The entanglement negativity in random spin chains
We investigate the logarithmic negativity in strongly-disordered spin chains in the random-singlet phase. We focus on the spin- random Heisenberg chain and the random chain. We find that for two arbitrary intervals the disorder-averaged negativity and the mutual information are proportional to the number of singlets shared between the two intervals. Using the strong-disorder renormalization group (SDRG), we prove that the negativity of two adjacent intervals grows logarithmically with the intervals length. In particular, the scaling behavior is the same as in conformal field theory, but with a different prefactor. For two disjoint intervals the negativity is given by a universal simple function of the cross ratio, reflecting scale invariance. As a function of the distance of the two intervals, the negativity decays algebraically in contrast with the exponential behavior in clean models. We confirm our predictions using a numerical implementation of the SDRG method. Finally, we also implement DMRG simulations for the negativity in open spin chains. The chains accessible in the presence of strong disorder are not sufficiently long to provide a reliable confirmation of the SDRG results.
Entanglement measures are nowadays accepted as powerful tools to characterize quantum many-body systems amico-2008 (); calabrese-2009 (); eisert-2010 (); laflorencie-2016 (). Arguably, the most popular and useful one is the entanglement entropy. Given a pure state and a bipartition into two parts and , the entanglement entropy of part is defined as
where is the reduced density matrix of . For a pure state it is clear that , reflecting the property that a good measure of entanglement is symmetric in and .
If a system is in a mixed state, for instance at finite temperature, a useful measure of the correlation between and is the mutual information , which is defined as the symmetrized combination . However, it is well known that the mutual information provides only an upper bound for the entanglement between and , as it is sensitive to both classical and quantum correlations. A similar issue arises when quantifying the entanglement between disconnected regions in pure states. For instance, given the tripartition of a system as (as illustrated in Fig. 1 (a) for a spin chain), with the region of interest, is not a measure of the mutual entanglement between and .
A computable measure of the mutual entanglement between two subsystems in a mixed state is provided by the so-called logarithmic negativity peres-1996 (); zycz-1998 (); zycz-1999 (); lee-2000 (); vidal-2002 (); plenio-2005 ()
Here is the partially transposed reduced density matrix with respect to . This is defined as , with and being bases for and , respectively.
Besides its interest in quantum information, recently the negativity became the focus of several interdisciplinary efforts to understand its role in many-body systems. For critical points described by a Conformal Field Theory (CFT) its scaling behavior has been derived analytically calabrese-2012 (); cct-neg-long (); calabrese-2013 (). This allowed to prove that, unlike the entanglement entropy, the negativity is scale invariant in gapless systems described by CFT, as it was already argued based on DMRG white-1992 (); uli1 (); uli2 () simulations hannu-2008 (), on semi-analytic results for the harmonic chain mrpr-09 (), and for mean-field exactly solvable spin models hannu-2010 (). Furthermore, for disjoint intervals the logarithmic negativity contains, in principle, complete information about the intricate structure of the underlying CFT calabrese-2012 () (as it does the entanglement entropy of two or more disjoint intervals 2int ()). Some of these results have been generalized to finite-temperature calabrese-2015 (), large central charge kpp-14 (), out of equilibrium situations ctc-14 (); hoogeveen-2015 (); eisler-2014 (); wen-2015 (), and holographicrr-15 () and massive quantum field theories fournier-2015 (). In topologically ordered phases, i.e., characterized by a non-zero topological entropy levin-2006 (); kitaev-2006 (), it has been shown that the negativity is sensitive only to the “quantum” contribution lee-2013 (); castelnovo-2013 (), while it is zero in classical topologically ordered states. Also, the negativity proved to be a useful tool to characterize Kondo-like behavior in spin chains bayat-2012 (); bayat-2014 (); abab-16 (). Surprisingly, the exact treatment of the partial transposition in free fermion models is an arduous task, and no exact results are known yet for the negativity, despite some recent progresses for the calculations of the moments of the partial transpose eisler-2014 (); coser-2015 (); ctc-16 (); ctc-16b (); chang-2016 (); hw-16 (). Oppositely, the negativity can be calculated analytically for free bosonic models audenaert-2002 (). for which few results are also available in higher dimensions eisler-2016 (); dct-16 (). Furthermore, some interesting results on the negativity in Chern-Simons theories have been provided recently wen-2016 (). Finally, from the numerical perspective, effective methods to calculate the negativity are available in the context of Tree Tensor Networks calabrese-2013 (), while the moments of the partially transposed reduced density matrix can be calculated using classical alba-2013 () and quantum chung-2014 () Monte Carlo techniques. When not analytically possible, numerical extrapolations can be used to obtain the negativity from the replica limit of the moments dct-15 ().
At the same time the study of the interplay between disorder and entanglement became a fruitful research area refael-2009 (); laflorencie-2016 (). For instance, in Ref. RefaelMoore2004, it has been shown that, for disordered spin chains exhibiting the random singlet (RS) phase, the scaling of the disorder averaged entanglement entropy is logarithmic with the subsystem size like in a CFT. This has been tested numerically in the random chainlaflorencie-2005 () (which is exactly solvable for each realization of the disorder), and in the random chain dechiara-2006 () using DMRG. Furthermore, the moments of the reduced density matrix have been also studied FagottiCalabreseMoore2011 (), as well as the spectrum py-13 (), and the entanglement in low-lying excited states ramirez-2014 (). Other disordered spin models raul-2006 (); refael-2007 (); hvlm-07 (); lir-07 (); by-07 (); bcmr-07 (); il-08 (); frbm-08 (); ysh-08 (); hlvv-11 (); ki-12 (); gah-16 (); rsrs-16 () have been also considered, obtaining similar results for the scaling of the entanglement entropy. The non-equilibrium features of the entanglement in these random spin chains are also under intensive investigation dechiara-2006 (); bo-07 (); isl-12 (); bpm-12 (); spa-13 (); va-14 (); prado-14 (); vm-15 (); zas-16 (). The behavior of entanglement related quantities in classical disordered spin systems has been explored alba-2016 ().
In this paper we investigate the disorder averaged logarithmic negativity in RS phases in the framework of the strong-disorder renormalization group (SDRG) Igloi-rev (). We focus on the spin- Heisenberg () chain with random antiferromagnetic couplings, and on the random chain. We consider both adjacent and disjoint intervals (see Fig. 1 (a)). We demonstrate that in a RS phase the negativity of two intervals is always proportional to the number of singlets shared between them. Surprisingly, due to the simple structure of the RS phase, this is also the case for the mutual information, which is given as . More quantitatively, we find that for two adjacent intervals embedded in an infinite chain the disorder-averaged negativity scales as
for large and (the lengths of the two intervals). Here is an additive constant. Interestingly, Eq. (3) has the same functional dependence on as in CFT calabrese-2012 (), but the prefactor of the logarithm is different. For two disjoint intervals (see Fig. 1 (a)), we obtain
Here are the intervals length, and their distance. Eq. (4) is scale invariant and for this reason the r.h.s. has been written as a function of the cross ratio
Again, this is similar to the CFT case, where depends in a more intricate way on calabrese-2012 (); cct-neg-long (). Interestingly, in the limit of two intervals far apart, from (4) one has the power-law decay , in contrast with the CFT case, where this decay is exponential calabrese-2012 (). Both (3) and (4) turn out to be in perfect agreement with exact results obtained from a numerical implementation of the SDRG for the random chain.
Finally, by generalizing the method of Ref. hannu-2008, , we discuss how to obtain the logarithmic negativity in DMRG simulations itensor () for arbitrary tripartitions of the spin chain. The computational cost of the algorithm is , with the maximal bond dimension of the matrix product state (MPS). As a byproduct of our analysis we numerically verify the CFT scaling of the negativity of two adjacent intervals in the clean chain. In the presence of randomness it is numerically challenging to ensure the convergence of DMRG, introducing a systematic error in the numerical data. The latter is negligible only for small chains and weak disorder. The accessible chain sizes are not sufficient to provide reliable numerical evidence for (3) and (4).
The manuscript is organized as follows. In section II we introduce the spin- chain with random antiferromagnetic couplings and summarize the strong disorder renormalization group (SDRG) method. In section III we provide the analytic expressions for the logarithmic negativity and the moments of the partially transposed reduced density matrix in RS phases. The scaling of the negativity is discussed in section IV for both adjacent and disjoint intervals. These results are verified in section V using a numerical implemtation of the SDRG method. Section VI focuses on the calculation of the logarithmic negativity in DMRG simulations. In section VII we present DMRG results for the negativity of two adjacent intervals in the random chain. Finally, we conclude in section VIII.
Ii The disordered Heisenberg spin chain and the strong disorder RG
The random antiferromagnetic spin- chain with open boundary conditions (OBC) is defined by the Hamiltonian
where are the spin components acting on site , is the length of the chain, the anisotropy parameter, and are uncorrelated positive random variables, drawn from a distribution . For periodic boundary conditions (PBC) one has an extra term in (6) connecting site with site . We focus on and , corresponding to the and the random chain, respectively. For generic , the latter can be treated analytically for each realization of the disorder, exploiting the mapping to free fermions (see Appendix A). Here we restrict ourselves to the family of distributions
with , and a parameter tuning the disorder strength. For one recovers the clean, i.e., without disorder, chain, whereas corresponds to the infinite-randomness fixed point (IRFP, see section II). The latter describes the low-energy physics of (6), irrespective of the chosen distribution . For , becomes the flat distribution (box distribution) in the interval .
We now briefly review the SDRG method for the chain. The main idea is to obtain a low-energy effective description of (6) by successively integrating out the strongest couplings, and renormalizing the remaining ones. Given an arbitrary coupling configuration , one starts by identifying the strongest bond . The interaction between the two spins coupled by (that we denote as and ) is given by the Hamiltonian as
The ground state of is the singlet state
The interaction between and , and their neighboring spins (here denoted as and , respectively) is described by the Hamiltonian as
Since by definition , one can treat as a perturbation. Within second-order perturbation theory, this leads to the effective Hamiltonian for as
Here the sum is over the triplet states of two spins . The corresponding energies are , whereas one has . In the last step in (11) we defined . The effective coupling between and reads
Note that (11) does not depend on anymore. Moreover, is still of the Heisenberg form (6) with the renormalized coupling . This process of decimating the spins connected by the strongest bond, renormalizing the remaining interactions, can be represented as
The low-energy properties of the model are asymptotically, i.e., after many iterations of (13), described by the so-called random singlet (RS) phase. This is illustrated in Fig. 1 (b). In the RS phase all the spins are paired (as stressed by the links in the figure) in a random fashion. Paired spins form singlet. From (13) it is clear that longer range singlets are generated at later steps of the SDRG.
The RS phase can be quantitatively characterized through the asymptotic distribution of the couplings . The iteration of the Dasgupta-Ma rule (13) leads to a flow for . It is convenient to introduce at the given SDRG step the variables and as
Here is the maximum coupling at the step . The physical interpretation is that quantifies the difference in energy scales between the initial step and step , while the measure the broadness of the energy scales at a fixed step . The equation describing the SDRG flow of is given as ReviewIgloi ()
It can be shown that (15) has a unique solution given by
Here represents the fixed point of the SDRG flow. This fixed point is known as infinite randomness fixed point (IRFP) ReviewIgloi (), to emphasize that the broadness of the distribution increases during the flow. This is reflected in being flat, which corresponds to being peaked at (cf. (14)). Note that this justifies the perturbative treatment Fisher1994 () of in (11). Remarkably, (16) does not depend on the initial distribution which is a manifestation of its universality.
Iii Logarithmic negativity in random singlet phases
For a generic realization of the disorder, assuming that the chain is in a RS phase, the logarithmic negativity between two subsystems of the chain can be calculated analytically. In this section we show that it is proportional to the number of singlets shared between the two intervals. Let us consider a partition of the chain as in Fig. 1 with the two intervals of interest and their complement. For later convenience let us define
Given any two blocks in the chain, we denote as the number of singlets shared between them.
Before considering the entire RS phase, it is instructive to write down the density matrix and its partial transpose for an isolated singlet (more generic situations have been also considered in the literature kor ()). The density matrix of two spins forming a singlet is
in the basis , , , and . The reduced density matrix for one of the spins is
In order to calculate the negativity, we need the partial transpose that is
with eigenvalues .
The density matrix for a chain in a random singlet phase is the tensor product of the density matrices of its constituent singlets. Thus, for the partition in Fig. 1, one can write
The reduced density matrix is obtained from (22) as
and its partial transpose
To compute the negativity one can exploit the additivity of the negativity on tensor products, i.e., that . From (24) one then has
Interestingly, the terms in the square brackets vanish because for any , and . As a consequence, depends only on the number of singlets shared between and . Physically, this could have been expected because the negativity is a measure of the mutual entanglement between and . In conclusion, we have
and using the explicit form of (21)
For a bipartition, i.e., for , with the complement of , Eq. (27) is equal to the entanglement entropy of (see Ref. RefaelMoore2004, ). This is just a consequence of the fact that in the RS phase the Rényi entropies with varying are all equalFagottiCalabreseMoore2011 (), while in the generic case the negativity for a bipartition is always equal to vidal-2002 (); cct-neg-long (). This is different from the clean bipartite system (which is conformal invariant) in which the Rényi entropies depend non trivially on the index in a well-known fashion cc-04 (); cc-rev ().
Finally, it is interesting to compare (27) with the mutual information between two intervals which can be readily obtained using . A straightforward calculation yields
iii.1 The moments of the partially transposed reduced density matrix
It is instructive to consider the moments of the partially transposed density matrix
The logarithmic negativity can be obtained from (29) via the analytic continuation restricted to the even , as it is routinely done in CFT calculations calabrese-2012 (); cct-neg-long (). Although are not entanglement measures, they encode universal information about critical systems. In particular, the moments contain more information than the entanglement negativity, and, at least in principle, they could be used to characterize the entire spectrum of on the same lines as it has been done for reduced density matrix calabrese-2008 ().
The computation of the moments starts by rewriting (23) as
Using that the partial transposition acts trivially on the terms in the curly brackets, one obtains
The only two non-zero eigenvalues of are
The corresponding degeneracies are
The moments can be readily written down as
In Eq. (35) it is evident that for generic , cannot be measures of the entanglement between and , since they depend on the number of singlets shared with . By analytically continuing Eq. (35) from the even sequence we recover the negativity (27), i.e., denoting with the even integers, we have
Iv Scaling of the negativity in the random singlet phase
In this section we derive the scaling properties of the logarithmic negativity between two intervals and in the random singlet phase. From the previous section, it is obvious that the only needed ingredient is the scaling of the average number of singlets shared between and (cf. (27)). Here denotes the disorder average. Remarkably, can be obtained from the result for a single interval RefaelMoore2004 (). Given an interval of length embedded in the infinite line, the average number of shared singlets scales for large as RefaelMoore2004 ()
where a non-universal constant.
In order to derive , let us consider a generic multipartition of the chain into blocks as , with (the case with is in Fig. 1 (a)). It is convenient to define the set of all possible compact subintervals of the chain. For instance, for one has . For each , one can decompose the number of singlets as the sum of all the singlets emerging from an arbitrary block and ending in an arbitrary block , i.e. in formula
After taking the disorder average, (38) gives
where we used that (37) is valid for any . One can obtain for any pair by solving the linear system of equations in (39) generated by considering all , as we are going to show in the following for two adjacent and disjoint intervals.
iv.1 Two adjacent intervals
iv.2 Two disjoint intervals
Interestingly, (43) does not depend on the non-universal additive constant indicating that the logarithmic negativity between two disjoint intervals is a universal scale invariant quantity, as in the clean case. In particular, from (43), can be rewritten as a function only of the cross ratio as in (4).
An intriguing consequence of (43) is that the entanglement between two disjoint intervals decays as , i.e., with a power law of their mutual distance, in stark contrast with the CFT case where this decays is exponential calabrese-2012 (). The result for adjacent intervals (41) is recovered from (43) by taking of the order of the lattice spacing (fixed to in all above formulas) and then considering much larger than that.
V Numerical SDRG
In this section we present numerical evidence confirming the analytical results for the average number of singlets between two arbitrary intervals (adjacent and disjoint) and consequently for the negativity and mutual information between them. In order to do so, we numerical implement the SDRG for finite-size spin chains which works according to the following the steps: a) for a given disorder realization, we apply the Dasgupta-Ma rule (13), i.e., we pair the spins interacting with the strongest bond to form a singlet; b) the two spins are then decimated, and the remaining couplings renormalized according to (12); c) this SDRG step is iterated until all the spins are paired in singlets. At every RG step we monitor the distribution of the renormalized couplings, as well as the spin configurations. In this section we only consider a flat probability distribution of the coupling between 0 and 1.
v.1 SDRG flow of the renormalized couplings
As a preliminary check of the numerical SDRG, we study the flow of the couplings in (6) as a function of the SDRG step . The results are reported in Fig. 2, where we plot the probability distribution , with (cf. (14)), and the maximum coupling at step . The data are for a finite chain with . The initial values of the couplings are drawn from the box distribution .
v.2 Single-interval entanglement entropy
Another important check of the validity of the numerical SDRG is provided by the scaling of the single block entanglement entropy. We consider the random chain with both open and periodic boundary conditions. For OBC we expect different scalings depending on whether the block is in the bulk of the chain or whether it touches the boundary. The entanglement entropy is obtained by determining the number of singlets between and for each realization of the disorder and then by multiplying the average by . We report the variance of the distribution as an estimate of statistical error.
Numerical SDRG results for are shown in Fig. 3 (a)-(d). In all panels the data are for chains of length and . The disorder average is over disorder realizations. We start by considering in panel (a) the block in a periodic chain. For , increases logarithmically as function of the interval length (note the logarithmic scale on the -axis). Instead, for finite-size effects are visible. The dash-dotted line in the figure is a fit to the expected SDRG result RefaelMoore2004 ()
with the only fitting parameter. For the data are in perfect agreement with (44).
Here is a symmetric function under , which ensures . It has been found that for a single block in a periodic chain, is well approximated by FagottiCalabreseMoore2011 ()
where the prefactor of the logarithm takes into account that subsystem shares only one edge with its complement. This formula indeed describes accurately the data for . The finite size and boundary effects can be taken into account again by replacing by a modified chord length (45). We find that the same function in Eq. (46) describes very accurately the data for OBC as shown in Fig. 3 (c). This is a non trivial result since there is no conformal invariance to guarantee the equality of the two finite size scaling functions (i.e. open and periodic) as in the clean case.
Finally in Fig. 3 (d) we consider a block of length in the center of an open chain. For we expect that the block does not feel significantly the presence of the boundary and (44) should describe accurately the data, as evident from the figure. It is instead not obvious how to modify the prediction to take into account finite-size and boundary effects: we only mention that replacing with the modified chord length (46) does not work. This does not come unexpected since also in clean systems the results is more complicated x11 ().
v.3 Logarithmic negativity: Two adjacent intervals
Using the numerical SDRG method, we now study the scaling behavior of the logarithmic negativity between two adjacent intervals and in the random Heisenberg chain. Specifically, we provide robust numerical evidence for (42). The negativity is just obtained from the statistics of the singlets between and and multiplying the resulting average by .
In Fig. 4 we report the SDRG data for for two adjacent intervals of the same length . Fig. 4 (a) shows the SDRG results for a periodic chain, while Fig. 4 (b) is for two intervals in the middle of an open chain. We plot the shifted negativity
as a function of so that data for different chain lengths are expected to collapse on a single scaling curve. This clearly happens for both open and periodic chains. In both cases increases logarithmically for , when finite-size effects kick in. Actually, for PBC one can take into account all the finite-size effects by replacing all lengths in (3) with modified chord lengths, obtaining
Conversely, for the OBC chain in Fig. 4 (b) there is not a simple modification of the result for infinite chain to take into account the boundary effects, as it was the case also for clean systems ctc-14 (). For this reason, in the figure we only report the one-parameter fit to the SDRG prediction for the infinite chain (42) valid for .
v.4 Logarithmic negativity: Two disjoint intervals
We finally move to the most interesting case of the logarithmic negativity between two disjoint intervals of equal length at distance . As usual in this section, we compute the negativity from the statistics of the singlets between the two intervals. Our results are reported in Fig. 5 for both periodic and open chains.
In the case of a periodic chain, we plot as a function of the cross-ratio (5) in which we have substituted all lengths by the corresponding modified chord lengths, i.e.
in order to take into account finite-size effects. The SDRG asymptotic prediction for the negativity is given by Eq. (4) which reads
In Fig. 5 (a) the data correspond to a fixed choice of and is running up to . It is clear from the figure that the prediction (52) describes incredibly well the numerical data, without any fitting parameter. Notice that close to corresponds to far away intervals and the decay of the negativity is algebraic, in constrast with the exponential behavior of clean systemscalabrese-2012 (); cct-neg-long (). Oppositely, the limit corresponds to very close intervals and the logarithmic divergence is needed to reproduce the adjacent intervals results.
For the case of an open chain, we limit ourselves to consider two intervals centered around the middle of the chain, i.e. and . We generate the data by fixing a value of and letting run up to the boundary. In this case, we do not have a prediction which takes into account finite size effects, so we expect Eq. (52) to describe correctly the data for , i.e. when the effects of the boundary can be neglected. For this reason in Fig. 5 (b) we report the negativity as function of the cross-ratio given by the infinite volume formula . It is evident from the figure that this prediction describes correctly the data for far from , as it should. With increasing , the effects of the boundaries becomes less and less relevant and the data get closer to the infinite volume result, as expected.
Vi Logarithmic negativity in DMRG simulations
In this section we describe how to calculate the logarithmic negativity in DMRG simulations. We consider a generic lattice of sites, restricting ourselves to open boundary conditions. On each site of the lattice we consider a local Hilbert space of finite dimension (e.g., for spin- systems one has ). A generic wavefunction of a pure state is written as a Matrix-Product-State (MPS) as
Here with is the so-called physical index representing the states in the local Hilbert space, whereas are the virtual indices, with the local bond dimension of the MPS. In (53) the sum over the repeated indices is assumed, and one has , with an element of the basis of the Hilbert space at site . At fixed , is a matrix, while and are vectors. The MPS in (53) can be pictorially represented as in Fig. 6 (a), where the boxes denote the tensors , and horizontal and vertical legs the virtual and physical indices, respectively. The full system density matrix is obtained as
where the bar in (and correspondingly in the boxes in Fig. 6) denotes the complex conjugation.
vi.1 The partially transposed reduced density matrix
In order to calculate the logarithmic negativity we now introduce a tripartition of the system (Fig. 6) (b) as , where and are the intervals of interest and the remaining parts. For simplicity we consider the case of two equal-length intervals with at mutual distance , and shifted from the left boundary by sites, i.e., , . This is illustrated in Fig. 6 (b) for , , and . The reduced density matrix is obtained by tracing over the degrees of freedom of in (54) as
where we defined
with , (cf. Fig. 6 (d) for the definition of the indices). A similar definition holds for . The term in (55) is because the virtual indices and are contracted. In (55) is the “transfer matrix” of the region connecting the two intervals ( in Fig.