Exactly solvable quantum impurity model with inverse-square interactions
EExactly solvable quantum impurity model with inverse-square interactions
Hong-Hao Tu ∗ and Ying-Hai Wu † Institut f¨ur Theoretische Physik, Technische Universit¨at Dresden, 01062 Dresden, Germany School of Physics and Wuhan National High Magnetic Field Center,Huazhong University of Science and Technology, Wuhan 430074, China
We construct an exactly solvable quantum impurity model which consists of spin-1/2 conductionfermions and the spin-1/2 magnetic moment. The ground state is a Gutzwiller projected Fermisea with non-orthonormal modes and its wave function in the site-occupation basis is a Jastrow-type homogeneous polynomial. The parent Hamiltonian has all-to-all inverse-square hopping termsbetween the conduction fermions and inverse-square spin-exchange terms between the conductionfermions and the magnetic moments. The low-lying energy levels, spin-spin correlation function,and von Neumann entanglement entropy of our model demonstrate that it exhibits the essentialaspects of spin-1/2 Kondo physics. The machinery developed in this work can generate many otherexactly solvable quantum impurity models.
Introduction — The interplay between the localizeddegree of freedom and itinerant electrons has been acentral subject of condensed matter physics in the pastdecades [1, 2]. The localized degree of freedom is usu-ally called impurity and the simplest example is spin-1/2magnetic moments due to unpaired d or f electrons insolids. The unpaired electron is screened by the itiner-ant electrons via spin-exchange interactions at low tem-peratures and a spin-singlet state emerges. Quantumimpurity physics has also been investigated in quantumdots [3, 4], magnetic adatoms on metallic surfaces [5],carbon nanotubes [6, 7], and quantum Hall devices [8–10]. The impurity may even be realized in a non-localmanner using Majorana zero modes in topological super-conductors [11–16].The theoretical understanding of quantum impurityphysics begins with two models proposed by Andersonand Kondo, which are related by the Schrieffer-Wolfftransformation [17–19]. To explain a mysterious changeof resistance in certain metallic samples with dilute mag-netic atoms, Kondo constructed a model with one mag-netic moment coupled to a bath of conduction electronsvia a spin-exchange term. The simplicity of this model ishighly deceptive as manifested by divergences in straight-forward perturbative calculations, which calls for drasti-cally different approaches. Based on the idea of renormal-ization, Wilson developed the numerical renormalizationgroup (NRG) method and brought out a rather completesolution of the Kondo problem [20–22]. The Kondo prob-lem elucidates the concept of asymptotic freedom andserves as a vivid demonstration of correlation effects inmany-body systems.Further insights into the Kondo physics were gath-ered along several directions. The NRG results helpedNozi`eres to formulate a Fermi liquid theory for the low-energy physics of the spin-1/2 Kondo model [23]. Themagnetic moment is only coupled to one particular modeof the conduction fermions and other modes in the con-duction bath remain free. Affleck and Ludwig appliedboundary conformal field theory (CFT) to study Kondo physics [24, 25]. This approach reconstructs the Fermiliquid picture for the spin-1/2 Kondo model, but its util-ity extends to more complicated quantum impurity sys-tems that are not amenable to the Fermi liquid descrip-tion. Andrei and Wiegmann independently constructeda Kondo model which can be solved exactly using theBethe ansatz [26–29]. It is defined on a semi-infinite con-tinuous chain populated by fermions with linear disper-sion relation. The Kondo physics has also been exploredin some other exactly solvable models [30–34].In this Letter, we propose an exactly solvable quan-tum impurity model in a one-dimensional open chainwith spin-1/2 conduction fermions and one spin-1/2 mag-netic moment at one end of the chain. This model isvery different from the one by Andrei and Wiegmann: itcan be defined on any finite-size lattices; the conductionfermions have a quadratic dispersion relation in a spe-cial basis; the lowest eigenstates in all cases with an oddnumber of conduction fermions are known. The low-lyingenergy levels, entanglement entropy, and spin-spin corre-lation function of our model clearly demonstrate that itexhibits spin-1/2 Kondo physics. It is further suggestedthat our model may be non-integrable based on the levelstatistics of the full energy spectrum.The most appealing property of our model is the ex-istence of highly structured eigenstates that can be ex-pressed in three different but equivalent ways: Gutzwillerprojected Fermi seas with non-orthogonal basis states,Jastrow-type homogeneous polynomial wave functions,correlators of the CFT for the system. The Gutzwillerapproach elucidates Nozi`eres’ Fermi liquid picture in ananalytically tractable manner. The CFT approach isreminiscent of the well-established connection betweenedge CFT and bulk wave function in quantum Hall sys-tems [35]. The machinery developed in this work opensup the exciting possibility of constructing a variety ofexactly solvable quantum impurity models. Wave function — The system of our interest is an openchain with L + 1 sites ( L can be even or odd) labeled by j = 0 , , · · · , L [Fig. 1(a)]. It is placed on the semi-circle a r X i v : . [ c ond - m a t . s t r- e l ] A ug with unity radius and projected onto the real line [ − , j -th site is θ j = πL ( j − / ≤ j ≤ L and θ j = 0 for j = 0, and the associated linearposition is u j = cos θ j . The motivation for this choice ofcoordinate will become clear later. The 1 ≤ j ≤ L sites arepopulated by spin-1/2 conduction fermions described bycreation (annihilation) operators c † j,σ ( c j,σ ) with σ = ↑ , ↓ being the spin index. The j = 0 site is occupied by a spin-1/2 magnetic moment described by operators S . Themagnetic moment can be represented using Abrikosovfermions as S = (cid:80) σσ (cid:48) c † ,σ (cid:126)τ σσ (cid:48) c ,σ (cid:48) with (cid:126)τ being thePauli matrices [2]. The redundance caused by this rep-resentation is removed by imposing the single-occupancyconstraint (cid:80) σ c † ,σ c ,σ = 1.The many-body state to be investigated is | Ψ (cid:105) = (cid:88) { n ↑ j } , { n ↓ j } Ψ( { n ↑ j } , { n ↓ j } ) ( c † , ↑ ) n ↑ · · · ( c † L, ↑ ) n ↑ L × ( c † , ↓ ) n ↓ · · · ( c † L, ↓ ) n ↓ L | (cid:105) . (1)The coefficientΨ( { n ↑ j } , { n ↓ j } ) = δ n =1 δ (cid:80) j n ↑ j = (cid:80) j n ↓ j = M × (cid:89) σ = ↑ , ↓ (cid:89) ≤ j 2) (except for θ = 0) and its projection on the horizontal axis is u j = cos θ j . which would be of great help when one searches for theparent Hamiltonian. The Fermi liquid picture of Nozi`eresis realized in an exact manner: the impurity forms aspin singlet with the ζ † ,σ modes and the other conduc-tion fermions form a Fermi sea in the non-orthogonal ζ † m> ,σ modes. If the state with a particular M is theground state, one can add (remove) conduction fermionsto (from) the ζ † m> ,σ modes to create excited states.However, this procedure can only give us some statesin which the numbers of spin-up and spin-down fermionsare equal. The Fermi liquid picture in other cases needsto be corroborated using numerical results. For the orig-inal Kondo problem, Yosida proposed a trial wave func-tion made of a Kondo singlet and a Fermi sea of con-duction fermions [42], but it does not work well [43].The structure of Eq. (3) is very similar to the proposalof Yosida, but the crucial additional insight is that thesingle-particle orbitals in the Kondo singlet and the Fermisea should be chosen properly. Parent Hamiltonian — The target state Eq. (1)is the exact ground state of the Hamiltonian H = π L ( H + H P + H K + H C ), where H = L − (cid:88) q =0 (cid:88) σ = ↑ , ↓ q d † q,σ d q,σ (4)[ d q,σ = (cid:112) (1 + δ ,q ) /L (cid:80) Lj =1 cos( qθ j ) c j,σ ] describes hop-ping processes of the conduction fermions, H P = 34 L (cid:88) j =1 (cid:88) σ = ↑ , ↓ cot θ j c † j,σ c j,σ (5)gives site-dependent energies to the conduction fermions,and H K = L (cid:88) j =1 cot θ j S · S j (6)stands for long-range spin-exchange interactions betweenthe conduction fermions and the magnetic moment, and H C = (cid:80) Lj =1 F ( L ) c † j,σ c j,σ is a L -dependent chemical po-tential [ F ( L ) = − (3 L / − 1) for odd L and F ( L ) = − (3 L / − L/ − / 4) for even L ]. The real space rep-resentation of H contains inverse-square hopping termsand site-dependent potential terms, as manifested by H = L (cid:88) j,k =1; j (cid:54) = k (cid:20) − j − k | z j − z k | − − j − k | z j − z ∗ k | (cid:21) c † j,σ c k,σ + L (cid:88) j =1 (cid:20) L + 12 − 32 sin θ j (cid:21) c † j,σ c j,σ , (7)where z j = exp( iθ j ) and z ∗ j = exp( − iθ j ) are the complexcoordinates of the site j and its mirror image with respectto the real axis [see Fig. (1)], respectively [37]. In thelimit of L → ∞ and j (cid:28) L , the Kondo coupling strengthalso has an inverse-square form as cot θ j ∼ ( j − / − .The ratio between the largest hopping strength andKondo coupling strength is 3 / L → ∞ . The par-ent Hamiltonian has three mutually commuting con-served quantities: the total number of fermions N f = (cid:80) Lj =0 (cid:80) σ c † j,σ c j,σ , the total spin S = ( (cid:80) Lj =0 S j ) , andits z -component S z = (cid:80) Lj =0 S zj .It is proved in [37] that Eq. (3) is an exact eigenstateof H . The key idea is as follows. When H K is acted on | Ψ (cid:105) , it creates particle-hole excitations (including spin-flipped ones) on top of the Fermi sea. The effect of H P isto compensate “unwanted” particle-hole excitations cre-ated by H K (similar to the cancellation of diffractive scat-tering in integrable models solved by Bethe ansatz [44]).The truly remarkable observation is that the remainingparticle-hole excitations in ( H K + H P ) | Ψ (cid:105) can be can-celed completely by H | Ψ (cid:105) , where H describes conduc-tion fermions with a quadratic dispersion relation anddiscrete Chebyshev polynomials are their single-particlebasis. The eigenvalue of | Ψ (cid:105) with respect to H + H P + H K is ( M − M − M − / M ) so the prefactor π / (4 L ) isadopted to ensure that H is physically valid. It will bedemonstrated below that H C selects the half-filled statesas the ground states [ M = ( L + 1) / L and M = ( L + 2) / L ]. Numerical results — The proof that | Ψ (cid:105) is an eigen-state of H is not sufficient because we have claimed thatit is actually the ground state. To this end, exact diago-nalizations are performed for various choices of L, N f , S z with L up to 14. For all the cases with even N f and S z = 0, the overlaps between numerically generated low-est eigenstates and Eq. (1) are exactly 1 (up to machineprecision). The half-filled states also turn out to be theground states. The density matrix renormalization group(DMRG) [45–48] method is employed to compute thelowest eigenstates for various choices of L, N f , S z with L up to 80. The lowest energy of the ( L, N f , S z ) sector isdenoted as E ( L, N f , S z ). The accuracy of our approachcan be checked by comparing numerical and analytical (80,0) (81,1/2)(79,1/2) (82,0) (e) (78,0) (80,0) (81,1/2) (82,0) (79,1/2) (78,0) (b) (c) (d) -97.36-97.38-97.40-97.42-97.44-97.46 (a) FIG. 2. (a) Energy spectrum of the L = 79 system. Eachcolumn corresponds to a sector whose quantum numbers areindicated as ( N f , S z ) in the panel. (b)-(d) Scaling relationsof three quantities in the energy spectrum. (e) Schematics ofthe Fermi liquid picture for the energy spectrum. values of E ( L, N f , 0) with even N f . For instance, ourDMRG results at L = 79 and N f = 78 , , 82 have ab-solute errors of the 10 − order, so they are almost ex-act. DMRG calculations also confirm that the half-filledstates are the ground states.The low-lying energy levels of the L = 79 systemare presented and analyzed in Fig. 2. The differ-ences in energy eigenvalues are characterized by ∆ = E ( L, L, / − E ( L, L + 1 , ( L ) = E ( L, L + 2 , / − E ( L, L + 1 , D ( L ) = ∆ ( L ) − ∆ ( L ) = E ( L, L +2 , / − E ( L, L, / and ∆ go to zero as1 / ( L + 1), and (ii) D ( L ) go to zero as 1 / ( L + 1) . Thesame L dependence is also observed in ∆ = E ( L, L − , − E ( L, L + 1 , , ∆ ( L ) = E ( L, L + 3 , − E ( L, L +1 , ( L ) − ∆ ( L ). The energy spectrum can beinterpreted using the Fermi liquid picture of Nozi`eres asillustrated in Fig. 2 (e): there is a Fermi level at ε = 0,the single-particle energy levels are equally spaced at ε m = ± πv F L +1 ( m − ) ( m = 1 , , · · · and v F is the Fermivelocity), the ground state is constructed by filling allthe single-particle states below the Fermi level, and theexcited states are obtained by adding and/or removingsome fermions from the ground state.The spin-spin correlation function Γ j = (cid:104) Ψ | S · S j | Ψ (cid:105) / (cid:104) Ψ | Ψ (cid:105) for the L = 79 system is shown inFig. 3 [37]. As an important signature of the Kondophysics, the decaying behavior of Γ j has been stud-ied extensively using CFT and numerical methods [49–52]. One generally observes a crossover from 1 /r de-cay in the Kondo screening cloud to 1 /r decay out-side the cloud. The latter is easy to understand be-cause the system behaves essentially as free fermions oncethe impurity is completely screened. For our model,the distance between the sites 0 and j is defined as R j = L +1) π sin θ j [53]. The log-log plot of | F j | versus site 20 40 60 800 (a) (b) FIG. 3. Spin-spin correlation function Γ of the L = 79 sys-tem. (a) Γ j on all sites. (b) Log-log plot of | Γ j | versus R j = L +1) π sin θ j . site 10 20 30 40 50 60 70 (a) GOEPoisson (b) FIG. 4. (a) von Neumann entanglement entropy of the groundstate of the L = 79 , N f = 80 , S z = 0 system. The red line isthe least square fit using the data points in 10 ≤ L A ≤ 70. (b)Energy level statistics of the L = 9 system in the subspacewith N f = 10 , S z = 0 , S = 0. The bars are the probabilitydensity of the ratio r m and the lines are the Poisson and GOEresults. R j in Fig. 3 (b) has a slope − . 000 with a coefficient ofdetermination higher than 99 . S ( L A ) = − Tr( ρ A ln ρ A ) with ρ A being the reduced density matrix for the subsystem0 ≤ j ≤ L A . It is expected that the Calabrese-Cardy for-mula S ( L A ) = c (cid:20) Lπ sin (cid:18) π L A L (cid:19)(cid:21) + g (8)is still applicable outside the Kondo screening cloud [54,55], where c is the CFT central charge and g is a non-universal number. For the L = 79 , N f = 80 , S z = 0 case,the data points in the range 10 ≤ L A ≤ 70 can be fittedusing c = 2 . 000 and g = 0 . 957 as shown in Fig. 4 (a).This is consistent with the fact that the system containstwo gapless modes. For the first few sites, there are obvi-ous deviations from the Calabrese-Cardy formula, whichshould reflect certain aspects of the coupling between theimpurity and conduction fermions, but it is unclear howto extract such information.It is natural to ask if our model is integrable giventhat a large class of inverse-square models, such as thecelebrated Haldane-Shastry model, has been proved tobe integrable [56–59]. One diagnostic of integrability islevel statistics of the energy spectrum [60, 61]. The eigen-values E m are sorted according to the conserved quan-tum numbers and arranged in ascending order, then the level spacing δ m = E m +1 − E m and the ratio r m =min( δ m , δ m +1 ) / max( δ m , δ m +1 ) can be obtained. Theprobability density of r m have two possible forms: a Pois-son distribution with P ( r ) = 2 / (1 + r ) for integrablemodels and a Gaussian orthogonal ensemble (GOE) with P ( r ) = [27( r + r )] / [4(1 + r + r ) / ] for non-integrablemodels. It is quite plausible that our model obeys theGOE distribution as shown in Fig. 4 (b). The expecta-tion value of r m is 0 . . 536 for GOE. This suggests that our model is notintegrable despite having explicit forms for the groundstates in all sectors with even N f . Conformal field theory formulation — The successfulconstruction of an exactly solvable model for the spin-1/2Kondo problem with polynomial wave function motivatesus to search for similar scenarios in more complicatedquantum impurity systems, such as the multichannel andSU(N) Kondo problems. A promising route along thisdirection manifests itself as soon as we establish a linkbetween the wave function and the associated boundaryCFT, which is very similar to the practice of construct-ing quantum Hall wave functions from edge CFT [35].The key observation is that the wave function (2) can berepresented as a CFT correlator [37]Ψ( { n ↑ j } , { n ↓ j } ) = (cid:104)O bg A n ↑ ,n ↓ ( u ) A n ↑ ,n ↓ ( u ) · · · A n ↑ L ,n ↓ L ( u L ) (cid:105) (9)with a background charge O bg controlling the total num-ber of fermions and vertex operators A n ↑ j ,n ↓ j ( u j ) = (cid:26) δ n j , : e i (cid:80) σ n σj φ σ ( u j ) : j = 0: e i (cid:80) σ n σj φ σ ( u j ) : j = 1 , · · · , L , (10)where φ σ ( u j ) is a two-component chiral bosonic field and: · · · : denotes normal ordering.This reformulation can be viewed as an infinite-dimensional matrix product state [62] and turns out to bevery illuminating in further analysis. If the vertex opera-tors for the impurity site were removed from Eq. (9), thewave function reduces to the ground state of H that re-alizes the free-fermion CFT with the free boundary con-dition on the lattice [63–66]. This is perhaps not toosurprising, since the vertex operators (10) for conduc-tion fermions all have conformal weight h = 1 / h = 1 / Conclusion and discussion — In summary, we haveconstructed an exactly solvable quantum impurity modelwith a highly structured ground state and a parentHamiltonian consists of inverse-square hopping and in-teraction terms. The low-energy physics of our modelis described by a boundary CFT and a particular cor-relator of this CFT reproduces the wave function. Thiswork establishes a connection between the wave functionand boundary CFT in quantum impurity models, whichpaves the way toward constructing wave functions andexactly solvable models for more complicated systemswith SU(N) fermions and/or multiple impurities [68].For Kondo problems that cannot be solved exactly, wemay define ζ † modes with unknown parameters and useEq. (3) as variational ansatz. The numerical prospect ofoptimizing these parameters with respect to some givenHamiltonians is left for future works. Acknowledgment — We are grateful to Jan von Delft,Biao Huang, Seung-Sup Lee, Fr´ed´eric Mila, Germ´anSierra, Andreas Weichselbaum, and Guang-Ming Zhangfor helpful discussions. This work was supported bythe DFG via project A06 (HHT) of SFB 1143 (project-id 247310070), the NSFC under Grant No. 11804107(YHW), and the startup grant of HUST (YHW). ∗ [email protected] † [email protected][1] A. C. 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Sierra, Phys. Rev. B , 041119 (2015).[64] B. Basu-Mallick, F. Finkel, and A. Gonz´alez-L´opez,Phys. Rev. B , 155154 (2016).[65] J.-M. St´ephan and F. Pollmann, Phys. Rev. B , 035119(2017).[66] A. Hackenbroich and H.-H. Tu, Nucl. Phys. B , 1(2017).[67] I. Affleck and A. W. W. Ludwig, J. Phys. A , 5375(1994).[68] Y.-H. Wu and H.-H. Tu, arXiv:1906.03464 (2019). APPENDIX A: WAVE FUNCTION This section provides more details about the properties of the ground-state wave functionΨ( { n ↑ j } , { n ↓ j } ) = δ n =1 δ (cid:80) j n ↑ j = (cid:80) j n ↓ j = M (cid:89) σ = ↑ , ↓ (cid:89) ≤ j We first prove Eq. (3) of the main text. To begin with, we can rewrite the ground state as | Ψ (cid:105) = (cid:88) { n ↑ j } , { n ↓ j } Ψ( { n ↑ j } , { n ↓ j } ) ( c † , ↑ ) n ↑ · · · ( c † L, ↑ ) n ↑ L ( c † , ↓ ) n ↓ · · · ( c † L, ↓ ) n ↓ L | (cid:105) = P G0 | (cid:101) Ψ (cid:105) . (A2)The Gutzwiller projector P G j enforces the single-occupancy constraint δ n =1 on the impurity site j = 0 and the state | (cid:101) Ψ (cid:105) is (cid:88) { n ↑ j } , { n ↓ j } δ (cid:80) j n ↑ j = (cid:80) j n ↓ j = M (cid:89) σ = ↑ , ↓ (cid:89) ≤ j This subsection aims to prove Eq. (10) of the main text. The chiral bosonic field for spin-up fermions is definedby φ ↑ ( u ) = φ ↑ − iπ ↑ ln u + i (cid:80) ∞ n (cid:54) =0 1 n a ↑ n u − n , where φ ↑ , π ↑ , and a ↑ n are operators satisfying the following commutationrelations: [ φ ↑ , π ↑ ] = i and [ a ↑ n , a ↑ m ] = nδ n + m, . Furthermore, π ↑ and a ↑ n> annihilate the vacuum | (cid:105) , i.e., π ↑ | (cid:105) = a ↑ n> | (cid:105) = 0. The corresponding chiral vertex operator for spin-up fermion is given by: exp [ iφ ↑ ( u )] := exp (cid:32) iφ ↑ + ∞ (cid:88) n =1 n a ↑− n u n (cid:33) exp (cid:32) π ↑ ln u − ∞ (cid:88) n =1 n a ↑ n u − n (cid:33) . (A8)For odd L , θ j =( L +1) / = π/ iφ ↑ ). For spin-down fermions, the expressions of the chiral bosonic field and vertex operators are similar, withthe spin index replaced by ↓ .The product of chiral vertex operators is obtained by normal ordering of operators: exp [ iφ ↑ ( u x )] : · · · : exp [ iφ ↑ ( u x M )] := exp (cid:32) iM φ ↑ + M (cid:88) m =1 ∞ (cid:88) n =1 n a ↑− n u nx m (cid:33) exp (cid:34) π ↑ ln( u x · · · u x M ) − M (cid:88) m =1 ∞ (cid:88) n =1 n a ↑ n u − nx m (cid:35) × (cid:89) ≤ i This section provides details about how to act the parent Hamiltonian on the ground state. The target state is asuperposition of two parts in which the spin on the j = 0 site points up and down, respectively. When acted upon byan operator, the result can still be written like this. To simplify subsequent analysis, we focus on the spin-up part.The spin-down part can be analyzed in the same way. For notational ease, we denote (cid:81) M − m =1 (cid:81) σ = ↑ , ↓ ζ † m,σ | (cid:105) as | PFS (cid:105) (PFS stands for partial Fermi sea). B1: The Kondo term The Kondo term is H K = S · Λ K = 12 S +0 Λ − K + 12 S − Λ +K + S z Λ z K (A12)with Λ K = L (cid:88) j =1 θ j − cos θ j S j = L (cid:88) j =1 cot θ j S j . (A13)As for usual spin operators, one can also define Λ +K , Λ − K , and Λ z K . Their commutators with the ζ modes are (cid:104) Λ − K , ζ † , ↑ (cid:105) = L (cid:88) j =1 θ j − cos θ j c † j, ↓ ≡ O K , ↓ , (cid:104) Λ − K , ζ † m, ↑ (cid:105) = 2 ζ † , ↓ − m − (cid:88) α =1 ζ † α, ↓ − ζ † m, ↓ for m > , (cid:104) Λ − K , ζ † m, ↓ (cid:105) = 0 for m ≥ , (cid:104) Λ z K , ζ † , ↑ (cid:105) = 12 L (cid:88) j =1 θ j − cos θ j c † j, ↑ ≡ O K , ↑ , (cid:104) Λ z K , ζ † m, ↑ (cid:105) = ζ † , ↑ − m − (cid:88) α =1 ζ † α, ↑ − ζ † m, ↑ for m > , (cid:104) Λ z K , ζ † , ↓ (cid:105) = − L (cid:88) j =1 θ j − cos θ j c † j, ↓ = − O K , ↓ , (cid:104) Λ z K , ζ † m, ↓ (cid:105) = − ζ † , ↓ + m − (cid:88) α =1 ζ † α, ↓ + 12 ζ † m, ↓ for m > . (A14)The action of H K on | Ψ (cid:105) gives | Ψ ↑ K (cid:105) + | Ψ ↓ K (cid:105) with | Ψ ↑ K (cid:105) = c † , ↑ (cid:18) − 12 Λ − K ζ † , ↑ + 12 Λ z K ζ † , ↓ (cid:19) | PFS (cid:105) = c † , ↑ (cid:18) − O K , ↓ − ζ † , ↑ Λ − K + 12 ζ † , ↓ Λ z K (cid:19) | PFS (cid:105) . (A15)The commutators listed above can be used to show that ζ † , ↑ Λ − K | PFS (cid:105) = 2 M − (cid:88) m =1 ζ † , ↑ ζ † , ↓ (cid:34) m − (cid:89) α =1 ζ † α, ↑ ζ † α, ↓ (cid:35) ζ † m, ↓ (cid:34) M − (cid:89) α = m +1 ζ † α, ↑ ζ † α, ↓ (cid:35) | (cid:105) ,ζ † , ↓ Λ z K | PFS (cid:105) = − M − (cid:88) m =1 ζ † , ↑ ζ † , ↓ (cid:34) m − (cid:89) α =1 ζ † α, ↑ ζ † α, ↓ (cid:35) ζ † m, ↓ (cid:34) M − (cid:89) α = m +1 ζ † α, ↑ ζ † α, ↓ (cid:35) | (cid:105) . (A16)The terms in Eq. (A15) are unwanted in the sense that they do not appear in the target state. However, the result isnot bad because the unwanted terms in Eq. (A16) are highly structured. The O K , ↓ term is somewhat special, so ournext step would be to eliminate it. B2: The site-dependent potential term The site-dependent potential term is H P = 34 L (cid:88) j =1 (cid:88) σ = ↑ , ↓ θ j − cos θ j c † j,σ c j,σ = 34 L (cid:88) j =1 (cid:88) σ = ↑ , ↓ cot θ j c † j,σ c j,σ . (A17)Its commutators with the ζ modes are (cid:104) H P , ζ † ,σ (cid:105) = 34 L (cid:88) j =1 θ j − cos θ j c † j,σ = 34 O K ,σ , (cid:2) H P , ζ † m,σ (cid:3) = 32 ζ † ,σ − m − (cid:88) α =1 ζ † α,σ − ζ † m,σ for m > . (A18)The action of H P on | Ψ (cid:105) gives | Ψ ↑ P (cid:105) + | Ψ ↓ P (cid:105) with | Ψ ↑ P (cid:105) = c † , ↑ H P ζ † , ↓ | PFS (cid:105) = c † , ↑ (cid:18) O K , ↓ + ζ † , ↓ H P (cid:19) | PFS (cid:105) . (A19)0The commutators listed above can be used to show that ζ † , ↓ H P | PFS (cid:105) = − M − (cid:88) m =1 ζ † , ↑ ζ † , ↓ (cid:34) m − (cid:89) α =1 ζ † α, ↑ ζ † α, ↓ (cid:35) ζ † m, ↓ (cid:34) M − (cid:89) α = m +1 ζ † α, ↑ ζ † α, ↓ (cid:35) | (cid:105) − 32 ( M − ζ † , ↓ | PFS (cid:105) . (A20)At this point, one can see why this choice of H P is promising: the O K , ↓ terms in Eqs. (A15) and (A19) cancel eachother; the unwanted terms in Eq. (A20) have the same form as those in Eq. (A16). It is obvious that | Ψ ↑ K (cid:105) + | Ψ ↑ P (cid:105) is c † , ↑ (cid:40) − M − (cid:88) m =1 ζ † , ↑ ζ † , ↓ (cid:34) m − (cid:89) α =1 ζ † α, ↑ ζ † α, ↓ (cid:35) ζ † m, ↓ (cid:34) M − (cid:89) α = m +1 ζ † α, ↑ ζ † α, ↓ (cid:35) | (cid:105) (cid:41) − 32 ( M − c † , ↑ ζ † , ↓ | PFS (cid:105) , (A21)where the first term represents particle-hole excitations on top of the Fermi sea. In order to find a parent Hamiltonian,these particle-hole excitations should be canceled by contributions from H | PFS (cid:105) . B3: The hopping term The hopping term is designed as H = L − (cid:88) q =0 (cid:88) σ = ↑ , ↓ q d † q,σ d q,σ , (A22)where d q,σ = √ L (cid:80) Lj =1 c j,σ for q = 0 (cid:113) L (cid:80) Lj =1 cos( qθ j ) c j,σ for 1 ≤ q ≤ L − . (A23)The angles are chosen to be θ j = πL ( j − ) to make sure that the fermionic modes d q,σ form an orthonormaland complete single-particle basis. This can be verified using the discrete orthogonality property of the Chebyshevpolynomials.Let us prove that H has an inverse-square form in the original fermion basis. It is obvious that H = L − (cid:88) q =1 L (cid:88) j,k L q cos( qθ j ) cos( qθ k ) c † j,σ c k,σ = L (cid:88) j =1 L − (cid:88) q =1 L q [1 + cos(2 qθ j )] c † j,σ c j,σ + L (cid:88) j,k =1; j (cid:54) = k L − (cid:88) q =1 L q [cos( qθ j − qθ k ) + cos( qθ j + qθ k )] c † j,σ c k,σ = L (cid:88) j =1 L − (cid:88) q =1 L q (cid:110) (cid:104) qπL (2 j − (cid:105)(cid:111) c † j,σ c j,σ + L (cid:88) j,k =1; j (cid:54) = k L − (cid:88) q =1 L q (cid:110) cos (cid:104) qπL ( j − k ) (cid:105) + cos (cid:104) qπL ( j + k + 1) (cid:105)(cid:111) c † j,σ c k,σ . (A24)For an integer X ∈ [0 , L ), we have the summation identity L − (cid:88) q =1 q cos (cid:16) qπL X (cid:17) = ( − X L (cid:20) ( πX L ) − (cid:21) X (cid:54) =0 L ( L − L − X = 0 . (A25)This helps us to show that H = L (cid:88) j,k =1; j (cid:54) = k ( − j − k (cid:18) | z j − z k | − | z j − z ∗ k | (cid:19) c † j,σ c k,σ + L (cid:88) j =1 (cid:20) L + 12 − 32 sin θ j (cid:21) c † j,σ c j,σ , (A26)1where z j = exp( iθ j ) and z ∗ j = exp( − iθ j ) are the complex coordinates of the site j and its mirror image, respectively.To compute the results of acting H on | Ψ (cid:105) , we need to know commutators of the [ H , ζ † m ] type. This requires lineartransformations between the two sets of single-particle orbitals defined by (the spin index σ is suppressed below forsimplicity) ζ † m = L − (cid:88) q =0 W mq d † q d † q = L − (cid:88) m =0 ( W − ) qm ζ † m , (A27)One can see that (cid:2) H , ζ † m (cid:3) = L − (cid:88) q,q (cid:48) =0 q W mq (cid:48) (cid:104) d † q d q , d † q (cid:48) (cid:105) = L − (cid:88) q =0 q W mq d † q = L − (cid:88) q =0 L − (cid:88) m (cid:48) =0 q W mq (cid:0) W − (cid:1) qm (cid:48) ζ † m (cid:48) = L − (cid:88) m (cid:48) =0 A mm (cid:48) ζ † m (cid:48) with A mm (cid:48) = L − (cid:88) q =0 q W mq (cid:0) W − (cid:1) qm (cid:48) . (A28)For the m = 0 case, we have (cid:104) H , ζ † (cid:105) = 0 because d † is a zero mode of H and ζ † = √ Ld † . This helps us to provethat H | Ψ (cid:105) = | Ψ ↑ (cid:105) + | Ψ ↓ (cid:105) with | Ψ ↑ (cid:105) = c † , ↑ H ζ † , ↓ | PFS (cid:105) = c † , ↑ ζ † , ↓ H | PFS (cid:105) = c † , ↑ (cid:40) − ζ † , ↑ ζ † , ↓ M − (cid:88) m =1 (cid:34) m − (cid:89) α =1 ζ † α, ↑ ζ † α, ↓ (cid:35) A m ζ † m, ↓ (cid:34) M − (cid:89) α = m +1 ζ † α, ↑ ζ † α, ↓ (cid:35) | (cid:105) + 2 ζ † , ↓ M − (cid:88) m =1 A mm | PFS (cid:105) (cid:41) , (A29)so the diagonal entries A mm and the first column A m of the matrix A mm (cid:48) are sufficient for our purpose.Let us consider the transformation matrix from d † q to ζ † m . The m = 0 case is simple since ζ † = (cid:80) Lj =1 c † j = √ Ld † .For m ≥ 1, we have ζ † m = L (cid:88) j =1 cos m − θ j (1 − cos θ j ) c † j = L (cid:88) j =1 (cid:34)(cid:18) e iθ j + e − iθ j (cid:19) m − − (cid:18) e iθ j + e − iθ j (cid:19) m (cid:35) c † j = L (cid:88) j =1 c † j m × (cid:80) m − α = − m − (cid:18) m − m − + α (cid:19) e i [( m − + α ) − ( m − − α )] θ j − (cid:80) m +12 α = − m − (cid:18) m m − + α (cid:19) e i [( m − + α ) − ( m +12 − α )] θ j odd m ≥ (cid:80) m α = − m +1 (cid:18) m − m − α (cid:19) e i [( m − α ) − ( m − α )] θ j − (cid:80) m α = − m (cid:18) m m + α (cid:19) e i [( m + α ) − ( m − α )] θ j even m ≥ L (cid:88) j =1 c † j m (cid:18) m − m − (cid:19) + 4 (cid:80) m − α =1 (cid:18) m − m − + α (cid:19) cos(2 αθ j ) − (cid:80) m +12 α =1 (cid:18) m m − + α (cid:19) cos(2 α − θ j odd m ≥ (cid:80) m α =1 (cid:18) m − m − α (cid:19) cos(2 α − θ j − (cid:18) m m (cid:19) − (cid:80) m α =1 (cid:18) m m + α (cid:19) cos(2 αθ j ) even m ≥ (cid:114) L m √ (cid:18) m − m − (cid:19) d † + 4 (cid:80) m − α =1 (cid:18) m − m − + α (cid:19) d † α − (cid:80) m +12 α =1 (cid:18) m m − + α (cid:19) d † α − odd m ≥ (cid:80) m α =1 (cid:18) m − m − α (cid:19) d † α − − √ (cid:18) m m (cid:19) d † − (cid:80) m α =1 (cid:18) m m + α (cid:19) d † α even m ≥ . (A30)2The matrix W mq is √ Lδ q m = 0 (cid:113) L m (cid:20) √ (cid:18) m − m − (cid:19) δ q + 4 (cid:80) m − α =1 (cid:18) m − m − + α (cid:19) δ q, α − (cid:80) m +12 α =1 (cid:18) m m − + α (cid:19) δ q, α − (cid:21) odd m ≥ (cid:113) L m (cid:20) −√ (cid:18) m m (cid:19) δ q + 4 (cid:80) m α =1 (cid:18) m − m − α (cid:19) δ q, α − − (cid:80) m α =1 (cid:18) m m + α (cid:19) δ q, α (cid:21) even m ≥ , (A31)which has a lower triangular form (i.e., W mq = 0 if q > m ) W = (cid:114) L √ √ − − √ − √ − 34 12 − − √ − 12 14 − √ − 58 12 − 516 18 − − √ − − 316 116 − · · · . . . . (A32)In particular, W mm = (cid:113) L ( √ δ m − m − δ m (cid:54) =0 ).Let us now consider the transformation matrix from ζ † m to d † q . It should also be a triangular matrix with ( W − ) qm =0 if m > q . The q = 0 case is simple since d † = (cid:113) L (cid:80) Lj =1 c † j = (cid:113) L ζ † . For q > 0, we have d † q> = (cid:114) L L (cid:88) j =1 cos( qθ j ) c † j = (cid:114) L q (cid:98) q/ (cid:99) (cid:88) k =0 ( − k ( q − k − k !( q − k )! 2 q − k L (cid:88) j =1 cos q − k θ j c † j (A33)= (cid:114) L q (cid:98) q/ (cid:99) (cid:88) k =0 ( − k ( q − k − k !( q − k )! 2 q − k (cid:32) ζ † − q − k (cid:88) p =1 ζ † p (cid:33) = q (cid:88) m =0 ( W − ) q> ,m ζ † m . (A34)The second equal sign is true because cos( qθ j ) = T q (cos θ j ), where T q ( x ) = q (cid:98) q/ (cid:99) (cid:88) k =0 ( − k ( q − k − k !( q − k )! (2 x ) q − k (A35)is the Chebyshev polynomial. The useful matrix elements are( W − ) q> ,q> = − (cid:114) L q − (A36)and ( W − ) q> , = (cid:114) L q (cid:98) q/ (cid:99) (cid:88) k =0 ( − k ( q − k − k !( q − k )! 2 q − k = (cid:114) L . (A37)The transformation matrices derived above can be used to prove that A m> ,m> = L − (cid:88) q =0 q W mq ( W − ) qm = 3 m W m> ,m> ( W − ) m> ,m> = 3 m (A38)3and A m> , = L − (cid:88) q =0 q W mq ( W − ) q = L − (cid:88) q =1 q W m> ,q ( W − ) q = L − (cid:88) q =1 q m (cid:20) √ (cid:18) m − m − (cid:19) δ q + 4 (cid:80) m − α =1 (cid:18) m − m − + α (cid:19) δ q, α − (cid:80) m +12 α =1 (cid:18) m m − + α (cid:19) δ q, α − (cid:21) odd m ≥ m (cid:20) −√ (cid:18) m m (cid:19) δ q + 4 (cid:80) m α =1 (cid:18) m − m − α (cid:19) δ q, α − − (cid:80) m α =1 (cid:18) m m + α (cid:19) δ q, α (cid:21) even m ≥ 2= 12 m (cid:80) m − α =1 α (cid:18) m − m − + α (cid:19) − (cid:80) m +12 α =1 α − (cid:18) m m − + α (cid:19) odd m ≥ (cid:80) m α =1 α − (cid:18) m − m − α (cid:19) − (cid:80) m α =1 α (cid:18) m m + α (cid:19) even m ≥ 2= 12 m (cid:40) m − m − − m m − m − m − − m m = − , (A39)where certain summation identities from Sec. B4 are used. For comparison, the matrix A computed using Mathematica is A = − − − − − − 15 21 48 − − 33 27 75 − − 57 33 108 · · · . . . . (A40)Finally, we arrive at | Ψ ↑ (cid:105) = c † , ↑ (cid:40) − ζ † , ↑ ζ † , ↓ M − (cid:88) m =1 (cid:34) m − (cid:89) α =1 ζ † α, ↑ ζ † α, ↓ (cid:35) A m ζ † m, ↓ (cid:34) M − (cid:89) α = m +1 ζ † α, ↑ ζ † α, ↓ (cid:35) | (cid:105) + 2 ζ † , ↓ M − (cid:88) m =1 A mm | PFS (cid:105) (cid:41) = c † , ↑ (cid:40) ζ † , ↑ ζ † , ↓ M − (cid:88) m =1 (cid:34) m − (cid:89) α =1 ζ † α, ↑ ζ † α, ↓ (cid:35) ζ † m, ↓ (cid:34) M − (cid:89) α = m +1 ζ † α, ↑ ζ † α, ↓ (cid:35) | (cid:105) + ζ † , ↓ M − (cid:88) m =1 m | PFS (cid:105) (cid:41) . (A41)Its first part and the unwanted particle-hole excitations in Eq. (A21) cancel each other. It is then obvious that( H + H P + H K ) | Ψ (cid:105) = (cid:34) M − (cid:88) m =1 m − 32 ( M − (cid:35) (cid:16) c † , ↑ ζ † , ↓ − c † , ↓ ζ † , ↑ (cid:17) | PFS (cid:105) , (A42)so the target state | Ψ (cid:105) is indeed an eigenstate with eigenvalue E ( M ) = M − (cid:88) m =1 m − 32 ( M − 1) = ( M − (cid:18) M − M − (cid:19) . (A43) B4: Some useful identities This subsection computes the sums in (A39). For odd m ≥ 1, we introduce an auxiliary function( x + x − ) m − = m − (cid:88) α = − m − (cid:18) m − m − + α (cid:19) x m − + α x − ( m − − α ) = m − (cid:88) α = − m − (cid:18) m − m − + α (cid:19) x α = (cid:18) m − m − (cid:19) + m − (cid:88) α =1 (cid:18) m − m − + α (cid:19) ( x α + x − α ) . (A44)4Its derivatives are ( m − x + x − ) m − (1 − x − ) = m − (cid:88) α =1 (cid:18) m − m − + α (cid:19) α (cid:0) x α − − x − α − (cid:1) (A45)and ( m − m − x + x − ) m − (1 − x − ) + ( m − x + x − ) m − x = m − (cid:88) α =1 (cid:18) m − m − + α (cid:19) α (cid:2) (2 α − x α − + (2 α + 1) x − α − (cid:3) . (A46)If x is chosen to be 1 in Eq. (A46), we obtain( m − m − = m − (cid:88) α =1 (cid:18) m − m − + α (cid:19) α . (A47)For odd m , we study( x + x − ) m = m +12 (cid:88) α = − m − (cid:18) m m − + α (cid:19) x m − + α x − ( m +12 − α ) = m +12 (cid:88) α = − m − (cid:18) m m − + α (cid:19) x α − = m +12 (cid:88) α =1 (cid:18) m m − + α (cid:19) ( x α − + x − α +1 ) . Its derivatives are m ( x + x − ) m − (1 − x − ) = m +12 (cid:88) α =1 (cid:18) m m − + α (cid:19) (cid:2) (2 α − x α − − (2 α − x − α (cid:3) , (A48)and m ( m − x + x − ) m − (1 − x − ) + m ( x + x − ) m − x = m +12 (cid:88) α =1 (cid:18) m m − + α (cid:19) (cid:2) (2 α − α − x α − + (2 α − αx − α − (cid:3) . (A49)If x is chosen to be 1 in Eq. (A49), we obtain m m − = m +12 (cid:88) α =1 (cid:18) m m − + α (cid:19) (2 α − . (A50)The sums with even m can be obtained from previous ones. In Eq. (A47), replacing m − m yields m m = m (cid:88) α =1 (cid:18) m m + α (cid:19) α . (A51)In Eq. (A50), replacing m + 1 with m yields( m − m − = m (cid:88) α =1 (cid:18) m − m − α (cid:19) (2 α − . (A52)5 APPENDIX C: SPIN-SPIN CORRELATION FUNCTION This section provides more details about how to compute the spin-spin correlation function of the ground state.To begin with, we compute the overlap of two generic free fermion states. For a set of fermionic operators c † j and c j satisfying the anticommutation relations, we define two sets of operators α † a and β † b α a = (cid:88) j U aj c j , β b = (cid:88) j V bj c j , (A53)where U aj and V bj are two matrices that may not be isometry. It is easy to see that (cid:104) | α a β † b | (cid:105) = (cid:88) jk U aj V ∗ bk (cid:104) | c j c † k | (cid:105) = (cid:88) j U aj V ∗ bj = (cid:88) j (cid:2) U V † (cid:3) ab = G ab (A54)with G = U V † . The overlap between α † α † · · · α † M | (cid:105) and β † β † · · · β † M | (cid:105) is found to be (cid:104) | α M · · · α α β † β † · · · β M | (cid:105) = (cid:104) | α β † | (cid:105)(cid:104) | α β † | (cid:105) · · · (cid:104) | α M β † M | (cid:105) + all possible permutations with signs= G G · · · G MM + all possible permutations with signs= det G (A55)using Eq. (A54) and Wick’s theorem.The numerator of the spin-spin correlation function can be simplified using the SU(2) spin symmetry as (cid:104) Ψ | S · S j | Ψ (cid:105) = (cid:104) Ψ | S · S j | Ψ (cid:105) = 3 (cid:104) Ψ | S x S xj | Ψ (cid:105) = 32 (cid:0) (cid:104) Ψ | S x S xj | Ψ (cid:105) + (cid:104) Ψ | S y S yj | Ψ (cid:105) (cid:1) = 32 (cid:104) Ψ | S +0 S − j | Ψ (cid:105) (A56)with (cid:104) Ψ | S +0 S − j | Ψ (cid:105) = (cid:104) | (cid:48) (cid:89) m = M − (cid:89) σ = ↓ , ↑ ζ m,σ ζ , ↑ c j, ↑ c † j, ↓ ζ † , ↑ M − (cid:89) m =1 (cid:89) σ = ↑ , ↓ ζ † m,σ | (cid:105) = −(cid:104) | (cid:48) (cid:89) m = M − ζ m, ↑ c j, ↑ ζ † , ↑ M − (cid:89) m =1 ζ † m, ↑ | (cid:105) · (cid:104) | (cid:48) (cid:89) m = M − ζ m, ↓ ζ , ↓ c † j, ↓ M − (cid:89) m =1 ζ † m, ↓ | (cid:105) , (A57)where (cid:81) (cid:48) m = M − (cid:81) σ = ↓ , ↑ ζ m,σ denotes reversed order of products. The denominator of the spin-spin correlation functionis the unnormalized overlap (cid:104) Ψ | Ψ (cid:105) = (cid:104) | (cid:48) (cid:89) m = M − ζ m, ↑ M − (cid:89) m =1 ζ † m, ↑ | (cid:105) · (cid:104) | (cid:48) (cid:89) m = M − ζ m, ↓ ζ , ↓ ζ † , ↓ M − (cid:89) m =1 ζ † m, ↓ | (cid:105) + (cid:104) | (cid:48) (cid:89) m = M − ζ m, ↑ ζ , ↑ ζ † , ↑ M − (cid:89) m =1 ζ † m, ↑ | (cid:105) · (cid:104) | (cid:48) (cid:89) m = M − ζ m, ↓ M − (cid:89) m =1 ζ † m, ↓ | (cid:105) ..