Even-odd entanglement in boson and spin systems
We examine the entanglement entropy of the even half of a translationally invariant finite chain or lattice in its ground state. This entropy measures the entanglement between the even and odd halves (each forming a “comb” of sites) and can be expected to be extensive for short range couplings away from criticality. We first consider bosonic systems with quadratic couplings, where analytic expressions for arbitrary dimensions can be provided. The bosonic treatment is then applied to finite spin chains and arrays by means of the random phase approximation. Results for first neighbor anisotropic couplings indicate that while at strong magnetic fields this entropy is strictly extensive, at weak fields important deviations arise, stemming from parity-breaking effects and the presence of a factorizing field (in which vicinity it becomes size-independent and identical to the entropy of a contiguous half). Exact numerical results for small spin chains are shown to be in agreement with the bosonic RPA prediction.
pacs:03.67.Mn, 03.65.Ud, 75.10.Jm
The entanglement properties of many-body systems are of great interest for both quantum information theory NC.00 () and condensed matter physics ON.02 (); AFOV.08 (); ECP.10 (). Their knowledge enables, on the one hand, to assess the potential of a given many-body system for quantum information processing tasks such as quantum teleportation BE.93 () and quantum computation NC.00 (); JLV.03 (); RB.01 (). On the other hand, it provides a deep understanding of quantum correlations and their relation with criticality ON.02 (); AFOV.08 (); ECP.10 (); OAFF.02 (); VLRK.03 (). In non-critical systems with short range couplings, i.e., local couplings in boson or spin lattices, ground state entanglement is believed to satisfy a general area law by which the entropy of the reduced state of a given region, which measures its entanglement with the rest of the system, scales as the area of its boundary as the system size increases ECP.10 (); PEDC.05 (). This behavior is quite different from that of standard thermodynamic entropy which scales as the volume. In one dimensional systems this statement has been quite generally and rigorously proved AEPW.02 (); ECP.10 () and simply means that the entropy of a contiguous section saturates, i.e., approaches a size independent constant, as the size increases. Violation of this scaling is therefore an indication of criticality VLRK.03 (); OAFF.02 (); CC.04 (). The exact expression of the entropy of a contiguous block in a one-dimensional XY spin 1/2 chain in the thermodynamic limit has been obtained FIJK.07 (); IJK.05 (); PE.04 () and confirms the previous behavior.
The conventional area law holds for contiguous subsystems. For non-contiguous regions it actually implies that the entropy is proportional to the number of couplings broken by the partition. For instance, for comb-like regions like the subset of all even sites in a chain, the entropy should scale as the total number of sites for first neighbor or short range couplings. This was in fact verified in AEPW.02 () for the harmonic cyclic chain, where the corresponding logarithmic negativity was calculated, and also verified numerically in CWZ.06 () for some spin arrays and a - half-filled Hubbard model, where the even entanglement entropy was computed. An exact treatment of general comb entropies for a large one-dimensional critical XX spin chain with first neighbor couplings was given in KMN.06 (), showing that they are indeed proportional to the size plus a logarithmic correction.
The aim of this work is to analyze in detail the entanglement entropy of all even sites in finite boson and spin arrays, both in one dimension as well as in general -dimensions. Such bipartition can be normally expected to be the maximally entangled bipartition at least for uniform nearest neighbor couplings, as it will there break all coupling links. We first analyze the bosonic case with general quadratic couplings, where a fully analytic treatment of this entropy is shown to be feasible and allows to derive simple general expressions in the weak coupling limit. Comparison with single site and block entropies is also made. The bosonic treatment is then applied to finite spin arrays with anisotropic ferromagnetic-type couplings in a uniform transverse field through the RPA approach MRC.10 (). This allows to predict in a simple way the main properties of the total even entropy in these systems. Comparison with exact numerical results indicate that the RPA prediction, while qualitatively correct, is also quite accurate outside the critical region already for low spin , representing the high spin limit. Results corroborate that for strong fields, the total even entropy in these systems is extensive, i.e., directly proportional to the total number of sites. However, for low fields , this entropy has an additive constant, which arises in the RPA from parity restoration MRC.10 (). Moreover, in the immediate vicinity of the factorizing field KTM.82 (); AA.06 (); RCM.08 (); GAI.08 (), extensivity is fully lost and the total even entropy reduces to this constant, which is the same as that for the block entropy and is exactly evaluated. The exact bosonic treatment is described in sec. II, whereas its application to spin systems is discussed in sec. III. Conclusions are finally drawn in IV.
Ii Entanglement entropy in bosonic systems
We start by considering a system of bosonic modes defined by boson creation operators (), interacting through a general quadratic coupling. The Hamiltonian can be written as
where , and the matrix is hermitian. The system is assumed stable, such that the matrix is positive definite. We may then also write (II) in the standard diagonal form
where are the symplectic eigenvalues of , i.e., the positive eigenvalues of the matrix , with , which come in pairs of opposite sign and are all real non-zero when is positive definite RS.80 (), and are the normal boson operators determined by the diagonalizing Bogoliubov transformation RS.80 () satisfying and . The ground state is the vacuum of the operators and is non-degenerate.
Ground state entanglement properties can be evaluated through the general Gaussian state formalism AEPW.02 (); CEPD.06 (); ASI.04 (), which we here recast in terms of the contraction matrix MRC.10 (); RS.80 ()
This hermitian matrix determines, through application of Wick’s theorem RS.80 (), the average of any many-body operator. In particular, the reduced state of a subsystem of modes ( denoting the complementary subsystem and the partial trace) is fully determined by the corresponding sub-matrix (Eq. (4) with ) and can be written as MRC.10 ()
where . Eq. (5) represents a thermal-like state of suitable independent modes determined by the effective Hamiltonian . The entanglement entropy of the partition, , is then determined by the symplectic eigenvalues of (i.e., the positive eigenvalues of the matrix , which has eigenvalues and ), and given by
For instance, the entanglement of a single mode with the rest of the system is just
where , the symplectic eigenvalue of the single mode contraction matrix , represents the deviation from minimum uncertainty of the mode: for , .
ii.1 Finite translationally invariant systems
Let us now associate each bosonic mode with a given site in a cyclic chain and consider a translationally invariant system of sites, such that and , with . We first consider for simplicity the one-dimensional case. Through a discrete Fourier transform , we can diagonalize analytically and obtain an explicit expression for the contractions . We will assume , in which case the energies in (2) adopt the simple form MRC.10 ()
where are the Fourier transforms of the couplings:
The contractions depend just on the separation and are given by
The symplectic eigenvalues of the full contraction matrix (4) are of course .
In the weak coupling limit , become small and up to lowest non-zero order we obtain
which leads to
At this order just sites linked by or its convolution are correlated. The eigenvalues of subsystem contraction matrices will depend up to lowest non-zero order on and , being then for . We can then use in (6) the approximation
such that .
On the other hand, it is seen from Eq. (9) that the present system is stable provided . For attractive couplings , with all of the same sign, the strongest condition is obtained for , so that stability occurs for
For , (while all other remain finite in a finite system), implying a divergence of (Eq. (12)):
plus terms . This entails in turn a divergence of the largest eigenvalue of a subsystem contraction matrix , with plus constant terms.
ii.2 Even-odd entanglement entropy
We now evaluate the entropy of the reduced state of all even sites, , which measures their entanglement with the complementary set of odd sites (Fig. 1 left). We will assume even, such that the even subsystem, defined by , is again translationally invariant. The ensuing contraction matrix can be obtained by removing contractions between even and odd sites in the full matrix (4) and extracting then the even part. This leads to elements
whose Fourier transforms are, using Eq. (11),
The final symplectic eigenvalues of then become
for . We then obtain
Whenever can be approximated by a smooth function of , we may replace (23) by the integral
In these cases, we may then expect extensive, i.e., proportional to the number of even sites. Let us remark, however, that this is not always the case: In a completely and uniformly connected system like the Lipkin model RS.80 (); BDV.06 (); WVB.10 (), the contraction matrix will have a single non-zero symplectic eigenvalue for any subsystem MRC.10 (), including the whole even set, and is no longer proportional to . A similar lack of extensivity holds in a finite system in the vicinity of the instability (, see below).
ii.3 First neighbor coupling
Let us now examine in detail the first neighbor case , where Eq. (10) becomes
Hence, in this limit is extensive, becoming times the single site entropy (29) minus a correction accounting for the interaction between even sites:
The last term represents the even mutual entropy , which is always a positive quantity and becomes here also extensive in this limit.
In contrast, the block entropy , where denotes a contiguous block of spins, rapidly saturates as increases AEPW.02 (). In the weak coupling limit, it is verified that the ensuing contraction matrix possesses, up to lowest non-zero order, just two positive non-zero symplectic eigenvalues for any , such that
for , i.e., it saturates already for . Hence, in this limit,
being then verified that plus a constant term up to leading order. Hence, in this limit .
As illustration, the left panels in Fig. 2 depict the single site, block and even-odd entanglement entropies for a ring of sites with , where .
ii.4 Even-Odd entropy in d-dimensions
The whole previous treatment can be directly extended to a translationally invariant cyclic array in dimensions. We should just replace by vectors , and , with . We will assume couplings satisfying , with . The same previous expressions (9)–(12) then hold, with
The subsystem of all even sites, like that formed by the blue sites in Fig. 1 right, is defined by
Its contraction matrix will then be the even block of
where . Assuming even , its Fourier transform is then given again by
where if . The symplectic eigenvalues of are then given again by Eq. (22) with , and the even-odd entanglement entropy reads
where in the sum and the integral is restricted to the unit cube and valid if is a smooth function of .
In the case of first neighbor couplings
where , Eq. (34) leads to
with . In the weak coupling limit we then obtain
Hence, the single site entropy is again while Eq. (38) yields
where is a geometric entropy factor:
(, ). In the isotropic case , we have , with (Eq. 31), and , approaching for large .
At fixed , and for , and hence both and increase as increases, reflecting the larger number of links. However, and assuming again , also increases, entailing that at fixed , (and so and ) decreases:
For example, the right panels in Fig. 2 depict and in an isotropic square lattice of sites, with the same previous ratio . At fixed , their values are verified to be roughly half that of the similar one-dimensional case (Eq. (45)). Their ratio is also slightly smaller due to the increase in the parameter in (43). On the other hand, for there is again a single vanishing energy , so that all entropies behave as up to leading order, with all ratios approaching .
We also depict there the entropy of a contiguous half-size block ( sites), which is now proportional to its boundary . For , it is verified that the number of non-zero positive eigenvalues of the corresponding contraction matrix is just the number of couplings “broken” by the partition (), being all approximately equal to up to leading non-zero order. We then obtain
whence in this limit, as verified in the right panels of Fig. 2.
Iii Application to spin systems
The previous bosonic formalism can be directly applied to interacting spin systems in an external magnetic field through the RPA approximation MRC.10 (). Denoting with the dimensionless spins at site , we will consider a cyclic translationally invariant finite array which can be described by an Hamiltonian of the form
where , and
We note that may in principle also denote local intrinsic axes at each site, in which case the field is assumed to be directed along the local axis. The scaling of the couplings ensures a spin-independent mean field and effective RPA boson Hamiltonian (see below).
Normal RPA. For sufficiently strong field , the lowest mean field state (i.e., the separable state with lowest energy) is the aligned state , where denotes the local state with maximum spin along the axis (). In such a case, RPA implies the approximate bosonization MRC.10 ()
which is similar to the Holstein-Primakoff bosoniztion RS.80 (); BDV.06 () and leads to the quadratic boson Hamiltonian (II) with the parameters (48) and . We may then directly apply all previous expressions.
The bosonic RPA scheme becomes exact for strong fields for any size , spin , geometry or interaction range, since for weak coupling it corresponds to the exact first order perturbative expansion of the ground state wave function MRC.10 (). As a check, in the case of the spin one-dimensional chain with first neighbor coupling, an analytic expression of the block entropy in the limit has been obtained in FIJK.07 (); IJK.05 (); PE.04 (). For , it is given in present notation by FIJK.07 ()
where , and is the elliptic integral of the first kind. An expansion of (50) for leads exactly to present Eq. (32), with given by (27). We can then expect the asymptotic expressions (30) and (43) for to be exact in this limit also in spin systems.
Parity breaking RPA. Considering now the anisotropic ferromagnetic-type case in (47a), the previous normal RPA scheme will hold, according to Eq. (16), for , i.e., when the corresponding boson system is stable.
For , the normal RPA becomes unstable ( becomes imaginary). The lowest mean field state corresponds here to degenerate states fully aligned along an axis forming an angle with the axis in the plane: , with . We are assuming here an anisotropic coupling such that commutes with the parity , but not with an arbitrary rotation around the axis (as in the case). Such states break then parity symmetry, satisfying . The angle is to be determined from MRC.10 ()
For we should also take into account the important effects from parity restoration for a proper RPA estimation of entanglement entropies MRC.10 (). The exact ground state in a finite array will have a definite parity outside crossing points RCM.08 (), implying that the actual RPA ground state should be taken as a definite parity superposition of the RPA spin states constructed around MRC.10 (). This leads to reduced RPA spin densities of the form if the complementary overlap can be neglected. If the subsystem overlap is also negligible, such that , then MRC.10 ()
where . The final effect is then the addition of a constant shift to the bosonic subsystem entropy for . This is applicable to both and if , and the block size are not too small.
For first neighbor couplings with anisotropy (if we just redefine the axes) as well as for arbitrary range couplings with a common anisotropy , another fundamental feature for is the existence of a transverse factorizing field where the mean field states become exact ground states KTM.82 (); AA.06 (); RCM.08 (); GAI.08 (). As seen from (52), at this field , so that the RPA vacuum remains the same as the mean field vacuum MRC.10 () and all contractions vanish, implying . All RPA entropies at reduce then to the correction term arising from parity restoration MRC.10 ().
This is essentially also the exact result at : The transverse factorizing field corresponds to the last ground state parity transition as increases from 0 RCM.08 () and the ground state side-limits for are actually the definite parity combinations of the mean field states RCM.08 (); RCM.09 (). These definite parity states have Schmidt number for any bipartition, implying that the side-limits of the exact entropy of the reduced state of any subsystem at do not approach 0 but rather the values RCM.09 ()