Emptiness and Depletion Formation Probability in spin models with inverse square interaction
We calculate the Emptiness Formation Probability (EFP) in the spin-Calogero Model (sCM) and Haldane-Shastry Model (HSM) using their hydrodynamic description. The EFP is the probability that a region of space is completely void of particles in the gro…
Authors: F. Franchini, M. Kulkarni
Emptiness and Depletion F ormation Probabilit y in spin mo dels with in v erse square in teration F abio F ran hini a , Manas Kulk arni b, a The A b dus Salam ICTP; Str ada Costier a 11, T rieste, 34100, Italy b Dep artment of Physis and Astr onomy, Stony Br o ok University, Stony Br o ok, NY 11794-3800 Dep artment of Condense d Matter Physis and Materials Sien e, Br o okhaven National L ab or atory, Upton, NY-11973 Abstrat W e alulate the Emptiness F ormation Probabilit y (EFP) in the spin-Calogero Mo del (sCM) and Haldane-Shastry Mo del (HSM) using their h ydro dynami de- sription. The EFP is the probabilit y that a region of spae is ompletely v oid of partiles in the ground state of a quan tum man y b o dy system. W e alu- late this probabilit y in an instan ton approa h, b y onsidering the more general problem of an arbitrary depletion of partiles (DFP). In the limit of large size of depletion region the probabilit y is dominated b y a lassial onguration in imaginary time that satises a set of b oundary onditions and the ation alu- lated on su h solution giv es the EFP/DFP with exp onen tial auray . W e sho w that the alulation for sCM an b e elegan tly p erformed b y represen ting the gradien tless h ydro dynamis of spin partiles as a sum of t w o spin-less Calogero olletiv e eld theories in auxiliary v ariables. In terestingly , the result w e nd for the EFP an b e asted in a form reminising of spin- harge separation, whi h should b e violated for a non-linear eet su h as this. W e also highligh t the onnetions b et w een sCM, HSM and λ = 2 spin-less Calogero mo del from a EFP/DFP p ersp etiv e. Key wor ds: EFP, DFP, In tegrable Mo dels, Calogero-Sutherland, Haldane-Shastry, Hydro dynamis, Colletiv e Field Theory P A CS: 71.10.Pm, 75.10.Pq, 02.30.Ik, 03.75.Kk, 71.27.+a Con ten ts 1 In tro dution 2 2 T w o-uid desription 5 Email addr esses: fabioitp.it (F abio F ran hini), kulkarnigrad.physis.sun ysb. edu (Manas Kulk arni) Pr eprint submitte d to Elsevier Otob er 25, 2018 3 The instan toni ation 7 4 Depletion F ormation Probabilit y 8 4.1 Asymptoti singlet state . . . . . . . . . . . . . . . . . . . . . . . 10 5 Emptiness F ormation Probabilit y 10 5.1 F ree fermions with spin . . . . . . . . . . . . . . . . . . . . . . . 10 5.2 Spin-less Calogero-Sutherland mo del . . . . . . . . . . . . . . . . 11 5.3 Probabilit y of F ormation of F erromagneti Strings . . . . . . . . 11 5.4 The freezing limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 5.5 Haldane-Shastry mo del . . . . . . . . . . . . . . . . . . . . . . . 12 6 Spin Depletion Probabilit y 13 7 Charge Depletion Probabilit y 15 8 Disussion of the results 16 9 Conlusions 17 A A tion for the DFP solution 19 B Linearized Hydro dynamis and DFP 22 1. In tro dution One-dimensional in tegrable mo dels ha v e an imp ortan t role in the study of strongly orrelated systems. When the redued dimensionalit y mak es in tera- tion una v oidable, p erturbativ e te hniques an qui kly lo ose appliabilit y and o v er the y ears more sophistiated to ols ha v e b een dev elop ed to ta kle these problems. These to ols learly in v olv e ertain appro ximations and the existene of an exat solution for some mo dels an allo w to he k their v alidit y . The on v en tional approa h in solving quan tum in tegrable mo del is kno wn as Bethe Ansatz (and its generalization). It is v ery suessful in onstruting the thermo dynamis of a system, but not v ery suitable to study its dynamis and the orrelation funtions, due to the inreasing omplexit y of its solutions. Ho w ev er, a v ery elegan t formalism w as dev elop ed using the Quan tum In v erse Sattering Metho d (QISM) [1 ℄ to express orrelation funtions as determinan ts of ertain in tegral op erators (F redholm determinan ts). In this formalism, the simplest orrelation funtion one an write is kno wn as the Emptiness F ormation Pr ob ability (EFP) and measures the probabilit y P ( R ) that a region of length 2 R is ompletely v oid of partiles. F or lattie mo dels, one is in terested in P ( n ) , the probabilit y that n onseutiv e lattie sites are empt y . In spin hain, taking adv an tage of the Jordan-Wigner mapping b et w een partiles and spins, the same quan tit y an b e though t of as the Pr ob ability of F ormation of F err omagneti Strings (PFFS), i.e. the probabilit y that n onseutiv e spins are aligned in the same diretion. 2 One should notie that the EFP is an n -p oin t orrelator and is, therefore, a m u h more ompliated ob jet ompared to the usual t w o-p oin t orrelation funtions one normally studies in ondensed matter ph ysis. Ho w ev er, due to the strongly in terating nature of the 1-D mo del, the QISM tells us that it is in fat no w orse than other orrelators b et w een t w o p oin ts a length n apart and ev en somewhat simpler and more natural. Moreo v er, the EFP is one of those extended ob jets lik e the V on Neumann En trop y , or the Ren yi En trop y , that in reen t y ears ha v e attrated a lot of in terest b eause of their abilit y to apture global prop erties that w ere not observ ed b efore from the study of 2 -p oin t orrelation funtions. The latter quan tities are of ourse motiv ated b y studies of en tanglemen t and quan tum omputation, while the EFP arises naturally in the on test of in tegrable theories. Despite the laimed simpliit y , the alulation of the EFP is b y no means an easy task. F or some mo dels, the sp ei struture of the solution has allo w ed to nd the asymptoti b eha vior of the EFP as n → ∞ . F or instane, the EFP in the whole of the phase-diagram of the X Y mo del w as alulated in [2, 3, 4 , 5℄ using the theory of T o eplitz determinan ts, while for the ritial phase of the X X Z spin hain the solution w as found in [ 6 , 7, 8℄ using a m ultiple-in tegral represen tation. The EFP has b een onsidered also for the 6 -v ertex mo del [9 , 10 , 11 ℄, for higher spins XXZ [12 ℄ and for dimer mo dels [13 ℄. W e also remark that high temp erature expansions of the EFP for Heisen b erg hains ha v e b een studied in [14 , 15 , 16 ℄. A reen t review of the EFP an b e found in [ 5℄ or [17℄. Field theory approa hes are normally most suited for the alulation of large distane asymptotis of orrelation funtions, but on v en tional te hniques lik e those inspired b y the Luttinger Liquid paradigm (i.e. b osonization) are not appropriate for extended ob jets lik e the EFP and only apture its qualitativ e b eha vior, while b eing quan titativ ely unreliable, as it w as sho w ed in [18 ℄. The reason for this failure is that Luttinger Liquid is appliable only to lo w-energy exitations around the F ermi p oin ts, where the linear sp etrum appro ximation is v alid, while orrelators lik e the EFP in v olv e degrees of freedom v ery deep in the F ermi sea, where the whole sp etrum with its urv ature is imp ortan t. F or this reason, the eld theory alulation of the EFP requires a non-linear generalization of on v en tional b osonization, i.e. a true h ydro dynami desrip- tion of the system. In [17 ℄ it w as sho wn that, with su h a non-linear olletiv e desription a v ailable, the alulation of the EFP is p ossible b y emplo ying, for instane, an instan ton approa h. In this pap er, w e will extend the ma hinery dev elop ed in [17 ℄ and apply it to the spin-Calogero Mo del (sCM), for whi h a (gradien tless) h ydro dynami desription w as reen tly onstruted from its Bethe Ansatz solution [19℄. The sCM is the spin − 1 / 2 generalization [20, 21 , 22 ℄ of the w ell-kno wn Calogero- Sutherland mo del [23 ℄ and is dened b y the Hamiltonian H = − ¯ h 2 2 N X j =1 ∂ 2 ∂ x 2 j + ¯ h 2 2 π L 2 X j 6 = l λ ( λ − P j l ) sin 2 π L ( x j − x l ) , (1) where P j l is the op erator that ex hanges the p ositions of partiles j and l . W e 3 hose to analyze this Hamiltonian assuming it ats on fermioni partiles, whi h means that the ex hange term selets an an ti-ferromagneti ground state [ 19 ℄. The oupling parameter λ is tak en to b e p ositiv e and N is the total n um b er of partiles. In [19 ℄, a olletiv e desription of the mo del w as deriv ed using four h ydro dy- nami elds: the densit y of partile with spin up/do wn ρ ↑ , ↓ and their v elo ities v ↑ , ↓ . The Hamiltonian in terms of these elds is v alid only for slo wly ev olv- ing ongurations, where terms with deriv ativ es of the densit y elds an b e negleted. This desription is referred to as a gr adientless hydr o dynamis . In [19 ℄, this theory w as used to sho w the non-linear oupling b et w een the spin and harge degrees of freedom b ey ond the Luttinger Liquid paradigm and it w as sho wn that, while a harge exitation an ev olv e without aeting the spin se- tor (for instane for a spin singlet onguration), a spin exitation arries also some harge with it, in a non-trivial w a y . The EFP for the sCM has not b een onsidered in the literature y et. F or the spin-less ase of the Calogero-Sutherland in teration, the asymptoti b eha vior of the EFP w as obtained using the form of the ground state w a v efuntion and thermo dynamial argumen ts [24 ℄ (see [5, 17 ℄ for details). It should b e noted that for ertain sp eial v alues of the oupling parameter λ , the spin-less theory is tigh tly link ed with Random Matrix Theory (RMT) and the EFP is the prob- abilit y of ha ving no energy eigen v alues in a giv en in terv al. F or these v alues of λ the EFP an b e alulated with m u h greater auray due to the additional struture pro vided b y RMT [26 ℄. If w e write the ground state of the system as Ψ G ( x 1 , x 2 , . . . , x N ) , the Empti- ness F ormation Probabilit y is dened as P ( R ) ≡ 1 h Ψ G | Ψ G i | x j | >R d x 1 . . . d x N | Ψ G ( x 1 , . . . , x N ) | 2 , (2) or, follo wing [1 ℄ P ( R ) = lim α →∞ h Ψ G | e − α R − R ρ c ( x ) d x | Ψ G i , (3) where ρ c ( x ) is the total partile densit y op erator ρ c ( x ) ≡ N X j =1 δ ( x − x j ) . (4) F or a mo del lik e the sCM, w e an also in tro due the EFPs for partiles with spin up or do wn separately P ↑ , ↓ ( R ) = lim α →∞ h Ψ G | e − α R − R ρ ↑ , ↓ ( x ) d x | Ψ G i , (5) whi h will allo w us to disuss the EFP as w ell as the PFFS. The approa h w e use to alulate the EFPs ( 5) in this w ork is similar to what w as explained in [17 ℄. The idea is to onsider the system as a quan tum uid 4 ev olving in imaginary time (Eulidean spae). Then the EFP an b e onsidered as the probabilit y of a rare utuation that will deplete the region − R < x < R of partile at a giv en imaginary time (sa y τ = 0 ). With exp onen tial auray , the leading on tribution to this probabilit y omes from the ation alulated on the saddle p oin t solution (instan ton) satisfying the EFP b oundary ondition. In setion 2 w e will rst review the results of [19 ℄ and transform them in to an in triguing form where the dynamis an b e deoupled in to t w o indep enden t uids of spin-less Calogero-Sutherland partiles. This t w o-uid desription is one of the in teresting observ ations of this pap er. In setion 3 w e will explain the instan ton approa h and form ulate the problem in this language. In setion 4 w e will onen trate on a generalization of the EFP , the Depletion F ormation Probabilit y (DFP) whi h w as in tro dued in [18℄. This orrelator will allo w us to alulate the dieren t EFPs v ery eien tly b y taking its dieren t limits in setion 5. Most notieably , w e will deriv e the PFFS for the Haldane-Shastry mo del as the freezing limit of the sCM. In setion 6 and 7 w e will onsider t w o additional DFP problems. Instead of sp eifying b oundary onditions for b oth the spin and harge setors of the uid as w e did in the previous setions, w e will no w relax these onditions and onstrain only one omp onen t at a time: this analysis suggests that an eetiv e spin- harge separation an b e onjetured for the EFP/DFP of the sCM. In setion 8 w e om bine all these results and suggest a ph ysial in terpretation of them. The nal setion on tains some onluding remarks. T o a v oid in terruptions in the exp osition, ertain te hnial formalities are mo v ed to the app endies and are organized as follo ws. In app endix A w e will revise and adapt the alulation of [17 ℄ to alulate the instan ton ation for our ases. In app endix B w e will rep eat this alulation in the linearized h ydro dynamis appro ximation or b osonization, to aid the disussions in setion 8 . 2. T w o-uid desription In [19 ℄ the gradien tless h ydro dynami desription for the sCM (1 ) w as de- riv ed in terms of densities and v elo ities of spin up and do wn partiles: ρ ↑ , ↓ ( t, x ) , v ↑ , ↓ ( t, x ) . Here, w e prefer to use densities and v elo ities of the majority and mi- nority spin: ρ 1 , 2 ( t, x ) , v 1 , 2 ( t, x ) , i.e. the subsript 1 ( 2 ) tak es the v alue ↑ or ↓ whi h ev er is most (least) abundan t sp eies: ρ 1 ≡ ρ ↑ + ρ ↓ + | ρ ↑ − ρ ↓ | 2 = ρ c + ρ s 2 , (6) ρ 2 ≡ ρ ↑ + ρ ↓ − | ρ ↑ − ρ ↓ | 2 = ρ c − ρ s 2 , (7) where w e in tro dued the harge and spin densit y ρ c ( t, x ) = ρ ↑ + ρ ↓ = ρ 1 + ρ 2 , (8) ρ s ( t, x ) = | ρ ↑ − ρ ↓ | = ρ 1 − ρ 2 . (9) 5 Please note that whatev er sp eies is ma jorit y or minorit y is deided dynamially in ea h p oin t in spae and time. Under the ondition [19 ℄ | v 1 − v 2 | < π ρ s , (10) the Hamiltonian is H = 1 12 π ( λ + 1) + ∞ −∞ d x k 3 R 1 − k 3 L 1 + 1 2 λ + 1 k 3 R 2 − k 3 L 2 , (11) where k R 1 ,L 1 ≡ v 1 ± ( λ + 1) π ρ 1 ± λπ ρ 2 , k R 2 ,L 2 = ( λ + 1) v 2 − λv 1 ± (2 λ + 1 ) πρ 2 (12) are the four dr esse d F ermi momenta . It turns out that an auxiliary set of h ydro dynami v ariables deouples the Hamiltonian (11) in to the sum of t w o indep enden t spin-less Calogero-Sutherland uids a and b : H = H a + H b = X α = a,b d x 1 2 ρ α v 2 α + π 2 λ 2 α 6 ρ 3 α , (13) where ρ a ≡ k R 1 − k L 1 2 π λ a = ρ 1 + λ λ + 1 ρ 2 , (14) ρ b ≡ k R 2 − k L 2 2 π λ b = 1 λ + 1 ρ 2 , (15) v a ≡ k R 1 + k L 1 2 = v 1 , (16) v b ≡ k R 2 + k L 2 2 = ( λ + 1) v 2 − λv 1 , (17) λ a ≡ λ + 1 , (18) λ b ≡ ( λ + 1) (2 λ + 1) . (19) W e remark that b oth the auxiliary v ariables and the real v ariables satisfy the anonial omm utation relations, i.e. [ ρ α ( x ) , v β ( y )] = − i¯ hδ α,β δ ′ ( x − y ) , α, β = { 1 , 2 } ; { a, b } . (20) The form of the Hamiltonian (13 ) is one of the in teresting observ ation of this pap er, sine it allo ws us to redue the spin Calogero-Sutherland mo del in to a sum of t w o spin-less theories. Ea h of the terms in square bra k ets in (13 ) is the gradien tless Hamiltonian of a spin-less CS system with oupling onstan ts λ a,b giv en b y (18 , 19 ). In [17 ℄ the gradien tless h ydro dynamis of spin-less partiles, lik e the ones in (13 ) w as used to alulate the EFP from the asymptotis of an instan ton solution. In the next setion w e review this approa h and w e lea v e the mathematial details to app endix A. 6 3. The instan toni ation Let us p erform a Wi k's rotation to w ork in imaginary time τ ≡ i t . Note that this mak es the v elo ities in (12) imaginary ( v → i v ) and the k 's omplex n um b ers. The x − t plane is mapp ed in to the omplex plane spanned b y z ≡ x + i τ . F ollo wing [17 ℄, w e will alulate the EFP as an instan ton onguration (i.e. a lassial solution in Eulidean spae) that satises the b oundary ondition ρ α ( τ = 0; − R < x < R ) = 0 , α = 1 , 2 , c , (21) in the limit R → ∞ , i.e. R m u h bigger than an y other length sale in the system. This limit guaran tees that the gradien t-less h ydro dynamis (13 ) is v alid in the bulk of the spae-time. One w e ha v e the lassial solution of the equation of motion φ EFP that satises (21 ), a saddle-p oin t alulation giv es the EFP with exp onen tial auray as the ation S alulated on this optimal onguration [17 ℄: P ( R ) ≃ e −S [ φ EFP ] . (22) Of ourse, to uniquely sp eify the problem, the b oundary onditions at innit y ha v e to b e pro vided as w ell and w e will tak e them to b e those of an equilibrium onguration: ρ 1 ( τ , x ) x,τ →∞ → ρ 01 , v 1 ( τ , x ) x,τ →∞ → 0 , ρ 2 ( τ , x ) x,τ →∞ → ρ 02 , v 2 ( τ , x ) x,τ →∞ → 0 . (23) When ρ 01 = ρ 02 w e ha v e an asymptoti singlet state (the AFM in zero magneti eld). The ondition ρ 01 6 = ρ 02 an b e a hiev ed via a onstan t external magneti eld whi h w ould result in a nite equilibrium magnetization. It is easy to implemen t these b oundary onditions in our t w o-uid desription using ( 14 -17 ). The k ey p oin t for the alulation is that w e an represen t the h ydro dynami elds in terms of the dressed F ermi momen ta k R 1 , k R 2 (whi h in Eulidean spae b eome omplex and omplex onjugated to k L 1 , k L 2 resp etiv ely) through ( 12): k R 1 = λ a π ρ a + i v a , k R 2 = λ b π ρ b + i v b . (24) In [19 ℄, it w as sho wn that these k -elds propagate indep enden tly aording to 4 deoupled Riemann-Hopf equations ∂ τ w − i w∂ x w = 0 , w = k R,L ;1 , 2 . (25) These equations ha v e the general (impliit) solution w = F ( x + i wτ ) (26) where F ( z ) is an analyti funtion to b e hosen to satisfy the b oundary ondi- tions. 7 Guided b y [17 ℄, the solution for an EFP problem is k R 1 = F a ( x + i k R 1 τ ) , k R 2 = F b ( x + i k R 2 τ ) , (27) with F a ( z ) ≡ λ a π ρ 0 a + λ a π η a z √ z 2 − R 2 − 1 , (28) F b ( z ) ≡ λ b π ρ 0 b + λ b π η b z √ z 2 − R 2 − 1 , (29) whi h automatially satisfy the onditions at innit y (23 ): ρ a ( τ , x → ∞ ) → ρ 01 + λ λ + 1 ρ 02 ≡ ρ 0 a , ρ b ( τ , x → ∞ ) → 1 λ + 1 ρ 02 , ≡ ρ 0 b , v a,b ( τ , x → ∞ ) → 0 , (30) while η a,b are t w o, p ossibly omplex, onstan ts that allo w to satisfy the EFP b oundary onditions (21). In app endix A w e sho w that the instan ton ation an b e expressed as a on tour in tegral where only the b eha viors of the solutions ( 27 ) at innit y and lose to the depletion region are needed, sa ving us the ompliation of solving the impliit equations in generalit y . Using the t w o-uid desription, the ation an b e written as the sum of t w o spin-less Calogero-Sutherland uids: from (96) w e ha v e S EFP = 1 2 π 2 R 2 X α = a,b λ α η α ¯ η α . (31) Before w e pro eed further, w e should men tion that the t w o-uid desrip- tion w e emplo y is v alid as long as the inequalit y ( 10 ) is satised. In fat, the solution (27 ) ould violate the inequalit y in a small region around the p oin ts ( τ , x ) = (0 , ± R ) . Ho w ev er, lose to these p oin ts the h ydro dynami desription is exp eted to b e somewhat pathologial, b eause gradien t orretions (whi h w e neglet) b eome imp ortan t. As it w as argued in [18 , 17 ℄, the on tributions that w ould ome to the EFP from these small regions are subleading and negligible, in the asymptoti limit R → ∞ w e onsider. Therefore, w e do not need to w orry ab out what happ ens near the p oin ts ( τ , x ) = (0 , ± R ) . Ho w ev er, a onsequene of the singular nature of these p oin ts is that, in our solution, the sp eies that onstitutes the ma jorit y (minorit y) spin in the region of depletion − R < x < R at τ = 0 , ould swit h and b eome minorit y (ma jorit y) at innit y . This ould b e imp ortan t to k eep in mind in in terpreting our form ulae, but our formalism already tak es that in to aoun t naturally . 4. Depletion F ormation Probabilit y It is more on v enien t to onsider a generalization of the EFP problem, alled Depletion F ormation Probabilit y (DFP) whi h w as in tro dued in [18℄. In h ydro- 8 dynami language the DFP b oundary onditions for the majority and minority spins are ρ 1 ( τ = 0; − R < x < R ) = ˜ ρ 1 , ρ 2 ( τ = 0; − R < x < R ) = ˜ ρ 2 . (32) The DFP is a natural generalization of the EFP (21 ) and it redues to it for ˜ ρ 1 , 2 = 0 . Of ourse, there is some am biguit y on the mirosopi denition of the DFP (see [18 , 17 ℄). One an, for instane, onsider it as the marosopi v ersion of the s -EFP in tro dued in [25 ℄. W e will rst alulate the DFP as the most general ase and later tak e the appropriate in teresting limits. In terms of the auxiliary elds w e in tro dued in (14 -17 ) to a hiev e the t w o- uids desription (13), the DFP b oundary onditions are ρ a ( τ = 0 , − R < x < R ) = ˜ ρ 1 + λ λ + 1 ˜ ρ 2 ≡ ˜ ρ a , ρ b ( τ = 0 , − R < x < R ) = 1 λ + 1 ˜ ρ 2 ≡ ˜ ρ b . (33) W e sp eify the parameters η 1 , 2 in (28 ,29 ) b y expressing the b oundary on- ditions (33) in terms of the dressed momen ta using ρ 1 = 1 π ( λ + 1) Re k R 1 − λ 2 λ + 1 Re k R 2 , ρ 2 = 1 π (2 λ + 1) Re k R 2 , v 1 = Im k R 1 , v 2 = 1 λ + 1 h Im k R 2 + λ Im k R 1 i . (34) This leads to λ a η a = λ a ( ρ 0 a − ˜ ρ a ) = ( λ + 1)( ρ 01 − ˜ ρ 1 ) + λ ( ρ 02 − ˜ ρ 2 ) , λ b η b = λ b ( ρ 0 b − ˜ ρ b ) = (2 λ + 1) ( ρ 02 − ˜ ρ 2 ) . (35) It is no w straigh tforw ard to obtain the DFP b y substituting ( 35 ) in to (31). After some simple algebra w e get P DF P ( R ) = e xp ( − π 2 2 h λ ( ρ 0 c − ˜ ρ c ) 2 + ( ρ 01 − ˜ ρ 1 ) 2 + ( ρ 02 − ˜ ρ 2 ) 2 i R 2 ) = exp ( − π 2 2 ( λ + 1 2 ) ( ρ 0 c − ˜ ρ c ) 2 + 1 2 ( ρ 0 s − ˜ ρ s ) 2 R 2 ) , (36) where w e in tro dued a notation in terms of the harge eld ( ρ 0 c = ρ 01 + ρ 02 , ˜ ρ c = ˜ ρ 1 + ˜ ρ 2 ) and of the spin eld ( ρ 0 s = ρ 01 − ρ 02 , ˜ ρ s = ˜ ρ 1 − ˜ ρ 2 ). Equation (36 ) is the main result of this w ork. T o understand it b etter, w e will onsider sev eral in teresting limits. 9 4.1. Asymptoti singlet state If no external magneti eld is applied, the equilibrium onguration of an an ti-ferromagneti system lik e the one w e onsider is in a singlet state. This means that in the b oundary onditions at innit y (23) w e should set ρ 01 = ρ 02 = ρ 0 . In this limit (36 ) redues to P singlet DF P ( R ) = e xp ( − π 2 2 h λ (2 ρ 0 − ˜ ρ c ) 2 + ( ρ 0 − ˜ ρ 1 ) 2 + ( ρ 0 − ˜ ρ 2 ) 2 i R 2 ) = exp ( − π 2 2 ( λ + 1 2 ) (2 ρ 0 − ˜ ρ c ) 2 + 1 2 ˜ ρ 2 s R 2 ) . (37) 5. Emptiness F ormation Probabilit y By taking the limit ˜ ρ 1 , 2 = 0 w e an use (36 ) to alulate the dieren t EFPs. The probabilit y to nd the region − R < x < R at τ = 0 ompletely empt y of partiles is therefore P E F P ( R ) = e xp ( − π 2 2 h λ ( ρ 01 + ρ 02 ) 2 + ρ 2 01 + ρ 2 02 i R 2 ) = exp ( − π 2 2 ( λ + 1 2 ) ρ 2 0 c + 1 2 ρ 2 0 s R 2 ) , (38) whi h b eomes P singlet E F P ( R ) = exp − π 2 2 ( λ + 1 2 )(2 ρ 0 ) 2 R 2 (39) for the asymptoti singlet state. This is equiv alen t to the EFP of a spin-less Calogero-Sutherland system with oupling onstan t λ ′ = λ + 1 / 2 , see ( 43). This is onsisten t with the phase-spae piture pro vided in [ 19 ℄, in whi h it is explained that for a singlet state ea h partile o upies an area of π ( λ + 1 / 2) due to the exlusion statistis, while it w ould o up y an area π ( λ + 1 ) if it w ere alone. Therefore, in this on text, the harge eld an b e though t of as desribing a spin-less Calogero system with oupling onstan t λ ′ = λ + 1 / 2 . 5.1. F r e e fermions with spin Setting the oupling parameter λ = 0 orresp onds to non-in terating (free) fermions with spins and this redues (36) to P free fermions DF P ( R ) = e xp ( − 1 4 [ π ( ρ 0 c − ˜ ρ c ) R ] 2 − 1 4 [ π ( ρ 0 s − ˜ ρ s ) R ] 2 ) = exp ( − 1 2 [ π ( ρ 01 − ˜ ρ 1 ) R ] 2 − 1 2 [ π ( ρ 02 − ˜ ρ 2 ) R ] 2 ) . (40) 10 This result is the same as the one obtained in [17 ℄. The EFP is then P free fermions E F P ( R ) = exp ( − 1 2 ( π ρ 01 R ) 2 − 1 2 ( π ρ 02 R ) 2 ) , (41) whi h agrees with the results obtained in the on text of Random Matrix Theory [26 , 24 ℄, where the subleading orretions w ere also found. 5.2. Spin-less Calo ger o-Sutherland mo del Of ourse, the prime he k to our form ula for the DFP/EFP of the sCM is to tak e its spin-less limit ˜ ρ 2 = ρ 02 = 0 , whi h giv es P spin-less DF P ( R ) = exp ( − ( λ + 1) 2 π 2 ( ρ 01 − ˜ ρ 1 ) 2 R 2 ) , (42) P spin-less E F P ( R ) = exp ( − ( λ + 1) 2 π 2 ρ 2 01 R 2 ) , (43) in p erfet agreemen t with [17, 24 ℄ for a spin-less Calogero-Sutherland system with oupling λ ′ = λ + 1 . 5.3. Pr ob ability of F ormation of F err omagneti Strings If w e require the minorit y spin partiles to ompletely empt y the region − R < x < R at τ = 0 , w e are left only with the ma jorit y spin and w e reated a (partially) p olarized state. W e an refer to this ase as the Probabilit y of F ormation of P artially F erromagneti Strings (PFPFS) [18 , 17 ℄. Setting ˜ ρ 2 = 0 in (36 ) and lea ving ˜ ρ 1 nite w e ha v e P P F P F S ( R ) = exp ( − π 2 2 h λ ( ρ 01 − ˜ ρ 1 + ρ 02 ) 2 + ( ρ 01 − ˜ ρ 1 ) 2 + ρ 2 02 i R 2 ) . (44) The ab o v e is the probabilit y of formation of ferromagneti strings aompanied b y a partial depletion of partiles, sine in the region of depletion w e ha v e ˜ ρ c = ˜ ρ 1 . W e an imp ose that the a v erage densit y of partiles is onstan t ev erywhere b y setting ˜ ρ 1 = ρ 01 + ρ 02 = ρ 0 c , while still requiring all partiles in the region − R < x < R at τ = 0 to b e ompletely p olarized (maximal magnetization: PFFS) P P F F S ( R ) = exp − [ πρ 02 R ] 2 . (45) Note that (45) is indep enden t of λ and exatly orresp onds to the Emptiness F ormation Probabilit y of a λ ′ = λ + 1 = 2 spin-less Calogero mo del with ba k- ground densit y giv en b y ρ 02 (43). In terestingly the same result (52 ) will b e deriv ed in the next setions as the EFP of minorit y spins, i.e. ˜ ρ 2 = 0 , in the Haldane-Shastry mo del (46 ). This is just another asp et of the w ell-kno wn re- lation b et w een spin-Calogero, Haldane-Shastry and λ ′ = 2 spin-less Calogero mo dels [27 , 19 ℄ as it will b e sho wn in the next setion. 11 5.4. The fr e ezing limit If w e tak e the λ → ∞ limit in the spin-Calogero mo del ( 1 ), the harge dynamis freezes (the partiles b eome pinned to a lattie) and only the spin dynamis surviv es. This fr e ezing limit w as sho wn b y P oly hronak os [ 28 ℄ to b e equiv alen t to the Haldane-Shastry mo del (HSM) [27 , 29 ℄: H HSM = 2 π 2 N 2 X j
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