(1)
where \(S_{\alpha }=\frac{1}{2}\sigma _{\alpha }\) and \(\sigma _{\alpha }\) are Pauli spin matrices, \(J_{ij}\) is the exchange coupling constant between ith and jth ions, \(g_{i}\) are the Landé g-factors corresponding to each metal ion, and h is an external magnetic field. The sign and magnitude of \(J_{ij}\) depend on the geometry, distance, and nature of the bridging ligands between the metal ions. Note that from now on, open boundary conditions are used for dinuclear systems, whereas periodic boundary conditions are used for trinuclear systems.
Given the energy spectrum \(E_{n}\) (\(n=\{0,1,2,\cdots \}\)), which represents the eigenvalue of the n-th quantum many body eigenstate, the partition function is defined as \(Z=\sum _{n}\textrm{e}^{-\beta E_{n}}\) with \(\beta ={1/k_{B}T}\).
First, let’s present the necessary definitions of some physical quantities51. Magnetization, defined as \(\text {M}=-\frac{\partial F}{\partial h},\)where F is free energy \(F=-k_{B}T\ln Z.\) Similarly, we can obtain magnetic susceptibility \(\chi\) of a material in terms of its magnetization M and the applied magnetic field h, which is \(\chi =\frac{\partial \text {M}}{\partial h}.\)
At thermal equilibrium, the state of the system is determined by the density matrix
$$\begin{aligned} \hat{\rho }(T)=\frac{\textrm{e}^{-\frac{H}{k_{B}T}}}{Z}=\sum _{n}\frac{\textrm{e}^{-\frac{E_{n}}{k_{B}T}}}{Z}|\psi _{n}\rangle \langle \psi _{n}|, \end{aligned}$$
(2)
where \(|\psi_n\rangle\) are corresponding eigenstates. Pairwise thermal entanglement is considered. Computable entanglement measures, such as concurrence C and entanglement entropy EE23,25,27, are used for dinuclear and trinuclear copper (II) complexes—systems with spin 1/2. The concurrence C, corresponding to the reduced density matrix \(\rho _{ab}(T)\), where \(\rho _{ab}(T)\) is a partial trace of the density matrix \(\rho (T)\), is defined as:
$$\begin{aligned} C=\max \{0,\sqrt{\lambda _{1}}-\sqrt{\lambda _{2}}-\sqrt{\lambda _{3}}-\sqrt{\lambda _{4}}\}, \end{aligned}$$
(3)
where the \(\lambda _{i}\) are the eigenvalues of the operator
$$\begin{aligned} \mathscr {R}_{ab}=\rho _{ab}(T)(\sigma _{y}\otimes \sigma _{y})\rho _{ab}^{*}(T)(\sigma _{y}\otimes \sigma _{y}). \end{aligned}$$
(4)
in descending order (\(\sigma _{y}\) are Pauli spin matrices).
The entanglement entropy EE is defined as:
$$\begin{aligned} EE= & -\left( \tfrac{1+\sqrt{1-C^{2}}}{2}\right) \log _{2}\left( \tfrac{1+\sqrt{1-C^{2}}}{2}\right) -\left( \tfrac{1-\sqrt{1-C^{2}}}{2}\right) \log _{2}\left( \tfrac{1-\sqrt{1-C^{2}}}{2}\right) . \end{aligned}$$
(5)
These measures are essential for understanding the quantum correlations present in the system, which can provide insights into the fundamental properties of quantum states and their potential applications in quantum information processing.
Dinuclear complexes
More explicitly, for dinuclear systems, the Hamiltonian (1) can be expressed as follows:
$$\begin{aligned} \hat{H}=-J\left( S_{1}^{x}S_{2}^{x}+S_{1}^{y}S_{2}^{y}+S_{1}^{z}S_{2}^{z}\right) -hg\mu _{B}(S_{1}^{z}+S_{2}^{z}), \end{aligned}$$
(6)
the eigenvalues and corresponding eigenstates of the Hamiltonian are
$$\begin{aligned} E_{1}=&-\frac{J}{4}-\mu _{B}gh,\quad & |\psi _{1}\rangle =|\uparrow \uparrow \rangle , \end{aligned}$$
(7)
$$\begin{aligned} E_{2}=&-\frac{J}{4}, & |\psi _{2}\rangle =\frac{1}{\sqrt{2}}\left( |\uparrow \downarrow \rangle +|\uparrow \downarrow \rangle \right) ,\end{aligned}$$
(8)
$$\begin{aligned} E_{3}=&\frac{3J}{4} & |\psi _{3}\rangle =\frac{1}{\sqrt{2}}\left( |\uparrow \downarrow \rangle -|\uparrow \downarrow \rangle \right) ,\end{aligned}$$
(9)
$$\begin{aligned} E_{4}=&-\frac{J}{4}+\mu _{B}gh, & |\psi _{4}\rangle =|\downarrow \downarrow \rangle . \end{aligned}$$
(10)
Energy spectrum for dinuclear and trinuclear complexes changing depending on magnetic field, when \(J=-1\): (a) for the dinuclear complexes given in Table 1. ; (b) for the trinuclear complexes given in Table 2.
In order to analyze dinuclear complexes, we consider the exchange interaction \(J=-1\) as an illustrative example because all complexes exhibit similar behavior. The dependence of the energy eigenvalues on the magnetic field, assuming a typical antiferromagnetic condition, is shown on the panel (a) of Fig. 1. There is a quantum phase transition occurring between the phases \(|\psi _{3}\rangle\) and \(|\psi _{1}\rangle\) in the magnetic field \(h=\frac{-J}{\mu _{B}g}\) .
To explore the thermodynamic properties, we need to obtain the partition function of the dinuclear Hamiltonian (6), which results in
$$\begin{aligned} Z=x\left( z^{2}+z^{-2}\right) +x^{-1}\left( x^{2}+x^{-2}\right) , \end{aligned}$$
(11)
where \(x=\textrm{e}^{\frac{J}{4k_{B}T}}\) and \(z=\textrm{e}^{\frac{g\mu _{B}h}{2k_{B}T}}\). Consequently, the magnetization for dinuclear complexes is given by
$$\begin{aligned} M=\frac{g\mu _{B}\left( -z^{4}+1\right) x^{4}}{\left( z^{4}+z^{2}+1\right) x^{4}+z^{2}}. \end{aligned}$$
(12)
Furthermore, the corresponding magnetic susceptibility becomes
$$\begin{aligned} \chi =-\frac{g^{2}\mu _{B}^{2}x^{4}z^{2}\left( \left( z^{4}+4z^{2}+1\right) x^{4}+z^{4}+1\right) }{T\left[ \left( z^{4}+z^{2}+1\right) x^{4}+z^{2}\right] ^{2}}. \end{aligned}$$
(13)
The concurrence using the relation (3) is given by
$$\begin{aligned} C=\textrm{max}(0,\frac{x^{-3}-3x}{Z}). \end{aligned}$$
(14)
Note that the numerator of the concurrence depends only on x (exchange interaction) and is proportional to \(Z^{-1}\). This result will be useful for obtaining the entanglement entropy (5) later on.
Trinuclear complexes
Here, we present our theoretical calculations of the critical points for trinuclear copper (II) complexes with different bridging ligands and compare them with experimental results obtained through magnetic susceptibility measurements. The theoretical magnetic characteristics of the copper(II) complexes are analyzed using the Heisenberg XXX model. This model, a spin Hamiltonian, provides insight into the structure and magnitude of the exchange interaction that occurs between the localized spins of the metal ions. The Hamiltonian for trinuclear copper (II) complexes can be expressed as follows
$$\begin{aligned} \hat{H}\!=\!-\!\sum _{j=1}^{3}\!\left\{ J\!\left( S_{j}^{x}S_{j+1}^{x}\!+S_{j}^{y}S_{j+1}^{y}\!+S_{j}^{z}S_{j+1}^{z}\!\right) \!+\!hg\mu _{B}S_{j}^{z}\right\} , \end{aligned}$$
(15)
assuming periodic boundary conditions \(S_{4}^{\alpha }=S_{1}^{\alpha }\).
Our findings provide insights into the magnetic properties of these complexes and can help us better understand their potential applications in various fields. We also reviewed other articles where similar experiments were conducted and good results were achieved. Our calculations yielded similar results, confirming the accuracy of our theoretical approach. The eigenvalues of the Hamiltonian 1 for trinuclear systems are obtained as follows:
$$\begin{aligned} E_{1,2}= -\frac{3J}{4}\pm \frac{3g\mu Bh}{2},\ \ E_{3,4}= -\frac{3J}{4}\pm \frac{g\mu Bh}{2},\ \ E_{5,6}= \frac{3J}{4}+\frac{g\mu Bh}{2},\ \ E_{7,8}= \frac{3J}{4}-\frac{g\mu Bh}{2}, \end{aligned}$$
(16)
and corresponding eigenstates are
$$\begin{aligned} |\psi _{1}\rangle = |\downarrow \downarrow \downarrow \rangle ,\;|\psi _{2}\rangle =|\uparrow \uparrow \uparrow \rangle ,\ \ |\psi _{3}\rangle = \tfrac{1}{\sqrt{3}}\left[ |\uparrow \downarrow \downarrow \rangle +|\downarrow \uparrow \downarrow \rangle +|\downarrow \downarrow \uparrow \rangle \right] ,\ \ |\psi _{4}\rangle = \tfrac{1}{\sqrt{3}}\left[ |\uparrow \uparrow \downarrow \rangle +|\uparrow \downarrow \uparrow \rangle +|\downarrow \uparrow \uparrow \rangle \right] ,\\|\psi _{5}\rangle = \tfrac{1}{\sqrt{3}}\left[ \tfrac{1}{\sqrt{2}}\left( |\downarrow \downarrow \uparrow \rangle +|\uparrow \downarrow \downarrow \rangle \right) -\sqrt{2}|\downarrow \uparrow \downarrow \rangle \right] ,\ \ |\psi _{6}\rangle = \tfrac{1}{\sqrt{3}}\left[ \tfrac{1}{\sqrt{2}}\left( |\downarrow \downarrow \uparrow \rangle +|\downarrow \uparrow \downarrow \rangle \right) -\sqrt{2}|\uparrow \downarrow \downarrow \rangle \right] , \\|\psi _{7}\rangle = \tfrac{1}{\sqrt{3}}\left[ \tfrac{1}{\sqrt{2}}\left( |\downarrow \uparrow \uparrow \rangle +|\uparrow \uparrow \downarrow \rangle \right) -\sqrt{2}|\uparrow \downarrow \uparrow \rangle \right] ,\ \ |\psi _{8}\rangle = \tfrac{1}{\sqrt{3}}\left[ \tfrac{1}{\sqrt{2}}\left( |\downarrow \uparrow \uparrow \rangle +|\uparrow \downarrow \uparrow \rangle \right) -\sqrt{2}|\uparrow \uparrow \downarrow \rangle \right] . \end{aligned}$$
The panel (b) in Fig. 1 illustrates the spectra for a theoretical value of \(J=-1\) as a function of magnetic field. We observe a quantum phase transition between the state \(|\psi _{7}\rangle\) and \(|\psi _{2}\rangle\), at a specific magnetic field value, \(h=-\frac{3J}{2\mu _{B}g}\).
This ground-state transition in the trinuclear system signals a quantum phase transition52 and predicts phase transitions in magnetic systems considered at zero temperature.
The partition function for an arbitrary complex is given by
$$\begin{aligned} Z=&z\,x^{3}+\frac{x^{3}}{z}+\frac{2z}{x^{3}}+\frac{x^{3}}{z^{3}}+x^{3}z^{3}+\frac{2}{z\,x^{3}} \left( z+z^{-1}\right) \left( x^{3}z^{2}+x^{3}z^{-2}+2x^{-3}\right) , \end{aligned}$$
(17)
where we use the same definition for \(x=\textrm{e}^{\frac{J}{4k_{B}T}}\) and \(z=\textrm{e}^{\frac{g\mu _{B}h}{2k_{B}T}}\).
One of the important aspects of the magnetic properties of trinuclear copper (II) complexes is the identification and characterization of ’critical points’ at low temperatures, where magnetization jumps occur and peak values of magnetic susceptibility are displayed.
In the case of trinuclear complexes, the magnetization M can be expressed as follows:
$$\begin{aligned} M=\frac{g\mu _{B}\left( z^{2}-1\right) \left( 3z^{4}x^{6}+4x^{6}z^{2}+3x^{6}+2z^{2}\right) }{6\left( z^{2}+1\right) \left( z^{4}x^{6}+x^{6}+2z^{2}\right) }, \end{aligned}$$
(18)
whereas the magnetic susceptibility \(\chi\) is given by
$$\begin{aligned} \chi = \frac{g^{2}\mu _{B}^{2}}{4T}\left\{ \frac{\left( 9z^{4}-8z^{2}+9\right) x^{6}+2z^{2}}{\left( \left( z^{4}+1\right) x^{6}+2z^{2}\right) }\right. \left. -\frac{\left( z^{2}-1\right) ^{2}\left( 3z^{4}x^{6}+4x^{6}z^{2}+3x^{6}+2z^{2}\right) ^{2}}{\left( z^{2}+1\right) ^{2}\left( z^{4}x^{6}+x^{6}+2z^{2}\right) ^{2}}\right\} . \end{aligned}$$
(19)
The reduced density operator for trinuclear complexes after performing partial trace has the following form
$$\begin{aligned} \rho =\left[ \begin{array}{cccc} \varrho _{1,1} & 0 & 0 & 0\\ 0 & \varrho _{2,2} & \varrho _{2,3} & 0\\ 0 & \varrho _{2,3} & \varrho _{2,2} & 0\\ 0 & 0 & 0 & \varrho _{4,4} \end{array}\right] , \end{aligned}$$
(20)
with the density operator elements given by
$$\begin{aligned} \varrho _{1,1} = Z^{-1}\left\{ z^{3}x^{3}+z\left( \tfrac{x^{3}}{3}+\tfrac{2}{3x^{3}}\right) \right\} ,\ \ \varrho _{2,2} = Z^{-1}\tfrac{\left( z^{2}+1\right) \left( x^{6}+2\right) }{3z\,x^{3}},\\ \varrho _{2,3} = Z^{-1}\tfrac{\left( z^{2}+1\right) \left( x^{6}-1\right) }{3z\,x^{3}},\ \ \varrho _{4,4} = Z^{-1}\left\{ \tfrac{x^{3}}{z^{3}}+\tfrac{1}{z}\left( \tfrac{x^{3}}{3}+\tfrac{2}{3x^{3}}\right) \right\} . \nonumber \end{aligned}$$
(21)
According to relation (3) and (4) the concurrence simply reduces to
$$\begin{aligned} C=\textrm{max}\left\{ 0,2\left( |\varrho _{2,2}|-\sqrt{\varrho _{1,1}\varrho _{4,4}}\right) \right\} . \end{aligned}$$
(22)