Abstract
Quantum magnetic phase transition in squareoctagon lattice was investigated by cellular dynamical mean field theory combining with continuous time quantum Monte Carlo algorithm. Based on the systematic calculation on the density of states, the double occupancy and the Fermi surface evolution of squareoctagon lattice, we presented the phase diagrams of this splendid many particle system. The competition between the temperature and the onsite repulsive interaction in the isotropic squareoctagon lattice has shown that both antiferromagnetic and paramagnetic order can be found not only in the metal phase, but also in the insulating phase. Antiferromagnetic metal phase disappeared in the phase diagram that consists of the anisotropic parameter λ and the onsite repulsive interaction U while the other phases still can be detected at T = 0.17. The results found in this work may contribute to understand well the properties of some consuming systems that have squareoctagon structure, quasi squareoctagon structure, such as ZnO.
Introduction
The discovery and classification of quantum phases of matters and the transition between these distinctive phases have been recurring theme in condensed matter physics for many years and still wheel the researchers' extensive interests^{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}. Notable quantum phases, such as superconductivity, quantum hall effect, Mott insulating phase and topological phase, have great significance in theoretical investigations and promising potential in applications. These exotic phases have been found in many quantum systems with quite common structure, such as the honeycomb lattice, the triangular lattice, the decorated honeycomb lattice, the kagomé lattice and so forth^{16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31}. Recently a unique quantum many particle lattice system named squareoctagon lattice have been investigated in theoretical way intensively and a plenty of meaningful results have been presented. Researchers have found topological phases and the transitions between these novel phases in the squareoctagon lattice that 1/4 and 3/4 filled with fermions under the framework of the tight binding model through considering the spinorbit coupling fermions^{32}. Another one theoretical joy models named Fully packed loop model also has been adopted to investigate the squareoctagon lattice^{33}. Additionally, researchers have found quasi squareoctagon structure in () surface of functional material ZnO by first principle calculations and aberrationcorrected transmission electron microscopy (ACTEM) observation experimentally during its pressure induced phase transition process^{34}.
However, few of the previous work related to the squareoctagon lattice considered the particles' onsite repulsive interactions that have crucial effect on the properties of the systems. Therefore in this work, the celebrated Hubbard model^{35,36} was used to describe this strongly correlated many particle systems for the purpose of understanding well the influences of interaction on the properties of the squareoctagon lattice with fermions. The cellular dynamical mean field theory (CDMFT)^{37,38,39,40,41,42}, which maps the lattice to a selfconsistent embedded cluster in real space, was adopted to deal with the Hubbard model and the continuous time quantum Monte Carlo (CTQMC)^{47} algorithm was used as a impurity solver to deal with the mean field equations. The CDMFT is proved to be more successful than the dynamical mean field theory and the CTQMC is more accurate than the general quantum Monte Carlo method. Based on the singleparticle Greens function given by the CDMFT and CTQMC, the singleparticle density of states and the double occupancy which play critical role in the identification of Mott metalinsulator transition have been calculated. The phase diagram which composed of the onsite interaction and the energy gap, the relationship between the interaction and magnetic properties of the systems also have investigated through defining the magnetic order parameter. We also presented phase diagram which consists of the competition between the temperature and onsite repulsive interaction for isotropic squareoctagon lattice and the the competition between the anisotropy and onsite repulsive interaction.
Results
Strongly correlated squareoctagon lattice system
The squareoctagon lattice is a bipartite lattice that can be thought of as a square lattice in which each vertex has been decorated with a tilted square, as shown in Fig. 1 (a) and its first Brillouin zone in Fig. 1 (b). It has the same coordination number as the honeycomb lattice has and its boundary shapes armchair. It enjoys the symmetry of the square lattice and symmetrically it satisfies C_{4} point group.
The standard Hubbard model is adopted to investigate the squareoctagon lattice and the Hamiltonian can be written as follows, where and c_{iσ} represent creation and annihilation operator of fermions with spin σ on site i respectively, while denote the particle number operator on lattice site i. The value of spin index σ is spin up or spin down. The first two terms in this Hamiltonian account for the kinetic energy of the system, which is characterized by the coefficient factor t_{1} and t_{2}. t_{1} represents the hopping between the nearest neighboring sites in the same square lattice and t_{2} is the hopping between the endpoints of the liking line of the two nearest neighboring square lattice. The third term describes the onsite repulsive interaction (U > 0) between fermions with opposite spin. Here we set t_{1} as energy unit (t_{1} = 1). μ is chemical potential and in order to reach half filled case μ should equals zero for this lattice system. We also defined an anisotropic parameter λ which equals to the ration t_{1}/t_{2} (λ = t_{1}/t_{2}).
For the case of U = 0 and μ = 0, the Hubbard model transmits to the tight binding model and the Hamiltonian in the momentum space is , in which Ψ_{k} = (c_{1k↑}, c_{2k↑}, c_{3k↑}, c_{4k↑}, c_{1k↓}, c_{2k↓}, c_{3k↓}, c_{4k↓})^{T}. The index i = 1, 2, 3,4 in creation and annihilation operators represent the four sites in each unit cell as illustrated in Fig. 1 (a) and k is the locations in the first Brillouin zone. ↑ and ↓ hint the spinup and spindown states respectively. takes the following form
Since H_{0} is decoupled in spin states, so is block diagonalized, i.e. two blocks representing spinup and spindown electrons are the same. The energy band of the first Brillouin zone of the squarelattice under the frame of the tight biding model has been obtained through diagonalizing and shown in Fig. 1 (c). The density of states of the squareoctagon lattice half filled with fermions without interaction at T = 0.2 for different anisotropic parameter λ in Fig. 1 (d).
In order to get the effect of anisotropic parameter λ and the value of hopping term t_{1} and t_{2} on the phase transitions, we presented the energy band along the line between the high symmetric points in the first Brillouin zone in Fig. 2 even has shown the 3dimensional energy band in the first Brillouin zone in Fig. 1 (c). The energy band Ek_{2} and Ek_{3} touch at Γ point and M point for λ = 2.0 in Fig. 2 (a). Energy band Ek_{2}, Ek_{3} and Ek_{4} cross at Γ point while Ek_{1}, Ek_{2} and Ek_{3} cross at M point for λ = 1, the system is in metallic states. It can be seen that with the decreasing of anisotropic parameter λ, Ek_{2} and Ek_{3} separate and meanwhile Ek_{1} and Ek_{2} contact at M point, Ek_{3} and Ek_{4} contact at Γ point while λ = 0.83. The system is still in metallic states. As Fig. 2 (d) shows that energy band Ek_{2} and Ek_{3} completely separated by the Fermi energy level while λ = 0.5 and the system turns into insulating states.
Phase diagrams of the squareoctagon lattice
With the increase of onsite repulsive interaction U the the probability of more than one fermions occupying the same lattice site will reduce and eventually only one fermion confined in per lattice site at certain large value of U. The confinement of fermion in one lattice site is described by double occupancy (Docc)^{43} which is an important quantity that used to characterize the critical point in Mott phase transitions and indicates the transition order, and also can be used to describe the localization of the electrons in strongly correlated electron systems. The formula of double occupancy is , where F is free energy. The double occupancy of isotropic squareoctagon lattice as a function of interaction for fixed temperature and as a function of temperature for fixed interaction have been shown in outer part and inner part of Fig. 3 respectively. It can be seen in the outer part of Fig. 3 that Docc decreases as the interaction increases due to the suppressing of the itinerancy of the atoms. When the interaction is stronger than the critical interaction of the Mott transition, the effect of the temperature on Docc is weakened and Docc for different temperatures consistently trend to zero, which shows the temperature does not affect the double occupancy distinctly. The continuity of the evolution of the double occupancy by interaction shows that it is a secondorder transition. We also have shown the relation between Docc and the temperature at different interaction in inner part of Fig. 3. From the inner part of Fig. 3 we can find that the double occupancy decreases with the increase of the temperature for fixed onsite repulsive interaction.
The density of states is one of the most important quantities in the characterization of the Mott metalinsulator phase transition of Hubbard model. For the purpose of investigating the Mott metalinsulator phase transition as the evolution of single particle spectral^{44}, we defined Density of states, the formula is where i is the lattice points index in the cluster. The Density of states can be derived from the imaginary time Greens function G(τ) by using the maximum entropy method^{45}. Fig. 4 (a) and (b) respectively shows the density of states of isotropic squareoctagon lattice for different temperature at U/t_{1} = 6 and the density of states for different repulsive interactions while T/t_{1} = 0.5. The inner part of Fig. 4 (a) is the density of states of system for U/t_{1} = 0 and T/t_{1} = 0.17. It can be evidently seen in Fig. 3 that the systems will change from metal state to Mott insulating state which characterized by the opened gap at ω = 0 with the increase of the repulsive interaction for fixed temperature and the decrease of the temperature for the fixed repulsive interactions. However, the evolution shape of the density of states with the change of frequency in this two cases is much different from each other. The critical point between paramagnetic metal state and Mott insulating state is (T/t_{1} = 0.17, U/t_{1} = 6), (T/t_{1} = 0.25, U/t_{1} = 7) and (T/t_{1} = 0.5, U/t_{1} = 8).
In order to describe the Fermi surface evolution, we defined the spectral function . A linear extrapolation of the first two Matsubara frequencies is used to estimate the selfenergy to zero frequency. The Fermi surface of isotropic squareoctagon lattice half filled with fermions for different interaction U/t_{1} at fixed temperature T/t_{1} = 0.1 is shown in Fig. 5 (a_ 2), (b_ 2) and (c_ 2). We also have shown the Fermi surface of anisotropic squareoctagon lattice in Fig. 5 for U/t_{1} = 4, 6, 8 while T/t_{1} = 0.1. With the decreasing of the λ for fixed interaction the amplitude of the spectral weight becomes bigger due to the localization of particles.
Based on the systematic calculations on the quantities mentioned above, we have presented the T  U phase diagram of isotropic squareoctagon lattice and the competition between anisotropic parameter λ and the onsite repulsive interaction (U) for fixed low temperature T/t_{1} = 0.17. We also studied the magnetic properties of each phase in the squareoctagon lattice by using the magnetic order parameter , where 〈n_{iσ}〉 is the electron density at lattice site i with spin index σ and sign(i) = 1 if i = 1, 3 and sign(i) = −1 if i = 2, 4 as shown in Fig. 1(a). From the definition of magnetic order it can be known that m = 0 correspond to paramagnetic phase while m ≠ 0 represents antiferromagnetic phase. Both paramagnetic and antiferromagnetic order as shown in Fig. 6 (a) and (b) have been not only found in insulating state but also in the metal state in the T  U phase diagram of isotropic squareoctagon lattice. Fermi surface evolution of isotropic squareoctagon lattice in paramagnetic metal state in Fig. 6 (c) and in antiferromagnetic metal state in Fig. 6 (d) for for U = 5.5 and T = 0.17. As shown in Fig. 6 (e) that only at low enough temperature or weak enough onsite repulsive scale the systems can transform to antiferromagnetic metal state. The narrow antiferromagnetic metal state region in Fig. 6 (e) means this state is sensitive to the temperature and the onsite repulsive interaction. This results have been confirmed further by the relation between the energy gap and onsite repulsive interaction and the magnetic order parameter m and the onsite repulsive interaction in Fig. 7. The antiferromagnetic metal state disappeared in the competition of anisotropic parameter λ and interaction diagram while other phases still exist at T = 0.17.
Discussion
In this work, we use standard Hubbard model to describe the squareoctagon lattice and present the quantum magnetic phases and the transition between these novel phases in this many particle systems. We have investigated not only the effect of onsite repulsive interaction of particles with the opposite direction spin on the same site, but also shown the influence of the Kinetic energy of the systems on the phase transitions. We also have studied the magnetic properties of the squareoctagon lattice through defining the magnetic order parameter m. We hope the results found in this study can be useful for understanding the property of this lattice and the real materials with this structure, even can be helpful for the research on the functional material ZnO with quasi squareoctagon lattice.
Methods
Cluster dynamical meanfield theory
The cellular dynamical meanfield theory (CDMFT) was used to investigate this many particle squareoctagon lattice. In comparison to the general dynamical mean field theory, the cellular dynamical mean field theory gives much more reliable simulation results for lowdimensional system with strong quantum fluctuations due to its efficient consideration of the nonlocal effect. In our case, the cellular dynamical mean field theory maps the original squareoctagon lattice onto a 4site effective cluster embedded in a selfconsistent bath field, as shown in Fig. 1 (a). At the beginning of the self consistent calculation process, we guess a mini selfenergy Σ(iω) which is independent of momentum^{46} and the Weiss field G_{0}(iω) can be obtained by the coarsegrained Dyson equation: where ω is Matsubara frequency, μ is the chemical potential, Σ_{K} is the summation all over the reduced Brillouin zone of the superlattice. t(K) is 4 dimensional hopping matrix of superlattice which drawn from the squareoctagon lattice under the framework of cluster dynamical mean field theory.
Continuoustime quantum MonteCarlo algorithm
The continuoustime quantum MonteCarlo (CTQMC) algorithm was used as impurity solver. The CTQMC is based on a series expansion for the partition function in the powers of interaction and the partition function is where T_{τ} is timeordering operator, and H_{1} is Hamiltonian in interaction picture, is the partition function for the unperturbed term. Through inserting H_{1} = UΣ_{i}n_{i}_{↑}n_{i}_{↓} into the partition function and using Wick's theorem further to reform ordering operators in partition functions. The ordering operators can be expressed by the determinants of matrix which consist of the noninteracting Green functions G^{0}. The new selfenergy Σ(iω) is recalculated by the Dyson equation:
The cluster Green's function G(iω) can be obtained by CTQMC and 1 × 10^{6} QMC sweeps are carried through for each CDMFT loop^{47}. The cluster Green's function both in imaginary time and at Matsubara frequencies: where G_{0}(iω) is a bare Green's function and M_{i}_{,j} is the elements of inverse matrix of matrix that composed of noninteracting Green's functions. The more details about CTQMC can be found in the reference herein^{47}.
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Acknowledgements
This work was supported by the NKBRSFC under grants Nos. 2011CB921502, 2012CB821305, NSFC under grants Nos. 61227902, 61378017, NSFC under grants Nos. 11174169, 11234007, and SPRPCAS under grants No. XDB01020300.
Author information
Affiliations
Laboratory of Advanced Materials, Department of Materials Science and Engineering, Tsinghua University, Beijing 100084, China
 An Bao
 & XiaoZhong Zhang
School of Mathematics, Physics and Biological Engineering, Inner Mongolia University of Science and Technology, Baotou 014010, China
 An Bao
Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
 HongShuai Tao
 , HaiDi Liu
 & WuMing Liu
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Contributions
A.B. performed calculations. A.B., H.S.T., H.D.L., X.Z.Z. and W.M.L. analyzed numerical results. A.B., X.Z.Z. and W.M.L. contributed in completing the paper.
Competing interests
The authors declare no competing financial interests.
Corresponding author
Correspondence to WuMing Liu.
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