Phase diagram of the spin-1/2 Heisenberg alternating chain in a magnetic field

TL;DR

This paper investigates the phase diagram of the spin-1/2 Heisenberg alternating chain in a magnetic field using tensor network methods, revealing various quantum phases and their entanglement properties.

Contribution

It provides a detailed phase diagram including Haldane, dimer, Luttinger liquid, and paramagnetic phases, with analysis of entanglement entropy and quantum coherence.

Findings

Identification of multiple quantum phases under magnetic field

Scaling of entanglement entropy with central charge c=1 in Luttinger liquid phase

Correlation between quantum coherence, string order, and quantum phase transitions

Abstract

By using the infinite time-evolving block decimation, we study quantum fidelity and entanglement entropy in the spin-1/2 Heisenberg alternating chain under an external magnetic field. The effects of the magnetic field on the fidelity are investigated, and its relation with the quantum phase transition (QPT) is analyzed. The phase diagram of the model is given accordingly, which supports the Haldane phase, the singlet-dimer phase, the Luttinger liquid phase and the paramagnetic phase. The scaling of entanglement entropy in the gapless Luttinger liquid phase is studied, and the central charge c = 1 is obtained. We also study the relationship between the quantum coherence, string order parameter and QPTs. Results obtained from these quantum information observations are consistent with the previous reports.

Equations

$$\begin{array}{rl}H=& \sum _{i=1}^{N/2}J(S_{2i-1}^x{S}_{2i}^x+S_{2i-1}^y{S}_{2i}^y+\Delta S_{2i-1}^z{S}_{2i}^z)\\ & +\sum _{i=1}^{N/2}\lambda (S_{2i}^x{S}_{2i+1}^x+S_{2i}^y{S}_{2i+1}^y+\Delta S_{2i}^z{S}_{2i+1}^z)\\ & -B\sum _{i=1}^N{S}_i^z,\end{array}$$

$$H_{DXX}$$

$$c_j^{†}=\frac{1}{2}\left[i{a}_{j+1}^{†}+a_j^{†}-(-1{)}^j(i{a}_{j+1}+a_j)\right],$$

$$H_{AXX}$$

$$H$$

$$d({\kappa }_1,{\kappa }_2)=\underset{N\to \infty }{\lim }{F}_N^{1/N}({\kappa }_1,{\kappa }_2)$$

$$C({\rho }_{12})=\max \{{\beta }_1-{\beta }_2-{\beta }_3-{\beta }_4,0\},$$

$$S_L({\rho }_{1\cdots L})=-\mathrm{Tr}({\rho }_L{\log }_2{\rho }_L).$$

$$\begin{array}{rl}& B=(1+\Delta )/2+R\sin \alpha ,\\ & \lambda =R\cos \alpha ,\end{array}$$

$$O_x^{even}(r)=-4⟨S_i^x\exp (i\pi \sum _{i<l<j}{S}_l^x)S_j^x⟩.$$

$$S_L\sim \frac{c}{6}{\log }_2[\frac{N}{\pi }\sin (\frac{\pi }{N}L)]+A,$$

$$C_{l_1}(\rho )=\sum _{i\ne j}⟨i|\rho |j⟩.$$

$$I(\rho ,K)=-\frac{1}{4}\text{Tr}([\rho ,K{]}^2),$$

$$H$$

$$\begin{array}{r}{S}_i^{+}=S_i^x+i{S}_i^y,S_i^{-}=S_i^x-i{S}_i^y,\end{array}$$

$$H$$

$$S_j^{+}$$

$$S_j^{-}$$

$$S_j^z$$

$$H$$

$$c_{2j-1}=\sqrt{\frac{2}{N}}\sum _k{e}^{-ikj}{a}_k,c_{2j}=\sqrt{\frac{2}{N}}\sum _k{e}^{-ikj}{b}_k,$$

$$k=\frac{2n\pi }{N},n=-(\frac{N}{2}-1),-(\frac{N}{2}-3),\dots ,\frac{N}{2}-1.$$

$$H=\sum _k(-\frac{1}{2}-\frac{\lambda }{2}{e}^{ik})b_k{a}_k^{+}+\sum _k(\frac{1}{2}+\frac{\lambda }{2}{e}^{ik})b_k^{+}{a}_k$$

$$+\sum _k\frac{B}{2}(1-2a_k^{+}{a}_k)+\sum _k\frac{B}{2}(1-2b_k^{+}{b}_k).$$

$$\begin{array}{rl}& {\Lambda }_k=\frac{1}{2}+\frac{\lambda }{2}{e}^{ik},\end{array}$$

$$H=\sum _k({\Lambda }_k{a}_k^{+}{b}_k+{\Lambda }_k^{\ast }{b}_k^{+}{a}_k)+\sum _{k}B(1-a_k^{+}{a}_k-b_k^{+}{b}_k).$$

$$H=\sum_{k}\left(\begin{array}[]{cc}a_{k}^{+}&b_{k}^{+}\\ \end{array}\right)\left(\begin{array}[]{cc}-B&\Lambda_{k}\\ \Lambda_{k}^{*}&-B\\ \end{array}\right)\left(\begin{array}[]{cc}&a_{k}\\ &b_{k}\end{array}\right)+\sum_{k}B.$$

$$\begin{array}{rl}& {\epsilon }_{\pm ,k}=\pm |{\Lambda }_k|-B.\end{array}$$

$$H=\sum _k[{\epsilon }_{+,k}({\gamma }_{1,k}^{+}{\gamma }_{1,k}-\frac{1}{2})+{\epsilon }_{-,k}({\gamma }_{2,k}^{+}{\gamma }_{2,k}-\frac{1}{2})],$$

$${\epsilon }_{+,k}=\frac{1}{2}\sqrt{1+{\lambda }^2+2\lambda \cos k}-B,$$

$${\epsilon }_{-,k}=-\frac{1}{2}\sqrt{1+{\lambda }^2+2\lambda \cos k}-B.$$

Full text

I introduction

Quantum phase transition (QPT) is a purely quantum process occurring in strongly correlated many-body systems at absolute zero temperature due to quantum fluctuations Sachdev . The spin chains attract a lot of attention since they give rise to many exotic properties in the ground state, such as bond alternating spin-1/2 Heisenberg chain Yu ; FHM ; Orignac ; Klyushina ; James ; Tennant ; xiong ; HTW . A number of compounds are discovered whose properties can be explained by invoking bond alternating chains. LiInCr4O8 was found to be spin-3/2 breathing pyrochlore antiferromagnet, which is an alternating array of small and large tetrahedra Okamoto17 . Recently, it is reported that the results from the dimer anisotropic XYZ model are relevant to a large number of quasi-one dimensional magnets Xu . In history, Bulaevskii predicted that a spin gap exists in the nonuniform antiferromagnetic (AFM) spin chains Bulaevskii . Kohmoto found the existence of the Haldane phase synonymous with hidden $D_{2}$ symmetry breaking in the AFM-ferromagnetic (FM) bond alternating spin-1/2 Heisenberg chain Kohmoto . Furthermore, it was pointed out that isotropic $S=1/2$ Heisenberg chain with alternating AFM and FM couplings can be mapped onto the isotropic $S=1$ AFM Heisenberg chain when the FM couplings tend to infinity Hida . The compounds like CuGeO3 Hase exhibiting spin-Peierls transitions belong to AFM-AFM bond alternating class, and DMACuCl3 Stone was claimed to fall into the $S=1/2$ AFM-FM bond alternating class. Additionally, the quantum simulation using ultracold atoms systems Bloch12 ; Sun14 ; Sun2014 and trapped polariton condensates Ohadi17 has made great progress in creating interesting quantum models motivated by solid-state physics. Geometrically frustrated magnets such as zigzag chains can be designed and tuned by the depth of the optical lattice, and thus nonuniform configurations in the ground state can be anticipated.

The magnetic phase transitions induced by applying a magnetic field in the low-dimensional magnets have attracted much interest recently from both experimental and theoretical points of view. When the magnetic interactions cannot be satisfied simultaneously owing to the existing competing orders, the magnetic systems become fertile ground for the emergence of exotic states. The ground-state magnetization plateaus appearing in polymerized Heisenberg chains under external magnetic fields was investigated, and the phase diagram of AFM bond alternating spin chain in homogeneous magnetic fields was presented Honecker ; Daniel . Moreover, the effects of temperature, magnetic field and dimerized interaction on the spin and heat transport in dimerized Heisenberg chains in a magnetic field are studied. It is noted that the spin and heat conductivity show different behaviors in different phases langer . For alternating spin-1/2 chains with anisotropic AFM-FM coupling under a transverse magnetic field, two successive phase transitions, i.e., from Haldane phase to stripe AFM phase and from stripe AFM phase to polarized paramagnetic phase, have been identified to be Ising tpye Saeed . The magnetization state and magnetic structure can be revealed through common techniques like neutron diffraction measurements and synchrotron X-ray scattering  Matsuda ; Matsuda2010 . However, the phase boundary of the spin-1/2 Heisenberg AFM-FM bond alternating chain in a magnetic field is still not clear, and needs to be discussed further. Fortunately, with the development of quantum information, various information measures, e.g., quantum coherence, entanglement entropy, and fidelity, can help us to study quantum critical phenomena in spin chains. It is found that the quantum critical points can be well characterized by both the ground-state entanglement and fidelity on large system Ren ; Chen ; You07 ; Gu ; You11 ; You15a ; You15b ; Ren02 . In this paper we study the entanglement, coherence and fidelity of the spin-1/2 Heisenberg alternating chain under a transverse magnetic field, and finally the phase diagram will be given.

II Hamiltonian and Measurements

The Hamiltonian of a one-dimensional (1D) spin chain is given by

[TABLE]

where $S_{i}^{\alpha}(\alpha=x,y,z)$ are spin operators on the $i$-th site and $N$ is the length of the spin chain. The parameter $J$ is the AFM coupling on odd bonds, and $J$=1 is assumed hereafter in the paper. $\lambda$ is considered to be either AFM ($\lambda>$0) or FM ($\lambda<$ 0) coupling strength on even bonds. The parameter $B$ is the strength of the magnetic field along the $z$-axis with the anisotropy $\Delta$. In the case of $\lambda=0$ the ground state is composed of local dimers on odd pairs $2j-1$ and $2j$. It is easy to obtain the energy per bond $E=1/4$, $-3/4$, $\Delta/4+B$, $\Delta/4-B$. As the increasing of $B$, the ground state of the system changes from a direct product of singlet pairs to the direct product of polarized qubits.

For $\Delta=0$, Eq.(1) reduces to the dimerized XX model, which can be solved by Jordan-Wigner transformation and Fourier-Bogoliubov transformation. It has been shown that the dimerized XX model is equivalent to anisotropic XY model in given parity blocks Venuti10 . It is more straightforward to see the equivalence between them in the fermionic form:

[TABLE]

Then, we use the local mapping under the assumption of even $N$ and periodic boundary condition:

[TABLE]

and thus Eq.(2) can be transformed into a generalized anisotropic model:

[TABLE]

where $J_{h}$=$(1+\lambda)/4$, $J_{p}$=$(\lambda-1)/4$. Eq. (4) can be traced back to the spin version:

[TABLE]

For $B=0$ the QPT occuring at $\lambda=1$ for the dimerized chain shares the same properties with the transition which occurs at $\lambda=1$ for the anisotropic XY chain separating the $x$-component phase from the $y$-component Néel phase. In the case of $\lambda=1$ dimerized XX model corresponds to the uniform XX chain. In the opposite limit $\lambda=0$ one arrives at a collection of isolated (uncoupled) XX dimers. $B$ term in Eq. (5) favors period-4 configurations $|\uparrow\leftarrow\downarrow\rightarrow\cdots\rangle$ or $|\downarrow\rightarrow\uparrow\leftarrow\cdots\rangle$, competing with $x$-component and $y$-component Néel orderings. The generic features of the spin-Peierls systems are analytically discussed in detail by Taylor and Müller Taylor85 . The details can be referred to Appendix A.

As we know, it is difficult to diagonalize the Hamiltonian Eq.(1) when $\Delta\neq 0$. The finite-size density matrix renormalization group (DMRG) would be the effective method to obtain the ground-state wavefunctions approximately white ; U01 . In this version of DMRG, an open chain is grown iteratively by adding two sites at a time to the center of the spin chain, and up to the sizes $N=400$. Then, we perform four sweepings, and the maximum number of the eigenstates kept is $m=200$ during the processing. Such truncation guarantees that the converging error is smaller than $10^{-7}$. With this accurate calculation, we can precisely analyze the QPTs through various theoretic measures.

A QPT taking place in this class of systems has been thoroughly investigated in the thermodynamic limit. However, both experimental and theoretical difficulties have boosted a high interest in finite-size systems, which show the forerunners of the points of QPT of the thermodynamic systems. In general, finite-size systems would exhibit many energy level crossings between physical and unphysical states. As a consequence, diverse theoretic measures would have some jumps. In order to avoid the finite-size effects, we also implement the infinite time-evolving block decimation (iTEBD) vidal01 ; vidal02 , which can be used to compute the ground-state wavefunctions for an infinite-size lattice in one or two dimensions with translational invariance. It can help us directly address physical quantities in the thermodynamic limit with high quality. Given a large bond dimensions $\chi$, the ground-state wavefunctions based on the matrix product state representations can be obtained by applying imaginary-time evolution gates $\exp(-\tau h)$ on a given initial random state $|\psi(0)\rangle$, until the latter converges to the variational ground state. Here $h$ is the local Hamiltonian, which is composed of two-site coupling terms on an odd bonds $h_{2i-1,2i}$ or even bonds $h_{2i,2i+1}$, and $\tau$ is the Trotter step length. In practice, we start from $\tau=0.1$ and gradually reduce it by $\tau=\tau/10$, and break the loop until $\tau<10^{-9}$. In the paper, $\chi=50$ is adopted, and we check our codes with the case $\lambda=1$, $B=0$, which is equal to well-known Heisenberg chain. The ground-state energy we obtain is $E_{0}=-0.443143049$, which is very close to the exact diagonalization result $E=1/4-\ln(2)=-0.4431471805$, and the error is smaller than $5.0\times 10^{-6}$.

As the external parameter varies across a critical point, the ground-state wavefunction undergoes a sudden change in the wake of QPT, accompanied by a rapid alteration in a variety of quantum measurements. Fidelity is one of the most effective measurements, which can detect the critical points zhou ; Wang ; Zhou2018 . It measures the similarity between the two closest states as the external parameter such as $\kappa$ is tuned, which is defined as $F_{N}(\kappa_{1},\kappa_{2})=|\langle\psi(\kappa_{1})|\psi(\kappa_{2})\rangle|$ in finite-size systems. In the thermodynamic limit, the ground-state fidelity per site

[TABLE]

can be calculated easily in iTEBD by transfer matrix Zhou2018 . The fidelity $d(\kappa_{1},\kappa_{2})$ should be equal to one when $\kappa_{1}=\kappa_{2}$, and QPTs may be detected through singularities exhibited in $d(\kappa_{1},\kappa_{2})$. Meanwhile, the concurrence is chosen as a measure of the pairwise entanglement between two qubits Wootters . The concurrence $C$ is defined as

[TABLE]

where the quantities $\beta_{i}(i=1,2,3,4)$ are the square roots of the eigenvalues of the operator $\varrho=\rho_{12}(\sigma_{1}^{y}\otimes\sigma_{2}^{y})\rho_{12}^{\ast}(\sigma_{1}^{y}\otimes\sigma_{2}^{y})$ and in descending order. The case of $C=1$ corresponds to the maximum entanglement between the two qubits, while $C=0$ means that there is no entanglement between the two qubits. The entanglement entropy is used as a measure of the bipartite entanglement. If $|\psi\rangle$ is the ground state of a chain of $N$ qubits, a reduced density matrix of contiguous qubits from $1$ to $L$ can be written as $\rho_{L}=\textrm{Tr}_{N-L}|\psi\rangle\langle\psi|$. The bipartite entanglement between the right-hand $L$ contiguous qubits and the rest of the system can be measured by the entropy

[TABLE]

One of the basic properties of the block entanglement entropy for a pure state can be given by $S_{L}=S_{N-L}$, since the spectrum of the reduced density matrix $\rho_{L}$ is the same as that of $\rho_{N-L}$ following from the Schmidt decomposition. This property implies the entanglement entropy is not extensive and boils down to a celebrated area law for non-critical ground states or finite temperature system, which states the leading term of the entanglement entropy is proportional to the boundary area between $L$ and $N-L$ qubits. Note that the area law is violated for highly excited states and for ground state of gapless systems.

III Results

The ground-state phase diagram of the Hamiltonian Eq. (1) for $\Delta=1$ is shown in Fig. 1. Note that the phase diagram on $B$-$\lambda$ plane has been shown in Refs.Honecker ; langer for $\lambda>0$. We calculated the fidelity per site to check our codes, which is shown in Fig. 2(a). It is seen that $d(B_{1},B_{2})=1$ when $|\psi(B_{1})\rangle,|\psi(B_{2})\rangle$ are both in the dimerized phase or both in the paramagnetic phase at the same time. The two pinch points which mean the critical points can be identified as $B_{c1}=0.64$, $B_{c2}=1.53$. The values agree with the results obtained by the spin gap Uhrig ; Yu ; Chitra . When $B_{c1}<B<B_{c2}$, the model will be in the Luttinger liquid phase, which is gapless. We also show the phase diagram for $\lambda<0$. It is obvious that the model would be in Haldane phase for $B=0$ and paramagnetic phase with large magnetic field. The Luttinger liquid phase separates these phases for intermediate $B$ sakai . The critical points between them can also be portrayed by the fidelity. The result is shown in Fig. 2(b). The two pinch points can be found $B_{c3}=0.61$, $B_{c4}=1.00$ for $\lambda=-1$.

Interestingly, all the critical lines cross at one point $B=1$, $\lambda=0$, which is a multicritical point. In order to better compare various measures, we inspect the QPTs along a loop path described by the radius $R$ and the angle $\alpha$:

[TABLE]

where $\alpha$ ranges from $\alpha=-\pi$ to $\alpha=\pi$. The entanglement entropy between two qubits and the rest of the system is shown in Fig.3 with $R=0.8$. When $\alpha/\pi\simeq-1$, the system is in the Luttinger liquid phase. With the increasing of $\alpha$, the system will be in the Haldane phase, and the critical point is uncovered by the discontinuity of entanglement entropy of even or odd bond. When $\alpha$ increases further, the entanglement entropy of even bond $S_{2i,2i+1}$ will rise, and entanglement entropy of odd bond $S_{2i-1,2i}$ will decline. When $\alpha=-\pi/2$, $S_{2i,2i+1}=1$ and $S_{2i,2i+1}=0$, which pinpoint the phase transition point between the Haldane phase and the dimer phase. For the Haldane phase, the system has $|\uparrow\downarrow\downarrow\uparrow\rangle_{1,2,3,4}\cdots\otimes|\uparrow\downarrow\downarrow\uparrow\rangle_{2i-1,2i,2i+1,2i+2}\cdots$, and $|\uparrow\downarrow\uparrow\downarrow\rangle_{1,2,3,4}\cdots\otimes|\uparrow\downarrow\uparrow\downarrow\rangle_{2i-1,2i,2i+1,2i+2}\cdots$ for singlet dimer. The singlet dimer-Luttinger liquid and Luttinger liquid- paramagnetic phase transitions can be captured by discontinuity of the entanglement entropy. It is noted that the Haldane-singlet dimer can be detected by either the peak of $S(\rho_{2i,2i+1})$ or the valley of $S(\rho_{2i-1,2i})$. This is because at this moment the odd bond is one of Bell states, and the concurrence of odd bond reaches the maximal value 1, which is shown in Fig. 3(b), so the entanglement between the odd bond and the rest of the system measured by entanglement entropy will go to zero. The counterpart for the even bond is opposite. Significantly, the concurrence for the even bond can not detect the Luttinger liquid-Haldane and Haldane-singlet dimer phase transition.

We also calculate the string order parameter (SOP)Hida1993 ; Lou ; Liu01 . The SOP characterizes the topological order in the Haldane phase of $S=1$ Heisenberg chain efficiently Ren03 . It is noted that the SOP will behave as an oscillation in dimerized model Liu ; Paul . In the paper, we adopt the version of SOP in terms of $S=1/2$ operators, which is given by

[TABLE]

Here we set $i$ = 1 and $j$ being an even number, and thus the distance $r=|i-j|$ is an odd integer. The results are presented in Fig. 4. It is seen that the SOP is almost one, and has little change with the distance $r$ when $0<B<0.61$. We can surmise that the SOP is not equal to zero when $r\rightarrow\infty$. As long as $B$ increases further, the SOP decays suddenly, indicating the system changes to Luttinger liquid phase. When $r$ is small, the SOP has a very small value, and decreases to zero rapidly, because the short-range correlations in Luttinger liquid phase are large and decreases exponentially with the increasing distance. When the system in the PM phase, the SOP is equal to zero for any $r$.

The entropy $S_{L}$ between contiguous $L$ qubits and the remaining $N-L$ qubits in the Luttinger liquid phase is plotted as a function of the subsystem length $L$ for $N=400$ in Fig. 5. One finds there are large odd-even effects for $\lambda=-1.0,B=0.8$. When $\lambda=0.5,B=1$, the oscillations become more complicated. Such phenomenon is caused by the open boundary condition and bond alternation. The entanglement entropy $S_{L}$ in the critical regime swells up with subsystem size $L$($L\leq N/2$), which is predicted by conformal field theory (CFT) as

[TABLE]

where $c$ is the central charge and $A$ is a non-universal constant Calabrese ; Chiara ; Nicolas ; Ren01 . The entropy $S_{L}$ is also plotted as a function of $\log_{2}[\frac{N}{\pi}\sin(\frac{\pi}{N}L)]$ in the inset of Fig. 5. It is shown that the entropy appears as a straight line, whose slope yields $c\simeq 1.0$ for both cases sakai . Such a multiplicative logarithmic scaling is known to violate the celebrated area law, which states the entanglement entropy between two subsystems scales with the boundary between them. As for non-critical ground states of one dimensional quantum systems, the area law yields a constant entanglement entropy, independent of the subsystem size.

Recently, quantum coherence is a resurgent concept in the quantum theory and acts as a manifestation of the quantum superposition principle. We also recall the $l_{1}$ norm of coherence Plenio , which can almost be estimated in experiments Zhang ; Zheng . For a density matrix $\rho$ in the reference basis $\{|i\rangle\}$, $l_{1}$ norm of coherence is given by

[TABLE]

Moreover, the local quantum coherence and the local quantum uncertainty, based on Wigner-Yanase skew information (WYSI), given by Girolami

[TABLE]

where the density matrix $\rho$ describes a quantum state, $K$ plays a role of an observable, and [.,.] denotes the commutator. The WSYI shows the capability of diagnosing the QPT in the anisotropic XY chain Karpat . Similar with the entanglement entropy, the quantum coherence is shown in Fig. 6. All the phase transitions can be identified by divergences or discontinuity of coherence for even bonds and odd bonds.

IV Conclusions

By using the infinite time-evolving block decimation, the ground-state properties of the spin-1/2 Heisenberg alternating chain in a magnetic field are calculated with very high accuracy. We numerically investigate the effects of magnetic field on the fidelity, which measures the similarity between two states, and then analyze its relation with the quantum phase transitions (QPTs). QPTs are intuitively accompanied by an abrupt change in the structure of the ground-state wave function, so generally a pinch point of the fidelity indicates a QPT and the location of the pinch point denotes the critical point. Based on the above analyisis, we obtain the phase diagram. This model supports the Haldane phase, the singlet-dimer phase, the Luttinger liquid and the paramagnetic phase. In the Luttinger liquid phase, we study the scaling of the entanglement entropy with the subsystem size $L$, and identified the central charge $c=1$. We also study the quantum coherence, whose anomalies detect all the phase transitions therein. In summary, conclusions drawn from these quantum information observables agree well with each other.

Acknowledgements.

This work is supported by the National Natural Science Foundation of China (NSFC) (Grants Nos.11384012, 11474211). J. R thanks Juan Felipe Carrasquilla for his discussion with iTEBD codes.

Appendix A Exact solution of dimerized XX model

Consider a chain of spin-1/2 operators interacting antiferromagnetically with their nearest neighbors, given by

[TABLE]

The dimerized XX model corresponds to $\Delta$=0 in Eq. (1). The Hamiltonian (14) can be exactly diagonalized following the standard procedure for 1D systems. In terms of the raising and lowering operators for spins,

[TABLE]

The Hamiltonian (14) then takes the form:

[TABLE]

The Jordan-Wigner transformation maps explicitly between quasispin operators and spinless fermion operators by

[TABLE]

where $c_{j}$ and $c_{j}^{\dagger}$ are annihilation and creation operators of spinless fermions at site $j$, which obey the standard anticommutation relations, $\{c_{i},c_{j}\}=0$, $\{c_{i}^{\dagger},c_{j}\}=\delta_{ij}$. By substituting Eq. (17) into Eq. (14), we find

[TABLE]

Next, utilizing translational invariance, discrete Fourier transformation for plural spin sites is introduced by

[TABLE]

with the discrete momentums as

[TABLE]

Let’s proceed with diagonalization of the dimerized XX model. We rewrite,

[TABLE]

Defining

[TABLE]

then Eq.(21) can be rewritten as

[TABLE]

Then we write the Hamiltonian in the matrix form:

[TABLE]

The eigenspectra can be obtained as:

[TABLE]

Consequently, the Hamiltonian can be written in the diagonal form:

[TABLE]

where

[TABLE]

As a magnetic field is turned on, the one-particle spectrum will simply moves to higher energies with its shape unchanged. Since the $\varepsilon_{-,k}$ is always negative, the gap closing can be identified by $\partial\varepsilon_{+,k}/\partial k=0$, which suggests that the boundaries are described by $B$=$|1+\lambda|/2$ for $k=0$ and $B$=$|1-\lambda|/2$ for $k=\pi$ as shown in Fig.7.