Driven-dissipative chiral condensate

Transcription

1 Driven-dissipative chiral condensate Masaru Hongo (RIKEN ithems) Recent Developments in Quark-Hadron Sciences, 018 6/14, YITP Based on ongoing collaboration with Yoshimasa Hidaka, and MH, Kim, Noumi, Ota, arxiv: [hep-th]

2 QCD matter in extreme situations Main characters in QCD matter QCD matter in several extreme situations High-µB High-T Strong-B

3 Outline Motivation: Time crystalline behavior and NG mode for chiral order parameter? Approach: Schwinger-Keldysh formalism for nonequilibrium open system Result: Noneq. Ginzburg-Landau expansion Schwinger-Keldysh based EFT

4 Time-evolution of QCD phase diagram Initial condition [Cabbibo-Parisi (1975)] Earliest QCD Phase diagram Time evolution [McLerran (003)] [Cabibbo-Parisi (1975)] [Baym et al. Long range plan (1983)]

5 QCD phase diagram in 010 s [Fukushima-Hatsuda (010)] Possibility for inhomogeneous chiral order parameter! [Nakano-Tatsumi (005), Nickel (009), Kojo et al.(010), Hidaka et al. (015), ]

6 Time crystal & No-Go theorem Inhomogeneous phase = SSB of spatial translation symmetry Lagrangian: Continuous translational symmetry SSB Crystalline state Only discrete translational symmetry (Classified by the space group!) Possibility of time crystal? [Wilczek (01)] Lagrangian: Continuous time-translational symmetry SSB? Time crystal? Only discrete symmetry No-Go theorem for equilibrium state [Watanabe-Oshikawa (015)],

7 Possibility in nonequilibrium system Discrete time crystal in periodically driven system [Yao et al. (017), ] Continuous time crystal in chemical reaction (BZ reaction)

8 '<latexit sha1_base64="lurcjgivwqfoedtxkiiklo/w=">aaacsnichve9t9tqfd1xs0kdlnaulbpejyi6ohcnihv1isrsericuzmoss0jpogvk6k1mof6i46dakjaxxv0rff6h/okj9qdqwsc0ovhsnei8k17hfeefccn6eru6b0a6jhsnnwcobrbppxzm5+4clidunpju90pupudmd0vlqu+ckutqgfmjbf3fwezummnupn6lz3z7wfony0hffs1l69hyxxpawfq7uxoy5og19fbiruprpujmgj60hifbu1s/prktqi1axkktelkv6gjptgw0iufarsbyxmafh4auefwmwshzm5jjonzgqeyro1yl+aojdld/nz410hym/erpx+rdf6lya/hyhwk9jvoaes/6cv9oas7vclyi8rs51ufa4xbxvz0vhp5r8rincdbjwpq5gd7wi+zss7uxkx0cos7338pkq+qrtcztqhv5z/mib0g9g9y6m0y1r+tilj85zovgo/w9jeuysfvuqqlulfpltmqg0xualxve0xqomd9hejdp8b0/ca6ulpekphjjviwvaj7hvinmp1cunwo=</latexit> <latexit sha1_base64="lurcjgivwqfoedtxkiiklo/w=">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</latexit> <latexit sha1_base64="lurcjgivwqfoedtxkiiklo/w=">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</latexit> <latexit sha1_base64="lurcjgivwqfoedtxkiiklo/w=">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</latexit> Driven-dissipative condensate Driven-dissipative environment in exciton-polariton system pump Total system Open System loss Environment [From Sieberer et al. (016)] Time-oscillating condensate ( time crystal) can be realized! Q. Do we have something special for the open system?

9 NG mode in open system? Spontaneous symmetry breaking & Nambu-Goldstone mode For some conserved charge ˆQ a 9 ˆ(x) such that h[i ˆQ a, ˆ i(x)]i Tr ˆ [i ˆQ a, ˆ i(x)] 6= 0 Spontaneous Symmetry Breaking (SSB) Massless mode = Nambu-Goldstone (NG) mode appears! In open system, charge can diffused to environment! Charge is no longer conserved! { - What is symmetry and its breaking? - Is there Nambu-Goldstone mode? pump Total system Open System [Minami-Hidaka (018)] loss Environment

10 QCD matter in extreme situations Main characters in QCD matter QCD matter in several extreme situations High-µB High-T Strong-B

11 QCD matter in extreme situations Main characters in QCD matter QCD matter in several extreme situations Driven High-µB High-T Strong-B dissipative

12 Outline Motivation: Time crystalline behavior and NG mode for chiral order parameter? Approach: Schwinger-Keldysh formalism for nonequilibrium open system Result: Noneq. Ginzburg-Landau expansion Schwinger-Keldysh based EFT

13 Need for real-time formalism QCD matter in several extreme situations Driven High-µB High-T Strong-B dissipative Equilibrium state: Imaginary-time formalism! Nonequilibrium (steady) state We need to use the real-time formalism of quantum theory!

14 Real-time formalism of Q uantum Field Theory in a Nutshell

15 Schwinger-Keldysh in a Nutshell QFT for T=0 - Observable Transition amp. hout Ô ini Non-equilibrium QFT - Observable Average value X hin Ô ini in Ex. S = h Ŝ i Ex. h ˆ'i =Tr(ˆ 0 ˆ') - Generating functional t i Z[J] =hvac ÛJ vaci J(t) t f Re t CTP generating functional ˆ Z[J 1,J ]=Tr 0 Û J Û J1 t i 0 J (t) Im t J 1 (t) t f Re t

16 Doubled symmetry structure Dynamical degrees of freedom Total system pump Total system = System {'} + Environment { } (G-inv.) <latexit sha1_base64="fbhro0drq5fdha7mukugyxgaue4=">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</latexit> Open{'} System <latexit sha1_base64="fbhro0drq5fdha7mukugyxgaue4=">aaacjhichvfnswjrfl1oxalvpugjsrfk7ktqleeuhat/uglngrmeurd+wjmfez8a+jrvautij/ye/yew/oroadcmrxfggsjj7vdmnxfepfed965sqtxelsbywr0bhwiobmamp4jr6kzcwxbafgkyyugalhhsmqzless73bhzuemxsrnvtmhxn919w+bzlk5or84lzmda1jv5xwusa5rxvk71jqss8zlnxi0jgn0ijyirb/ewy+0exeepyaaqo0qamgojievzdapq8iiicyxb1dmzilepfgxqgrnogzthkkiit079kq6lp6rra9qewqftvbowkwowjc94jz18xgd8xc8/a7w9gq6xfs1yx8vmcursiffxr0qjyhatqozwcqsof55etd9bj3fkpf3zy96ous8vtfbzfn/j/g118ohvozxfllsoyl0p8yotfby/4uxmdolcwedehzplx1i7fqcaswhksujfwiqx7kia8957ncaxxqlhiclvcdj9vcpiaefgrwt4xd5uuxa==</latexit> <latexit sha1_base64="wws+5x6hr4zlsuy8afkwwrwswa=">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</latexit> Stot [', ] = Ssys ['] + Senv [ ] + Sint [', ] <latexit sha1_base64="wtjvdaeuevvj1p3shkh+kmoqo3k=">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</latexit> { } loss Environment <latexit sha1_base64="wws+5x6hr4zlsuy8afkwwrwswa=">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</latexit> Doubled symmetry structure in total & open systems Z i(stot ['1, 1 ] Stot [', ]) Z = D'1 D' D 1 D e 0 (', ) Z i(ssys ['1 ] Ssys [' ])+i ['1,' ] = N D'1 D' e 0 (') <latexit sha1_base64="g+618t7bcirg8bhj7dhnqed6p64=">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</latexit> <latexit sha1_base64="oxvfq1euoz5av3uyo048tpzst4q=">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</latexit> Stot ['1, <latexit sha1_base64="ifzfpwzehlr/tibyyrdrpaeowq=">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</latexit> Ssys ['1 ] <latexit sha1_base64="zkilfxqwdnjzrsznhmgqyn6ystu=">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</latexit> 1] Stot [', ] is invariant under G1 G = Gr Ga <latexit sha1_base64="zaza7kzgourvoothnzg3c10z/4=">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</latexit> <latexit sha1_base64="fxs1tmnklzjnn7a9sd5wmnk7hz8=">aaacj3ichvg7sgnbfd1z3+9eg8emgbsrcfeeiiuelakdgqocj7c7jjq4l3ynaq3+gd+gybxbquz9art/wmjpeesfgwtvnguiot5hzs6cuefoga7pzjuri8jraw1rbjs6u7p7evfyczglwlvupgizllv6wyrqhskurskoqwz4gtac0xbr5uf84369kojqeu6qovlftmpsu3jpwoziaqdq1tnbsjoitbfkk+vkhriurfon0goqqrxlxviww9ifbwsvobbworjbmbdyiqogs/cnmrmbyxkdc9wgm7wvjhlcibb7cgv+3zajfmxz4ayas+bwbz8dknmboga7phe7php7o/ddatahgw8sr7ztk/zywolw8e1flco7wsgn6k/pcnuyjrxk9u5htomxvlnfpt57kc6sjnxg6zkex+dhumof+bwx6rzbfy8ycfk7cchv07834cdymszpl9ewpth4ublqnrjckce5gdnksyaklrh7ghhvcaiktp81q+waqlog1q/gsuihl6ouuq==</latexit> Ssys [' ] + ['1, ' ] can be only invariant under Gr <latexit 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17 <latexit sha1_base64="p98ttp/fsnuwsfbqe/fvrhi+ft0=">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</latexit> <latexit sha1_base64="p98ttp/fsnuwsfbqe/fvrhi+ft0=">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</latexit> <latexit sha1_base64="p98ttp/fsnuwsfbqe/fvrhi+ft0=">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</latexit> <latexit sha1_base64="p98ttp/fsnuwsfbqe/fvrhi+ft0=">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</latexit> Simple example: Brownian motion Dynamical degrees of freedom Total system (G-inv.) = Brownian particle + Surrounding particles {X} {x n } S tot [X, x n ]=S sys [X]+S env [x n ]+S int [X, x n ] Total system Key point - Generating functional: Z = N - Effective action (in a simplified case): - Z DX 1 DX e is eff [X 1,X ] 0 (X) is e [X 1,X ]=is sys [X 1 ] is sys [X ] Z 1 dt hi (X 1 Ẋ X Ẋ 1 )+ T (X 1 X ) i ( X 1 (t)! X1(t) 0 =X 1 (t + R ) is only invariant under X (t)! X(t) 0 =X (t + R ) S e [X 1,X ] Energy of Brownian particle is not conserved, but we still have symmetry!

18 Towards driven-dissipative QCD Lesson from the simple example - Doubled dynamical degrees of freedom: {'1, ' } Driven <latexit sha1_base64="k5re/iv/y/baljxeuqsjufyh1o=">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</latexit> - Only one (vector-like) symmetry: Gr Ga <latexit sha1_base64="pj60dwegubmst3yu3imxopcnx+0=">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</latexit> - Driven-dissipative coupling: ( < 0) <latexit sha1_base64="v5txaebmwxripcrwprepcxqi1i=">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</latexit> ( > 0) for the Brownian particle dissipative <latexit sha1_base64="9oximsrf4zigqvxhlphptm7i6zs=">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</latexit> (In general, coupling could be complex!) We can discuss SSB of surviving Gr-symmetry!! [Minami-Hidaka (018)] <latexit sha1_base64="kjlocm4dgah/i4lzuve/y1d3mj4=">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</latexit> Driven-dissipative -flavor QCD in chiral limit - Symmetry: (SU ()R SU ()L )r (SU ()R SU ()L )a = O(4)r O(4)a <latexit sha1_base64="zcnw1seteijyo4i5prjos/9nay=">aaacmxichvg7sgnbfdur/iocilybioitbgrauuqfdri4yngbzwwu050cf/stgia/qf/wmjkwuk09gds/aelp0esfwwsvnksiip6l9k598w9d85wtd+wosj6rnpqgxqbmhmtrw3thz1dye6etdarbzbiw57tbrumeqpbuikvpllfhh8iwzftsw7uz1xp18sickxnrqodxw7xq4ri9iyffofzn9uaneko1oiultkoikszuyhmaymrzh6cfqypbhhkpe8xrz4mfccq4excjgngye/g1cb8fnbhsv5gjgmjoxoeyra0tcjbjcyhaf/7ucbcasy3m1zxiplb7f5hwwmouheqareqf7uqynev+1vyxqufvywltz0wq/0hxsn3v7v+xwrrd3qfrts0ire5fxyd79ikm+wqrpy4enl7njlahkmf3qm/s/p0e64xe45vfrclmsnp3hxqvxzwe/fswfok1syxogx05m56ejqevwaagmcltgmc05rgephc/wgwucamnadpavlzqk9xqyk0vvosw+war1za5</latexit> <latexit sha1_base64="trd4y84mggxgx/cvh3ku+7am13m=">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</latexit> Possible SSB for (SU ()R SU ()L )r = O(4)r <latexit sha1_base64="efyl0mwcqswndkuxudxg90llu=">aaac0hichvhps9xafh7gwq3autql0etwuvyoy0smrgqctjcektvdv4xdjstpub3ml5lzrruw0qv/qa89tdbd6vhop+cl/0ape9wteftw4se3ybu0b4wme /ds0owxqozkfpbh4pdj0zhhl9+mwsnz6xgqetygfvj3cdanuyuzyn1uffy7bdinmebbltuzd1/j8q8imaf+htgpwcozmj7f4y4lidjzq0k9bvklmnyjweliwrxuzl/eimjj0zuj9jixz3qklwpbnzxlal1wj8vql307a0bqsvquyejqmxn5qzsh3gvabvkqxvqq+w11ea8bonacdxj4iai7yefmxw00qaija0bcxesip+cmojbmhzvmaqwinlf5oywsb6lmuecap6baxvkrkfabxl/7ec/ydv/aur+7tlaq9pjd9ueloxmmfk5fk/ko9ar9uva96fradc6lxtt7dljgvchr69scv55wl8nqyg9/xjpx/wy4e0qv89oxzy5vvz7gxyyvcjza7whcbzt6ucoitm7kv15lgxqcfzafbzpgcvbgdaxblbofqheo4uqpk3vkj+vzr1tpyztp4z9qdq4b6nqn0q==</latexit>

19 Outline Motivation: Time crystalline behavior and NG mode for chiral order parameter? Approach: Schwinger-Keldysh formalism for nonequilibrium open system Doubled symmetry structure and its breaking in open system Result: Noneq. Ginzburg-Landau expansion Schwinger-Keldysh based EFT

20 Two (theoretically) simple situations Near (nd-order) phase transition point Ginzburg-Landau theory (Linear sigma model) Deep inside the ordered-phase Effective field theory/lagrangian (Nonlinear sigma model)

21 Towards driven-dissipative TDGL Driven-dissipative -flavor QCD in chiral limit - Symmetry: (SU ()R SU ()L )r (SU ()R SU ()L )a = O(4)r O(4)a <latexit sha1_base64="zcnw1seteijyo4i5prjos/9nay=">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</latexit> <latexit sha1_base64="trd4y84mggxgx/cvh3ku+7am13m=">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</latexit> Possible SSB for (SU ()R SU ()L )r = O(4)r 1 - Building blocks: 1 =, = 1 <latexit sha1_base64="efyl0mwcqswndkuxudxg90llu=">aaac0hichvhps9xafh7gwq3autql0etwuvyoy0smrgqctjcektvdv4xdjstpub3ml5lzrruw0qv/qa89tdbd6vhop+cl/0ape9wteftw4se3ybu0b4wme /ds0owxqozkfpbh4pdj0zhhl9+mwsnz6xgqetygfvj3cdanuyuzyn1uffy7bdinmebbltuzd1/j8q8imaf+htgpwcozmj7f4y4lidjzq0k9bvklmnyjweliwrxuzl/eimjj0zuj9jixz3qklwpbnzxlal1wj8vql307a0bqsvquyejqmxn5qzsh3gvabvkqxvqq+w11ea8bonacdxj4iai7yefmxw00qaija0bcxesip+cmojbmhzvmaqwinlf5oywsb6lmuecap6baxvkrkfabxl/7ec/ydv/aur+7tlaq9pjd9ueloxmmfk5fk/ko9ar9uva96fradc6lxtt7dljgvchr69scv55wl8nqyg9/xjpx/wy4e0qv89oxzy5vvz7gxyyvcjza7whcbzt6ucoitm7kv15lgxqcfzafbzpgcvbgdaxblbofqheo4uqpk3vkj+vzr1tpyztp4z9qdq4b6nqn0q==</latexit> <latexit sha1_base64="lirctwv+tr+thd+l3rbbxc5dcxu=">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</latexit> - Symmetry property 1: ( <latexit sha1_base64="/jp7qjm0qr7xej1mpzo8iwtldk=">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</latexit> 1 <latexit sha1_base64="t1nt4o/osnuy4ok4wwqhps1rim=">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</latexit> 0 1 0!! = R1 = R 1 with R1 = R O(4)r <latexit sha1_base64="xk1anpip/nllmk/vni6gtpwy1kq=">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</latexit> - Symmetry property : NO Lorentz invariance for mixing terms - Invariants: { LO 3 t 1 <latexit sha1_base64="9izep17s8+uwp3pdkrwywtqxovi=">aaachhichvg7sgnbfdugum7aipybiniiegudxqlew0snteqmbb1zeo7ovdtubdfkcw1cjkwuls/qebf8dctxblbrslbzylosf4h5k5c+aeoe4umtkpyb6blpaozpjxfhunt6+/ohbxndwtu+upupkdmd0vf1d84upbzelzgckxdctmqwbykc/wqvf71se50vh3gqoxvgwtjitd6shbuxl87ysjlkupjcszucnqaprbdije+sxdwcgyragycngbekdzmpkggucwvumfmyyfbeoiye1py5s3cgxuwrryu+7uwszed6tt9ug/ykydnjzrit9es39eapdecv9plnrwpyo+7lmhe9orvucfb0npvxr8ripcdht6ql5wahway9svbuhkz9f0zdxzm5emsuzsaqk3rnr+z/ip7pgx9gv96nm0ruwzhrcvnw6p+rszzwb7jq1swtcs6srokywzjmojulgaf69hajquxcizzxcgxzvqzveybqupbpbnbj1cwvwchrzcp</latexit> <latexit <latexit sha1_base64="f34qiha9rfiks1iapf5va+136bw=">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</latexit> NLO <latexit sha1_base64="9izep17s8+uwp3pdkrwywtqxovi=">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</latexit> <latexit <latexit sha1_base64="mgy/bvwuthcgx0o+iglzhs0vscw=">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</latexit> <latexit sha1_base64="6zwe0cq8rauu10lajzush9wjwce=">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</latexit> t 1, t µ 1@ t <latexit sha1_base64="/a6pqfgaydvx8cldhk7v/vx8s=">aaaczxicsyriyswumtc4ycjezmlkxs7bycxnw8vhlyaoffacx1qunbqanj+txxsrlficmpozlxpaklmskxprujsamjuukxqelo0mkg8vsy0qzszpcympleinzu1mz8tmy0xolaekxqt46drvx4cnis5kt4qtntcztdazmjhtmdazmda3ndiemyzmtyxnammcmjljdenkfmami506cb1xapgojrdqbmkcaqvmhiguocwabqh5atsyyhhsgpizkhlkgxizuhnygeqa7bygriziiixmmgqwycgaisuyvapfiocstlb8kkmtaxdqbylqvspqrsjqnbtipgn50vdrpcafzgyxwhcy0jycic4c6lrgudw4ards4lpbcypvbi8n/ua0qxpsbsgtlua6cai3tscev0si+dtbxblauoqha6elr5tlgniylmbuzqs6vqasavjfmkr/wdx0z8fwqarvagaldf4d3b/q4kbbyaap8sq+jc8ntaajcc9suc3gklhed0ymblhrnqgbnqggsbkdk7qiojgkgzqytaaxoy5gwodb0maqyjq9a0m1xjumzxg8mcqzaphqomozwke6hfmqafm7qb/uqkt</latexit> t 1 t, t ( 1, ( t ) 1, ( 1, t 1 1) t 1, ( t ) ),,

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sha1_base64="xae6cfd0iolyw1dnjh61nyb8c1k=">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</latexit> Driven-dissipative Time-Dep. GL Driven-dissipative TDGL action in ra-basis [Talk by Hayata last week] Z S e = d 4 x t 0 + r (1 apple t r r)@ t m g t r r r + with r(x) 1 (x)+ 0 + (1 apple )@ 0 + m + g (t) =0, a(x) 1 (x) (x) with h r (x)i = Linear analysis in the trivial phase: (t) =0+ (t) (t) 0 Solution: (t) =c 1 e 1 ( +p 4m )t + c e 1 ( p 4m )t If < 0 (driven nature), trivial phase is linearly unstable!! Non-linear terms will be important!

23 <latexit sha1_base64="uuv9iwqvvpez0jozn6fqzbdw+ba=">aaacnichvg/axrbfp6y/orrj1y0ewwwj8gf5xgrgkeqgjawl8tlbtlhnlmnkpf7tzx8xlgju1fsysfcze3s7kxn/airwvqfuegwvf7ikbumbzuabb9735hueteodg6k9kevi0wpht0yfte+dpnnpjf7bjvpbpmvun4sjtma9hiv6lh1jtahwksz5uuyvd5c6e87w1vluskvmdu7ureugst7tvgab6jz7obgunnzp3bjmkcrqczfgnsn0agylxaer5wipg5zxrjgi1s1zvciogi3nk5pxyzcecbakn39p+40mtakk5ybwa9behzk8qycm0jgy4aicjem4xaech7rcefimdtawvzgsff3cmpyrb1wluimj9kdxgm+rddszoeyzl6pfx4l5jmx0secfabxte/v6tv9oh//rfvunuovu7zlival/zlhf1a+/1cv8w6w/vt1qgedlsxuxjv7tyum/iu/0q/vp91fubk8v1ymf/sz/t+npxrhp4ih3/yxsr5sf+jhsp+p+3yydypva6wu3s9uxi7btq0luiswtyng1jexxtq5epv8rff8nus1gprofv4kmpn1zrz+cosjz8bllswzq==</latexit> <latexit 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sha1_base64="3+zauutqcc6p006zjl7msujqm10=">aaacg3ichvg7sgnbfdur/ioggylaeidyvgciiqrvlgi0kgsluooll9sxujbbdfkbs1cjkwuls/qebf8dctxblbrslbzylokg9w8ycoxppntnc3bpmqbi9dsidxd09vbg+/ohboegr+ojydubwfepkdndy/v1dc4rloiintwmjxc8xmq1bykcvrzxvd6rcd0zxzi1t+rtresyrdpqjfnzwxbicupsgfptqi1aalfk3pgd9neafwyqschgqdkochgsqcvbi+5porm+yzm8f6ggx7wvjhlcibgbjnxep/itbhc7nmekonfsxi6bnycjp0sdf0sg90s8/08wutelij6axgu97scq8wcjyx+f6vyuzd4vbl9adniskwqq8me/dcpvklo6wvhp/bi5nz+qzdeuv7p+snuief+bu34zrdzg9+mopzl4a3b71zzpawfzcuqwkupfkpfejrsuwiwnmctcwkcy6mshx9sjociozpvuzvxauvctv6yg04/gwysonzcmqtq==</latexit> <latexit 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sha1_base64="3+zauutqcc6p006zjl7msujqm10=">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</latexit> <latexit sha1_base64="3+zauutqcc6p006zjl7msujqm10=">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</latexit> SSB of time-translational symmetry /m =0. > 0, g /(m apple)=0.1 /m = 1 > 0, g /(m apple)=0.1 p apple (t) mt Trivial phase due to the dissipation p apple (t) mt Oscillating χ condensate due to drive+dissipation NG mode (x) associated with SSB of time translation t + (t, x) (x) = EOM 0 Dispersion relation:! = idk Diffusive NG mode!!

24 Two (theoretically) simple situations Near (nd-order) phase transition point Ginzburg-Landau theory (Linear sigma model) Deep inside the ordered-phase Effective field theory/lagrangian (Nonlinear sigma model)

25 EFT for time-dependent condensate Time-translationally breaking & Nambu-Goldstone field 9 (t, x) such that h (t, x)i = (t) with (t) 6= 0 <latexit sha1_base64="vypaj3souelzhghg6htsmtmg0bc=">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</latexit> <latexit sha1_base64="xpcu6q+sejpragwa5ifg//pfnci=">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</latexit> <latexit sha1_base64="vcdexpursi5wzsqasnnf4yx1ck=">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</latexit> (t, x) = t + (t, x) Introducing (t, x) as embedding: <latexit sha1_base64="9tpxrrismoanepevxnjnhmd5ks=">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</latexit> <latexit sha1_base64="ra8xkewbqzmcpfr1gfh3wc6pvlu=">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</latexit> Ex. Driven-dissipative condensate, Dynamics of inflaton, Synchronization - Symmetry properties: Time-translation = Nonlinearly realized 0 0 (t, x)! (t, x) = (t +, x + ) + 0 <latexit sha1_base64="6tv9ilrpqu5ygks7af0g6sma04s=">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</latexit> - Invariants: t + (t, x) and its derivatives e.g. t + (t, x) <latexit sha1_base64="umwafrkmqg9fjc9fi+carz7bcsm=">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</latexit> <latexit sha1_base64="grxjdzmcdhskznfxh0owwjyx8o=">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</latexit> - General effective Lagrangian for in-out formalism: µ Le ) = 0 (t + ) + 1 (t + )Pµ P + <latexit sha1_base64="iixhjmxdsnihw0olrvpkhautb4=">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</latexit> X n µ n (t + )(Pµ P + 1) n

26 <latexit sha1_base64="p6iqivfgclloclejwmc0vgscg4y=">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</latexit> <latexit sha1_base64="p6iqivfgclloclejwmc0vgscg4y=">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</latexit> <latexit sha1_base64="p6iqivfgclloclejwmc0vgscg4y=">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</latexit> <latexit sha1_base64="p6iqivfgclloclejwmc0vgscg4y=">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</latexit> Noneq. EFT and diffusive NG mode ra-basis NG fields and transformation properties r (t, x) 1(t, x)+ (t, x), a (t, x) 1 (t, x) (t, x) ( r (t, x)! r(t, 0 x) = r (t + 0 r, x + r )+ 0 r which transforms as a (t, x)! a(t, 0 x) = a (t + 0 r, x + r ) Part of effective Lagrangian in slow-roll limit ( (t) = const. ) L () e =( 1 ) a r i i r 1 a r h +i 1 a ( 4 ) a + (@ i a ) i In the low-energy limit, c s k { - One gapped mode - One diffusive NG mode:! ' i c s k