Static and dynamical quantum correlations in phases of an alternating field XY model
We investigate the static and dynamical patterns of entanglement in an anisotropic XY model with an alternating transverse magnetic field, which is equivalent to a two-component one-dimensional Fermi gas on a lattice, a system realizable with current technology. Apart from the antiferromagnetic and paramagnetic phases, the model possesses a dimer phase which is not present in the transverse XY model. At zero temperature, we find that the first derivative of bipartite entanglement can detect all the three phases. We analytically show that the model has a “factorization line” on the plane of system parameters, in which the zero temperature state is separable. Along with investigating the effect of temperature on entanglement in a phase plane, we also report a non-monotonic behavior of entanglement with respect to temperature in the anti-ferromagnetic and paramagnetic phases, which is surprisingly absent in the dimer phase. Since the time dynamics of entanglement in a realizable physical system plays an important role in quantum information processing tasks, the evolutions of entanglement at small as well as large time are examined. Consideration of large time behavior of entanglement helps us to prove that in this model, entanglement is always ergodic. We observe that other quantum correlation measures can qualitatively show similar features in zero and finite temperatures. However, unlike nearest-neighbor entanglement, the nearest-neighbor information theoretic measures can be both ergodic as well as non-ergodic, depending on the system parameters.
Quantum many-body systems have been established to be a possible candidate for the implementation of quantum information protocols amico-rmp (); lewenstein-rev () such as one-way quantum computation one-way-qc () and network quantum communication comun-spin (). Also, laboratory realization of model Hamiltonians in various substrates, including optical lattice coldatom (); xxz-exp (); optical-lattice (), ion traps lewenstein-rev (); lab_xy_ion (), solid state systems solid_xy (), and NMR nmr (), have made possible the testing of properties of several information theoretic measures of quantum correlations, belonging to both of the entanglement-separability horodecki-rmp () and information-theoretic modi-rmp () domains. On the other hand, tools developed in, and with the help of, quantum information theory have been found to be useful in the analysis of the ground and excited states of such many-body systems sim-comp (); dmrg-usual (); dmrg-info (). Moreover, development of topological quantum computation and especially topological quantum memories indicate the importance of quantum many-body systems in the goal of practical realization of a quantum computer topo-q-comp (). Consequently, in recent years, characterization of quantum many-body systems from quantum information theoretic perspectives have become a vibrant field of research.
Although most of such studies are restricted to the “static” properties of quantum correlations in the zero-temperature and thermal states, the time evolution of the system is also extremely important in quantum information processing tasks like in one way quantum computation one-way-qc (). In the static case, the traditional approach to study a quantum many-body system is to recognize appropriate order-parameters defining the phases occurring in the system, and to investigate the response of these order parameters to external perturbations. The ground state of such a system is usually represented by a complex multipartite quantum state, characterized by the classical as well as the quantum correlations present between its constituting parts. A quantum phase transition (QPT) qpt-books (); dutta-qpt-books (), which occurs at zero temperature and solely due to quantum fluctuations, brings about a qualitative change in the ground state of a quantum many-body system, when a system parameter is varied. Quantum correlations having quantum information theoretic origins, are shown to be useful in characterizing various phases and corresponding QPTs in a large spectrum of quantum many-body systems XXZ-qpt (); latorre-jphysa (); eisert-jphysa (); xy-qpt-ent (); other-qpt-ent (); other-nonmono (); other-qpt-ent-2 (); syl (); xy-qpt-disc (); other-qpt-disc () (see also amico-rmp (); modi-rmp (), and the references therein). Among all these models, a prominent one is the one-dimensional (1d) Fermi gas of spinless fermions in an optical lattice – a system realizable in ultracold atom substrate, by using a Fermi-Bose mixture in the strong-coupling limit 1dfermigas (). In the spin language, the model can be described by an anisotropic XY model in a transverse magnetic field lsm (); bmpapers (); qpt-books (); dutta-qpt-books ().
Manipulation of cold atoms in the laboratory has allowed the realization of physical systems such as dilute atomic Fermi and Bose gases, in different spatial dimensions, thereby providing excellent opportunities to apply quantum information theoretic concepts in these systems bose-review (); fermi-review (). Recent experimental evidences of superfluid, metallic, and Mott-insulating phases fermi-old-exp (); fermi-recent-exp () motivates one to investigate a Fermi gas of spinless fermions in an 1d optical lattice, where the fermions are of two types, distinguished by different chemical potentials. Considering the two types of fermions to be located on two different sublattices, one of which contains all the “even” sites and the other one holds all the “odd” ones, the fermionic model, via a Jordan-Wigner transformation, can be shown to be equivalent to an 1d anisotropic XY model in the presence of a uniform, and an alternating transverse magnetic field that alternates its direction from to depending on whether the lattice site is even, or odd dutta-qpt-books (); kogut (); alt-field-old (); alt-field-defect (); diep (). The model offers a rich phase diagram. While only two phases, viz. a “paramagnetic” (PM) phase and an “antiferromagnetic” (AFM) phase occur in the ground state of the XY model in a uniform transverse field, qpt-books (); bmpapers (); lsm (), an additional “dimer” (DM) phase emerges due to the introduction of the local site-dependent alternating field in the present model dutta-qpt-books (); kogut (); alt-field-old (); alt-field-defect (); diep (). Although the properties of several quantum information theoretic measures of quantum correlations have been extensively studied and reported for different phases and corresponding QPTs in the former case xy-qpt-ent (); xy-qpt-disc (); syl () (see also references in amico-rmp (); modi-rmp ()), it is interesting to see how the new phase structure, formed due to the introduction of the alternating field, can be characterized using quantum correlations. In this paper, we characterize the static as well as dynamic properties of quantum correlations in the 1d anisotropic XY model in a uniform and an alternating field. As the measures of quantum correlations, we focus on bipartite measures, and use logarithmic negativity (LN) neg_group () from the entanglement-separability genre, and quantum discord (QD) disc_group (); total_corr () from the information-theoretic domain. We show that irrespective of the values of the anisotropy parameter, first derivative of bipartite entanglement can detect all the three phases in this model. Moreover, the finite-size scaling analysis of the system near the QPTs is performed to distinguish phase boundaries between the AFM and the PM, and between the AFM and the DM. Similar investigations are also carried out for quantum discord, which also faithfully indicate the quantum critical points. Like the factorization point in the XY model factor_old (); factor_new (), we here prove the existence of a line in the space of the system parameters, which we call as the “factorization line” (FL), on which the ground state of the system is separable, having a Néel-type order.
The change of phase diagram with finite temperature has both fundamental and experimental importance due to the technological limitations of reaching absolute zero temperature. In this scenario, we discuss the weathering of the landscapes of quantum correlations over the phase-plane of the system parameters, chosen to be the strengths of the uniform and the alternating transverse field, with increasing temperature. We point out that bipartite entanglement is the most fragile in the AFM phase, while it is robust in the DM phase against increasing temperature. We identify the phases in which nonmonotonicity of entanglement with the increase of temperature is observed. Specifically, we perform a non-monotonicity cartography, and map, on the plane of the chosen system parameters, the regions in which the thermal quantum correlations exhibit non-monotonic variation with temperature. We show that for LN and for high values of anisotropy parameter, most of the non-monotonicity occurs in the AFM region, while QD is found to be nonmonotonic in the PM phase for low anisotropy. Interestingly, we discover that the temperature variation of LN is found to be monotonic in the entire DM phase, while for QD, non-monotonicity occurs at a very small region of the DM phase.
As already stated, the time dynamics of quantum correlations in any physical system is extremely relevant for implementation of quantum information processing tasks. In this paper, we find both the small and large time quantum correlation patterns of the evolved state. We observe that although entanglement dies quickly compared to QD, it possesses larger value than QD, which ensures the possibility of implementing several information tasks requiring high values of entanglement. The study of large time behaviour of quantum correlations also helps us to settle issues like the ergodicity ergodic_def (); dyn-ergo (); dyn-ergo-2 (); dyn-ergo-3 (); dyn-ergo-xy (); dyn-ergo-xy-dual (); dyn-ergo-xy-sen (); dyn-ergo-xy-group (); dyn-ergo-other () of LN and QD, quantified by the ergodicity scores. We find that, up to our numerical accuracy, entanglement always remains ergodic, while QD shows nonergodicity in different phases of the model. We point out that the region of nonergodicity of QD increases with an increase in the anisotropy in the system. Therefore with respect to transverse field parameter, we show that QD undergoes a nonergodic to ergodic transition which is absent for entanglement upto our numerical accuracy, irrespective of the anisotropy parameter and initial temperature.
The paper is organized as follows. In Sec. II, the Hamiltonian describing the anisotropic XY model in the presence of a uniform, and an alternating transverse field, and its relation to a two-component 1d Fermi gas are discussed. Brief descriptions on the diagonalization of the model Hamiltonian, and the different phases occurring in the ground state of the model are provided in the same section. Sec. III contains the definitions of the canonical equilibrium state and the time-evolved state of the system. The determination of the single-site and two-site reduced density matrices from the canonical equilibrium state and the time-evolved state of the model is also presented in this section. The static properties of the quantum correlations, including the different types of QPTs, finite-size scaling analysis, determination of the factorization line, and thermal quantum correlations are discussed in Sec. IV. Sec. V reports the ergodicity of quantum correlations and short-time dynamics of entanglement as well as QD. Sec. VI contains the concluding remarks.
Ii The model
Let us consider a family of models describing a system of spins of magnitude on an 1d lattice consisting of sites. We assume that an external transverse magnetic field of site-dependent strength , being the site index, acts on the spins at time . The magnetic field can be interpreted as the resultant of a uniform transverse field, , and a transverse field, , which reverses its direction from to , depending on whether the lattice site is even, or odd. The Hamiltonian describing the system is given by
Here, the system parameter represents the strength of the exchange interaction, while is the anisotropy present in the system. We assume periodic boundary condition (PBC), and an even number of lattice sites, such that , where .
ii.1 Relation to one-dimensional Fermi gas
can be mapped onto a two-component Fermi gas of spinless fermions, on an 1d optical lattice consisting of two sublattices. Here, , where the operators are related to the Pauli operators via the relations , , and . One of the two sublattices in the fermionic model is constituted of the “odd” lattice sites, while the other contains the “even” ones. One of the two components of the fermions is situated on the odd sublattice, while the other is located on the even sublattice. The two components are distinguished by two different time-dependent chemical potentials, and , and the corresponding creation operators are denoted by and , respectively, following the usual fermionic anticommutation relations , and . Here, or , depending on whether , the site indices, are odd or even, respectively.
Applying the transformation in Eq. (LABEL:jwt), the form of the Hamiltonian representing the 1d two-component Fermi gas of spinless fermions at every time instant , up to an additive constant energy , can be written as
where the operators , , , and describe the interactions between the spinless fermions belonging to the odd and the even sublattices, with and being the corresponding number operators. Here, is the fermionic tunneling strength between a pair of even and odd sites, and is the total number of lattice sites. Note that the existence of the two types of magnetic field (uniform and alternating) in the original model is reflected by the existence of the two sublattices in the fermionic model, differentiated by the chemical potentials and thereby leading to two types of fermionic operators, and .
For general , the Hamiltonian given in Eq. (LABEL:ham_ferm) can be written as , with
via the Fourier transformations given by
Here , , and () is fermionic operators. Since , the above Fourier transformation decomposes the space upon which acts into non-interacting subspaces. These subspaces, each having a dimension sixteen, do not allow transitions within themselves, irrespective of the values of the system parameters , , and . The diagonalization of the Hamiltonian is thereby reduced to the diagonalization of , acting on the subspace, which can be achieved by a convenient choice of the basis (see Appendix A). We note that the lowest eigenvalue of , is given by . The ground state energy per site, , of the Hamiltonian can be obtained as .
We now briefly discuss the patterns of different phases, and the corresponding QPTs, present in the model described by the Hamiltonian in Eq. (LABEL:ham_ferm). We choose the strength of the transverse fields, uniform and alternating, as the tuning parameters. Information about the phase-boundaries can be obtained from the second-order derivatives of the ground state energy, , with respect to and , where we take and . For , the system undergoes two different second-order QPTs, namely, a transition from a paramagnetic (PM) to an antiferromagnetic (AFM) phase, and a transition from the AFM to a dimer (DM) phase. Fig.1(a) and (b) depict the spectrum of at critical points corresponding to AFM PM (), and AFM DM () QPTs, respectively, for (see Appendix A for the expressions of the eigenvalues as functions of and the system parameters). Note that the vanishing of the energy gap in the spectrum occurs at for the AFM PM transition, and at for AFM DM transition. On the other hand, Fig. 1(c) depicts the variation of the spectrum of as a function of for . This point on the plane belongs to the factorization line, which is discussed in Sec. IV.2. One of our aims in this paper is to detect such transitions by using quantum information quantities. The phase boundaries corresponding to these transitions are given by the lines , and , respectively. It is interesting to note that there exists a set of duality relations, given by , by virtue of the unitary transformation , which indicates that both AFM PM and AFM DM transitions belong to the same universality class, namely, the Ising universality class dutta-qpt-books (). One must also note that for , the model reduces to the well-known anisotropic XY model in a uniform transverse magnetic field of magnitude .
Iii Canonical-equilibrium and time-evolved states: Local density matrices
In this paper, we intend to study the statistical mechanical properties of the model in terms of bipartite quantum correlations. We now briefly introduce the notions of canonical equilibrium states and time-evolved states corresponding to the Hamiltonian given in Eq. (1), and describe how two-spin reduced density matrices corresponding to such states can be obtained. For our purpose, we consider the situation where the time-dependent magnetic fields and , are chosen as
The canonical equilibrium state (CES) of the system at time is given by
where is the partition function, and is given in Eq. (1). Here, , is the absolute temperature, and is the Boltzmann constant. In all our calculations, we set . For the purpose of this paper, we consider a system which is in contact with a heat bath at temperature for a long time up to the instant that we call , so that a thermal equilibrium between the system and the heat bath have developed. The equilibrium is in the canonical sense, allowing exchange of energy between the bath and the system with the usual average energy constraint, but forbidding exchange of particle. To study quantum correlations in the evolution, we choose the canonical equilibrium state as an initial state. When the magnetic fields are switched off, the CES starts evolving in time following the Schrödinger equation dictated by the Hamiltonian in Eq. (1). At any time , the time-evolved state (TES), , is given by
where represents the Hamiltonian given in Eq. (1) at .
iii.1 Local density matrices
To investigate the behaviour of bipartite quantum correlation measures of the CES and TES, computation of the single-site and the two-site reduced density matrices of the entire state is necessary. Since we consider the system with periodic boundary condition, all the nearest neighbour bipartite state are same and hence their two-spin correlation functions would be independent of the choice of the pairs of spins, while the single-site magnetizations depends on whether the lattice site is even, or odd. A general single-site density matrix, given by , can be obtained by tracing out all the spins except the spin at the lattice site , which is “” for the odd site, and “” for the even site. Here, is the identity operator in the qubit Hilbert space. For CES corresponding to a real Hamiltonian, , implying with the complex conjugation being taken in the computational basis. Also, the Hamiltonian possesses a global phase-flip symmetry, such that , implying . Hence, the single-site reduced density matrix corresponding to the CES is given by . On the other hand, corresponding to the evolved state is not necessarily equal to its complex conjugation, and the existence of the global phase-flip symmetry is a complicated issue due to the time dependence of the Hamiltonian. However, use of the Wick’s theorem leads to the same form of , when TES is considered instead of the CES.
Let us now consider the two-site reduced density matrix , corresponding to the spins at the lattice sites and , and obtained by tracing out all the other spins except those at the positions and . In the present case, we restrict ourselves to nearest-neighbor pairs of spins, such that . To keep the notations uncluttered, from now on, we shall discard the lattice indices, and denote the nearest neighbor two-spin density matrix by , where we assume that the lattice site belongs to the even sublattice without any loss of generality. The two-party state , of the CES and TES in this system, can be written as
where are the two-site spin correlation tensor. In the case of CES, by using arguments similar to that in the case of the single-site density matrix, and by applying the Wick’s theorem, one can show that only diagonal elements of the correlation tensor, given by , , remain. On the other hand, in the case of TES, and remain non-zero in addition to the diagonal correlators. For brevity, from now onward, we discard the site indices while mentioning the two-spin correlators.
iii.2 Quantum correlations between two modes of a 1d Fermi gas
We now demonstrate that the quantum correlation between a nearest-neighbour spin pair chosen from the anisotropic XY model in a uniform and an alternating transverse magnetic field is the same as that present between two fermionic modes located at the two nearest-neighbour lattice sites in the fermionic model given in Eq. (LABEL:ham_ferm). Without any loss of generality, the two-site density matrix of a nearest-neighbour pair of lattice sites, denoted by “eo”, can be written as , where , and . Here, depending on whether . The coefficients, , are given by . Expanding and applying Wick’s theorem as in Sec. III.1, the TES corresponding to a pair of fermionic modes on the “eo” site pair for the fermionic model is given by
With a convenient choice of basis given by , where represents the vaccume state, the individual terms in Eq. (10) can be expressed in their respective matrix forms. A comparison with the matrix forms of the operators implies that in matrix form, can be expressed as
Here, are the Pauli matrices, where e.g. in the basis and where e.g. in the basis. Note that is connected to the TES in the spin model via a local unitary transformation given by , thereby implying no change in the values of the chosen measure of bipartite quantum correlation.
Iv Static behaviour of quantum correlations
In this section, we discuss the behaviour of bipartite quantum correlation measures of the reduced density matrix of the nearest-neighbour qubit pair, obtained from the zero-temperature and the thermal states of the model. Since the model is not evolving, we call the states as static states. For our purpose, we consider logarithmic negativity (LN), denoted by , and quantum discord (QD), denoted by , in the ground and thermal states of the model. The former belong to the entanglement separability paradigm, while the latter is from the quantum information theoretic regime of quantum correlations. Short descriptions of these measures are provided in Appendix B. While computing QD in the entire paper, we always perform local rank- projection measurement on the “even” qubit. We choose two different types of quantum correlation quantities since they behave differently as demonstrated in the XY as well as XXZ model xy-qpt-ent (); xy-qpt-disc (); XXZ-qpt ().
iv.1 Quantum correlations at zero temperature
In the limit , we now investigate the behavior of LN and QD, as functions of the system parameters and , in the thermodynamic limit. For this system, and all the non-zero classical correlations can be obtained analytically by diagonalizing , following the similar prescription for the XY model (see Appendix C) and hence the exact computation of LN and QD is possible, as depicted in the top horizontal pannels of Fig. 2. To keep the notation uncluttered, from now on, we denote LN by , and QD by . In this paper, all the analysis are carried out for unless specified otherwise. The qualitative feature of the entire investigation remains same for . Note that the LN has a high value in the DM and PM phases, while the value is low in the AFM phase. On the other hand, the value of QD is moderate in the PM region. In the AFM region, the QD has a low value except in the cases where the values of and are comparable. Along the line , QD is vanishingly small. Note that the situation is reversed if one performs measurement on the odd qubit while determining QD. In that case, low values of QD are found along the line. Hence, the asymmetry imposed into the model due to the introduction of the alternating field is captured from the distribution of QD values over the AFM region in the parameter space of , but not by LN. However in the case of LN, there exist two zero-entanglement lines in the AFM phase, as depicted in the left pannel of the top horizontal row of Fig. 2, which represent fully separable ground states. These lines, which we refer to as the “factorization lines”, are discussed in detail in the subsequent section.
One must note here that the introduction of only a local parameter, i.e. the alternating transverse field, in the well-known transverse-field XY Hamiltonian lsm (); bmpapers (), gives rise to the DM phase, which is not present in the transverse-field XY model. It is interesting to investigate how the QPTs occurring at the AFM DM phase boundaries can be characterized using entanglement and information-theoretic quantum correlation measures, and whether such characteristic behaviours are similar to those observed in the case of the AFM PM transition in this model as well as in the usual transverse-field XY model lsm (); bmpapers (). In the thermodynamic limit, and for the latter case, the QPT is found to be signaled by a divergence in the first derivative of entanglement as well as in the information theoretic measures with respect to the system parameters, and (see the bottom panel of Fig. 2). We find that, similar to the AFM PM QPT, other transitions can also be detected by the first derivative of appropriate measures of quantum correlations. As an example, in Fig. 2 (bottom horizontal pannels), we plot (left pannel) and (right pannel) as functions of , and for , , and . From the figures, we can clearly see that both (left pannel) and (right pannel) diverge at the AFM DM and AFM PM boundaries. We plot the absolute values of the first derivatives of LN for a better representation of the divergence, as the actual first derivative can tend to both positive as well as negative infinity, depending on the variation of LN with respect to and . Note here that there exists two lines, one vertical () and the other horizontal () in the variations of the first derivative of LN, as depicted in Fig. 2, over which the value of LN remains almost constant. This is indicated by the low value of the first derivative of LN over those lines. Note also that there exists several models in which bipartite entanglement cannot detect quantum phase transitions bipartite-failed (); amico-rmp (); lewenstein-rev (). Such example includes the spin liquid-dimer transition in 1D model J1-J2 (). The results obtained here show that this is not the case for the XY model with uniform and alternating transverse field.
Finite-size scaling analysis
Advancement of experimental techniques has made the laboratory realization of several quantum many-body systems of finite size, such as the quantum anisotropic XY model with a transverse alternating magnetic field, possible lab_xy_ion (); solid_xy (), which highlights the importance of studying the behavior of quantum correlations in the context of QPTs in system of finite number of spins. Towards this aim, we present the finite size scaling analysis of the system using the bipartite quantum correlations, and determine the scaling exponents. More specifically, we discuss the finite-size scaling of the system at the QPTs corresponding to (i) AFM PM and (ii) AFM DM transitions.
(i) AFM PM transitions: At , the model reduces to the widely studied anisotropic XY model in a uniform transverse magnetic field of strength . As and , the model undergoes a QPT, between the quantum PM phase and the AFM phase, at . It is well known that this QPT is signaled by a non-analyticity in the first-derivatives of the quantum correlation measures, , with respect to the system parameter, xy-qpt-ent (); xy-qpt-disc (). With the introduction of the transverse alternating field , the QPT point changes according to the line , which denotes the phase-boundary between the AFM and the quantum PM phase in the present model. As shown in Fig. 2, the AFM PM transition is also signalled by a non-analyticity in the first-derivative of LN, or QD, with respect to , when is kept fixed as . In the case of a system of finite size, the QPT is signalled by a maximum or a minimum in the variation of the first-derivative of LN and QD with respect to , for fixed values of (see Fig. 3). The position of the maximum or minimum denotes the position of the critical point on the axes of the respective system parameter. The maximum or minimum sharpens with increasing system size, and the position of the QPT approaches the QPT point as , denoted by , as
Here, are dimensionless constants, and are the scaling exponents.
Fig. 3(a) depicts the variation of derivative of LN and QD, as functions of , as , for fixed value of i.e. with . The approach of the QPT points, , at finite , towards the QPT point in the thermodynamic limit, , are depicted in the insets. Fitting the numerical data with Eq. (12), one can estimate the values of and . Table 1(a) contains the values of and , in the case of both LN and QD, when the value of is kept fixed at and (). Note that the values of and change with although the qualitative feature remains invariant.
(ii) AFM DM transition: Similar to the case of AFM PM transition, in the thermodynamic limit, the AFM DM transition is signaled by a non-analyticity in the first-derivative of LN, or QD, with respect to either of and . It is interesting to investigate how the position of the QPT point, as determined by the position of the sharp peak in the variation of the derivatives of LN and QD with respect to either , or , changes with a variation in the system size. In order to do so, one may try to determine the canonical equilibrium state at zero temperature by using the same methodology as in the case of the AFM PM transition. However, due to the approximations involved in determining the zero-temperature state, in the present case, LN and QD, as functions of either of and , exhibit finite jumps in values at the QPT point, thereby forbidding a finite-size analysis in a similar fashion as in the previous case (for a discussion on the behaviour of the finite jumps, and a figure, see Appendix E). Therefore, we employ the exact diagonalization technique in the present case, and determine the non-degenerate ground state of the Hamiltonian given in Eq. (1) by using Lanczos algorithm lanczos (). The reduced density matrix correspponding to a nearest-neighbour even-odd spin pair, labeled by “”, can be determined by tracing out all the other spin variables from the ground state. Using the reduced density matrix, the nearest-neighbour LN and QD can be computed. Here, for the purpose of discussions, the first-derivatives of LN and QD, with respect to , by keeping fixed at , are plotted in Fig. 3(b). We find that in the case of the AFM DM transition also, the position of the QPT at a finite approaches the actual QPT point at according to an equation similar to Eq. (12), where the constants are denoted by and . For example, for , the corresponding values of these fitting parameters are , (for LN), and , (for QD).
iv.2 Factorization line: Separable ground state
We now discuss the occurrence of the separable ground state in the AFM phase of the model which can observed by considering the variation of bipartite as well as multipartite entanglement as functions of and (Fig. 2). The symmetry of the Hamiltonian (Eq. (1)) under PBC motivates one to look for a separable eigenstate of the form
having a Néel type order, where are the states of the spins on the odd (even) site . The Hamiltonian can be written as , where is the two-site Hamiltonian given by
with , defined on an even-odd pair of sites, and can be obtained from straightforwardly by interchanging the site indices. Using Eq. (13), the lowest separable eigenenergy can be obtained as
where we have used the fact that and are energetically equivalent. This leads to a minimum separable energy per site, , given by . Without any loss of generality, one can choose the states to be
where and are real parameters such that and . The Two-spin reduced density matrix , corresponding to the odd-even pair of spins, is then given by , where . Since the Hamiltonian in Eq. (1) is a real one, we expect , leading to
where the optimization over the states is reduced to an optimization over the real parameter space of and . The minimum is achieved for
However, the state (Eq. (13)) would be the ground state of the Hamiltonian if , the ground state energy of the two-spin Hamiltonian factor_old (); factor_new (). We find that the ground state of is nondegenerate, with a ground state energy given by . Determination of using the values of , and equating to leads to the following condition
equivalently , which represents a line on the plane for fixed values of . The ground state of the Hamiltonian, at every point on this line on the plane, is separable, represented by a line of vanishing entanglement (Fig. 2). We call this line as factorization line. At any point on this line, the minimum eigenvalue of , given by , becomes independent of , as demonstrated for () with in Fig. 1(c). This feature is in contrast to the dependence of at the QPT points (see Figs. 1(a) and (b)).
iv.3 Effect of temperature on quantum correlations
Quantum correlations are known to be fragile quantities, and are expected to decay with increasing thermal noise in the system. Moreover, absolute zero temperature is hard to be achieved in a real experiment. It is therefore interesting to investigate the effect of thermal fluctuations on the bipartite quantum correlations corresponding to the Hamiltonian (Eq. (1)). The patterns of LN and QD as functions of and , for (Fig. 4(a)) and (Fig. 4(b)) are plotted in Fig. 4. In the case of LN, we observe that starting from the factorization line at , a zero-entanglement region grows with increasing temperature, and spans the entire AFM phase at sufficiently high temperature. Note here that the zero-entanglement region at can also be found in the PM phase, while it is absent in the DM phase even at high temperature. A few interesting features emerge from these results.
The rate of spreading of the vanishing entanglement region with increasing temperature is found to be much slower towards the PM phase compared to that inside the AFM phase. This can be easily perceived from the fact that with the increase of temperature from to , the entanglement vanishes in the entire AFM phase, but covers only a small region in the PM phase. It implies that bipartite entanglement is more fragile in the AFM region compared to the other phases.
Remarkably, bipartite entanglement in the DM phase is the most robust against increasing thermal noise among the three phases.
The effect of thermal noise on QD is less drastic compared to that in the case of LN, as observed from Fig. 4. With increasing temperature, the minimum value of QD along the line increases. However, the qualitative distribution of QD over the plane remains unchanged.
Remark 1. We choose and treat as high temperature since bipartite entanglement of the AFM phase has been destroyed at this temperature. However, if one increases temperature beyond , LN in the entire region of plane becomes zero.
Remark 2. For the purpose of demonstration, we have kept the anisotropy parameter constant to a fixed value . One must remember that the definition of “high” depends, along with the other system parameters, on the anisotropy parameter also. However, the qualitative features, such as the robustness of bipartite entanglement in the DM phase compared to other phases, or the fragility of LN in the AFM phase remain unchanged with a change in the value of the anisotropy parameter.
Monotonicity vs. non-monotonicity
Up to now we have discussed the variation in the pattern of entanglement and QD with the increase of temperature. We now report the existence of non-monotonic variation of LN and QD as functions of temperature in this model. Such non-monotonicity is known for other quantum many-body Hamiltonians, including the transverse-field XY model other-nonmono (); other-qpt-ent (); other-qpt-ent-2 (). Since the model under consideration possess different phase diagram than the XY model, non-monotonicity of quantum correlations, specially entanglement with temperature may reveal some new feature. We will show that this is indeed the case. For fixed choices of , typical variation profiles exhibiting non-monotonicity of LN and QD with temperature, as shown in Fig 5. The importance of non-monotonic behavior of bipartite quantum correlation lies in the fact that even at high temperature, which is much easier to attain in the laboratory, a higher value of quantum correlations is obtained compared to the state with lower temperature. This has potential applicability in the realization of those quantum protocols in the laboratory, which use quantum correlations as resources.
It is therefore necessary to map the occurrence of non-monotonic variations of bipartite quantum correlations over the phase plane of the model, so that the useful regions at finite temperature can be recognized. Let us consider a set of values in the space of the system parameters, denoted by , which results in a non-monotonic variation of the bipartite quantum correlation measure, , with the variation of temperature. We call such a set as the “non-monotonicity generator” (NG). Fig. 6 exhibits the NGs for different values of , specially , , , and , on the -plane, when LN and QD are considered to be the bipartite quantum correlation measures. We observe that in the case of LN, for low values of , the NGs are confined to the AFM phase, and narrow regions inside the PM phase, in the vicinity of the AFM PM QPT line. At , the factorization lines, denoted by the solid line on the plane, almost coincides with the AFM PM QPT line, which is represented by the dashed lines. With increasing value of , the factorization lines get separated from the AFM PM transition lines, and the NGs span the region confined by these lines, as can be seen in the case of and . At , which represents the Ising model in transverse-uniform and transverse-alternating field, the factorization lines meet each other, and almost entire AFM phase is filled by the NGs. Remarkably, the DM phase remains completely free from NGs for all values of .
The behaviors of QD and LN, with respect to non-monotonicity, are somewhat complementary to each other for low and high values of the anisotropy parameter. At , NGs for QD span the PM phase, which is in contrast to the case of LN, where NGs can be found in the PM phase only in the vicinity of the AFM PM phase boundary. On the other hand, for , in the case of LN, NGs fill almost entire AFM phase while being absent in the PM and the DM phase while in the case of QD, non-monotonicity occurs in a very small region of the AFM and DM phase. In Fig. 7, we map, on the -plane, the regions where LN increases with decreasing the value of which confirms the findings in Fig. 6. To generate the figures corresponding to the -plane, we have kept the value of fixed.
Note. In a system of finite number of spin- particles, use of the open boundary condition (OBC) instead of the PBC changes the phase boundaries only slightly, and the AFM region on the plane shrinks. With an increase in the system size, the difference between the phase portraits corresponding to the PBC and the OBC reduces. Note here that each of the pairs of nearest-neighbour spins in the quantum spin model described by Eq. (1) consist of an even, and an odd spin. In the case of the PBC, there is a special type of translational symmetry in the model, such that , where is, say, an odd site. Hence, LN is same for all the nearest-neighbour spin pairs, while due to this property, quantum discord is same only when measurement is performed on the same type of spin (even or odd) in all nearest-neighbour spin pairs. This implies that under PBC, investigation of the bipartite quantum correlations belonging to any one of the nerest-neighbour spin pairs suffice. On the other hand, for complete characterization of the static and dynamical behaviour of nearest-neighbour bipartite quantum correlations in a system of spins under OBC, computation of bipartite quantum correlation measures corresponding to nearest-neighbour pairs, depending on whether is even (odd), is necessary. However, the broad qualitative features of the factorization line and the phase boundaries, as reported in this paper, remain unaltered even under OBC for finite-sized systems.
V Dynamics of quantum correlations
So far, we have considered the static characteristics of quantum correlations in different phases of the 1d anisotropic XY model in uniform and alternating transverse field. In this section, we aim to study the behaviour of quantum correlations and their statistical mechanical properties under time evolution. In order to compute nearest neighbour LN and QD of TES, the two-spin reduced density matrix has to be determined, which, in turn, requires the evaluation of single-site magnetizations and two-site spin correlation functions. This can be done by utilizing the fact that the evolutions of the subspaces in the momentum space (see Sec. II and Appendix A) are independent of each other. This leads to , where is the Hamiltonian in the momentum subspace at , and . The time-evolved single-site magnetizations and two-site spin correlation functions are given by