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The one-dimensional spin-$s$ ${\rm SU}(2)$ ferromagnetic Heisenberg model, as a paradigmatic example for spontaneous symmetry breaking (SSB) with type-B Goldstone modes (GMs), is expected to exhibit an abstract fractal underlying the ground state subspace. This intrinsic abstract fractal is here revealed from a systematic investigation into the entanglement entropy for a linear combination of factorized (unentangled) ground states on a fractal decomposable into a set of the Cantor sets. The entanglement entropy scales logarithmically with the block size, with the prefactor being half the fractal dimension of a fractal, as long as the norm for the linear combination scales as the square root of the number of the self-similar building blocks kept at each step $k$ for a fractal, under an assumption that the maximum absolute value of the coefficients in the linear combination is chosen to be around one, and the coefficients in the linear combination are almost constants within the building blocks. Actually, the set of the fractal dimensions for all the Cantor sets forms a {\it dense} subset in the interval $[0,1]$. As a consequence, the ground state subspace is separated into a disjoint union of countably infinitely many regions, each of which is labeled by a decomposable fractal. Hence, the interpretation of the prefactor as half the fractal dimension is valid for any support beyond a fractal, which in turn leads to the identification of the fractal dimension with the number of type-B GMs for the orthonormal basis states. Our argument may be extended to any quantum many-body systems undergoing SSB with type-B GMs., Comment: 23 pages, 8 figures

Exact matrix product state representations for a type of scale-invariant states are presented, which describe highly degenerate ground states arising from spontaneous symmetry breaking with type-B Goldstone modes in one-dimensional quantum many-body systems. As a possible application, such a representation offers a convenient but powerful means for evaluating the norms of highly degenerate ground states. This in turn allows us to perform a universal finite system-size scaling analysis of the entanglement entropy. Moreover, this approach vividly explains why the entanglement entropy does not depend on what types of the boundary conditions are adopted, either periodic boundary conditions or open boundary conditions. Illustrative examples include the ${\rm SU}(2)$ spin-$s$ Heisenberg ferromagnetic model, the ${\rm SU}(2s+1)$ ferromagnetic model, and the staggered ${\rm SU}(3)$ spin-1 ferromagnetic biquadratic model., Comment: 12 pages, 5 figures, 3 tables

Spontaneous symmetry breaking (SSB) with type-B Goldstone modes is investigated in the macroscopically degenerate phase for a quantum spin-1 many-body system with competing dimer and trimer interactions. The SSB involves three distinct patterns. The first occurs at the dimer point, with the pattern from staggered ${\rm SU}(3)$ to ${\rm U}(1)\times{\rm U}(1)$. The second occurs at the trimer point, with the pattern from uniform ${\rm SU}(3)$ to ${\rm U}(1)\times{\rm U}(1)$. The third occurs in the dimer-trimer regime, with the pattern from uniform ${\rm SU}(2)$ to ${\rm U}(1)$. The number of type-B Goldstone modes is thus two, two and one for the three patterns, respectively. The ground state degeneracies arising from the three patterns are exponential with the system size, which may be recognized as sequences of integers relevant to self-similar logarithmic spirals. This in turn is attributed to the presence of an emergent symmetry operation tailored to a specific degenerate ground state. As a consequence, the residual entropy is non-zero, which measures the disorder present in a unit cell of highly degenerate ground state generated from a generalized highest weight state. An exact Schmidt decomposition exists for the highly degenerate ground states, thus exposing the self-similarities underlying an abstract fractal, described by the fractal dimension. The latter is extracted from performing a universal finite system-size scaling analysis of the entanglement entropy, which is identical to the number of type-B Goldstone modes. The model under investigation thus accommodates an exotic scale invariant quantum state of matter., Comment: 14 pages, 3 figures, 2 tables

The ground-state phase diagram is mapped out for an alternative anisotropic extension of quantum spin-1 ferromagnetic biquadratic model, which accommodates twelve distinct phases: three degenerate fractal phases, six Luttinger liquid phases and three symmetry-protected trivial phases. It is found that distinct types of quantum phase transitions are involved between them. In particular, one type arises from an instability of a Luttinger liquid towards a degenerate fractal phase, and the other type describes spontaneous symmetry breaking with type-B Goldstone modes from one degenerate fractal phase to another degenerate fractal phase, with the fractal dimension $d_f$ being identical to the number of the type-B Goldstone modes, both of which turn out to be one. In addition, quantum phase transitions from the Luttinger liquid phases to the symmetry-protected trivial phases are identified to be in the Kosterlitz-Thouless universality class, with central charge being one., Comment: 11 pages, 8 figures, 2 tables

An emergent gapless Goldstone mode originates from continuous spontaneous symmetry breaking, which has become a doctrine since the pioneering work by Goldstone [J. Goldstone, Nuovo Cimento \textbf{19}, 154 (1961)]. However, we argue that it is possible for a continuous symmetry group $\rm{U}(1)$ to make an exceptional case, simply due to the well-known mathematical result that a continuous symmetry group $\rm{U}(1)$ may be regarded as a limit of a discrete symmetry group $Z_q$ when $q$ tends to infinity. As a consequence, spontaneous symmetry breaking for such a continuous symmetry group $\rm{U}(1)$ does not necessarily lead to any gapless Goldstone mode. This is explicitly explained for an anisotropic extension of the ferromagnetic spin-1 biquadratic model. In a sense, this model provides an illustrative example regarding the dichotomy between continuity and discreteness., Comment: 13 pages, 9 figures, 9 tables

It is found that the staggered $\rm SU(4)$ ferromagnetic spin-orbital model accommodates highly degenerate ground states arising from spontaneous symmetry breaking with type-B Goldstone modes. The spontaneous symmetry breaking pattern is ${\rm SU(4)} \rightarrow {\rm U(1)} \times {\rm U(1)} \times {\rm U(1)}$, with three type-B Goldstone modes. An abstract fractal underlies the ground state subspace with the fractal dimension identified with the number of type-B Goldstone modes. This connection is established by evaluating the entanglement entropy, which exhibits a universal finite system-size scaling behaviour. The latter in turn implies that the entanglement entropy scales logarithmically with the block size in the thermodynamic limit. In addition, the ground state degeneracies, depending on the boundary conditions adopted, constitute the two Fibonacci-Lucas sequences. In the limit of large system size their asymptotic forms become a self-similar logarithmic spiral. As a result, the model has a non-zero residual entropy $S_{\!r} = -2 \ln R $, where $R=(\! \sqrt{6}-\!\sqrt{2})/2$., Comment: 16 pages, 5 figures, 1 table. arXiv admin note: substantial text overlap with arXiv:2302.13126

A generic scheme is proposed to perform a finite-entanglement scaling analysis for scale-invariant states, which appear to be highly degenerate ground states arising from spontaneous symmetry breaking with type-B Goldstone modes. This allows us to extract the number of type-B Goldstone modes and the dynamical critical exponent, in combination with a finite block-size scaling analysis, from numerical simulations of quantum many-body systems in the context of tensor network representations. The number of type-B Goldstone modes is identical to the fractal dimension, thus reflecting an abstract fractal underlying the ground state subspace. As illustrative examples, we investigate the spin-$s$ Heisenberg ferromagnetic model, the $\rm{SU}(3)$ ferromagnetic model and the $\rm{SO}(4)$ spin-orbital model., Comment: 14 pages, 24 figures, 11 tables

A universal finite system-size scaling analysis of the entanglement entropy is presented for highly degenerate ground states arising from spontaneous symmetry breaking with type-B Goldstone modes in exactly solvable one-dimensional quantum many-body systems. These states appear to be scale-invariant, but not conformally invariant. Our findings are based on a physical argument, imposing three constraints on the entanglement entropy, in addition to further confirmation from an asymptotic analysis of the entanglement entropy for the ${\rm SU}(2)$ spin-$1/2$ ferromagnetic states. The resulting universal scaling form is demonstrated for three fundamental models -- the ${\rm SU}(2)$ spin-$s$ Heisenberg ferromagnetic model, the ${\rm SU}(N+1)$ ferromagnetic model, and the staggered ${\rm SU}(3)$ spin-1 ferromagnetic biquadratic model. The results point towards a classification for distinct types of scale-invariant states, relevant to a complete classification of quantum states of matter., Comment: 8 pages, 3 figures

Ferromagnetic ground states have often been overlooked in comparison to seemingly more interesting antiferromagnetic ground states. However, both the physical and mathematical structure of ferromagnetic ground states are particularly rich. We show that the highly degenerate and highly entangled ground states of the ferromagnetic spin-1 biquadratic model are scale invariant, originating from spontaneous symmetry breaking from ${\rm SU}(3)$ to ${\rm U}(1)\times {\rm U}(1)$ with two type-B Goldstone modes. The ground state degeneracies are characterized as the Fibonacci-Lucas sequences -- an ancient mathematical gem, under open and periodic boundary conditions, with the residual entropy being non-zero. This implies that the ground state degeneracies for this model are asymptotically the golden spiral. In addition, sequences of degenerate ground states generated from highest and generalized highest weight states are constructed to establish that the entanglement entropy scales logarithmically with the block size in the thermodynamic limit, with the prefactor being half the number of type-B Goldstone modes. The latter in turn is identified to be the fractal dimension., Comment: 20 pages, 3 figures,1 table, additional references to earlier related work added, a minor change

An extensive investigation, both numerical and analytical, is performed for an anisotropic extension of the ferromagnetic spin-1 biquadratic model. The ground state phase diagram accommodates three symmetry-protected trivial phases, three coexisting fractal phases and six Luttinger liquid phases. A novel universality class arises from an instability of a Luttinger liquid towards an exotic quantum state of matter with infinitely degenerate ground states. The latter in turn is a scale-invariant quantum state of matter, which may be attributed to the coexistence of ${\rm SU}(2)$ spontaneous symmetry breaking with one type-B Goldstone mode on the characteristic line: $J_y=J_z$, and ${\rm U}(1)$ spontaneous symmetry breaking without any gapless Goldstone mode on the characteristic line $J_x/J_z=0$, together with their cyclic permutations with respect to $x$, $y$ and $z$., Comment: 11+ pages, 12 figures and 2 tables

It is argued that highly degenerate ground states arising from symmetry spontaneous breaking with a type-B Goldstone mode are scale invariant, with a salient feature that the entanglement entropy $S(n)$ scales logarithmically with the block size $n$ in the thermodynamic limit. As it turns out, the prefactor is half the number of the type-B Goldstone modes $N_B$. This is achieved by performing an exact singular value decomposition of the degenerate ground states, thus unveiling their self-similarities - the essence of a fractal. Combining with a field-theoretic prediction [O. A. Castro-Alvaredo and B. Doyon, Phys. Rev. Lett. \textbf{108}, 120401 (2012)], we are led to the identification of the fractal dimension $d_f$ with the number of the type-B Goldstone modes $N_B$ for quantum many-body systems undergoing symmetry spontaneous breaking., Comment: 11 pages, 3 figures

The absence of a critical nematic phase in the vicinity of the $\rm {SU}(3)$ ferromagnetic point for the one-dimensional spin-1 bilinear-biquadratic model is demonstrated by means of the tensor network algorithms. As it turns out, the phase transition from the ferromagnetic phase to the dimerized phase at the $\rm {SU}(3)$ ferromagnetic point is direct, but not of the first-order. The transition point features highly degenerate ground states, which are scale but not conformally invariant, with the fractal dimension being equal to 2. The conceptual developments in effective field theories - the fractal dimension and the counting rule of the Goldstone modes - play a pivotal role in clarifying the numerical artifacts arising from the finiteness of the bond dimension in the tensor network simulations, which are attributed to a proximity effect to a highly entangled scale or conformally invariant ground state., Comment: 15 pages, 21 figures, 7 tables

Five duality transformations are unveiled for the quantum XYZ model with arbitrary spin $s$ in one spatial dimension. The presence of these duality transformations drastically reduces the entire ground-state phase diagram to two {\it finite} regimes - the principal regimes, with all the other ten regimes dual to them. Combining with the determination of critical points from the conventional order parameter approach and/or the fidelity approach to quantum phase transitions, we are able to map out the ground-state phase diagram for the quantum XYZ model with arbitrary spin $s$. This is explicitly demonstrated for $s=1/2,1,3/2$ and 2. As it turns out, all the critical points, with central charge $c=1$, are self-dual under a respective duality transformation for half-integer as well as integer spin $s$. However, in the latter case, the presence of the so-called symmetry protected topological phase, i.e., the Haldane phase, results in extra lines of critical points with central charge $c=1/2$, which is not self-dual under any duality transformation., Comment: 4+ pages, 5 figures

Fidelity mechanics is formalized as a framework to investigate quantum critical phenomena in quantum many-body systems. This is achieved by introducing fidelity temperature to properly quantify quantum fluctuations, which, together with fidelity entropy and fidelity internal energy, constitute three basic state functions in fidelity mechanics, thus enabling us to formulate analogues of the four thermodynamic laws and Landauer's principle at zero temperature. Fidelity flows are defined and may be interpreted as an alternative form of renormalization group flows. Thus, both stable and unstable fixed points are characterized in terms of fidelity temperature and fidelity entropy: divergent fidelity temperature for unstable fixed points and zero fidelity temperature and (locally) maximal fidelity entropy for stable fixed points. In addition, an inherently fundamental role of duality is clarified, resulting in a canonical form of the Hamiltonian in fidelity mechanics. Dualities, together with symmetry groups and factorizing fields, impose the constraints on a fidelity mechanical system, thus shaping fidelity flows from an unstable fixed point to a stable fixed point. A detailed analysis of fidelity mechanical state functions is presented for the quantum XY model, the transverse field quantum Ising chain in a longitudinal field, the spin-$1/2$ XYZ model and the XXZ model in a magnetic field., Comment: 56 pages, 23 figures and 2 tables

The fully anisotropic two-leg spin-1/2 $XXZ$ ladder model is studied in terms of an algorithm based on the tensor network representation of quantum many-body states as an adaptation of projected entangled pair states to the geometry of translationally invariant infinite-size quantum spin ladders. The tensor network algorithm provides an effective method to generate the groundstate wave function, which allows computation of the groundstate fidelity per lattice site, a universal marker to detect phase transitions in quantum many-body systems. The groundstate fidelity is used in conjunction with local order and string order parameters to systematically map out the groundstate phase diagram of the ladder model. The phase diagram exhibits a rich diversity of quantum phases. These are the ferromagnetic, stripe ferromagnetic, rung singlet, rung triplet, N$\rm \acute{e}$el, stripe N$\rm \acute{e}$el and Haldane phases, along with the two $XY$ phases $XY1$ and $XY2$., Comment: 23 pages, 15 figures. Revised version with additional text and figures. New Format

The finite temperature phase diagram is obtained for an infinite honeycomb lattice with spin-$1/2$ Ising interaction $J$ by using thermal-state fidelity and von Neumann entropy based on the infinite projected entangled pair state algorithm with ancillas. % The tensor network representation of the fidelity, which is defined as an overlap measurement between two thermal states, is presented for thermal states on the honeycomb lattice. % We show that the fidelity per lattice site and the von Neumann entropy can capture the phase transition temperatures for applied magnetic field, consistent with the transition temperatures obtained via the transverse magnetizations, which indicates that a continuous phase transition occurs in the system. In the temperature-magnetic field plane, the phase boundary is found to have the functional form $(k_BT_c)^2 + h_c^2/2 = a J^2$ with a single numerical fitting coefficient $a = 2.298$, where $T_c$ and $h_c$ are the critical temperature and field with the Boltzmann constant $k_B$. For the quantum state at zero temperature, this phase boundary function gives the critical field estimate $h_c = \sqrt{2a} J \simeq 2.1438 J$, consistent with the known value $h_c = 2.13250(4)\, J$ calculated from a Cluster Monte Carlo approach. The critical temperature in the absence of magnetic field is estimated as $k_BT_c = \sqrt{a}J \simeq 1.5159\, J$, consistent with the exact result $k_BT_c = 1.51865...\, J$., Comment: 9 pages, 8 figures, Typos corrected and references added

Geometric entanglement(GE), as a measure of multipartite entanglement, has been investigated as a universal tool to detect phase transitions in quantum many-body lattice models. We outline a systematic method to compute GE for two-dimensional (2D) quantum many-body lattice models based on the translational invariant structure of infinite projected entangled pair state (iPEPS) representations. By employing this method, the $q$-state quantum Potts model on the square lattice with $q \in \{2, 3 ,4, 5\}$ is investigated as a prototypical example. Further, we have explored three 2D Heisenberg models, such as the spin-$\frac{1}{2}$ XXX and antiferromagnetic anisotropic XYX models in an external magnetic field, and the spin-1 antiferromagnetic XXZ model. We find that continuous GE does not guarantee a continuous phase transition across a phase transition point. We observe and thus classify three different types of continuous GE across a phase transition point: (i) GE is continuous with maximum value at the transition point and the phase transition is continuous, (ii) GE is continuous with maximum value at the transition point but the phase transition is discontinuous, and (iii) GE is continuous with non-maximum value at the transition point and the phase transition is continuous. For the models under consideration we find that the second and the third types are related to a point of dual symmetry and a fully polarized phase, respectively., Comment: 11 pages, 9 figures, minor changes, additional figure and references, and clarification of relation to arXiv:1106.2129

Recently, emergent symmetry is one of fast-growing intriguing issues in many-body systems. Its roles and consequential physics have not been well understood in quantum phase transitions. Emergent symmetry of degenerate groundstates is discussed in possible connection to spontaneous symmetry breaking within the Landau theory. For a clear discussion, a quantum spin-$1/2$ plaquette chain system is shown to have rich emergent symmetry phenomena in its groundstates. A covering symmetry group over all emergent symmetries responsible for degenerate groundstates in the plaquette chain system is found to correspond to a largest common symmetry group of constituent Hamiltonians describing the plaquette system. Consequently, this result suggests that, as a guiding symmetry principle in quantum phase transitions, {\it degenerate groundstates are induced by a spontaneous breaking of symmetries belonging to a largest common symmetry group of continent Hamiltonians describing a given system but can have more symmetries than the largest common symmetry}., Comment: 18 pages,18 figures

We propose a method to construct universal order parameters for quantum phase transitions in many-body lattice systems. The method exploits the $H$-orthogonality of a few near-degenerate lowest states of the Hamiltonian describing a given finite-size system, which makes it possible to perform finite-size scaling and take full advantage of currently available numerical algorithms. An explicit connection is established between the fidelity per site between two $H$-orthogonal states and the energy gap between the ground state and low-lying excited states in the finite-size system. The physical information encoded in this gap arising from finite-size fluctuations clarifies the origin of the universal order parameter.We demonstrate the procedure for the one-dimensional quantum formulation of the $q$-state Potts model, for $q=2,3,4$ and 5, as prototypical examples, using finite-size data obtained from the density matrix renormalization group (DMRG) algorithm., Comment: 4 pages, 4 figures

The geometric entanglement per lattice site, as a holistic measure of the multipartite entanglement, serves as a universal marker to detect quantum phase transitions in quantum many-body systems. However, it is very difficult to compute the geometric entanglement due to the fact that it involves a complicated optimization over all the possible separable states. In this paper, we propose a systematic method to efficiently compute the geometric entanglement per lattice site for quantum many-body lattice systems in two spatial dimensions in the context of a newly-developed tensor network algorithm based on an infinite projected entangled pair state representation. It is tested for quantum Ising model in a transverse magnetic field and anisotropic spin 1/2 anti-ferromagnetic XYX model in an external magnetic field on an infinite-size square lattice. In addition, the geometric entanglement per lattice site is able to detect the so-called factorizing field. Our results are in a quantitative agreement with Quantum Monte Carlo simulations., Comment: 4+ pages, 4 figures

We investigate a two-leg Heisenberg spin ladder with cyclic four-spin exchange by exploiting a newly-developed tensor network algorithm. The algorithm allows to efficiently compute the ground-state fidelity per lattice site, which enables us to establish the ground-state phase diagram for quantum lattice many-body systems. The latter is based on the observation that, for an infinite-size system, any singularity on a ground-state fidelity surface characterizes a critical point, at which the system undergoes a phase transition. For the two-leg Heisenberg spin-1/2 ladder with cyclic four-spin exchange, six different phases are identified: the ferromagnetic phase, the rung singlet phase, the staggered dimer phase, the scalar chirality phase, the dominant vector chirality region, and the dominant collinear spin region. Our findings are in a good agreement with the previous studies from the exact diagonalization and the density-matrix renormalization group., Comment: 4 pages, 2 figures

An efficient algorithm is developed for quantum spin tubes in the context of the tensor network representations. It allows to efficiently compute the ground-state fidelity per lattice site, which in turn enables us to identify quantum critical points, at which quantum spin tubes undergo quantum phase transitions. As an illustration, we investigate the isosceles spin 1/2 antiferromagnetic three-leg Heisenberg tube. Our simulation results suggest that two Kosterlitz-Thouless transitions occur as the degree of asymmetry of the rung interaction is tuned, thus offering an alternative route towards a resolution to the conflicting results on this issue arising from the density matrix renormalization group., Comment: 5 pages, 5 figures

We establish an intriguing connection between quantum phase transitions and bifurcations in the ground-state fidelity per lattice site, and construct the universal order parameter for quantum Ising model in a transverse magnetic field on an infinite-size square lattice in two spatial dimensions, a prototypical model with symmetry breaking order. This is achieved by computing ground-state wave functions in the context of the tensor network algorithm based on the infinite projected entangled-pair state representation. Our finding is applicable to any systems with symmetry breaking order, as a result of the fact that, in the conventional Landau-Ginzburg-Wilson paradigm, a quantum system undergoing a phase transition is characterized in terms of spontaneous symmetry breaking captured by a local order parameter. In addition, a bifurcation in the reduced fidelity between two different reduced density matrices is also discussed., Comment: 4+ pages, 3 figures

The ground-state phase diagram of the two-dimensional t-J model is investigated in the context of the tensor network algorithm in terms of the graded Projected Entangled-Pair State representation of the ground-state wave functions. There is a line of phase separation between the Heisenberg anti-ferromagnetic state without hole and a hole-rich state. For both J=0.4t and J=0.8t, a systematic computation is performed to identify all the competing ground states for various dopings. It is found that, besides a possible Nagaoka's ferromagnetic state, the homogeneous regime consists of four different phases: one phase with charge and spin density wave order coexisting with a p_x (p_y)-wave superconducting state, one phase with the symmetry mixing of d+s-wave superconductivity in the spin-singlet channel and p_x (p_y)-wave superconductivity in the spin-triplet channel in the presence of an anti-ferromagnetic background, one superconducting phase with extended s-wave symmetry, and one superconducting phase with p_x (p_y)-wave symmetry in a ferromagnetic background., Comment: 4+ pages, 3 figures, and 1 table

For any translation-invariant quantum lattice system with a symmetry group G, we propose a practical and universal construction of order parameters which identify quantum phase transitions with symmetry-breaking order. They are defined in terms of the fidelity between a ground state and its symmetry-transformed counterpart, and are computed through tensor network representations of the ground-state wavefunction. To illustrate our scheme, we consider three quantum systems on an infinite lattice in one spatial dimension, namely, the quantum Ising model in a transverse magnetic field, the quantum spin-1/2 XYX model in an external magnetic field, and the quantum spin-1 XXZ model with single-ion anisotropy. All these models have symmetry group Z_2 and exhibit broken-symmetry phases. We also discuss the role of the order parameters in identifying factorized states., Comment: 4+ pages, 5 figures

An algorithm to find a graded Projected Entangled-Pair State representation of the ground state wave functions is developed for translationally invariant strongly correlated electronic systems on infinite-size lattices in two spatial dimensions. It is tested for the two-dimensional t-J model at and away from half filling, with truncation dimensions up to 6. We are able to locate a line of phase separation, which qualitatively agrees with the results based on the high-temperature expansions. We find that the model exhibits an extended s-wave superconductivity for J=0.4t at quarter filling. However, we emphasize that the currently accessible truncation dimensions are not large enough, so it is necessary to incorporate the symmetry of the system into the algorithm, in order to achieve results with higher precision., Comment: 5 pages, 5 figures

We establish an intriguing connection between quantum phase transitions and bifurcations in the reduced fidelity between two different reduced density matrices for quantum lattice many-body systems with symmetry-breaking orders. Our finding is based on the observation that, in the conventional Landau-Ginzburg-Wilson paradigm, a quantum system undergoing a phase transition is characterized in terms of spontaneous symmetry breaking that is captured by a local order parameter, which in turn results in an essential change of the reduced density matrix in the symmetry-broken phase. Two quantum systems on an infinite lattice in one spatial dimension, i.e., quantum Ising model in a transverse magnetic field and quantum spin 1/2 XYX model in an external magnetic field, are considered in the context of the tensor network algorithm based on the matrix product state representation., Comment: 4+ pages, 4 figures

The global geometric entanglement is studied in the context of newly-developed tensor network algorithms for finite systems. For one-dimensional quantum spin systems it is found that, at criticality, the leading finite-size correction to the global geometric entanglement per site behaves as $b/n$, where $n$ is the size of the system and $b$ a given coefficient. Our conclusion is based on the computation of the geometric entanglement per spin for the quantum Ising model in a transverse magnetic field and for the spin-1/2 XXZ model. We also discuss the possibility of coefficient $b$ being universal., Comment: 5 pages, 2 figures, and 3 tables.

Realizable spin models are investigated in a two superconducting flux qubit system. It is shown that a specific adjustment of system parameters in the two flux qubit system makes it possible to realize an artificial two-spin system that cannot be found naturally. For the artificial two-spin systems, time evolution of a prepared quantum state is discussed to quantify quantum entanglement dynamics. The concurrence and fidelity as a function of time are shown to reveal a characteristic entanglement dynamics of the artificial spin systems. It is found that the unentangled input state can evolute to be a maximally entangled output state periodically due to the exchange interactions induced by two-qubit flipping tunneling processes while single-qubit flipping tunneling processes plays a role of magnetic fields for the artificial spins., Comment: 5 pages, 2 figures