complex plane Singular drift

29 <latexit sha1_base64="rvf7k144ohjbuqzsfuai4pnjvfk=">aaab9hicjvdlsgmxfl3js9zx1awbybfcldijqu1cklpxjrucw2jhkkkzbwisgzkmwob+hxsxkm79ghf+jeljoalggquhc+7lhk6yckan6344c/mli0vluzx86tr6xmzha/tax6ki1ccxj1uzxjpyjqlvmog0msikrchpixycjf3glvwaxflkdbmacnytlgiegyvdxhtu0qmqlfc1hdyvuyh6zxcc9dcpwgz1tug93y1jkqg0hgotw56bmcddyjdc6sjftjvnmbnghm1zkrggosgmqudo3ypdfmxkjjroon69yldqeihcuymw6euf3lj8zwuljjoomiat1fbjpo+ilcmto3efqmsujyyplcfemzsvkt5wmbhbvp5/jfih5wrzvtwq1k5nbergf/bgadyoqa3ooq4+efdwae/w7nw5j86l8zpdnxnmnzvwdc7bjzxhj+g=</latexit> <latexit sha1_base64="z5yayb6iyfzuymsen5ldocls4du=">aaacd3icjvblswmxgmzwv62vvy9egkwschvbbpugfl14robqov2wbjq2oul2ycnslv4el/4vlx5uvhr15r8x3fagoubaygtmg5jmldcqtod9olmz2bn5hfxiywl5zxxnxd+4vrgrmpg4zrfsregrrgxxndwmnbjjei8yuyn652p/5pzirwnxpycjctjqctqhggkrhe5uixt4ckt2c1smqvlga7gps2m7hohm2duohm6xuvyywl9jeuxrd913m8ege6exq0o1k16igxrjttejo0llkjig3edd0rruie5ukgyfgsedq7rhj5z2cq0z9wsirvypiy/sjee6p356y/e3r2l05zhiquimjgjpluoybnumx+3anpueaza0bgfj7vsh7igjslydfv5xgl8tn5s9y8ni7wzarh5sgw1qahvwbgrgatsbdzc4aw/gctw7986j8+k8tkzzzjszcb7befsezpibxa==</latexit> <latexit sha1_base64="xn6grqklosce1k8wpnkbnbg0gdk=">aaab8nicjvdlsgnbeoz1gemr6thlyba8hv0rhwch6mwtrhbnilue2clsmmt2wuypejb8hhcpkl79gm/+jzphquxbgoaiqpsuksql0oi6h87c/mli0njppby6tr6xwdnavtozuyz7ljozakvucyls7qnayvu54jsjjg9gg8ux37znsossvcvhzsoe9lirc0brssf1j0bqydnxjt1operv3ani36qkmzq6lfegmzgt8bszpfq3ptfhskakbzn8va6m5jlla9rjbuttmnadfppmi7jvls6jm2unrtjrv14unnf6mer2m6hy1z+9sfib1zyyn4afshodpgxtr7grbdmylob0heim5dasypswwqnru0uz2prk/yvbp6yd1dybo2r9ytzgcxzhdw7agxoowxu0wacgotzaezw7xnl0xpzx6eqcm7vzgw9w3j4bdgkqlw==</latexit> <latexit sha1_base64="umgqeqpn2kzum1t1ojthvs3q5qg=">aaacdxicjvblswmxgmzwv62vvy9egqvqd9ztedsduptisyjrhxzzsmm2dc0msx6wuvolvphxvhhq8erdm//gdnudioidgcnmn3zjrcmjsnveh5obm19yxmovf1zw19y33m2taywmxmthggl5eyffgoxe11qzcpnkgpkikwbup5/4zvsifrx8sg9teisoy2lmmdjwct1sd57ccjsxydukseoxgpswu3beggfc3ketdivvipcb/k2kyizg6l7bpdyj4rozpfsr6qu6gcgpkwzkxggbrvke+6hlwpzylbavjllvjghjkh0yc2kp1zbtvyzgkffqmer2mkg6p356e/e3r2v0fbymke+njhxpf8wgqs3gpbvyozjgzyawicypfsvepsqr1rbbwv9k8guvk4p3evisn83ayimdsavkoaqoqb1cgabwaqz34ae8gwfn3nl0xpzx6wjomww2wtc4b5+6r5rc</latexit> <latexit sha1_base64="pt/z4v2kkihmgfihn6gdvyrwmai=">aaacgxicjvdnssnagnzuv1r/oh69lbbbg5rebpugfaxxifkhsyumhm1m0y7dbmluriyhz+hfv/hiqcwjnnwbt2kffquhphhm5upbnsblvcrlejdku9mzs3pl+crc4tlyirm6diwttgdi4iqloh0gsrjlxffumdjobufxwegr6j+m/ny1ezimvkkgkffi1ou0ohgplfmmbcmj6blgofum3gyr39z5zkeffq5df/5neftnql2zcsc/srvm0pdnvzdmcbytrjbdunzsk1vejosimjfhxc0ksrhuoy7pamprtksxf18bwi2thdbkhb6uykf+3chrlougdnqyrqonf3oj8tevk6nowmsptznfob4fijigvqjhpcgqcoivg2icskd6rrd3keby6tyr/yvb2a0d1qzlvwr9enjggwyatbanblap6uamniadmlgf9+arpbl3xopxblymoyvjsrmovsf4+wb4qz4/</latexit> <latexit sha1_base64="sjnqgnwoi6tai6c5vzv6qjpbbju=">aaacd3icjzbns8mwhmbt+tbrw9wjl+aqpy1oheltkijhcasbrkwkwbqfjwlnumgufqqvfhuvhls8evxmtzf7axufhwj8+t3pnyrpldkqtot+wiw5+yxfpekyvbk6tr7hbg5dqystmhg4yylsrugrrgxxnnwmtfjjei8yaub985hfvcvs0uq09calauddqwokktyodpz9zsidfdagrygnn7dq+/ymxszgueiukmv3lpj3uajt1upn3e8kooneamyquu2km+ogr1jtzmjq9jnfuot7qevazhsiexxk4w8n4z4hhrgn0hyh4zh+3cgrv2rai5pkspfut28ef/pamy5pgpyknnne4mlfccagtucohdihkmdnbmzawflzvoh7sckstyf2/0rwdsunzffqufq7m7zrbdtgfxyacqicgrgedeabdo7aa3gcz9a99wi9wk+tamga7mydb7lepgh7a5t7</latexit> <latexit sha1_base64="/cdlilv33l3xnrhogpihyz3rqyu=">aaab6xicjvdlsgnbeoynrxhfuy9ebopgkwxeug9blx4jukziqpid9czdzmexmv4hlpkelx5uvpph3vwbj4+dioifduvvn91dyaqkjd//8aoli0vlk8xv0tr6xuzwexvn1iazeriircxmluqwldqykcsfd6lbhockm+hwyui379fymegbgqxyixlfy0gktk661l2/w67uqv4u7g9sgtka3fj7u5eileznqnfrwzu/pu7odumhcfxqzxztloa8jy1hny/rdvlpqwn24jqeixljshobql8nch5bo4pd1xlzgtif3kt8zwtlfj12cqntjfcl2aiou4wsnvmb9arbqwrkcbdguluzghddbbl0sv8litiqnlx9q+nk/xyerhh2yb8ooqynuidlaeaaavrwae/w7cnv0xvxxmetbw8+swvf4l19amzqjws=</latexit> Results First, a few definitions: h =(µ " µ # )/2 µ =(µ " + µ # )/2 n 0 : noninteracting, unpolarized density Lattice parameters: N x =7, 9, 11 Continuum limit N = 160 1=` <latexit sha1_base64="xn6grqklosce1k8wpnkbnbg0gdk=">aaab8nicjvdlsgnbeoz1gemr6thlyba8hv0rhwch6mwtrhbnilue2clsmmt2wuypejb8hhcpkl79gm/+jzphquxbgoaiqpsuksql0oi6h87c/mli0njppby6tr6xwdnavtozuyz7ljozakvucyls7qnayvu54jsjjg9gg8ux37znsossvcvhzsoe9lirc0brssf1j0bqydnxjt1operv3ani36qkmzq6lfegmzgt8bszpfq3ptfhskakbzn8va6m5jlla9rjbuttmnadfppmi7jvls6jm2unrtjrv14unnf6mer2m6hy1z+9sfib1zyyn4afshodpgxtr7grbdmylob0heim5dasypswwqnru0uz2prk/yvbp6yd1dybo2r9ytzgcxzhdw7agxoowxu0wacgotzaezw7xnl0xpzx6eqcm7vzgw9w3j4bdgkqlw==</latexit> T, F L = N x` T ' 7 F ' 3

30 Results: Density EoS (check)

31 Results: Compressibility

32 <latexit sha1_base64="esvuoclnt0ztb+w5x4raqh3y7sg=">aaacdnicbvdlssnafj34rpuvdelmscguskhk0boqim5cvjc20iyymd62qycpzizccf0dn/6kgxcqbl2782+ctflo64u5hm65lzv3+dfnutn2t7g0vlk6tl7ykg5ube/smnv79zjkbawxrjwslz9i4cwevzhforuliihpoemprjo/+qbcsii8u+myviamqtznlcgtdc2tjg+k4cg+xlzllzvum6ivswnvmjgr44pld82stqeff4mtkxlkq9e1vzq9icybhipyimxbswplpuqorjlmip1eqkzoiaygrwliapbeor1ngo+10sp9sogxkjxvf0+kjjbyhpi6mybqkoe9tpzpayeqx/nsfsajgpdofvutjlwes3bwjwmgio81ivqw/vdmh0qqqnserr2cm3/yiner1ovl31zl9as8jqi6refofdnohnxrdwogf1h0ij7rk3oznowx4934mluugfnmafptxucpctiv3w==</latexit> Results: Density EoS h =0.0, 0.4, 0.8, 1.2, 1.6, 2.0

33 Results: Magnetization EoS

34 Results: Susceptibility

35 Summary & conclusions Polarized nonrelativistic matter, in particular the unitary Fermi gas (UFG), will typically have a sign problem. Complex stochastic quantization (CL) provides a way to carry out nonperturbative calculations with complex actions. However, its mathematical underpinnings remain uncertain. We have carried out multiple explorations of polarized matter at finite temperature in 1D, 2D, 3D (UFG), and other situations that have a sign problem when using AFQMC. Our calculations for the UFG heal to known answers (virial, mid/low-t) in the range explored (T > h/2). We see little or no variation in the location of the superfluid critical point, but much more needs to be done to reach that regime.

36 Thank you!

Pairing properties, pseudogap phase and dynamics of vortices in a unitary Fermi gas

Pairing properties, pseudogap phase and dynamics of vortices in a unitary Fermi gas Piotr Magierski (Warsaw University of Technology/ University of Washington, Seattle) Collaborators: Aurel Bulgac (Seattle)