Equipartition of the Entanglement Entropy
The entanglement in a quantum system that possess an internal symmetry, characterized by the -magnetization or -charge, is distributed among different sectors. The aim of this letter is to gain a deeper understanding of the contribution to the entanglement entropy in each of those sectors for the ground state of conformal invariant critical one dimensional systems. We find surprisingly that the entanglement entropy is equally distributed among the different magnetization sectors. Its value is given by the standard area law violating logarithmic term, that depends on the central charge , minus a double logarithmic correction related to the zero temperature susceptibility. This result provides a new method to estimate simultaneously the central charge and the critical exponents of -symmetric quantum chains. The method is numerically simple and gives precise results for the spin- quantum XXZ chain. We also compute the probability distribution of the magnetization in contiguous sublattices.
pacs:05.70.Jk, 03.67.Mn, 37.10.Jk, 71.10.Pm, 75.10.Pq
Introduction. In recent years the study of entanglement in quantum many body systems, and in quantum field theory, has been carried out intensively. As a result of it, many links have been established among previously disconnected areas of Physics, Computer Science and Mathematics. These studies have led to a quantum information perspective of phase transitions and topological order, topics that belonged traditionally to Condensed Matter Physics and Statistical Mechanics ent (). For most of the quantum critical systems in one spatial dimension, a precise characterization of entanglement has been achieved thanks to the powerful methods of Conformal Field Theory (CFT). In these systems, the area law of the von Neumann entanglement entropy (EE) of the ground state (GS), in a single interval area (), develops a logarithmic violation parameterized by the central charge of the underlying CFT CW94 ()-K04 ().
Employing ultra-cold atoms loaded in optical lattices, it is nowadays possible to simulate many one-dimensional quantum systems review-OL (). Quite recently, a measurement of entanglement was done using a one-dimensional optical lattice composed of a few Rb atoms exp-EE (). Since the number of atoms involved in these experiments is small, finite-size effects play an important role in measuring the EE. Fortunately, CFT predicts the leading finite-size correction of the Rényi entropy of the GS of a chain of sites, which is given by CW94 ()-K04 ()
where is the size of the subsystem , for periodic/free boundary conditions (PBC/FBC); and is a non universal constant. The EE corresponds to the choice .
Besides the central charge , the entanglement properties of a quantum chain can also depend on the critical exponents or the operator content of the underlying CFT. This dependence was previously observed in the entanglement for multiple intervals 2int-refs (), scaling corrections corrections (), parity effects parity (), and in the primary states and descendants of the CFT exc-stat ().
The aim of this Letter is to split the total entanglement into the contributions coming from disjoint symmetry sectors. We carry out this analysis for the critical quantum chains that have a symmetry that, in the scaling limit, develops a Kac-Moody algebra (KM). Those are the models with a critical line with continuously varying exponents KB (). As a byproduct of our calculation, we present herewith a simple new method to evaluate simultaneously the central charge and the critical exponents for this class of quantum chains. We hope, that these results could be tested in ultracold atoms experiments as the ones performed in reference exp-EE ().
Let us start with a general quantum chain with sites, whose Hamiltonian , commutes with the magnetization operator , where are spin matrices. Let be a common eigenstate of and , with eigenvalues and respectively. We split the chain into disjoints blocks and , and compute the reduced density matrix ). The magnetization operator also splits into the sum . Then, tracing over the Hilbert subspace of the block in the equation yields . This implies
where , is a density matrix with eigenvalue of , and is the probability of finding in a measurement of .
The decomposition (2) is implemented normally in numerical methods, like the DMRG W92 () and MPS, MERA, etc MPS () to reduce the memory resources needed for high precision results. The latter equation implies
where , , and . Equation (3) means that the quantum entropy in the subsystem is greater, in general, to the weighted sum of the entropies of the different magnetization sectors. This fact expresses the holistic nature of quantum entanglement. Actually, Eq. (3) can be seen as a special case of the general Holevo theorem in quantum information theory holevo (); bookNielsen (), that states that the maximum information we can extract from a general mixed state, , is given by the difference . In the case of Eq. (3) the maximum information is given by the Shannon entropy.
The critical chains studied in this Letter, reveals surprisingly that the contributions to the entropy , are equal for the low values of the magnetization . We call this situation equipartition of entanglement entropy.
Analytic predictions. The previous discussion applies to any quantum lattice system with a symmetry. In the following we shall derive analytic predictions of and , for critical spin- quantum chains, like the spin- XXZ model. In a block with sites, Eq. (2) can be inverted to obtain
where the sum projects the density matrix into the sector with . In the limit , Eq. (4) becomes
Taking the trace over the states in , and using , gives the probability distribution
Similarly, the power of Eq. (5) yields
The reduced density matrix in CFT is given by
where is the entanglement Hamiltonian and is a component of the stress tensor CT16 (). is the conformal map from the euclidean space-time, with a cut along the interval and two boundaries, into an annulus of width and height . Taking the trace of the power in (8) yields
where is the euclidean partition function of an sheeted cover of the original space-time with conical singularities around the end points of . We propose the following extension of Eq. (9) to CFTs with a KM symmetry:
where is the zero mode of the current . is the partition function, given in Eq. (9), but with fugacity . Since the eigenvalues of are given by , where are the dimensions of the boundary operators CT16 () and is the eigenvalue of , we obtain
where and is the degeneracy of the boundary operator . In the case of the ground state of the CFT, with periodic/free boundary conditions, the width should be fixed to [recall Eq. (1)]:
For the thermal state at temperature , CT16 ().
The first application of the analytic formula (10) is the Luttinger liquid which is a CFT with and a symmetry generated by the current operator , where is a chiral boson and a constant. A state with charge (with is associated to the vertex operator , and has conformal weight . The partition function (10) reads in this case
where , is the Dedekind eta function, and is a Jacobi theta function with characteristics. In the limit , one has and therefore , so that a large number of terms contribute to Eq. (12). However, using the modular transformation NS (),
and , we obtain
For special values of , the CFT is rational and becomes a finite sum (e.g. if is an even number yellow ())
where are non-negative integers, that depend on the boundary conditions on the annulus. The coefficients are denoted non-specialized characters that are labelled by the representation of an extended KM algebra. Their modular transformations non-spe (); yellow (), have been used to study the correlators in the multichannel Kondo model AL94 () and bulk susceptibilities sus (); B83 ().
The second application we report deals with the spin -isotropic exactly solvable model hspin (). This is a critical system described by the Wess-Zumino-Witten (WZW) model at level , and central charge . The model contains a similar current operator, which is now . The primary fields are labelled by the total spin . The partition function (15) is a linear combination of the non-specialized characters of and using their modular transformations yellow (); non-spe (), we obtain future ()
We can summarize both previous applications as
where or for the Luttinger liquid and the spin- chain, respectively. Quite remarkably, this parameter satisfies the universal relation , where is the spin wave velocity, and is the zero field susceptibility at zero temperature of these distinct spin chains sus (); B83 (). The case in Eq. (17) coincides with the full counting statistics (FCS) for the subsystem magnetization FCS (); ent-fluc (). Taking provides a generalized FCS where the entanglement properties are taken into account.
The constant comes from the lattice cutoff in the chains, that has not been included in (17). The highest probability corresponds to ,
where is the error function. can be approximated by replacing the integration limits in (18) by ,
which is a distribution whose Shannon entropy
quantifies our knowledge after measurement of sublattice magnetization. For a half block it will go as , a remarkably slow increase with . It is interesting to observe that the relation obtained in Refs. sus, and B83, , can be derived from Eq. (20). The zero field susceptibility is given by , where is the magnetization of a region of lenght and is the temperature. Using Eq. (20), one finds , where [Notice that the expression of , defined in Eq. (18) is for , where ]. In the limit one finds , that gives the relation . The same sort of computation provides, for example, the Gibbs entropy for the subsystem of size , that is given by . Note also that since Eq. (17) is related with the zero field susceptibility (which is related with the spin fluctuations), we would expect a connection between the entanglement and spin fluctuations. This is very interesting, since measurement of fluctuations are easier to do than the entanglement ones. Indeed, recently some authors have made this connection ent-fluc () (see also Ref. suscep-vedral-zoller, ).
Hence, the EE of the density matrix is dominated by the EE of the full density matrix , with a reduction that is independent of the quantum number . This is the equipartition of the EE mentioned above.
Numerical tests. We have considered the spin-1/2 XXZ Hamiltonian with PBC
in the critical regime , whose low energy is described by a Luttinger liquid with parameter KB (); XXZ (). Using the DMRG method we obtained the GS and the reduced density matrices and . We consider system sizes up to under PBC and keeping up to =3000 states per block in the final sweep. We have done sweeps, and the discarded weight was typically at that final sweep. To verify Eq. (17), we write it as
where are non universal constants (), and
In our opinion Eqs. (25)-(26) give us the most simple and numerically easier method to evaluate the central charge and the Luttinger parameter from reduced density matrices. The evaluation of with the DMRG does not require any additional numerical effort because it is already calculated in the evaluation of . For , Eq. (25) yields a -extension of the trace that provides the Luttinger parameter (, ), and for it gives the central charge .
The DMRG data show clearly, for each , the linear dependence on in Eq. (25). We illustrate in Fig. 1(a) the function obtained from Eq. (25) for several values of and . Table 1 summarizes the results for the estimated values of and . Notice the excellent agreement between the numerical and theoretical results.
We also test Eq. (19) for the XXZ spin-1/2 chain using as a fitting parameter. In the case of the XX model, the exact value is given by future (). Fig. 1(b) illustrates the excellent agreement between the numerical and the analytical prediction (19) for as function of for three values of .
Finally, we present the results for the Rényi-2 entropy . We found that future (), where
and . The asymptotic behavior was already shown in (23). Fig. 1(c) shows the DMRG data for in the XXZ spin-1/2 chain with , where we use the values of and found in Fig. 1(a), shown in the insets of Figs. 1(b,c).
|0||0.00 (0)||0.25 (0.25)||0.44 (0.444…)|
|0||1.99 (2)||0.99 (1)||0.66 (0.666…)|
|0.5||0.00 (0)||0.25 (0.25)||0.44 (0.444…)|
|0.5||1.48 (1.5)||0.75 (0.75)||0.48 (0.5)|
|0.00 (0)||0.25 (0.25)||0.44 (0.444…)|
|1.13 (1.1428)||0.57 (0.5714)||0.35 (0.3805)|
Twist fields. Although we have derived the analytic results using the modular properties of non-specialized characters, we think that the twist field method of references CC04 (); CCD08 () can be extended to this case. This is suggested by Eq. (17), whose r.h.s. is proportional to . The first factor comes from the correlator of the twist field with scaling dimensions , and the second factor corresponds to the correlator of a field with scaling dimensions . The field is a generalized string-order parameter with angle DR89 (), which for and has the two point correlator described above AH92 (). We expect that the generalized string-order fields provide an extension of the twist fields, that reminds the ones used in non unitary CFTs where the ground state is not the CFT vacuum nu1 (). Double log corrections to the EE have been discussed in the context of non unitary CFTs nu1 (); nu2 (), and in the non compact Liouville theory with nu2 ().
Conclusions. We have shown that for critical Hamiltonians, with a KM symmetry, the bipartite entanglement of the projected states exhibits universal properties related to the underlying CFT such as the Luttinger parameter , or the level of the KM algebra . The numerical determination of the parameter using entanglement measures are quite difficult and imprecise. We have presented here a simple way to compute together with the central charge , through the projected density matrices. We have also derived the probabilities of measuring a given magnetization in a part of the system, a problem that is related to the full counting statistics which we generalize to deal with entanglement effects.
We believe that the results obtained in this Letter can be measured in experiments with ultracold atoms. For that it is necessary to measure . In principle, this quantity can be measured using two different schemes, proposed recently in Refs. measure-opt1, and measure-opt2, . In the scheme of Ref. measure-opt1 (), it is necessary to build copies of the state . Since , where is the shift operator measure-opt1 (), we only need to measure the expectation value on n copies, for a fixed value of . Note that expectation values can be measured in optical lattices measure-opt1 (). On the other hand, the scheme proposed in Ref. measure-opt2, uses a random measurement protocol in a single copy and for the re-construction it explores the decomposition of the density matrix into disjoint blocks with different quantum numbers. This scheme seems to be a natural route to measure for a fix value of .
Note that the generalization of our approach to systems with higher rank KM algebras like is straightforward and will be reported elsewhere future (). Finally, we would also to point out that the results obtained in this article apply only to critical theories. They can be extended to the massive theories, obtained by adding relevant perturbation to the critical ones. The reduced density matrix for an interval whose size is smaller that the correlation length coincides with the critical one, except that the cord length is now replaced by the ratio , where is the lattice spacing. The equipartition of the entanglement entropy, will also holds for this more general class of models.
Acknowledgements.GS would like to thanks A. Ludwig for a useful discussion. We also acknowledge conversations with L. Balents, E. Fradkin, J.I. Latorre, E. López, J. Rodríguez-Laguna, M. Srednicki, H. Tu, W. Witczak-Krempa and G. Vidal. We acknowledge financial support from the Brazilian agencies FAPEMIG, FAPESP, and CNPq, the grants FIS2015-69167-C2-1-P, QUITEMAD+ S2013/ICE-2801 and SEV-2016-0597 of the ”Centro de Excelencia Severo Ochoa” Programme. This research was also supported in part by the Grant No. NSF PHY17-48958.
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