Classical Correlation and Quantum Discord in Critical Systems
We discuss the behavior of quantum and classical pairwise correlations in critical systems, with the quantumness of the correlations measured by the quantum discord. We analytically derive these correlations for general real density matrices displaying symmetry. As an illustration, we analyze both the XXZ and the transverse field Ising models. Finite-size as well as infinite chains are investigated and the quantum criticality is discussed. Moreover, we identify the spin functions that govern the correlations. As a further example, we also consider correlations in the Hartree-Fock ground state of the Lipkin-Meshkov-Glick model. It is then shown that both classical correlation and quantum discord exhibit signatures of the quantum phase transitions.
pacs:03.65.Ud, 03.67.Mn, 75.10.Jm
The concept of correlation, i.e., information of one system about another, is a key element in many-body physics. Indeed, the properties of a many-body system are strongly affected by changes in the correlations among its constituents. These changes are responsible for the occurrence of remarkable phenomena such as a quantum phase transition (QPT), which is a critical change in the ground state of a quantum system due to level crossings in its energy spectrum. QPTs occur at low temperatures where the de Broglie wavelength is greater than the correlation length of the thermal fluctuations (effectively ) (1); (2). Striking examples of QPTs include metal-insulator transitions in strongly correlated electronic materials, magnetic transitions in quantum spin lattices, superfluid-Mott insulator transitions in atomic gases induced by a Bose-Einstein condensation, among others.
Correlations can be both from classical and quantum sources. The existence of genuinely quantum correlations can be usually inferred by the presence of entanglement among parts of a system. Indeed, entanglement displays a rather interesting behavior at QPTs (3), being able to indicate a quantum critical point (QCP) through nonanalyticities inherited from the ground state energy (4); (5). Moreover, for one-dimensional critical systems, ground state entanglement entropy exhibits a universal logarithmic scaling governed by the central charge of the Virasoro Algebra associated with the underlying conformal field theory (6); (7); (8). Remarkably, the logarithmic scaling is robust against disorder (9); (10); (11); (12). In higher dimensions, entanglement usually scales following an area law for noncritical systems (see, e.g., Ref. (13)). For critical models, violations of the area law have been found (14); (15), with logarithmic-type corrections appearing. More recently, it has been observed that an area law is generally implied by a finite correlation length when measured in terms of the mutual information (16). This remarkable behavior of entanglement at criticality is indeed a consequence of the correlation pattern exhibited by the ground state of the system.
Nevertheless, although entanglement provides a route to find out the existence of quantum correlations, it can be shown that they can appear even when entanglement is absent (17); (18). Quantum correlations, which can be measured by the quantum discord (17), often arise as a consequence of coherence in a quantum system, being present even for separable states. Moreover, entangled states commonly involve more than only quantum correlations, i.e., they usually carry classical correlations among their parts. In a multiparty mixed-state scenario, the possibility of nonseparable states with purely quantum correlations (with no supporting background of classical correlations) have also been investigated (19), but genuine indicators of multiparticle classical correlations are still under debate (20). By focusing on condensed-matter systems, the aim of this work is, starting from an arbitrary -symmetric model, to investigate pairwise correlations, explicitly splitting up their classical and quantum contributions and analyzing their behavior at QPTs. As an illustration, ground states of both XXZ and transverse field Ising spin chains will be shown to share quantum as well as classical correlations for nearest-neighbor spin pairs, with both of them signalling the critical behavior of the system. An analysis of such correlations for the thermodynamic limit of XXZ and Ising chains has recently appeared in Ref. (21). Here, we generalize this analysis by providing analytical expressions for the correlations in each model and also by discussing their finite-size behavior. As a further illustration, we describe the correlations in the Hartree-Fock ground state of many-body systems, taking the Lipkin-Meshkov-Glick (LMG) model as an example.
The paper is organized as follows. In Sec. II we review the evaluation of correlations from the point of view of information theory, describing in particular the concept of quantum discord. In Sec. III, we discuss the computation of classical and quantum correlations for general models displaying symmetry. Sec. IV, Sec. V, and Sec. VI are devoted to illustrating our results for the XXZ chain, the transverse field Ising chain, and the LMG model, respectively. Conclusions are then presented in Sec. VII.
Ii Classical correlations and quantum discord
In classical information theory, the information obtained, on average, after knowing the value of a random variable , which takes values within a set of probabilities , can be quantified by its Shannon entropy . We use the symbol as denoting logarithm at base 2 throughout the text. By taking two such random variables and , we can measure the correlation between them by their mutual information
where is the joint entropy for and . By introducing the conditional entropy
which quantifies the ignorance (on average) about the value of X given Y is known, we can rewrite Eq. (1) as
In order to generalize the above equations to the quantum domain, we replace classical probability distributions by density matrices. Denoting by the density matrix of a composite system and by and the density matrices of parts and , respectively, the quantum mutual information can be defined as
where is the von Neumann entropy for subsystem and
is a quantum generalization of the conditional entropy for and . A remarkable observation realized in Ref. (17) is that the conditional entropy can be introduced by a different approach which, although classically equivalent to Eq. (2), yields a result in the quantum case that differs from Eq. (5). Indeed, let us consider a measurement performed locally only on part . This measurement can be described by a set of projectors . The state of the quantum system, conditioned on the measurement of the outcome labelled by , becomes
where denotes the probability of obtaining the outcome and denotes the identity operator for the subsystem . The conditional density operator given by Eq. (6) allows for the following alternative definition of the quantum conditional entropy:
Therefore, following Eq. (3), the quantum mutual information can also be alternatively defined by
Eqs. (4) and (8) are classically equivalent but they are different in the quantum case. The difference between them is due to quantum effects on the correlation between parts and and provides a measure for the quantumness of the correlation, which has been called quantum discord (17). In fact, following Refs. (17); (22), we can define the classical correlation between parts and as
with the quantum correlation accounted by the quantum discord, which is then given by
Iii Pairwise correlations for -symmetric quantum spin lattices
We will consider here an interacting pair of spins-1/2 in a spin lattice, which is governed by a Hamiltonian that is both real and exhibits symmetry, i.e. invariance under -rotation around a given spin axis. By taking this spin axis as the direction, this implies the commutation of with the parity operator , where denotes the total number of spins and is the Pauli operator along the -axis at site . Note that a number of spin models are enclosed within these requirements as, for instance, the XXZ spin chain and the transverse field Ising model. Disregarding spontaneous symmetry breaking (see, e.g., Refs. (23); (24); (25) for a treatment of spontaneously broken ground states), the two-spin reduced density matrix at sites labelled by and in the basis , with and denoting the eigenstates of , will be given by
In terms of spin correlation functions, these elements can be written as
where is the magnetization density at site and () denote two-point spin-spin functions at sites and , with the expectation value taken over the quantum state of the system. Note that, in case of translation invariance, we will have that () and, therefore, . Moreover, observe also that the density operator given in Eq. (11) can be decomposed as
In particular, for translation invariant systems, we have that . In order to determine classical and quantum correlations, we first evaluate the mutual information as given by Eq. (4). The eigenvalues of read
Therefore, the mutual information is given by
with , , , and . Classical correlations can be obtained by following a procedure that is similar to those of Refs. (18); (21), but applying it now for the case of the general density matrix given by Eq. (11). We first introduce a set of projectors for a local measurement on part given by , where is the set of projectors on the computational basis ( and ) and . Note that the projectors represent therefore an arbitrary local measurement on . We parametrize as
where and . Note that and can be interpreted as the azimuthal and polar angles, respectively, of a qubit over the Bloch sphere. By using Eq. (6) and the equation , with denoting the Kronecker symbol, we can show that the state of the system after measurement will change to one of the states
Therefore, the classical correlation for the spin pair at sites and will be given by
where . For some cases, the maximization in Eq. (25) can be worked out and an expression purely in terms of the spin correlation functions can be obtained (e.g., the XXZ and Ising chains below). In general, however, has to be numerically evaluated by optimizing over the angles and . Once classical correlation is obtained, insertion of Eqs. (16) and (25) into Eq. (10) can be used to determine the quantum discord.
Iv The XXZ spin chain
Let us illustrate the discussion of classical and quantum correlations between two spins by considering the XXZ spin chain, whose Hamiltonian is given by
where periodic boundary conditions are assumed, ensuring therefore translation symmetry. We will set the energy scale such that and will be interested in a nearest-neighbor spin pair at sites and . Concerning its symmetries, the XXZ chain exhibits invariance, namely, , which provides a stronger constraint over the elements of the density matrix than the symmetry. Indeed, invariance ensures that the element of the reduced density matrix given by Eq. (11) vanishes. Moreover, the ground state has magnetization density (), which implies that
where, due to translation invariance, we write (). Due to the fact that , we will have that , which considerably simplifies the computation of classical and quantum correlations. Moreover, we will have that and . Then, the maximization procedure in Eq. (25) can be analytically worked out (18), yielding
with . For the mutual information we obtain
where is the ground state energy density
with denoting the ground state of . Eqs. (31) and (32) hold for a chain with an arbitrary number of sites, allowing the discussion of correlations either for finite or infinite chains. Indeed, ground state energy as well as its derivatives can be exactly determined by Bethe Ansatz technique (26), which allows us to obtain the correlation functions , , and . In Fig. 1, we plot classical and quantum correlations between nearest-neighbor pairs for an infinite XXZ spin chain.
Note that, in the classical Ising limit , we have a fully polarized ferromagnet. The ground state is then a doublet given by the vectors and , yielding the mixed state
Indeed, this is simply a classical probability mixing, with and . The same applies for the antiferromagnetic Ising limit , where a doubly degenerate ground state arises. Moreover, observe that the classical (quantum) correlation is a minimum (maximum) at the infinite order QCP . On the other hand, both correlations are discontinuous at the first-order QCP . This is indeed in agreement with the usual behavior of entanglement both at infinite and first-order QPTs. For an infinite-order QCP, entanglement commonly display a maximum at the QCP (30); (31); (32), while for a first-order QCP, entanglement usually exhibits a jump at the QCP (33); (34). Nevertheless, we note that in the specific case of the ferromagnetic QCP and for pairwise entanglement measures such as concurrence (35) and negativity (36), no jump is detected, being hidden by the operation (37). It is interesting to observe the behavior of the functions and that govern the classical and quantum correlations. For , we have that , which means that the classical correlation is governed by . For , we have that , with the crossing occurring exactly at the infinite-order QCP. Therefore, the correlations are governed by different parameters in different phases. For , we obtain and , which implies that and . These results are shown in Fig. 2 below.
V The transverse field Ising model
Let us consider now the Ising chain in a transverse magnetic field, whose Hamiltonian is given by
with periodic boundary conditions assumed, namely, . As before, we will set the energy scale such that and will be interested in a nearest-neighbor spin pair at sites and . This Hamiltonian is -symmetric and can be exactly diagonalized by mapping it to a spinless free fermion model with single orbitals. This is implemented through the Jordan-Wigner transformation
where and are the creation and annihilation fermion operators at site , respectively. By rewriting Eq. (34) in terms of and we obtain
In order to diagonalize we consider fermions in momentum space
where and are creation and annihilation fermion operators with momentum , respectively, and is the fermion position at site . The wave vectors satisfy the relation , where denotes the distance between two nearest-neighbor sites and , with and taken, for simplicity, as an even number. Then, by inverting Eq. (37) and inserting the result in Eq. (36), we obtain
Diagonalization is then obtained by eliminating the terms and from the Hamiltonian given by Eq. (38), which do not conserve the particle number. This is indeed achieved through the Bogoliubov transformation in which new fermion operators and are introduced as linear combination of and
where and are real numbers parametrized by and . This parametrization naturally arises as a consequence of the fermionic algebra , , with standing for the Kronecker delta symbol. Moreover, to recast the Hamiltonian in a diagonal form we define by demanding that . Therefore, by expressing in terms of Bogoliubov fermions and by imposing the trace invariance of the Hamiltonian, Eq. (38) becomes
with . Hamiltonian (40) is diagonal, with ground state given by the -fermion vacuum. The procedure above also applies for the evaluation of the matrix elements of the reduced density operator given by Eq. (12), which amounts for the computation of the magnetization density and the two-point functions . This can be achieved by using that (27) and by expressing the remaining correlation functions as
Hence, we exactly determine the two-spin reduced density matrix. Classical and quantum correlations can then be directly obtained from Eqs. (9) and (10). By numerically computing the classical correlation in Eq. (25) for nearest-neighbor spin pairs at sites and , we can show that the maximization is achieved for any by the choice and . Then, the measurement that maximizes is given by , with and denoting the up and down spins in the direction, namely, . This numerical observation implies that , . Therefore, Eq. (25) is ruled by the spin functions and , i.e.
where is the binary entropy
We plot and in Fig. 3 for a chain with sites. Note that, for the system is a classical Ising chain, whose ground state is a doublet given by the vectors and . Therefore, the system is in the mixed state
with and . On the other hand, in the limit the system is a paramagnet (vanishing magnetization in the direction), with all spins in state . Therefore the system will be described by the density operator
which is a pure separable state, containing neither classical nor quantum correlations.
The QPT from ferromagnetic to paramagnetic state is a second-order QPT and occurs at . Signatures of this QPT can be found out by looking at the derivatives of either classical or quantum correlations. Indeed, the QPT can be identified as a pronounced minimum of the first derivative of the classical correlation, which is exhibited in Fig. 4. Note that the minimum logarithmically diverges at as the thermodynamic limit is approached (see inset of Fig. 4). In the case of quantum correlations, its first derivative shows an inflexion point around , as displayed in Fig. 5. Indeed, by looking at its second derivative in Fig. 6, the QPT is identified by a pronounced maximum, which shows quadratic logarithmic divergence at as the thermodynamic limit is approached (see inset of Fig. 6).
The behavior of the quantum discord is therefore rather different from the entanglement behavior, whose first derivative is already divergent at the QCP. Remarkably, the scaling of pairwise entanglement derivative in this case (see e.g. Refs. (4); (5)) is much closer to the scaling of the classical correlation derivative (as given by Fig. 4) than that of the quantum correlation derivative (as given by Fig. 5). As in the case of the XXZ model, it is interesting to observe that the spin functions and , which govern the correlations in the Ising chain [see Eqs. (43)-(45)], exhibit a crossing at the QCP. This is shown in Fig. 7 for a chain with sites.
Vi The LMG model
The discussion of correlations above can also be applied in collective systems. As an illustration, we will consider here the LMG model (38), which describes a two-level Fermi system , with each level having degeneracy . The Hamiltonian for LMG model is given by
The operators and create a particle in the upper and lower levels, respectively. This Hamiltonian can be taken as describing an effective model for many-body systems, with one level just below the Fermi level and and the other level just above, with the level below being filled with particles (39). Alternatively, the LMG model can be seen as a one-dimensional ring of spin-1/2 particles with infinite range interaction between pairs. Indeed, the Hamiltonian can be rewritten as
where and (39). The system undergoes a second-order QPT at . As , the ground state, as given by the Hartree-Fock (HF) approach, reads
where we have introduced new levels labelled by and governed by the operators
In Eq. (51), is a variational parameter to be adjusted in order to minimize energy, which is achieved according to the choice
Despite being an approximation, the HF ground state provides the exact description of the critical point (for recent discussions of the exact spectrum of the LMG model, see Refs. (40); (41)). The pairwise density operator for general modes and is given by
where and , with . By evaluating the matrix elements of for the HF ground state, we obtain
Note that Eq. (53) displays symmetry and, therefore, classical and quantum correlations can be computed by using Eq. (25). Note also that, for , the density matrix is diagonal and the state is completely pairwise uncorrelated. On the other hand, for , there is an equal amount of classical and quantum correlations between the modes. These correlations vanish for , which is the fully polarized state. The result is plotted in Fig. 8. We can then observe that the derivatives of both classical correlation and quantum discord exhibit a signature of the QPT (see inset of Fig. 8). These signatures are in agreement with the caracterizations in terms of entanglement (42); (43) and Fisher information (44).
In conclusion, we have investigated the behavior of pairwise correlations in general symmetric systems, splitting up their classical and quantum contributions. This allowed for the treatment of spin systems such as the XXZ model and the transverse field Ising chain, where the maximization required for the evaluation of the correlations has been analytically worked out. Moreover, we have identified the spin functions that govern the correlations and also discussed their behavior for finite size chains. As a further application, we have used our approach to investigate the case of a collective model, given by the LMG Hamiltonian.
As it was shown, both classical correlation and quantum discord display signatures of the critical behavior of the system for the cases of first-order, second-order, and infinite-order QPTs. For first-order QPTs, as illustrated by the ferromagnetic point of the XXZ spin chain, we found that both classical correlation and quantum discord display a jump at the critical point, which closely resembles the behavior of entanglement. For second-order QPTs, although both kind of correlations exhibit signatures of the QPTs, the derivatives of the quantum discord show a scaling that is rather different from the entanglement behavior in the transverse field Ising model (see, e.g., Refs. (4); (5)). It is remarkable that, for this model, the first derivative of pairwise entanglement with respect to the parameter that drives the QPT exhibits a scaling that is much closer to the scaling of the first derivative of the classical correlation. For infinite-order QPTs, as given by the antiferromagnetic point of the XXZ spin chain, classical correlation is a minimum at the QCP while quantum discord is a maximum. Whether or not this might be a general feature of infinite-order QPTs is still under analysis.
Further investigations including correlations between blocks of particles and the effect of temperature may be interesting to establish a precise comparison between (classical and quantum) correlations and entanglement at QPTs. Moreover, dynamics in open quantum systems (45); (46); (47) may also provide an interesting scenario for the discussion of the properties of the correlations and its implications for phase transitions. Such topics are left for a future research.
This work was supported by the Brazilian agencies MCT/CNPq and FAPERJ.
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