Properties of the simplest inhomogeneous and homogeneous Tree-Tensor-States for Long-Ranged Quantum Spin Chains with or without disorder
PProperties of the simplest inhomogeneous and homogeneous Tree-Tensor-Statesfor Long-Ranged Quantum Spin Chains with or without disorder
C´ecile Monthus
Institut de Physique Th´eorique, Universit´e Paris Saclay, CNRS, CEA, 91191 Gif-sur-Yvette, France
The simplest Tree-Tensor-States (TTS) respecting the Parity and the Time-Reversal symmetriesare studied in order to describe the ground states of Long-Ranged Quantum Spin Chains withor without disorder. Explicit formulas are given for the one-point and two-point reduced densitymatrices that allow to compute any one-spin and two-spin observable. For Hamiltonians containingonly one-body and two-body contributions, the energy of the TTS can be then evaluated andminimized in order to obtain the optimal parameters of the TTS. This variational optimization ofthe TTS parameters is compared with the traditional block-spin renormalization procedure basedon the diagonalization of some intra-block renormalized Hamiltonian.
I. INTRODUCTION
The entanglement between the different regions of many-body quantum systems (see the reviews [1–6] and refer-ences therein) has emerged as an essential physical property that should be taken into account in their descriptions.In the field of Tensor Networks (see the reviews [7–17] and references therein), the ground-state wavefunction is de-composed into elementary small tensors that can be assembled in various ways in order to adapt to the geometry,to the symmetries, and to the entanglement properties of the problem under focus. In particular, various previ-ous real-space renormalization procedures for the ground states of quantum spin chains have been reinterpreted andpossibly improved within this new perspective. For instance the Density-Matrix-Renormalization-Group [18–20] wasreformulated as a variational problem based on Matrix-Products-States that are well adapted to describe non-criticalstates displaying area-law entanglement. The traditional block-spin renormalization for critical points corresponds toscale-invariant Tree-Tensor-States, and has been improved via the multi-scale-entanglement-renormalization-ansatz(MERA) [21, 22], where ’disentanglers’ between blocks are introduced besides the block-coarse-graining isometriesalready present in Tree-Tensor-States. Finally in the field of disordered spin chains, the Strong Disorder Renormaliza-tion approach (see the reviews [23, 24]) has been reformulated either as a Matrix-Product-Operator-Renormalizationor as a self-assembling Tree-Tensor-Network, and various improvements have been proposed [25–30].However, even in the second example where the ’old’ block-spin renormalization procedure and the ’new’ Tree-Tensor-State variational approach share the same entanglement architecture, the precise choice of the elementaryisometries remains different. Indeed in the traditional block-spin renormalization, the isometries are determined viathe diagonalization of some ’intra-block’ Hamiltonian involving a few renormalized spins, so that one can usuallyobtain explicit RG flows for the parameters of the renormalized Hamiltonian. The two main criticisms levelledagainst this procedure can be summarized as follows : (i) at the level of principles, the choice of the ground stateof the ’intra-block Hamiltonian’ does not take at all into account the ’environment’ of the neighboring blocks; (ii) inpractice, there is usually some arbitrariness in the decomposition of the Hamiltonian into the ’intra-block’ and the’extra-block’ contributions that can lead to completely different outputs, so that the quality of the results stronglydepends on the cleverness of the choice of the intra-block Hamiltonian. In the Tree-Tensor-Network perspective, oneconsiders instead the whole ground-state wavefunction as a variational tree-tensor-state involving isometries, and theoptimization of each isometry is based on the minimization of the total energy of the Tree-Tensor-State. At the levelof principles, the theoretical advantage is clearly that the output corresponds to the optimal Tree-Tensor-State, i.e.to the best renormalization procedure within the class of all renormalization procedures of a given dimension. Inpractice, the drawback is that this global optimization is more complicated and can usually be done only numerically,unless the isometries are completely fixed by the very strong quantum symmetries of the model [31].In the present paper, the goal is to analyze the explicit properties of the simplest Tree-Tensor-States of the smallestbond dimension D = 2 in the context of Long-Ranged quantum spin chains with or without disorder, in orderto analyze more precisely the improvement given by the global optimization of the isometries with respect to thetraditional block-spin procedure.The paper is organized as follows. In section II, we introduce the notations for Long-Ranged quantum spin chainswith Parity and Time-Reversal symmetries. In section III, we describe the simplest inhomogeneous Tree-Tensor-Statesrespecting these two symmetries and write the corresponding ascending and descending superoperators. In sectionIV, the explicit forms of the one-point and two-point reduced density matrices are derived in order to analyze thestructure of magnetizations and two-points correlations. In section V, we focus on the energy of the Tree-Tensor-Statefor disordered Long-Ranged Spin Chains in order to discuss the optimization with respect to the Tree-Tensor-Statesparameters. In section VI, we turn to the case of pure Long-Ranged Spin Chains in order to take into account the a r X i v : . [ c ond - m a t . d i s - nn ] J a n supplementary symmetries in the Tree-Tensor-States. Finally in section VII, we study the properties of the scale-invariant Tree-Tensor-States for the critical points of pure models. Our conclusions are summarized in section VIII.The appendix A contains the traditional block-spin determination of the parameters of the Tree-Tensor-State, in orderto compare with the variational optimization discussed in the text. II. LONG-RANGED SPIN CHAINS WITH PARITY AND TIME-REVERSAL SYMMETRIES
Within the Tensor-Network perspective, the symmetries play an essential role in order to restrict the form of thepossible isometries. In the present paper, we focus on quantum spin chains with Parity and Time-Reversal Symmetries.
A. Parity and Time-Reversal operators
For a quantum spin chain of N spins described by Pauli matrices σ a =0 ,x,y,zn , the Parity operator P = N (cid:89) n =1 σ zn (1)and the Time-Reversal operator T whose action can be defined via T i T − = − i T σ xn T − = σ xn T σ yn T − = − σ yn T σ zn T − = σ zn (2)are among the most important possible symmetries. It is then useful to decompose the space of operators O intosectors that commute P = +1 or anticommute P = − P , and that commute T = +1 oranticommute T = − TPO = P OPT O = T OT (3)Let us now describe the classification of one-body and two-body operators with respect to these four symmetry sectors( P = ± , T = ± B. Classification of one-body operators σ a =0 ,x,y,zn with respect to the four sectors ( P = ± , T = ± The four operators σ a =0 ,x,y,zn of the Pauli basis can be classified as follows :(1) the sector ( P = +1 , T = −
1) is empty(2) the sector ( P = − , T = +1) contains only σ xn (3) the sector ( P = − , T = −
1) contains only σ yn (4) the sector ( P = +1 , T = +1) contains the two operators σ n and σ zn . C. Classification of two-body operators σ a =0 ,x,y,zn σ b =0 ,x,y,zn (cid:48) with respect to the four sectors ( P = ± , T = ± The 16 two-body operators σ a =0 ,x,y,zn σ b =0 ,x,y,zn (cid:48) of the Pauli basis can be classified as follows :(1) the sector ( P = +1 , T = −
1) contains the two operators σ xn σ yn (cid:48) , σ yn σ xn (cid:48) (4)(2) the sector ( P = − , T = +1) contains the four operators σ xn σ n (cid:48) , σ n σ xn (cid:48) , σ xn σ zn (cid:48) , σ zn σ xn (cid:48) (5)(3) the sector ( P = − , T = −
1) contains the four operators σ yn σ n (cid:48) , σ n σ yn (cid:48) , σ yn σ zn (cid:48) , σ zn σ yn (cid:48) (6)(4) the sector ( P = +1 , T = +1) contains the six operators σ n σ n (cid:48) , σ zn σ n (cid:48) , σ n σ zn (cid:48) , σ zn σ zn (cid:48) , σ xn σ xn (cid:48) , σ yn σ yn (cid:48) (7) D. Hamiltonians containing only one-body and two-body terms respecting the Parity and Time-Reversal
The Hamiltonians commuting with the Parity and Time-Reversal operators belong to the sector ( P = +1 , T = +1).If they contain only one-body and two-body terms, the above classification yields that they can be parametrized interms of fields h n and in terms of couplings J a = x,y,zn,n (cid:48) as H N = − N (cid:88) n =1 h n σ zn − (cid:88) ≤ n Dyson hierarchical models are based on the following binary tree structure. The generation g = 0 contains the singlesite called the root. The generation g = 1 contains its two children labelled by the index i = 1 , 2. The generation g = 2 contains the two children i = 1 , i = 1 , g = 1, and so on. So the generation g contains N g = 2 g sites labelled by the g binary indices ( i , i , .., i g ) that indicate the whole line of ancestors up to theroot at g = 0.The Dyson hierarchical version of the Long-Ranged Hamiltonian of Eqs 8 and 9 is then defined for a chain of N G = 2 G spins σ I labelled by the positions I = ( i , i , ..., i G ) of the last generation G of the tree structure by H [ G ] = − (cid:88) I =( i ,i ,..,i G ) h I σ zI − (cid:88) I =( i ,i ,..,i G )
1) indices I c = ( i , .., i G − k − ), while they have different indices i G − k = 1 and i G − k = 2 at generation( G − k ), are separated by the distance r ( I = ( I c , i G − k = 1 , F ) , I (cid:48) = ( I c , i G − k = 2 , F (cid:48) )) ≡ k (14)for any values of the remaining indices F = ( i G − k +1 , .., i G ) and F (cid:48) = ( i (cid:48) G − k +1 , .., i (cid:48) G ). The minimal value k = 0corresponds to the distance r = 2 = 1 between spins that have the same ancestor at position I c = ( i , i , .., i G − ) ofthe generation ( G − 1) while they differ i G = 1 , G (here F and F (cid:48) are empty). The maximal value k = G − r = 2 G − = N G between any spin belonging to the first half i = 1 and anyspin belonging to the second half i = 2 (here I c is empty and their Last Common ancestor is the root 0). As aconsequence, the Dyson Hamiltonian of Eq. 12 can be rewritten more explicitly as a sum over the index k = 0 , ..., G − r k = 2 k as H [ G ] = − (cid:88) I =( i ,i ,..,i G ) h I σ zI − G − (cid:88) k =0 (cid:88) I c =( i ,i ,..,i G − k − ) (cid:88) F =( i G − k +1 ,..,i G ) (cid:88) F (cid:48) =( i (cid:48) G − k +1 ,..,i (cid:48) G ) (cid:88) a = x,y,z J aI c F,I c F (cid:48) k (1+ ω a ) σ aI c F σ aI c F (cid:48) (15)The Dyson hierarchical version of the pure quantum Ising model of Eq. 10 has been already studied via block-spinrenormalization in order to analyze its critical properties [42] and its entanglement properties for various bipartitepartitions [43]. The block-spin renormalization has also been used to analyze the Dyson random transverse field Isingmodel [42] and the quantum spin-glass in uniform transverse field [44].More generally, the Dyson hierarchical versions of many long ranged models have been considered since the originalDyson hierarchical classical ferromagnetic Ising model [45] that has been much studied by both mathematicians [46–49] and physicists [50–53], including the properties of the dynamics [54, 55]. In the field of classical disordered systems,equilibrium properties have been analyzed for random fields Ising models [56, 57] and for spin-glasses [58–63], whilethe dynamical properties are discussed in Refs [55, 64]. Finally, let us mention that Dyson hierarchical models havebeen also considered for Anderson localization models [65–72] and for Many-Body-Localization [73]. III. SIMPLEST TREE-TENSOR-STATES WITH PARITY AND TIME-REVERSAL SYMMETRIES In this section, the goal is to construct the simplest inhomogeneous Tree-Tensor-States for disorder spin chainswith Parity and Time-Reversal symmetries, while the case of homogeneous Tree-Tensor-States for pure spin chains ispostponed to the sections VI and VII where their supplementary symmetries will be taken into account. A. Isometries based on blocks of two spins preserving the ( P, T ) symmetries The traditional block-spin renormalization procedure based on blocks of two sites can be summarized as follows interms of the tree notations introduced in the subsection II E. The initial chain of N G = 2 G spins σ [ G ] i ,i ,..,i G belongingto the last generation G will be first renormalized into a chain of N G − = 2 G − = N G spins σ [ G − i ,i ,..,i G − of generation( G − . This procedure will be then iterated up to the last RG step where there will be a single spin σ [ g =0] at theroot corresponding to generation g = 0.The basic block-spin RG step is implemented by the elementary coarse-graining isometry w [ g,I ] , where I = ( i , .., i g )labels the possible positions at generation g , between the two-dimensional Hilbert space of the renormalized spin | σ [ g ] zI = ±(cid:105) and the two relevant states | ψ [ g +1] ± I ,I (cid:105) that are kept out of the four-dimensional Hilbert space of its twochildren | σ [ g +1] zI = ± , σ [ g +1] zI = ±(cid:105) w [ g,I ] ≡ | ψ [ g +1]+ I ,I (cid:105) (cid:104) σ [ g ] zI = + | + | ψ [ g +1] − I ,I (cid:105) (cid:104) σ [ g ] zI = −| (16)So the product (cid:16) w [ g,I ] (cid:17) † (cid:16) w [ g,I ] (cid:17) = | σ [ g ] zI = + (cid:105) (cid:104) σ [ g ] zI = + | + | σ [ g ] zI = −(cid:105) (cid:104) σ [ g ] zI = −| = σ [ g ]0 I (17)is simply the identity operator σ [ g ]0 I of the Hilbert space of the ancestor spin, while the product (cid:16) w [ g,I ] (cid:17) (cid:16) w [ g,I ] (cid:17) † = | ψ [ g +1]+ I ,I (cid:105) (cid:104) ψ [ g +1]+ I ,I | + | ψ [ g +1] − I ,I (cid:105) (cid:104) ψ [ g +1] − I ,I | ≡ Π ψ [ g +1] ± I ,I (18)corresponds to the projector Π ψ [ g +1] ± I ,I onto the subspace spanned by the two states ψ [ g +1] ± I ,I that are kept out of thefour-dimensional Hilbert space of the two children.In order to preserve the Parity, the normalized ket | ψ [ g +1]+ I ,I (cid:105) will be chosen as some linear combination of the twostates of positive parity | σ [ g +1] zI = + , σ [ g +1] zI = + (cid:105) and | σ [ g +1] zI = − , σ [ g +1] zI = −(cid:105) for the block of the two children.Since the Time-Reversal-Symmetry imposes real coefficients, the parametrization involves only a single angle θ [ g,I ]+ | ψ [ g +1]+ I ,I (cid:105) = cos( θ [ g,I ]+ ) | σ [ g +1] zI = + , σ [ g +1] zI = + (cid:105) + sin( θ [ g,I ]+ ) | σ [ g +1] zI = − , σ [ g +1] zI = −(cid:105) (19)Similarly, the ket | ψ [ g +1] − I ,I (cid:105) will be chosen as some linear combination of the two states of negative parity | σ [ g +1] zI = + , σ [ g +1] zI = −(cid:105) and | σ [ g +1] zI = − , σ [ g +1] zI = + (cid:105) for the block of the two children and involves only anothersingle angle θ [ g,I ] − | ψ [ g +1] − I ,I (cid:105) = cos( θ [ g,I ] − ) | σ [ g +1] zI = + , σ [ g +1] zI = −(cid:105) + sin( θ [ g,I ] − ) | σ [ g +1] zI = − , σ [ g +1] zI = + (cid:105) (20)From the ascending block-spin renormalization perspective, Eqs 19 and 20 parametrize the representative statesthat are kept in each two-dimensional parity sector P = ± respectively. From the descending perspective, Eqs 19 and20 can be interpreted as the Schmidt decompositions of the kept state of parity P = ± in terms of the states of itstwo children | ψ [ g +1] PI ,I (cid:105) = (cid:88) α = ± Λ [ g,I ] Pα | σ [ g +1] zI = α (cid:105) ⊗ | σ [ g +1] zI = αP (cid:105) (21)where the two Schmidt singular values are given byΛ [ g,I ] Pα =+ = cos( θ [ g,I ] P )Λ [ g,I ] Pα = − = sin( θ [ g,I ] P ) (22)while the kets | σ [ g +1] zI = α (cid:105) and | σ [ g +1] zI = αP (cid:105) correspond to the associated Schmidt eigenvectors of the first childand the second child respectively. Indeed, the partial traces over a single child of the projector associated to the stateof parity P of Eq. 21 is diagonal for these eigenvectorsTr { I } (cid:16) | ψ [ g +1] PI ,I (cid:105) (cid:104) ψ [ g +1] PI ,I | (cid:17) = (cid:88) α = ± (cid:16) Λ [ g,I ] Pα (cid:17) | σ [ g +1] zI = α (cid:105) (cid:104) σ [ g +1] zI = α | Tr { I } (cid:16) | ψ [ g +1] PI ,I (cid:105) (cid:104) ψ [ g +1] PI ,I | (cid:17) = (cid:88) α = ± (cid:16) Λ [ g,I ] Pα (cid:17) | σ [ g +1] zI = αP (cid:105) (cid:104) σ [ g +1] zI = αP | (23)and (cid:16) Λ [ g,I ] Pα = ± (cid:17) are the two common weights normalized to unity as it should (cid:88) α = ± (cid:16) Λ [ g,I ] Pα (cid:17) = cos ( θ [ g,I ] P ) + sin ( θ [ g,I ] P ) = 1 (24) B. Local ascending and descending super-operators A [ g,I ] and D [ g,I ] The local ascending superoperator A [ g,I ] describes how the the 16 two-spin operators σ [ g +1] a =0 ,x,y,zI σ [ g +1] a =0 ,x,y,zI of the two children are projected onto the four Pauli operators σ [ g ] a =0 ,x,y,zI of their ancestor via the isometry w [ g,I ] A [ g,I ] (cid:104) σ [ g +1] a I σ [ g +1] a I (cid:105) ≡ ( w [ g,I ] ) † (cid:16) σ [ g +1] a I σ [ g +1] a I (cid:17) w [ g,I ] = (cid:88) a =0 ,x,y,z F [ g,I ] aa ,a σ [ g ] aI (25)where the fusion coefficients F [ g,I ] aa ,a = 12 Tr { I } (cid:16)(cid:104) A [ g,I ] [ σ [ g +1] a I σ [ g +1] a I ] (cid:105) σ [ g ] aI (cid:17) = 12 Tr { I } (cid:16)(cid:104) ( w [ g,I ] ) † (cid:16) σ [ g +1] a I σ [ g +1] a I (cid:17) w [ g,I ] (cid:105) σ [ g ] aI (cid:17) (26)can be rewritten as F [ g,I ] aa ,a = 12 Tr { I ,I } (cid:16) σ [ g +1] a I σ [ g +1] a I (cid:104) w [ g,I ] σ [ g ] aI ( w [ g,I ] ) † (cid:105)(cid:17) (27)As a consequence, the local descending superoperator D [ g,I ] that translates the four ancestor spin operators σ [ g ] a =0 ,x,y,zI into operators for its two children involves the same fusion coefficients D [ g,I ] (cid:104) σ [ g ] I (cid:105) ≡ w [ g,I ] σ [ g ] aI (cid:16) w [ g,I ] (cid:17) † = 12 (cid:88) a =0 ,x,y,z (cid:88) a =0 ,x,y,z F [ g,I ] aa ,a σ [ g +1] a I σ [ g +1] a I (28)Since the isometry w [ g,I ] preserves the Parity and the Time-Reversal symmetries, the fusion rules respect the foursymmetry sectors ( P = ± , T = ± 1) of operators described in the subsections II B and II C. As a consequence, thetwo operators of Eq. 4 corresponding to the sector ( P = +1 , T = − 1) are projected out A [ g,I ] (cid:104) σ [ g +1] xI σ [ g +1] yI (cid:105) = 0 A [ g,I ] (cid:104) σ [ g +1] yI σ [ g +1] xI (cid:105) = 0 (29)and will never be produced by D [ g,I ] .The fusion rules in the three other non-trivial symmetry sectors are described in the following subsections in termsof the two angles φ [ g,I ] ≡ π − θ [ g,I ]+ − θ [ g,I ] − ˜ φ [ g,I ] ≡ − θ [ g,I ]+ + θ [ g,I ] − (30)with the following notations for their cosinus and sinus c [ g,I ] ≡ cos (cid:16) φ [ g,I ] (cid:17) s [ g,I ] ≡ sin (cid:16) φ [ g,I ] (cid:17) ˜ c [ g,I ] ≡ cos (cid:16) ˜ φ [ g,I ] (cid:17) ˜ s [ g,I ] ≡ sin (cid:16) ˜ φ [ g,I ] (cid:17) (31)in order to obtain simpler explicit expressions. C. Local fusion rules for operators in the symmetry sector ( P = − , T = +1) The action of the ascending superoperator A [ g,I ] on the four operators of Eq. 5 corresponding to the symmetrysector ( P = − , T = +1) can only involve the operator σ [ g ] xI and the explicit computation yields the fusion coefficients A [ g,I ] (cid:104) σ [ g +1] xI σ [ g +1]0 I (cid:105) = c [ g,I ] σ [ g ] xI A [ g,I ] (cid:104) σ [ g +1]0 I σ [ g +1] xI (cid:105) = ˜ c [ g,I ] σ [ g ] xI A [ g,I ] (cid:104) σ [ g +1] xI σ [ g +1] zI (cid:105) = ˜ s [ g,I ] σ [ g ] xI A [ g,I ] (cid:104) σ [ g +1] zI σ [ g +1] xI (cid:105) = s [ g,I ] σ [ g ] xI (32)Reciprocally, the four operators of Eq. 5 will appear in the application of the descending superoperator D [ g,I ] to σ [ g ] xI with the same fusion coefficients given by the duality of Eq. 28 D [ g,I ] (cid:104) σ [ g ] xI (cid:105) = 12 (cid:104) c [ g,I ] σ [ g +1] xI σ [ g +1]0 I + ˜ c [ g,I ] σ [ g +1]0 I σ [ g +1] xI + ˜ s [ g,I ] σ [ g +1] xI σ [ g +1] zI + s [ g,I ] σ [ g +1] zI σ [ g +1] xI (cid:105) (33)while the partial traces over a single child reduce toTr { I } (cid:16) D [ g,I ] (cid:104) σ [ g ] xI (cid:105)(cid:17) = c [ g,I ] σ [ g +1] xI Tr { I } (cid:16) D [ g,I ] (cid:104) σ [ g ] xI (cid:105)(cid:17) = ˜ c [ g,I ] σ [ g +1] xI (34)It is thus convenient to introduce the following notation λ [ g,I ] xi g +1 ≡ c [ g,I ] δ i g +1 , + ˜ c [ g,I ] δ i g +1 , (35)to denote the local scaling property of the single child operator σ [ g +1] xIi g +1 with respect to its ancestor operator σ [ g ] xI . D. Local fusion rules for operators in the symmetry sector ( P = − , T = − Similarly, the action of the ascending superoperator A [ g,I ] on the four operators of Eq. 6 corresponding to thesymmetry sector ( P = − , T = − 1) can only involve the operator σ [ g ] yI and the explicit computation yields the fusioncoefficients A [ g,I ] (cid:104) σ [ g +1] yI σ [ g +1]0 I (cid:105) = ˜ s [ g,I ] σ [ g ] yI A [ g,I ] (cid:104) σ [ g +1]0 I σ [ g +1] yI (cid:105) = s [ g,I ] σ [ g ] yI A [ g,I ] (cid:104) σ [ g +1] yI σ [ g +1] zI (cid:105) = c [ g,I ] σ [ g ] yI A [ g,I ] (cid:104) σ [ g +1] zI σ [ g +1] yI (cid:105) = ˜ c [ g,I ] σ [ g ] yI (36)Reciprocally, the four operators of Eq. 6 will appear in the application of the descending superoperator to σ [ g ] yI withthe same fusion coefficients given by the duality of Eq. 28 D [ g,I ] (cid:104) σ [ g ] yI (cid:105) = 12 (cid:104) ˜ s [ g,I ] σ [ g +1] yI σ [ g +1]0 I + s [ g,I ] σ [ g +1]0 I σ [ g +1] yI + c [ g,I ] σ [ g +1] yI σ [ g +1] zI + ˜ c [ g,I ] σ [ g +1] zI σ [ g +1] yI (cid:105) (37)while the partial traces over a single child reduce toTr { I } (cid:16) D [ g,I ] (cid:104) σ [ g ] yI (cid:105)(cid:17) = ˜ s [ g,I ] σ [ g +1] yI Tr { I } (cid:16) D [ g,I ] (cid:104) σ [ g ] yI (cid:105)(cid:17) = s [ g,I ] σ [ g +1] yI (38)Again it is convenient to introduce the following notation λ [ g,I ] yi g +1 ≡ ˜ s [ g,I ] δ i g +1 , + s [ g,I ] δ i g +1 , (39)to denote the local scaling property of the single child operator σ [ g +1] yIi g +1 with respect to its ancestor operator σ [ g ] yI . E. Local fusion rules for operators in the symmetry sector ( P = +1 , T = +1) The action of the ascending superoperator A [ g,I ] on the six operators of Eq. 7 corresponding to the symmetrysector ( P = +1 , T = +1) can only involve the two operators σ [ g ] a =0 ,zI . The identity σ [ g +1]0 I σ [ g +1]0 I of the childrenspace is projected onto the identity σ [ g,I ]0 of the ancestor space as a consequence of Eq 17 A [ g,I ] (cid:104) σ [ g +1]0 I σ [ g +1]0 I (cid:105) = σ [ g ]0 I (40)while the parity σ [ g +1] zI σ [ g +1] zI of the block of the two children is projected onto the parity σ [ g ] zI of the ancestor A [ g,I ] (cid:104) σ [ g +1] zI σ [ g +1] zI (cid:105) = σ [ g ] zI (41)The remaining operators are projected onto the following linear combinations of the two operators σ [ g ]0 I and σ [ g ] zI A [ g,I ] (cid:104) σ [ g +1] zI σ [ g +1]0 I (cid:105) = s [ g,I ] ˜ c [ g,I ] σ [ g ]0 I + c [ g,I ] ˜ s [ g,I ] σ [ g ] zI A [ g,I ] (cid:104) σ [ g +1]0 I σ [ g +1] zI (cid:105) = c [ g,I ] ˜ s [ g,I ] σ [ g ]0 I + s [ g,I ] ˜ c [ g,I ] σ [ g ] zI A [ g,I ] (cid:104) σ [ g +1] xI σ [ g +1] xI (cid:105) = c [ g,I ] ˜ c [ g,I ] σ [ g ]0 I − s [ g,I ] ˜ s [ g,I ] σ [ g ] zI A [ g,I ] (cid:104) σ [ g +1] yI σ [ g +1] yI (cid:105) = s [ g,I ] ˜ s [ g,I ] σ [ g ]0 I − c [ g,I ] ˜ c [ g,I ] σ [ g ] zI (42)The duality of Eq. 28 yields that the application of the descending superoperator D [ g,I ] to σ [ g ] zI involves fiveoperators of the list of Eq. 7 (only the block identity σ [ g +1]0 I σ [ g +1]0 I does not appear) D [ g,I ] (cid:104) σ [ g ] zI (cid:105) = 12 (cid:104) c [ g,I ] ˜ s [ g,I ] σ [ g +1] zI σ [ g +1]0 I + s [ g,I ] ˜ c [ g,I ] σ [ g +1]0 I σ [ g +1] zI (cid:105) + 12 (cid:104) σ [ g +1] zI σ [ g +1] zI − s [ g,I ] ˜ s [ g,I ] σ [ g +1] xI σ [ g +1] xI − c [ g,I ] ˜ c [ g ] n σ [ g +1] yI σ [ g +1] yI (cid:105) (43)and the partial traces over a single child reduce toTr { I } (cid:16) D [ g,I ] (cid:104) σ [ g ] zI (cid:105)(cid:17) = c [ g,I ] ˜ s [ g,I ] σ [ g +1] zI Tr { I } (cid:16) D [ g,I ] (cid:104) σ [ g ] zI (cid:105)(cid:17) = s [ g,I ] ˜ c [ g,I ] σ [ g +1] zI (44)Again it is convenient to introduce the following notation λ [ g,I ] zi g +1 ≡ c [ g,I ] ˜ s [ g,I ] δ i g +1 , + s [ g,I ] ˜ c [ g,I ] δ i g +1 , (45)to denote the local scaling property of the single child operator σ [ g +1] zIi g +1 with respect to its ancestor operator σ [ g ] zI .The duality of Eq. 28 yields that the application of the descending superoperator D [ g,I ] to the identity σ [ g ]0 I of theancestor space involves five operators of the list of Eq. 7 (only the block parity σ [ g +1] zI σ [ g +1] zI does not appear) D [ g,I ] (cid:104) σ [ g ]0 I (cid:105) = 12 (cid:104) σ [ g +1]0 I σ [ g +1]0 I + s [ g,I ] ˜ c [ g,I ] σ [ g +1] zI σ [ g +1]0 I + c [ g,I ] ˜ s [ g,I ] σ [ g +1]0 I σ [ g +1] zI (cid:105) + 12 (cid:104) c [ g,I ] ˜ c [ g,I ] σ [ g +1] xI σ [ g +1] xI + s [ g,I ] ˜ s [ g,I ] σ [ g +1] yI σ [ g +1] yI (cid:105) (46)and the partial traces over a single child reduce toTr { I } (cid:16) D [ g,I ] (cid:104) σ [ g ]0 I (cid:105)(cid:17) = σ [ g +1]0 I + s [ g,I ] ˜ c [ g,I ] σ [ g +1] zI Tr { I } (cid:16) D [ g,I ] (cid:104) σ [ g ]0 I (cid:105)(cid:17) = σ [ g +1]0 I + c [ g,I ] ˜ s [ g,I ] σ [ g +1] zI (47)Although the meaning is different from the three scaling factors λ [ g,I ] a = x,y,zi g +1 introduced above, it will be convenientto introduce λ [ g,I ]0 i g +1 ≡ s [ g,I ] ˜ c [ g,I ] δ i g +1 , + c [ g,I ] ˜ s [ g,I ] δ i g +1 , (48)to denote the local scaling property of the single child operator σ [ g +1] zIi g +1 with respect to its ancestor identity σ [ g ]0 I . F. Assembling the elementary isometries to build the whole Tree-Tensor-State of parity P = + The correspondence between the ket | Ψ [ g ] (cid:105) for the chain of generation g containing N g = 2 g spins and the ket | Ψ [ g +1] (cid:105) for the chain of generation ( g + 1) containing N g +1 = 2 g +1 spins | Ψ [ g +1] (cid:105) = W [ g ] | Ψ [ g ] (cid:105) (49)is described by the global isometry W [ g ] made of the tensor product over the 2 g positions I = ( i , i , .., i g ) of theelementary isometries of Eq. 16 W [ g ] = (cid:89) I =( i ,i ,..,i g ) w [ g,I ] (50)At generation g = 0, the state of the single spin σ [0] at the root of the tree represents the Parity of the whole chain.We will focus on the positive parity sector P = + corresponding to the initial ket | Ψ [0] (cid:105) = | σ [0] z = P = + (cid:105) (51)The iteration of the rule of Eq. 49 will then generate a Tree-Tensor-State of parity P = + for the chain of generation g containing N g = 2 g spins | Ψ [ g ] (cid:105) = W [ g − | Ψ [ g − (cid:105) = W [ g − W [ g − ...W [1] W [0] | Ψ [0] (cid:105) (52)Since the elementary isometry w [ g (cid:48) ,I (cid:48) ] at generation g (cid:48) and position I (cid:48) involves only the two angles θ [ g (cid:48) ,I (cid:48) ] ± , the globalisometry W [ g (cid:48) ] for the 2 g (cid:48) sites I (cid:48) of the generation g (cid:48) involves 2 × g (cid:48) angles, except for the generation g (cid:48) = 0 whereonly the angle θ [0] ± will appear for the initial condition of Eq. 51. So the total number of parameters involved in theTree-Tensor-State | Ψ [ g ] (cid:105) of generation g containing N g = 2 g spins grows only linearly with respect to N g N P arametersg = 1 + 2 × g − (cid:88) g (cid:48) =1 g (cid:48) = 2(2 g − − N g − θ [ g (cid:48) ,I (cid:48) ] ± parametrize the hierarchical entanglement at different levelslabelled by the generation g (cid:48) and different positions labelled by the positions I (cid:48) . The consequences of this tree-tensorstructure for the entanglement of various bipartite partitions have been studied in detail in [43] on the specific caseof the pure Dyson quantum Ising model. In the following section, we will thus focus instead on the consequences forthe one-point and two-point reduced density matrices that allow to compute any one-spin and two-spin observable. IV. EXPLICIT FORMS FOR ONE-POINT AND TWO-POINT REDUCED DENSITY MATRICES The hierarchical structure of the inhomogeneous Tree-Tensor-States described in the previous section allows towrite simple recursions for the corresponding one-point and two-point reduced density matrices. A. Recursion for the full density matrices via the descending super-operator D [ g ] The full density matrix for the chain at generation gρ [ g ] ≡ | Ψ [ g ] (cid:105) (cid:104) Ψ [ g ] | (54)satisfies the recurrence involving the global descending superoperator D [ g ] ρ [ g +1] = W [ g ] ρ [ g ] ( W [ g ] ) † ≡ D [ g ] (cid:104) ρ [ g ] (cid:105) (55)while the initial condition at generation g = 0 reads (Eq 51) ρ [0] ≡ | Ψ [0] (cid:105) (cid:104) Ψ [0] | = | σ [0] z = + (cid:105) (cid:104) σ [0] z = + | = σ [0]0 + σ [0] z ρ [ g ] of generation g can be expanded in the Pauli basis of the 2 g spins σ [ g ] a I =0 ,x,y,zI , one justneeds to know how to apply the descending superoperator to products of Pauli matrices D [ g ] (cid:89) I =( i ,I ,..,i g ) σ [ g ] a I I = (cid:89) I =( i ,I ,..,i g ) D [ g,I ] (cid:104) σ [ g ] a I I (cid:105) (57)where the properties of the local descending superoperator D [ g,I ] of Eq. 28 have been described in detail in theprevious section. B. Parametrization of one-spin and two-spins reduced density matrices In order to compute all the one-spin and two-spins observables, one just needs the one-spin and two-spins reduceddensity matrices. Since the initial condition of Eq. 56 belongs to the symmetry sector ( P = + , T = +), the fulldensity matrices of Eq. 55 obtained by the successive application of the global descending superoperator D [ g ] are alsoin the sector ( P = + , T = +), and the partial traces over some spins will also preserve this symmetry sector. As aconsequence, the single-spin reduced density matrices can be parametrized as ρ [ g ] I = σ [ g ]0 I + m [ g ] I σ [ g ] zI / σ [ g ]0 I is fixed by the normalizationTr { I } ( ρ [ g ] I ) = 1 (59)while m [ g ] I represents the magnetization at site I of generation gm [ g ] I = Tr { I } ( σ [ g ] zI ρ [ g ] I ) (60)Similarly, the reduced density matrices ρ [ g ] I,I (cid:48) of two spins at positions I and I (cid:48) of generation g can only involve the sixtwo-spin operators of Eq. 7 of the sector ( P = + , T = +) and can be thus parametrized as ρ [ g ] I,I (cid:48) = σ [ g ]0 I σ [ g ]0 I (cid:48) m [ g ] I σ [ g ] zI σ [ g ]0 I (cid:48) m [ g ] I (cid:48) σ [ g ]0 I σ [ g ] zI (cid:48) C [ g ] zI,I (cid:48) σ [ g ] zI σ [ g ] zI (cid:48) C [ g ] xI,I (cid:48) σ [ g ] xI σ [ g ] xI (cid:48) C [ g ] yI,I (cid:48) σ [ g ] yI σ [ g ] yI (cid:48) C [ g ] a = x,y,zI,I (cid:48) represent the two-points xx , yy and zz correlations C [ g ] aI,I (cid:48) = Tr { I,I (cid:48) } ( σ [ g ] aI σ [ g ] aI (cid:48) ρ [ g ] I,I (cid:48) ) (62) C. Recursions for the one-point magnetizations and the two-point correlations The application of the local descending superoperator D [ g,I ] to the reduced density matrix ρ [ g ] I of the single site I of generation g of Eq. 58 produces the following reduced density matrix of its two children ( I , I 2) of generation( g + 1) using Eqs 43 and 46 ρ [ g +1] I ,I = D [ g,I ] [ ρ [ gP ] I ] = 12 (cid:104) D [ g,I ] [ σ [ g ]0 I ] + m [ g ] I D [ g,I ] [ σ [ g ] zI ] (cid:105) (63)= σ [ g +1]0 I σ [ g +1]0 I (cid:16) s [ g,I ] ˜ c [ g,I ] + m [ g ] I c [ g,I ] ˜ s [ g,I ] (cid:17) σ [ g +1] zI σ [ g +1]0 I (cid:16) c [ g,I ] ˜ s [ g,I ] + m [ g ] I s [ g,I ] ˜ c [ g,I ] (cid:17) σ [ g +1]0 I σ [ g +1] zI m [ g ] I σ [ g +1] zI σ [ g +1] zI (cid:16) c [ g,I ] ˜ c [ g,I ] − m [ g ] I s [ g,I ] ˜ s [ g,I ] (cid:17) σ [ g +1] xI σ [ g +1] xI (cid:16) s [ g,I ] ˜ s [ g,I ] − m [ g ] I c [ g,I ] ˜ c [ g,I ] (cid:17) σ [ g +1] yI σ [ g +1] yI λ [ g,I ] a =0 ,zi g +1 introduced in Eqs 45 and 48 m [ g +1] Ii g +1 = λ [ g,I ]0 i g +1 + λ [ g,I ] zi g +1 m [ g ] I (64)1and gives how the correlations between the two children of the same ancestor appear in terms of the coefficients λ [ g,I ] a = x,yi g +1 introduced in Eqs 35 and 39 C [ g +1] xI ,I = λ [ g,I ] x λ [ g,I ] x − λ [ g,I ] y λ [ g,I ] y m [ g ] I C [ g +1] yI ,I = λ [ g,I ] y λ [ g,I ] y − λ [ g,I ] x λ [ g,I ] x m [ g ] I C [ g +1] zI ,I = m [ g ] I (65)The application of the descending superoperator D [ g ] to the reduced density matrix ρ [ g ] I,I (cid:48) of two different sites I (cid:54) = I (cid:48) of generation g of Eq. 61 will produce the four-sites reduced density matrix for their children ( I , I 2) and ( I (cid:48) , I (cid:48) g + 1) ρ [ g +1] PI ,I ,I (cid:48) ,I (cid:48) = D [ g ] [ ρ [ g ] PI,I (cid:48) ]= 14 (cid:104) D [ g,I ] [ σ [ g ]0 I ] D [ g,I (cid:48) ] [ σ [ g ]0 I (cid:48) ] + m [ g ] I D [ g,I ] [ σ [ g ] zI ] D [ g,I (cid:48) ] [ σ [ g ]0 I (cid:48) ] + m [ g ] I (cid:48) D [ g,I ] [ σ [ g ]0 I ] D [ g,I (cid:48) ] [ σ [ g ] zI (cid:48) ] (cid:105) + 14 (cid:104) C [ g ] zI,I (cid:48) D [ g,I ] [ σ [ g ] zI ] D [ g,I (cid:48) ] [ σ [ g ] zI (cid:48) ] + C [ g ] xI,I (cid:48) D [ g,I ] [ σ [ g ] xI ] D [ g,I (cid:48) ] [ σ [ g ] xI (cid:48) ] + C [ g ] yI,I (cid:48) D [ g,I ] [ σ [ g ] yI ] D [ g,I (cid:48) ] [ σ [ g ] yI (cid:48) ] (cid:105) (66)One then needs to take the trace over one child in each block to obtain the reduced density matrices of the tworemaining children ρ [ g +1] I ,I (cid:48) = Tr { I ,I (cid:48) } (cid:16) D [ g ] [ ρ [ g ] I,I (cid:48) ] (cid:17) ρ [ g +1] I ,I (cid:48) = Tr { I ,I (cid:48) } (cid:16) D [ g ] [ ρ [ g ] I,I (cid:48) ] (cid:17) ρ [ g +1] I ,I (cid:48) = Tr { I ,I (cid:48) } (cid:16) D [ g ] [ ρ [ g ] I,I (cid:48) ] (cid:17) ρ [ g +1] I ,I (cid:48) = Tr { I ,I (cid:48) } (cid:16) D [ g ] [ ρ [ g ] I,I (cid:48) ] (cid:17) (67)Using the partial traces over a single child in each block computed before in Eqs 34 38 44 47, one obtains the followingrules for the two-point correlations between the children of different blocks. The xx and yy correlations are governedby the following multiplicative factorized rules as long as I (cid:54) = I (cid:48) C [ g +1] xIi g +1 ,I (cid:48) i (cid:48) g +1 = λ [ g,I ] xi g +1 λ [ g,I (cid:48) ] xi (cid:48) g +1 C [ g ] xI,I (cid:48) (68) C [ g +1] yIi g +1 ,I (cid:48) i (cid:48) g +1 = λ [ g,I ] yi g +1 λ [ g,I (cid:48) ] yi (cid:48) g +1 C [ g ] yI,I (cid:48) (69)The recursions for the zz correlations involve four terms C [ g +1] zIi g +1 ,I (cid:48) i (cid:48) g +1 = λ [ g,I ]0 i g +1 λ [ g,I (cid:48) ]0 i (cid:48) g +1 + λ [ g,I ] zi g +1 λ [ g,I (cid:48) ]0 i (cid:48) g +1 m [ g ] I + λ [ g,I ]0 i g +1 λ [ g,I (cid:48) ] zi (cid:48) g +1 m [ g ] I (cid:48) + λ [ g,I ] zi g +1 λ [ g,I (cid:48) ] zi (cid:48) g +1 C [ g ] zI,I (cid:48) (70)so that it is more convenient to consider the simpler multiplicative factorized rule satisfied by the connected correlationsfor I (cid:54) = I (cid:48) using Eq. 64 (cid:16) C [ g +1] zIi g +1 ,I (cid:48) i (cid:48) g +1 − m [ g +1] Ii g +1 m [ g +1] I (cid:48) i (cid:48) g +1 (cid:17) = λ [ g,I ] zi g +1 λ [ g,I (cid:48) ] zi (cid:48) g +1 (cid:16) C [ g ] zI,I (cid:48) − m [ g ] I m [ g ] I (cid:48) (cid:17) (71) D. Explicit solutions for the one-point magnetizations The initial condition for the magnetization at generation g = 0 is given by the parity P = + (Eq 56) m [0] = +1 (72)The first iterations of the affine recursion of Eq. 64 give for the generations g = 1 and g = 2 m [1] i = λ [0]0 i + λ [0] zi m [2] i ,i = λ [1 ,i ]0 i + λ [1 ,i ] zi (cid:16) λ [0]0 i + λ [0] zi (cid:17) (73)2More generally, one obtains that the magnetization at position ( i , ..., i g ) of generation g reads m [ g ] i ,...,i g = λ [ g − , ( i ,...,i g − )]0 i g + g − (cid:88) g (cid:48)(cid:48) =0 g − (cid:89) g (cid:48) = g (cid:48)(cid:48) +1 λ [ g (cid:48) , ( i ,...,i g (cid:48) )] zi g (cid:48) +1 λ [ g (cid:48)(cid:48) , ( i ,..,i g (cid:48)(cid:48) )]0 i g (cid:48)(cid:48) +1 + g − (cid:89) g (cid:48) =0 λ [ g (cid:48) , ( i ,...,i g (cid:48) )] zi g (cid:48) +1 (74)The first term involving a single scaling factor λ [ g − , ( i ,...,i g − )]0 i g is already enough to produce a finite magnetization,while the last term involving the g scaling factors up to the initial condition of the root will be exponentially small. E. Explicit solutions for the two-points correlations The xx correlations between two sites ( I, , i G − k +1 , .., i G ) and ( I, , i (cid:48) G − k +1 .., i (cid:48) G ) that have their Last CommonAncestor at the position I = ( i , .., i G − k − ) of generation g = G − k − r k = 2 k (Eq14) on the tree satisfy the recursion of Eq 68 as long as they are apart C [ G ] x ( I, ,i G − k +1 ,..,i G ) , ( I, ,i (cid:48) G − k +1 ,..,i (cid:48) G ) = G − (cid:89) g (cid:48) = G − k λ [ g (cid:48) , ( I, ,i G − k +1 ,..,i g (cid:48) )] xi g (cid:48) +1 λ [ g (cid:48) , ( I, ,i (cid:48) G − k +1 ,..,i (cid:48) g (cid:48) )] xi g (cid:48) +1 C [ G − k ] x ( I , ( I (75)while the remaining correlation at generation ( G − k ) is given by Eq. 65 in terms of the magnetization m [ G − k − I =( i ,..,i G − k − ) of their Last Common Ancestor C [ G − k ] xI ,I = λ [ G − k − ,I ] x λ [ G − k − ,I ] x − λ [ G − k − ,I ] y λ [ G − k − ,I ] y m [ G − k − I (76)and will thus be finite. As a consequence, the decay of the correlation of Eq. 75 with respect to the distance r = 2 k will be governed by the two strings of the k scaling factors λ [ g (cid:48) ,. ] x. leading to their Last Common Ancestor.Similarly, the yy correlations are given by C [ G ] y ( I, ,i G − k +1 ,..,i G ) , ( I, ,i (cid:48) G − k +1 ,..,i (cid:48) G ) = G − (cid:89) g (cid:48) = G − k λ [ g (cid:48) , ( I, ,i G − k +1 ,..,i g (cid:48) )] yi g (cid:48) +1 λ [ g (cid:48) , ( I, ,i (cid:48) G − k +1 ,..,i (cid:48) g (cid:48) )] yi g (cid:48) +1 C [ G − k ] y ( I , ( I (77)with C [ G − k ] yI ,I = λ [ G − k − ,I ] y λ [ G − k − ,I ] y − λ [ G − k − ,I ] x λ [ G − k − ,I ] x m [ G − k − I (78)Again, the decay of the correlation of Eq. 77 with respect to the distance r = 2 k will be governed by the two stringsof the k scaling factors λ [ g (cid:48) ,. ] y. leading to their Last Common Ancestor.Finally, the zz connected correlations satisfying Eq 71 read (cid:16) C [ G ] z ( I, ,i G − k +1 ,..,i G ) , ( I, ,i (cid:48) G − k +1 ,..,i (cid:48) G ) − m [ G ]( I, ,i G − k +1 ,..,i G ) m [ G ]( I, ,i (cid:48) G − k +1 ,..,i G ) (cid:48) (cid:17) = G − (cid:89) g (cid:48) = G − k λ [ g (cid:48) , ( I, ,i G − k +1 ,..,i g (cid:48) )] zi g (cid:48) +1 λ [ g (cid:48) , ( I, ,i (cid:48) G − k +1 ,..,i (cid:48) g (cid:48) )] zi g (cid:48) +1 (cid:16) C [ G − k ] z ( I , ( I − m [ G − k ] I m [ G − k ] I (cid:17) (79)where the remaining connected correlation at generation ( G − k ) is given by Eqs 65 and 64 C [ G − k ] zI ,I − m [ G − k ] I m [ G − k ] I = m [ G − k − I − (cid:16) λ [ G − k − ,I ]01 + λ [ G − k − ,I ] z m [ G − k − I (cid:17) (cid:16) λ [ G − k − ,I ]02 + λ [ G − k − ,I ] z m [ G − k − I (cid:17) (80)in terms of the magnetization m [ G − k − I =( i ,..,i G − k − ) of their last common ancestor. V. ENERGY OF THE TREE-TENSOR-STATE AND OPTIMIZATION OF ITS PARAMETERS Up to now, we have only used the Parity and the Time-Reversal symmetries to build the simplest inhomogeneousTree-Tensor-States and analyze its general properties. In the present section, we take into account the specific formof the Hamiltonian, in order to evaluate the energy of the Tree-Tensor-State and to optimize its parameters.3 A. Energy of the Tree-Tensor-State in terms of the magnetizations and the correlations at generation G For the Dyson Hamiltonian H [ G ] of Eq. 12 that contains only one-body and two-body terms, the energy of theTree-Tensor-State | Ψ [ G ] (cid:105) E [ G ] ≡ (cid:104) Ψ [ G ] | H [ G ] | Ψ [ G ] (cid:105) = Tr { G } ( H [ G ] ρ [ G ] ) (81)involves only the one-body and the two-body reduced density matrices of Eqs 58 and 61. It can be thus rewritten interms of the magnetizations m [ G ] I and of the two-point correlations C [ G ] a = x,y,zI,I (cid:48) as E [ G ] = − (cid:88) I =( i ,i ,..,i G ) h I m [ G ] I − (cid:88) I =( i ,i ,..,i G )
The renormalization rules for the parameters of the renormalized Hamiltonian of Eq. 86 can be derived via theapplication of the ascending superoperator (Eq. 85). Here to stress the duality with the recursions for the magneti-zations and correlations derived in the previous section, it will be more instructive the use instead the identification4between the energy computed at generation g with Eq. 87 and the energy computed at generation ( g + 1) E [ G ] = E [ g +1] = E [ g +1] − (cid:88) I =( i ,i ,..,i g ) (cid:88) i g +1 h [ g +1] Ii g +1 m [ g +1] Ii g +1 − (cid:88) I =( i ,i ,..,i g ) (cid:88) a = x,y,z J [ g +1] aI ,I C [ g +1] aI ,I − (cid:88) I =( i ,i ,..,i g )
2) and ( I (cid:48) , I (cid:48) 2) of generation ( g + 1) J [ g ] aI,I (cid:48) = (cid:88) i g +1 =1 , (cid:88) i (cid:48) g +1 =1 , λ [ g,I ] ai g +1 λ [ g,I (cid:48) ] ai (cid:48) g +1 ω a J [ g +1] aIi g +1 ,I (cid:48) i (cid:48) g +1 (90)As a consequence, if some coupling component a = x, y, z is not present in the initial Hamiltonian, it will not begenerated via renormalization.The renormalized field h [ g ] I involves local terms coming from the two fields h [ g +1] I and h [ g +1] I of its children and thethree couplings between its two children J [ g +1] a = x,y,zI ,I , but also long-ranged terms coming from all the z-couplingsbetween one child ( I 1) or ( I 2) with other children from other blocks I (cid:48) (cid:54) = I : h [ g ] I = h [ g +1] I λ [ g,I ] z + h [ g +1] I λ [ g,I ] z + J [ g +1] zI ,I − J [ g +1] xI ,I λ [ g,I ] y λ [ g,I ] y − J [ g +1] yI ,I λ [ g,I ] x λ [ g,I ] x + (cid:88) I (cid:48) =( i (cid:48) ,i (cid:48) ,..,i (cid:48) g ) >I (cid:88) i g +1 (cid:88) i (cid:48) g +1 J [ g +1] zIi g +1 ,I (cid:48) i (cid:48) g +1 [2 r ( I, I (cid:48) )] ω z λ [ g,I ] zi g +1 λ [ g,I (cid:48) ]0 i (cid:48) g +1 + (cid:88) I (cid:48) =( i (cid:48) ,i (cid:48) ,..,i (cid:48) g )
2) of the blocks, as well as the z-couplings between spins ( I , I 2) and ( I (cid:48) , I (cid:48) I < I (cid:48) E [ g ] = E [ g +1] − (cid:88) I =( i ,i ,..,i g ) (cid:16) h [ g +1] I λ [ g,I ]01 + h [ g +1] I λ [ g,I ]02 + J [ g +1] xI ,I λ [ g,I ] x λ [ g,I ] x + J [ g +1] yI ,I λ [ g,I ] y λ [ g,I ] y (cid:17) − (cid:88) I =( i ,i ,..,i g )
1, the dependence on the parameters of the Tree-Tensor-State of theenergy E [ g ] of Eq. 87 is divided in two parts : the magnetizations m [ g ] I and the correlations C [ g ] aI,I (cid:48) of generation g only involve the Tree-Tensor-State parameters of smaller generations g (cid:48) = 0 , .., g − 1, while the renormalized param-eters ( E [ g ] , h [ g ] I , J [ g ] aI,I (cid:48) ) of the Hamiltonian of generation g only involve the Tree-Tensor-State parameters of biggergenerations g (cid:48) = g, .., G − g can be seen inEq. 89 via the scaling factors λ [ g,I ] a =0 ,x,y,zi g +1 =1 , of generation g , while the dependence with respect to smaller generations g (cid:48) = 0 , .., g − m [ g ] I and the correlations C [ g ] aI,I (cid:48) of generation g , and the dependencewith respect to bigger generations g (cid:48) = g + 1 , .., G − g + 1).Since the general case with the the z-couplings leads to somewhat heavy expressions for the disordered modelsdescribed by inhomogeneous Tree-Tensor-States, it is more instructive to focus now on the simpler models withoutz-couplings, while we will return to the general case with the three type of couplings a = x, y, z in the next sectionsconcerning pure models. E. Optimization of the parameters of the Tree-Tensor-State for the case without z-couplings For the case without z-couplings, Eq. 89 yields the following optimization equation with respect to the angle φ [ g,I ] of Eqs 300 = ∂ E [ g +1] ∂φ [ g,I ] = − h [ g +1] I (cid:32) ∂λ [ g,I ]01 ∂φ [ g,I ] + ∂λ [ g,I ] z ∂φ [ g,I ] m [ g ] I (cid:33) − h [ g +1] I (cid:32) ∂λ [ g,I ]02 ∂φ [ g,I ] + ∂λ [ g,I ] z ∂φ [ g,I ] m [ g ] I (cid:33) − J [ g +1] xI ,I (cid:32) ∂ ( λ [ g,I ] x λ [ g,I ] x ) ∂φ [ g,I ] − ∂ ( λ [ g,I ] y λ [ g,I ] y ) ∂φ [ g,I ] m [ g ] I (cid:33) − J [ g +1] yI ,I (cid:32) ∂ ( λ [ g,I ] y λ [ g,I ] y ) ∂φ [ g,I ] − ∂ ( λ [ g,I ] x λ [ g,I ] x ) ∂φ [ g,I ] m [ g ] I (cid:33) − (cid:88) i g +1 ∂λ [ g,I ] xi g +1 ∂φ [ g,I ] (cid:88) I (cid:48) >I (cid:88) i (cid:48) g +1 J [ g +1] xIi g +1 ,I (cid:48) i (cid:48) g +1 C [ g ] xI,I (cid:48) [2 r ( I, I (cid:48) )] ω x λ [ g,I (cid:48) ] xi (cid:48) g +1 + (cid:88) I (cid:48) I (cid:88) i (cid:48) g +1 J [ g +1] yIi g +1 ,I (cid:48) i (cid:48) g +1 C [ g ] yI,I (cid:48) [2 r ( I, I (cid:48) )] ω y λ [ g,I (cid:48) ] yi (cid:48) g +1 + (cid:88) I (cid:48) I (cid:88) i (cid:48) g +1 J [ g +1] xI ,I (cid:48) i (cid:48) g +1 C [ g ] xI,I (cid:48) [2 r ( I, I (cid:48) )] ω x λ [ g,I (cid:48) ] xi (cid:48) g +1 + (cid:88) I (cid:48) I (cid:88) i (cid:48) g +1 J [ g +1] yI ,I (cid:48) i (cid:48) g +1 C [ g ] yI,I (cid:48) [2 r ( I, I (cid:48) )] ω y λ [ g,I (cid:48) ] yi (cid:48) g +1 + (cid:88) I (cid:48) I (cid:88) i (cid:48) g +1 J [ g +1] xI ,I (cid:48) i (cid:48) g +1 C [ g ] xI,I (cid:48) [2 r ( I, I (cid:48) )] ω x λ [ g,I (cid:48) ] xi (cid:48) g +1 + (cid:88) I (cid:48) I (cid:88) i (cid:48) g +1 J [ g +1] yI ,I (cid:48) i (cid:48) g +1 C [ g ] yI,I (cid:48) [2 r ( I, I (cid:48) )] ω y λ [ g,I (cid:48) ] yi (cid:48) g +1 + (cid:88) I (cid:48) I (cid:88) i (cid:48) g +1 J [ g +1] xI ,I (cid:48) i (cid:48) g +1 C [ g ] xI,I (cid:48) [2 r ( I, I (cid:48) )] ω x λ [ g,I (cid:48) ] xi (cid:48) g +1 + (cid:88) I (cid:48) I (cid:88) i (cid:48) g +1 J [ g +1] yI ,I (cid:48) i (cid:48) g +1 C [ g ] yI,I (cid:48) [2 r ( I, I (cid:48) )] ω y λ [ g,I (cid:48) ] yi (cid:48) g +1 + (cid:88) I (cid:48) I (cid:88) i (cid:48) g +1 J [ g +1] xI ,I (cid:48) i (cid:48) g +1 C [ g ] xI,I (cid:48) [2 r ( I, I (cid:48) )] ω x λ [ g,I (cid:48) ] xi (cid:48) g +1 + (cid:88) I (cid:48) I (cid:88) i (cid:48) g +1 J [ g +1] yI ,I (cid:48) i (cid:48) g +1 C [ g ] yI,I (cid:48) [2 r ( I, I (cid:48) )] ω y λ [ g,I (cid:48) ] yi (cid:48) g +1 + (cid:88) I (cid:48)
Up to now, we have considered inhomogeneous Tree-Tensor-States for disordered spin chains. In this section, weturn to the case of pure Dyson models, where their supplementary symmetries need to be taken into account in theTree-Tensor description. A. Supplementary symmetries of the pure Dyson models When the fields h I and the couplings J aI,I (cid:48) of the Dyson Hamiltonian (Eqs 12 13) are uniform h I = hJ aI,I (cid:48) = J a (104)one needs to take into account two supplementary symmetries for the choice of the isometries w [ g,I ] of Eq. 16. Thefirst symmetry concerns the equivalence between the various branches of the tree, so that the two angles θ [ g,I ] ± ofEqs 19 and 20 will only depend on the generation g but not on the position I = ( i , .., i g ) anymore θ [ g,I ] ± = θ [ g ] ± (105)The second symmetry concerns the equivalence of the two children of a given ancestor. In the parity sector P = +,the ket of Eq. 19 is symmetric with respect to the two children for any value of the angle θ [ g ] , + . However in the paritysector P = − , the ket of Eq. 20 is symmetric with respect to the two children only for the value θ [ g ] , − = π G of generations for the homogeneous Tree-Tensor-Statescorresponding to pure models N P arameterspure = G − (cid:88) g =0 G (107)The two symmetries of Eq. 105 and 106 yields that the two angles of Eq. 30 now coincide and do not depend onthe position I anymore φ [ g ] = ˜ φ [ g ] = π − θ [ g ]+ (108)8so the coefficients of Eq. 31 reduce to c [ g ] ≡ cos (cid:16) φ [ g ] (cid:17) ≡ ˜ c [ g ] s [ g ] ≡ sin (cid:16) φ [ g ] (cid:17) ≡ ˜ s [ g ] (109)and the scaling factors of Eqs 35 39 45 48 simplify into λ [ g ] x ≡ c [ g ] λ [ g ] y ≡ s [ g ] λ [ g ] z ≡ c [ g ] s [ g ] λ [ g ]0 ≡ c [ g ] s [ g ] (110) B. Explicit solution for the one-point magnetizations The magnetization now only depends on the generation g . The recursion of Eq. 64 simplifies into m [ g +1] = c [ g ] s [ g ] (1 + m [ g ] ) (111)and the solution of Eq. 74 reduces to m [ g ] = g − (cid:88) g (cid:48)(cid:48) =0 g − (cid:89) g (cid:48) = g (cid:48)(cid:48) c [ g (cid:48) ] s [ g (cid:48) ] + g − (cid:89) g (cid:48) =0 c [ g (cid:48) ] s [ g (cid:48) ] = c [ g − s [ g − + c [ g − s [ g − c [ g − s [ g − + ... + g − (cid:89) g (cid:48) =1 c [ g (cid:48) ] s [ g (cid:48) ] + 2 g − (cid:89) g (cid:48) =0 c [ g (cid:48) ] s [ g (cid:48) ] (112) C. Explicit solutions for the two-point correlations The two-point correlations between two sites of generation G now only depend on the generation g = G − k − r k = 2 k on the tree. Eqs 65 give the values of thecorrelations at distance r = 2 = 1 as a function of the magnetization given in Eq. 112 C [ g +1] x ( r =1) = ( c [ g ] ) − ( s [ g ] ) m [ g ] C [ g +1] y ( r =1) = ( s [ g ] ) − ( c [ g ] ) m [ g ] C [ g +1] z ( r =1) = m [ g ] (113)while Eqs 68, 69 71 correspond to the following recursions for k ≥ C [ g +1] x (2 r k =2 k +1 ) = ( c [ g ] ) C [ g ] x ( r k =2 k ) C [ g +1] y (2 r k =2 k +1 ) = ( s [ g ] ) C [ g ] y ( r k =2 k ) (cid:16) C [ g +1] z (2 r k =2 k +1 ) − ( m [ g +1] ) (cid:17) = ( c [ g ] s [ g ] ) (cid:16) C [ g ] z ( r k =2 k ) − ( m [ g ] ) (cid:17) (114)The solutions of Eqs 75 77 and 79 reduce to C [ G ] x ( r k =2 k ) = G − (cid:89) g (cid:48) = G − k ( c [ g (cid:48) ] ) (cid:104) ( c [ G − k − ) − ( s [ G − k − ) m [ G − k − (cid:105) C [ G ] y ( r k =2 k ) = G − (cid:89) g (cid:48) = G − k ( s [ g (cid:48) ] ) (cid:104) ( s [ G − k − ) − ( c [ G − k − ) m [ G − k − (cid:105)(cid:104) C [ G ] z ( r k =2 k ) − ( m [ G ] ) (cid:105) = G − (cid:89) g (cid:48) = G − k ( c [ g (cid:48) ] s [ g (cid:48) ] ) (cid:20) m [ G − k − − ( c [ G − k − s [ G − k − ) (cid:16) m [ G − k − (cid:17) (cid:21) (115)9 D. Energy of the homogeneous Tree-Tensor-State and optimization of its G parameters For the pure Dyson model of Eq. 15 where the magnetization m [ G ] depends only on the generation and where thecorrelation depends only on the generation G and on the distance r k = 2 k , the energy of the Tree-Tensor-State of Eq.82 becomes E [ G ] = − G hm [ G ] − G − G − (cid:88) k =0 (cid:88) a = x,y,z J a kω a C [ G ] a ( r k =2 k ) (116)while the equivalent computation of the energy at any generation g (Eq 87) reads similarly E [ g ] = E [ g ] − g h [ g ] m [ g ] − g − g − (cid:88) k =0 (cid:88) a = x,y,z J [ g ] a kω a C [ g ] a ( r k =2 k ) (117)in terms of the parameters ( E [ g ] , h [ g ] , J [ g ] a ) of the renormalized Hamiltonian.The dependence with respect to the parameter of the generation g can be obtained from the energy computed atgeneration ( g + 1) when the magnetization and the correlations of generation ( g + 1) are written in terms of themagnetizations and the correlations of generation g via the recursions of Eqs 111, 113 and 114 E [ g +1] = E [ g +1] − g +1 h [ g +1] m [ g +1] − g (cid:88) a = x,y,z J [ g +1] a C [ g +1] a ( r =1) − g g − (cid:88) k =0 (cid:88) a = x,y,z J [ g +1] a ω a kω a C [ g +1] a ( r k +1 =2 k +1 ) (118)= E [ g +1] − g +1 h [ g +1] c [ g ] s [ g ] (1 + m [ g ] ) − g (cid:104) J [ g +1] x (cid:16) ( c [ g ] ) − ( s [ g ] ) m [ g ] (cid:17) + J [ g +1] y (cid:16) ( s [ g ] ) − ( c [ g ] ) m [ g ] (cid:17) + J [ g +1] z m [ g ] (cid:105) − g − g − (cid:88) k =0 (cid:20) − ω x ( c [ g ] ) J [ g +1] x kω x C [ g ] x ( r k =2 k ) + 2 − ω y ( s [ g ] ) J [ g +1] y kω y C [ g ] y ( r k =2 k ) + 2 − ω z ( c [ g ] s [ g ] ) J [ g +1] z kω z (cid:16) m [ g ] + C [ g ] z ( r k =2 k ) (cid:17)(cid:21) The identification with Eq. 87 yields the renormalization rules for the couplings (instead of Eq. 90) J [ g ] x = 2 − ω x ( c [ g ] ) J [ g +1] x J [ g ] y = 2 − ω y ( s [ g ] ) J [ g +1] y J [ g ] z = 2 − ω z ( c [ g ] s [ g ] ) J [ g +1] z (119)for the field (instead of Eq. 91) h [ g ] = 2 c [ g ] s [ g ] h [ g +1] − ( s [ g ] ) J [ g +1] x − ( c [ g ] ) J [ g +1] y + J [ g +1] z (cid:34) − ω z ( c [ g ] s [ g ] ) g − (cid:88) k =0 kω z (cid:35) (120)and for the constant term (instead of Eq. 92) E [ g ] = E [ g +1] − g (cid:34) c [ g ] s [ g ] h [ g +1] + ( c [ g ] ) J [ g +1] x + ( s [ g ] ) J [ g +1] y + 2 − ω z ( c [ g ] s [ g ] ) J [ g +1] z g − (cid:88) k =0 kω z (cid:35) (121)Eq 118 also gives the explicit dependence of the energy with respect to the angle φ [ g ] associated to the generation g − g E [ g +1] = 2 − g E [ g +1] − h [ g +1] (1 + m [ g ] ) sin(2 φ [ g ] ) − J [ g +1] z m [ g ] − J [ g +1] x (cid:18) (1 − m [ g ] ) + (1 + m [ g ] ) cos(2 φ [ g ] )2 (cid:19) − J [ g +1] y (cid:18) (1 − m [ g ] ) − (1 + m [ g ] ) cos(2 φ [ g ] )2 (cid:19) − − − ω x [1 + cos(2 φ [ g ] )] J [ g +1] x g − (cid:88) k =0 C [ g ] x ( r k =2 k ) kω x − − − ω y [1 − cos(2 φ [ g ] )] J [ g +1] y g − (cid:88) k =0 C [ g ] y ( r k =2 k ) kω y − − − ω z sin (2 φ [ g ] ) J [ g +1] z g − (cid:88) k =0 (cid:16) m [ g ] + C [ g ] z ( r k =2 k ) (cid:17) kω z φ [ g ] reads0 = ∂ (2 − g E [ g +1] ) ∂φ [ g ] = (1 + m [ g ] ) (cid:104) − h [ g +1] cos(2 φ [ g ] ) + ( J [ g +1] x − J [ g +1] y ) sin(2 φ [ g ] ) (cid:105) + sin(2 φ [ g ] ) − ω x J [ g +1] x g − (cid:88) k =0 C [ g ] x ( r k =2 k ) kω x − − ω y J [ g +1] y g − (cid:88) k =0 C [ g ] y ( r k =2 k ) kω y − cos(2 φ [ g ] ) sin(2 φ [ g ] )2 − − ω z J [ g +1] z g − (cid:88) k =0 (cid:16) m [ g ] + C [ g ] z ( r k =2 k ) (cid:17) kω z (122)If one neglects the contributions of the second and third lines, the first line allows to recover the usual criterion basedon the diagonalization of the intra-Hamiltonian in each block (Eq A5) for the angle (Eq 108)2 φ [ g ] = π − θ [ g ]+ (123) VII. SCALE-INVARIANT TREE-TENSOR-STATES FOR THE CRITICAL PURE DYSON MODELS In this section, we focus on the possible critical points of pure Dyson models, where the corresponding homogeneousTree-Tensor-State of the last section becomes in addition scale invariant. A. Supplementary symmetry : scale invariance At the critical points of the pure Dyson models discussed in the previous section, the scale invariance means thatthe isometries do not even depend on the generation g anymore, so that the only remaining parameter is the angle θ + or the angle φ = π − θ + (124)so the parameters c [ g ] and s [ g ] of the previous section do not depend of g anymore c ≡ cos ( φ ) s ≡ sin ( φ ) (125) B. Explicit solutions for the one-point magnetizations The magnetization of Eq. 112 reduces to m [ g ] = cs − cs + ( cs ) g (cid:18) − cs − cs (cid:19) (126)The dependence with respect to the generation g comes only from the finite size and from the initial condition m [0] = +1 at generation g = 0. In the thermodynamic limit g → + ∞ , the influence of this initial condition disappearsand the asymptotic magnetization is simply m [ ∞ ] = cs − cs (127)1 C. Explicit solutions for the two-point correlations The two-point correlations of Eq. 115 simplify into C [ G ] x ( r k =2 k ) = (cid:0) c (cid:1) k (cid:16) c − s m [ G − k − (cid:17) C [ G ] y ( r k =2 k ) = (cid:0) s (cid:1) k (cid:16) s − c m [ G − k − (cid:17)(cid:104) C [ G ] z ( r k =2 k ) − ( m [ G ] ) (cid:105) = (cid:0) c s (cid:1) k (cid:20) m [ G − k − − ( cs ) (cid:16) m [ G − k − (cid:17) (cid:21) (128)Again the dependence with respect to the generation G comes only from the finite size via the magnetization m [ G − k − of the Last Common Ancestor. In the thermodynamic limit G → + ∞ where the asymptotic magnetizationis given by Eq. 127, the two-point-correlations become simple power-laws with respect to the distance r k = 2 k C [ ∞ ] x ( r k =2 k ) = (cid:0) c (cid:1) k (cid:16) c − s m [ ∞ ] (cid:17) = (cid:0) c (cid:1) k c ( c − s )1 − cs ≡ A x r x k C [ ∞ ] y ( r k =2 k ) = (cid:0) s (cid:1) k (cid:16) s − c m [ ∞ ] (cid:17) = (cid:0) s (cid:1) k s ( s − c )1 − cs ≡ A y r y k (cid:104) C [ ∞ ] z ( r k =2 k ) − ( m [ ∞ ] ) (cid:105) = (cid:0) c s (cid:1) k (cid:20) m [ ∞ ] − ( cs ) (cid:16) m [ ∞ ] (cid:17) (cid:21) = (cid:0) c s (cid:1) k cs (1 − cs )(1 − cs ) ≡ A z r z k (129)where the scaling dimensions ∆ a that govern the power-law decay with respect to the distance r k = 2 k ∆ x = − ln | c | ln 2∆ y = − ln | s | ln 2∆ z = − ln | cs | ln 2 (130)and the amplitudes A x = c ( c − s )1 − csA y = s ( s − c )1 − csA z = cs (1 − cs )(1 − cs ) (131)depend only on the angle φ . D. Scale-invariance of the renormalized Hamiltonian with the dynamical exponent z The renormalization rules for the couplings (Eqs 119) become J [ g ] x = 2 − ω x c J [ g +1] x J [ g ] y = 2 − ω y s J [ g +1] y J [ g ] z = 2 − ω z ( cs ) J [ g +1] z (132)while the renormalization rule for the field (Eq 120) reads (when the thermodynamic limit is taken in the last sum) h [ g ] = 2 csh [ g +1] − s J [ g +1] x − c J [ g +1] y + J [ g +1] z (cid:34) − ω z ( cs ) ∞ (cid:88) k =0 kω z (cid:35) (133)2At the critical point, the field and the couplings that do not vanish in the renormalized scale-invariant Hamiltonianshould all have the same scaling dimension given by the dynamical exponent zJ [ g ] a (cid:39) − z J [ g +1] a h [ g ] (cid:39) − z h [ g +1] (134)As a consequence, the ratios K [ g ] a ≡ J [ g ] a h [ g ] associated to the couplings surviving in the scale-invariant renormalizedHamiltonian should take fixed point values independent of the generation gK [ g ] a = J [ g ] a h [ g ] = K [ g +1] a ≡ K a (135)The optimization equation of Eq. 122 can be then rewritten in the thermodynamic limit g → + ∞ as0 = (1 + m [ ∞ ] ) [ − φ ) + ( K x − K y ) sin(2 φ )] + sin(2 φ ) − ω x K x ∞ (cid:88) k =0 C [ ∞ ] x ( r k =2 k ) kω x − − ω y K y ∞ (cid:88) k =0 C [ ∞ ] y ( r k =2 k ) kω y − cos(2 φ ) sin(2 φ )2 − − ω z K z ∞ (cid:88) k =0 (cid:0) m [ ∞ ] (cid:1) + (cid:16) C [ ∞ ] z ( r k =2 k ) − ( m [ ∞ ] ) (cid:17) kω z (136)where the magnetization m [ ∞ ] of Eq. 127 and the correlations of Eq. 129 only depend on the angle φ . E. Critical points of the pure Dyson quantum Ising model ( K x (cid:54) = 0 and K y = 0 = K z ) Let us consider the critical points where the y-couplings and the z-couplings vanish in the scale-invariant renormal-ized Hamiltonian. This will occur either if the y-couplings and the z-couplings already vanish in the initial condition,i.e. if the initial condition corresponds to the pure Dyson quantum Ising model, or if their scaling dimensions inEq. 132 make the two ratios J [ g ] a = y,z J [ g ] x converge towards zero via renormalization. Then the scale invariance with thedynamical exponent z of Eq. 134 yields the two conditions from Eq. 132 and 1332 − z = J [ g ] x J [ g +1] x = 2 − ω x c − z = h [ g ] h [ g +1] = 2 cs − s K x (137)while the optimization equation of Eq. 136 gives the constraint0 = − φ ) + K x sin(2 φ ) − ω x (1 + m [ ∞ ] ) ∞ (cid:88) k =0 C [ ∞ ] x ( r k =2 k ) kω x (138)If one neglects the correlations C [ ∞ ] x ( r k =2 k ) → − φ ) + K x sin(2 φ ) (139)based on diagonalization of the intra-Hamiltonian in each block (see Appendix A), and the properties of the corre-sponding critical point have been discussed in Refs [42, 43] as a function of the power-law exponent ω x (called σ inRefs [42, 43]).When the correlations C [ ∞ ] x ( r k =2 k ) are not neglected, the line of critical points is parametrized by the four variables( φ, z, ω x , K x ) related by the three equations (the two Eqs of 137 and Eq 138). F. Critical points where K x (cid:54) = 0 and K y (cid:54) = 0 while K z = 0 Let us now consider the case where both the x-coupling K x (cid:54) = 0 and the y-coupling K y (cid:54) = 0 survive in therenormalized scale invariant Hamiltonian, while the z coupling vanishes K z = 0.3Then the scale invariance with the dynamical exponent z of Eq. 134 yields the following three conditions from Eq.132 and 133 2 − z = J [ g ] x J [ g +1] x = 2 − ω x c − z = J [ g ] y J [ g +1] y = 2 − ω y s − z = h [ g ] h [ g +1] = 2 cs − s K x − c K y (140)while the optimization equation of Eq. 136 gives the constraint0 = (1 + m [ ∞ ] ) [ − φ ) + ( K x − K y ) sin(2 φ )] + sin(2 φ ) − ω x K x ∞ (cid:88) k =0 C [ ∞ ] x ( r k =2 k ) kω x − − ω y K y ∞ (cid:88) k =0 C [ ∞ ] y ( r k =2 k ) kω y (141)As a consequence of the two first equations of Eq. 137, the angle φ is now completely fixed by the difference betweenthe exponents ω x and ω y tan ( φ ) = s c = 2 ω y − ω x (142)Then z is fixed by the two first equations of Eq. 137, then the fixed-point values K x,y for the x-coupling and they-coupling are given by the solutions of the two remaining equations, namely the third equation of 137 and Eq. 141. VIII. CONCLUSIONS In this paper, we have analyzed the simplest Tree-Tensor-States (TTS) respecting the Parity and the Time-Reversalsymmetries in order to describe the ground states of Long-Ranged Quantum Spin Chains with or without disorder.We have first focused on inhomogeneous TTS for disordered Long-Ranged spin-chains. Explicit formulas have beengiven for the one-point and two-point reduced density matrices as parametrized by the magnetizations and the two-point correlations. We have then analyzed how the total energy of the TTS depend on each parameter of the TTSin order to obtain the optimization equations and to compare them with the traditional block-spin renormalizationprocedure based on the diagonalization of some intra-block renormalized Hamiltonian.We have then considered the pure Long-Ranged spin-chains in order to include the supplementary symmetries inthe TTS description, both for the off-critical region where the homogeneous TTS is made of isometries that onlydepend on the generation, and for critical points where the homogeneous TTS becomes scale invariant with isometriesthat do not depend on the generation anymore.Further work is needed to investigate whether the variational optimization with respect to parameters can be alsowritten explicitly for other types of Tensor-States based on different entanglement architectures. Appendix A: Comparison with the isometries determined by the intra-block Hamiltonians In this Appendix, we recall the usual block-spin RG rules based on the diagonalization of the intra-Hamiltonianin each block in order to compare with the variational optimization of the isometries discussed in the text. Therenormalized intra-Hamiltonian associated to the block of the two children ( I , I 2) of generation ( g + 1) having thesame ancestor I at generation g reads H intraI ,I = − h [ g +1] I σ [ g +1] zI − h [ g +1] I σ [ g +1] zI − J [ g +1] zI ,I σ [ g +1] zI σ [ g +1] zI − J [ g +1] xI ,I σ [ g +1] xI σ [ g +1] xI − J [ g +1] yI ,I σ [ g +1] yI σ [ g +1] yI (A1) 1. Diagonalization in the parity sector P = + In the parity sector σ [ g +1] zI σ [ g +1] zI = +, the diagonalization of the Hamiltonian of Eq. A1 H intraI ,I | ++ (cid:105) = − ( h [ g +1] I + h [ g +1] I + J [ g +1] zI ,I ) | ++ (cid:105) − ( J [ g +1] xI ,I − J [ g +1] yI ,I ) |−−(cid:105) H intraI ,I |−−(cid:105) = − ( J [ g +1] xI ,I − J [ g +1] yI ,I ) | ++ (cid:105) + ( h [ g +1] I + h [ g +1] I − J [ g +1] zI ,I ) |−−(cid:105) (A2)4leads to the two eigenvalues e [ P =+] ± = − J [ g +1] zI ,I ± (cid:113) ( h [ g +1] I + h [ g +1] I ) + ( J [ g +1] xI ,I − J [ g +1] yI ,I ) (A3)The eigenvector associated to the lowest eigenvalue e [ P =+] − is the kept state | ψ [ g +1]+ I ,I (cid:105) of Eq. 19 | ψ [ g +1]+ I ,I (cid:105) = cos( θ [ g,I ]+ ) | ++ (cid:105) + sin( θ [ g,I ]+ ) |−−(cid:105) (A4)where the angle θ [ g,I ]+ is fixed by the parameters of the renormalized intra-Hamiltoniancos(2 θ [ g,I ]+ ) = h [ g +1] I + h [ g +1] I (cid:113) ( h [ g +1] I + h [ g +1] I ) + ( J [ g +1] xI ,I − J [ g +1] yI ,I ) sin(2 θ [ g,I ]+ ) = J [ g +1] xI ,I − J [ g +1] yI ,I (cid:113) ( h [ g +1] I + h [ g +1] I ) + ( J [ g +1] xI ,I − J [ g +1] yI ,I ) (A5) 2. Diagonalization in the parity sector P = − In the parity sector σ [ g +1] zI σ [ g +1] zI = − , the diagonalization of the Hamiltonian of Eq. A1 H intraI ,I | + −(cid:105) = ( − h [ g +1] I + h [ g +1] I + J [ g +1] zI ,I ) | + −(cid:105) − ( J [ g +1] xI ,I + J [ g +1] yI ,I ) |− + (cid:105) H intraI ,I |− + (cid:105) = − ( J [ g +1] xI ,I + J [ g +1] yI ,I ) | + −(cid:105) + ( h [ g +1] I − h [ g +1] I + J [ g +1] zI ,I ) |− + (cid:105) (A6)leads to the two eigenvalues e [ P = − ] ± = J [ g +1] zI ,I ± (cid:113) ( h [ g +1] I − h [ g +1] I ) + ( J [ g +1] xI ,I + J [ g +1] yI ,I ) (A7)The eigenvector associated to the lowest eigenvalue e [ P = − ] − is the kept state | ψ [ g +1] − I ,I (cid:105) of Eq. 20 | ψ [ g +1] − I ,I (cid:105) = cos( θ [ g,I ] − ) | + −(cid:105) + sin( θ [ g,I ] − ) |− + (cid:105) (A8)where the angle θ [ g,I ] − is fixed by the parameters of the renormalized intra-Hamiltoniancos(2 θ [ g,I ] − ) = h [ g +1] I − h [ g +1] I (cid:113) ( h [ g +1] I − h [ g +1] I ) + ( J [ g +1] xI ,I + J [ g +1] yI ,I ) sin(2 θ [ g,I ] − ) = J [ g +1] xI ,I + J [ g +1] yI ,I (cid:113) ( h [ g +1] I − h [ g +1] I ) + ( J [ g +1] xI ,I + J [ g +1] yI ,I ) (A9) [1] L. 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