Non-local scaling operators with entanglement renormalization
NNon-local scaling operators with entanglement renormalization
G. Evenbly, P. Corboz, and G. Vidal School of Mathematics and Physics, the University of Queensland, QLD 4072, Australia (Dated: October 29, 2018)The multi-scale entanglement renormalization ansatz (MERA) can be used, in its scale invariantversion, to describe the ground state of a lattice system at a quantum critical point. From the scaleinvariant MERA one can determine the local scaling operators of the model. Here we show that,in the presence of a global symmetry G , it is also possible to determine a class of non-local scalingoperators. Each operator consists, for a given group element g ∈ G , of a semi-infinite string Γ (cid:47)g witha local operator ϕ attached to its open end. In the case of the quantum Ising model, G = Z , theycorrespond to the disorder operator µ , the fermionic operators ψ and ¯ ψ , and all their descendants.Together with the local scaling operators identity I , spin σ and energy (cid:15) , the fermionic and disorderscaling operators ψ , ¯ ψ and µ are the complete list of primary fields of the Ising CFT. Thefore thescale invariant MERA allows us to characterize all the conformal towers of this CFT. PACS numbers: 03.67.–a, 05.50.+q, 11.25.Hf
The multi-scale entanglement renormalization ansatz(MERA) [1, 2] is a tensor network introduced to effi-ciently represent ground states and low energy subspacesof quantum many-body systems on a lattice. It is basedon a real-space renormalization group (RG) techniqueknown as entanglement renormalization [1], that employsunitary tensors ( disentanglers ) to remove short-range en-tanglement from the system at each RG iteration. Theremoval of entanglement is a key difference with otherreal-space RG techniques, such as Wilson’s ground break-ing numerical RG (NRG) for the Kondo problem [3] orWhite’s extremely succesful density matrix RG (DMRG)[4] for arbitrary one-dimensional systems. At a fixedpoint of the RG flow, it produces a representation thatis explicitly scale invariant: the scale invariant
MERA.This ansatz is characterized by only a small number oftensors and can be used to describe systems with topo-logical order (at a non-critical RG fixed point) [5] as wellas continuous quantum phase transitions (correspondingto a critical RG fixed point) [1, 2, 6–10].Here we shall be concerned with the characterization ofa (scale invariant) quantum critical point with the scaleinvariant MERA [1, 2, 6–10]. Evidence that the scale-invariant MERA is capable of describing critical groundstates was first presented in Ref. [1] for the quantumIsing model and in Ref. [6] for non-interacting systemsof fermions and bosons. On the other hand it was arguedthat this ansatz naturally reproduces two important as-pects of critical ground states: the logarithmic scaling forthe entanglement entropy of a block of contiguous sites(in one-dimensional systems) and the power-law decay ofcorrelations [2]. The latter was seen to follow from thefact that a two-point correlator C ( s , s ) between twopoints separated a distance r = | s − s | is obtained af-ter O (log( r )) applications of a fixed superoperator thatintroduces a constant factor z < C ( s , s ) ≈ z log( r ) = r − q , q ≡ − log z. (1) This result was formalized in Ref. [7] by relating thepossible values of the factor z with the eigenvalues ofthat superoperator, and by identifying the scaling opera-tors of the theory with the corresponding eigenoperators.A connection between the scale invariant MERA and theconformal field theory (CFT) underlying a quantum crit-ical point was then established in Ref. [8], including away to extract the conformal data: central charge, pri-mary fields, and their scaling dimensions and operatorproduct expansion (OPE) [12]. However, the analysis ofRefs. [7, 8] was only concerned with local scaling opera-tors. For instance, for the quantum Ising model, Ref. [8]identified the primary fields identity I , energy (cid:15) and spin σ , as well as some of their descendants, all of which wereexpressed as an operator acting on two contiguous sitesof a coarse-grained lattice. Instead, non-local scaling op-erators were not considered. One reason is that entan-glement renormalization, being based on locally coarse-graining the system, has a computational cost that growsexponentially with the size of the support of the opera-tors under consideration.In this paper we show that the scale invariant MERAcan be used to characterize a whole class of non-localscaling operators of a critical lattice model. We considera quantum spin chain whose Hamiltonian H is invariantunder a symmetry group G ,Γ g H Γ † g = H, ∀ g ∈ G , (2)where Γ g ≡ · · · V g ⊗ V g ⊗ V g · · · is an infinite string ofcopies of a matrix V g , with V g a unitary representationof G . We shall see that, by incorporating the symme-try G into the MERA, it is possible to study non-localoperators that have a semi-infinite string of V g ’s. Impor-tant examples of such non-local operators are the disorderoperator µ and the fermionic operators ψ and ¯ ψ of thequantum Ising model, which are associated to the lowenergy spectrum of a chain with anti-periodic boundaryconditions. a r X i v : . [ c ond - m a t . s t r- e l ] N ov FIG. 1. (i) Scale invariant MERA, characterized by a dis-entangler u and an isometry w that are copied throughoutthe ansatz. (ii) These tensors are isometric, meaning that u † u = I , w † w = I . (iii) We choose u and w to be symmetric,Eq. 3. (iv) As a result, the infinite string Γ (cid:47)g commutes witha layer of disentanglers and isometries. In other words, Γ (cid:47)g isinvariant under coarse-graining. Local scaling operators. — Recall that the scale invari-ant MERA is made of copies of a unique pair of bulktensors, namely a disentangler u and an isometry w , dis-tributed in layers according to Fig. 1(i). In the presenceof the symmetry (2), we choose these tensors to be in-variant under G [11],( V g ⊗ V g ) u ( V g ⊗ V g ) † = u, ( V g ⊗ V g ⊗ V g ) w ( V g ) † = w (3)where V g acting on different indices may actually denotedifferent (in general, reducible) representations of G . Thelayers of disentanglers and isometries define a real spaceRG transformation and a sequence of increasingly coarse-grained lattices { (cid:32)L , (cid:32)L (cid:48) , (cid:32)L (cid:48)(cid:48) , · · · } . Under coarse-graining,a local operator o transforms according to the scalingsuper-operator S of Fig. 2(v) for g = I , o S −→ o (cid:48) S −→ o (cid:48)(cid:48) · · · (4)The scaling operators φ α and scaling dimensions ∆ α areobtained from the eigenvalue decomposition of the scalingsuperoperator S [7, 8], S ( φ α ) = λ α φ α , ∆ α ≡ − log λ α . (5) Non-local scaling operators. — In this work we considerthe coarse-graining of non-local operators o (cid:47)g of the form o (cid:47)g = Γ (cid:47)g ⊗ o, Γ (cid:47)g ≡ · · · V g ⊗ V g ⊗ V g (cid:124) (cid:123)(cid:122) (cid:125) ∞ (6) FIG. 2. (i) Coarse-graining of a non-local operator o (cid:47) = Γ (cid:47)g ⊗ o . (ii) Most of the string Γ (cid:47)g of V g ’s commutes with the coarse-graining thanks to Eq. 3. (iii) Then we can remove mostof disentanglers and isometries using Fig. 1(ii). (iv) o (cid:48) isdefined in term of o , u , w and V g . (v) Scaling superoperator S g , o (cid:48) = S g ( o ), for the local part of a non-local operator o (cid:47)g in Eq. 6, see Eqs. 7-8. Notice the average over the threepossible ways in which o can be coarse-grained. In the caseof g = I , we have V g = I , so that that o (cid:47) I is simply a localoperator and we recover the ‘usual’ scaling superoperator S of Eqs. 4-5 (see Fig. 1 of Ref. [8] for further details). where Γ (cid:47)g is a semi-infinite string made of copies of V g and o is a local operator attached to the open end of Γ (cid:47)g .Notice that, under coarse-graining, o (cid:47)g is mapped intoanother non-local operator o (cid:47)g (cid:48) of the same type, o (cid:47)g = Γ (cid:47)g ⊗ o −→ o (cid:47)g (cid:48) = Γ (cid:47)g ⊗ o (cid:48) , (7)since the semi-infinite string Γ g commutes with thecoarse-graining everywhere except at its open end, as il-lustrated in Fig. 2, where we exploit that the disentangler u and isometry w have been chosen to be symmetric, Eq.3. In other words, we can study the sequence of coarse-grained non-local operators o (cid:47)g −→ o (cid:47)g (cid:48) −→ o (cid:47)g (cid:48)(cid:48) · · · by justcoarse-graining the operator o with the modified scalingsuperoperator S g of Fig. 2, o S g −→ o (cid:48) S g −→ o (cid:48)(cid:48) · · · (8)In particular, by diagonalizing this scaling superoperator, S g ( φ g,α ) = λ g,α φ g,α , ∆ g,α ≡ − log λ g,α , (9)we obtain non-local scaling operators φ (cid:47)g,α of the form φ (cid:47)g,α = Γ (cid:47)g ⊗ φ g,α . (10)Notice that for g = I we recover the local scaling opera-tors φ α of Refs. [7, 8]. Quantum Ising model. —As a first example, we use theabove formalism to identify the non-local operator con-tent of the Ising CFT starting from the Ising quantumspin chain, as described by the Hamiltonian H Ising ≡ ∞ (cid:88) r = −∞ ( X ( r ) X ( r + 1) + Z ( r + 1)) , (11)where X and Z are Pauli matrices. This model preservesparity, G = Z , so that g ∈ { +1 , − } , with V +1 = I and V − = Z , andΓ − H Ising Γ †− = H Ising , Γ − ≡ ∞ (cid:79) m = −∞ Z. (12)Each index i of tensors u and w decomposes as i =( p, α p ), where p labels the parity ( p = 1 for even par-ity and p = − α p labels the dis-tinct values of i with parity p . Then the tensors u , w are chosen to be parity preserving, e.g. u j ,j i ,i = 0 if p ( i ) p ( i ) p ( j ) p ( j ) = −
1. An operator O acting on thespin chain has parity p if (Γ − ) O (Γ − ) † = p O .We have used the algorithm of Refs. [8, 13] to obtaina scale invariant MERA approximation for the groundstate of H Ising , with a computational effort that scales as O ( χ ˜ χ ) with the dimension χ (and ˜ χ ) of the lower (andupper) indices of the disentangler u . The present resultscorrespond to χ = 36 and ˜ χ = 20 and required one weekon a 3 GHz dual core desktop with 8 Gb of RAM. Thescaling superoperators S and S − were diagonalized ineach parity sector. The resulting non-local scaling oper-ators are of the form φ (cid:47) − ,α = · · · Z ⊗ Z ⊗ Z ⊗ φ − ,α . (13)Table I contains a few scaling dimensions extracted from S − . The second and fifth columns are for scaling op-erators with even and odd parity, respectively, and re-produce the exact results with several digits of accuracy.Fig. 3 shows scaling dimensions for both local and non-local operators. Local scaling operators with even parityform the two conformal towers [12] of the primary fieldsidentity I and energy (cid:15) of the Ising CFT, whereas thosewith odd parity form the conformal tower of the primaryfield spin σ . Non-local scaling operators with even par-ity form the conformal tower of the disorder operator µ ,and those with odd parity are organized according to twotowers corresponding to the fermion operators ψ and ¯ ψ .We have also computed the coefficients C αβγ of theoperator product expansion (OPE) [12] for all primaryfields, by analysing three-point correlators as explainedin Ref. [8]. Notice that a three-point correlator (cid:104) φ (cid:47)g α φ (cid:47)g β φ (cid:47)g γ (cid:105) will vanish unless (i) the product of par-ities of the three operators is +1 (since the ground stateis invariant under parity) and (ii) g g g = I ∈ G (sinceotherwise the product φ (cid:47)g α φ (cid:47)g β φ (cid:47)g γ is a non-local op-erator o (cid:47) , which must decompose as a sum of non-local ∆ exact ∆ MERA χ = 36 error ∆ exact ∆ MERA χ = 36 error( µ ) 1/8 0.1250002 0.0002% ( ψ ) 1/2 0.5 < − %1+1/8 1.124937 0.006 % 1+1/2 1.49999 < − %1+1/8 1.124985 0.001 % 2+1/2 2.49931 0.028 %2+1/8 2.123237 0.083 % 2+1/2 2.50118 0.047 %2+1/8 2.124866 0.006 %2+1/8 2.125487 0.023 %TABLE I. Scaling dimensions of a few non-local operators ofthe quantum Ising model. The conformal towers of ψ and ¯ ψ have identical scaling dimensions.FIG. 3. A few scaling dimensions of local (left) and non-local (right) scaling operators of the quantum Ising model,organized in its six conformal towers. scaling operators φ (cid:47) , and (cid:104) φ (cid:47) (cid:105) = 0 since all non-localscaling dimensions ∆ − ,α are larger than zero, so that (cid:104) o (cid:47) (cid:105) = 0). Table II shows a numerical estimate of allnon-vanishing OPE coefficients C αβγ . Again, the resultsmatch the exact solution with several digits of accuracy. C exact C MERA χ = 36 error C (cid:15),σ,σ = 1 / C (cid:15),µ,µ = − / C ψ,µ,σ = e − iπ/ √ . e − iπ/ √ C ¯ ψ,µ,σ = e iπ/ √ . e iπ/ √ C (cid:15),ψ, ¯ ψ = i . i C (cid:15), ¯ ψ,ψ = − i − . i Thus, not only have we been able to identify the en-tire field content { I , (cid:15), σ, ψ, ¯ ψ, µ } of the Ising CFT from asimple and rather unexpensive analysis of a quantum spinchain, but we can now also identify all possible subsetsof primary fields that close a subalgebra by inspectingTable II. Indeed, it follows that we have the followingfusion rules (cid:15) × (cid:15) = I , σ × σ = I + (cid:15), σ × (cid:15) = σ, (14) µ × µ = I + (cid:15), µ × (cid:15) = µ, (15) ψ × ψ = I , ¯ ψ × ¯ ψ = I , (16) ψ × ¯ ψ = (cid:15), ψ × (cid:15) = ¯ ψ, ¯ ψ × (cid:15) = ψ, (17)(as well as other, such as σ × µ = ψ + ¯ ψ , etc) from wherewe see that { I , (cid:15) } and { I , (cid:15), σ } close subalgebras of localprimary fields, whereas { I , (cid:15), µ } and { I , (cid:15), ψ, ¯ ψ } close sub-algebras that contain both local and non-local primaryfields, where locality is relative to the spin variables. FIG. 4. (i) Some scaling dimensions for local operators ofthe quantum XX model. [Sectors with particle numbers + | n | and −| n | yield the same scaling dimensions.] (ii) Some scalingdimensions for non-local operators with V θ = Z . (iii) Scalingdimensions ∆ θ,α as a function of θ , see Eq. 19. The scalingdimensions for n = 0 , , Quantum XX model .—As a second example we studythe quantum spin chain with Hamiltonian H XX ≡ ∞ (cid:88) r = −∞ ( X ( r ) X ( r + 1) + Y ( r ) Y ( r + 1)) , (18)where X and Y are Pauli matrices. This model is invari-ant under rotations V θ = e − iθZ/ on all spins. Therefore G = U (1), group elements g can be labeled by an angle θ ∈ [0 , π ), andΓ θ H XX Γ † θ = H XX , Γ θ ≡ ∞ (cid:79) m = −∞ V θ . (19)To simplify the analysis, we regard each site as containingtwo spins, so that the on-site ˆ z -component of the spin cantake the values 0 and ±
1. Then each index i of a tensordecomposes as i = ( n, α n ), where n ∈ Z is the ‘particlenumber’ (z spin component) and α n labels the distinctvalues of i with particle number n . Tensors u and w are chosen to be invariant under U (1), e.g. u j j i i = 0 if n ( i )+ n ( i ) (cid:54) = n ( j )+ n ( j ). An operator O acting on thespin chain has particle number n if (Γ θ ) O (Γ θ ) † = e − inθ . The optimization of a scale invariant MERA with χ =54 and ˜ χ = 32 took one week (by exploiting the blockstructure of the tensors [11]). For several values of θ ∈ [0 , π ), we diagonalized the scaling superoperator S θ inthe particle number sectors n = 0 , ± , ± , · · · . Fig. 4(i)-(ii) show the resulting scaling dimensions ∆ θ,α for θ = 0and π —that is for V = I (local operators) and V π = Z ,which also appear clearly organized in conformal towers.Finally, Fig. 4(iii) shows the scaling dimensions as afunction of θ . They are seen to accurately approximatethe expression (denoted ’exact’ in Fig. 4(iii))∆ θ,α = ∆ ,α + (cid:18) θ π + q (cid:19) − q , q = 0 , ± , (20)which is consistent with previous results [14]. Up to ashift and a rescaling factor, the scaling dimensions ∆ θ,α reproduce the low energy spectrum of the XX chain withtwisted boundary conditions with twisting angle θ .In summary, we have explained how to use the scale in-variant MERA to characterize non-local scaling operatorsof a critical quantum spin chain. For the quantum Isingmodel, we have identified all non-local primary fields andobtained remarkably accurate estimates of their scalingdimensions and OPE coefficients. For the quantum XXmodel, we have obtained continuous families of non-localoperators associated to twisted boundary conditions.Support from the Australian Research Council (APA,FF0668731, DP0878830) is acknowledged. [1] G. Vidal, Phys. Rev. Lett. , 220405 (2007).[2] G. Vidal, Phys. Rev. Lett. , 110501 (2008).[3] K.G. Wilson, Rev. Mod. Phys. , 773 (1975).[4] S. R. White, Phys. Rev. Lett. , 2863 (1992). U. Scholl-woeck, Rev. Mod. Phys. 77, 259 (2005)[5] M. Aguado, G. Vidal, Phys. Rev. Lett. , 070404(2008). R. Koenig, B. Reichardt, G. Vidal, Phys. Rev.B , 195123 (2009).[6] G. Evenbly, G. Vidal, Phys. Rev. B , 235102 (2010);ibid, New J. Phys. 12, 025007 (2010).[7] V. Giovannetti, S. Montangero, R. Fazio, Phys. Rev.Lett. , 180503 (2008).[8] R. N. C. Pfeifer, G. Evenbly, G. Vidal, Phys. Rev. A (4), 040301(R) (2009).[9] S. Montangero et al., Phys. Rev. B , 113103 (2009).V. Giovannetti et al., Phys. Rev. A , 052314 (2009).[10] G. Evenbly et al, arXiv:0912.1642.[11] S. Singh, R. N. C. Pfeifer, G. Vidal, arXiv:0907.2994.[12] P. Di Francesco, P. Mathieu, and D. Senechal, ConformalField Theory (Springer, 1997). M. Henkel,
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