The Bethe ansatz equation is solved to obtain analytically the leading finite-size correction of ... more The Bethe ansatz equation is solved to obtain analytically the leading finite-size correction of the spectra of the asymmetric XXZ chain and the accompanying isotropic 6-vertex model near the antiferromagnetic phase boundary at zero vertical field. The energy gaps scale with size N as N^-1/2 and its amplitudes are obtained in terms of level-dependent scaling functions. Exactly on the phase boundary, the amplitudes are proportional to a sum of square-root of integers and an anomaly term. By summing over all low-lying levels, the partition functions are obtained explicitly. Similar analysis is performed also at the phase boundary of zero horizontal field in which case the energy gaps scale as N^-2. The partition functions for this case are found to be that of a nonrelativistic free fermion system. From symmetry of the lattice model under π /2 rotation, several identities between the partition functions are found. The N^-1/2 scaling at zero vertical field is interpreted as a feature ar...
We investigate the thermodynamic and critical properties of an interacting domain wall model whic... more We investigate the thermodynamic and critical properties of an interacting domain wall model which is derived from the triangular lattice antiferromagnetic Ising model with the anisotropic nearest and next nearest neighbor interactions. The model is equivalent to the general five-vertex model. Diagonalizing the transfer matrix exactly by the Bethe Ansatz method, we obtain the phase diagram displaying the commensurate and incommensurate (IC) phases separated by the Pokrovsky-Talapov transitions. The phase diagram exhibits commensurate phases where the domain wall density q is locked at the values of 0, 1/2 and 1. The IC phase is a critical state described by the Gaussian fixed point. The effective Gaussian coupling constant is obtained analytically and numerically for the IC phase using the finite size scaling predictions of the conformal field theory. It takes the value 1/2 in the noninteracting limit and also at the boundaries of q = 0 or 1 phase and the value 2 at the boundary of q = 1/2 phase, while it varies smoothly throughout the IC region.
We investigate the interacting domain-wall model derived from the triangularlattice antiferromagn... more We investigate the interacting domain-wall model derived from the triangularlattice antiferromagnetic Ising model with two next-nearest-neighbor interactions. The system has commensurate phases with a domain-wall density q = 2/3 as well as that of q = 0 when the interaction is repulsive. The q = 2/3 commensurate phase is separated from the incommensurate phase through the Kosterlitz-Thouless (KT) transition. The critical interaction strength and the nature of the KT phase transition are studied by the Monte Carlo simulations and numerical transfer-matrix calculations. For strongly attractive interaction, the system undergoes a first-order phase transition from the q = 0 commensurate phase to the incommensurate phase with q = 0. The incommensurate phase is a critical phase which is in the Gaussian model universality class. The effective Gaussian coupling constant is calculated as a function of interaction parameters from the finite-size scaling of the transfer matrix spectra .
We present a model for evolving population which maintains genetic polymorphism. By introducing r... more We present a model for evolving population which maintains genetic polymorphism. By introducing random mutation in the model population at a constant rate, we observe that the population does not become extinct but survives, keeping diversity in the gene pool under abrupt environmental changes. The model provides reasonable estimates for the proportions of polymorphic and heterozygous loci and for the mutation rate, as observed in nature.
By the use of duality and star-triangle transformations together with a uniqueness assumption, th... more By the use of duality and star-triangle transformations together with a uniqueness assumption, the exact transition temperature of the Potts model with q states per site is obtained for the triangular and honeycomb lattices.
In the reaction-diffusion process A + B → ∅ on random scale-free (SF) networks with the degree ex... more In the reaction-diffusion process A + B → ∅ on random scale-free (SF) networks with the degree exponent γ, the particle density decays with time in a power law with an exponent α when initial densities of each species are the same. The exponent α is α > 1 for 2 < γ < 3 and α = 1 for γ ≥ 3. Here, we examine the reaction process on fractal SF networks, finding that α < 1 even for 2 < γ < 3. This slowly decaying behavior originates from the segregation effect: Fractal SF networks contain local hubs, which are repulsive to each other. Those hubs attract particles and accelerate the reaction, and then create domains containing the same species of particles. It follows that the reaction takes place at the non-hub boundaries between those domains and thus the particle density decays slowly. Since many real SF networks are fractal, the segregation effect has to be taken into account in the reaction kinetics among heterogeneous particles.
Physica A: Statistical Mechanics and its Applications, 2005
We introduce a growing network model which generates both modular and hierarchical structure in a... more We introduce a growing network model which generates both modular and hierarchical structure in a self-organized way. To this end, we modify the Baraba´si-Albert model into the one evolving under the principles of division and independence as well as growth and preferential attachment (PA). A newly added vertex chooses one of the modules composed of existing vertices, and attaches edges to vertices belonging to that module following the PA rule. When the module size reaches a proper size, the module is divided into two, and a new module is created. The karate club network studied by Zachary is a simple version of the current model. We find that the model can reproduce both modular and hierarchical properties, characterized by the hierarchical clustering function of a vertex with degree k, CðkÞ; being in good agreement with empirical measurements for real-world networks.
The energy gap between the ground state and the first excited state of the one-dimensional attrac... more The energy gap between the ground state and the first excited state of the one-dimensional attractive Bose-Hubbard Hamiltonian is investigated in connection with directed polymers in random media. The excitation gap ∆ is obtained by exact diagonalization of the Hamiltonian in the twoand three-particle sectors and also by an exact Bethe Ansatz solution in the two-particle sector. The dynamic exponent z is found to be 2. However, in the intermediate range of the size L where U L ∼ O(1), U being the attractive interaction, the effective dynamic exponent shows an anomalous peak reaching high values of 2.4 and 2.7 for the two-and the three-particle sectors, respectively. The anomalous behavior is related to a change in the sign of the first excited-state energy. In the two-particle sector, we use the Bethe Ansatz solution to obtain the effective dynamic exponent as a function of the scaling variable U L/π. The continuum version, the attractive delta-function Bose-gas Hamiltonian, is integrable by the Bethe Ansatz with suitable quantum numbers, the distributions of which are not known in general. Quantum numbers are proposed for the first excited state and are confirmed numerically for an arbitrary number of particles.
Transfer matrix spectra can be used to establish a connection between two dimensional critical la... more Transfer matrix spectra can be used to establish a connection between two dimensional critical lattice models and conformal field theories. Affine-D models are a class of spin models with (L+3) state per site (L=3, 4, ...) generalizing the multicritical line of the magnetic hard square model. There are two continuous parameters characterizing the models; crossing parameter lambda with which critical exponents vary continuously and spectral parameter u which controls anisotropy of the interactions. Their transfer matrix spectra combined with finite-size scaling theory show that the models in the positive u regime belong to the universality class of the Ashkin-Teller model and are related to the c=1 conformal field theory by r={2L2(1-lambda/pi)}-1/2, where r is the orbifold radius. The models in the negative u regime are found to be related to those in the positive u regime by a certain transformation and also belong to the Ashkin-Teller universality class with the correspondence r={2L2lambda/pi}-1/2.
During the last 10 years since the publication of pioneering papers on small-world and scale-free... more During the last 10 years since the publication of pioneering papers on small-world and scale-free networks, more than 5,500 distinct researchers produced more than 4,000 research papers on complex networks, setting an unprecedented example in the history of science. Based on the dataset of published papers on complex networks during the years 1998-2007, here we study the complete evolution of the co-authorship network in network science. This dataset allows us to study the complete trail of social network evolution from the inception, in particular in the early transient stage, which has not been addressed empirically in previous studies. We find that distinct patterns in network topology emerge during the evolution: A fractal, tree-like giant cluster forms in the early stage through the cluster aggregation process, akin to the pattern near the percolation point, then followed by the entanglement process due to appearance of large-scale loops in later times. This evolution pattern is also observed in the co-authorship network on string theory. The giant cluster is found to be dynamic yet robust upon removal of obsolete inactive links, providing the core part underneath the further developed network. Finally, based on the empirical observations, we introduce a network evolution model, successfully reproducing the observed patterns.
Transfer matrix methods are used to locate accurate phase boundary of the triangular lattice anti... more Transfer matrix methods are used to locate accurate phase boundary of the triangular lattice antiferromagnetic Ising model in magnetic field. Universal quantities such as the central charge and the first few scaling dimensions are obtained along the phase boundary except near the zero field point where the crossover effect degrades convergence. Numerical results are fully consistent with the operator content of the 3-state Potts model indicating that whole phase boundary belongs to the 3-state Potts universality class.
Exact analyses are given for two three-dimensional lattice systems: A system of close-packed dime... more Exact analyses are given for two three-dimensional lattice systems: A system of close-packed dimers placed in layers of honeycomb lattices and a layered triangular-lattice interacting domain wall model, both with nontrivial interlayer interactions,. We show that both models are equivalent to a 5-vertex model on the square lattice with interlayer vertex-vertex interactions. Using the method of Bethe ansatz, a closed-form expression for the free energy is obtained and analyzed. We deduce the exact phase diagram and determine the nature of the phase transitions as a function of the strength of the interlayer interaction.
We investigate the localization of two interacting particles in one-dimensional random potential.... more We investigate the localization of two interacting particles in one-dimensional random potential. Our definition of the two-particle localization length, ξ, is the same as that of v. Oppen et al. [Phys. Rev. Lett. 76, 491 (1996)] and ξ's for chains of finite lengths are calculated numerically using the recursive Green's function method for several values of the strength of the disorder, W , and the strength of interaction, U. When U = 0, ξ approaches a value larger than half the single-particle localization length as the system size tends to infinity and behaves as ξ ∼ W −ν 0 for small W with ν0 = 2.1 ± 0.1. When U = 0, we use the finite size scaling ansatz and find the relation ξ ∼ W −ν with ν = 2.9 ± 0.2. Moreover, data show the scaling behavior ξ ∼ W −ν 0 g(b|U |/W ∆) with ∆ = 4.0 ± 0.5.
Bifractal is a highly anisotropic structure where planar fractals are stacked to form a 3dimensio... more Bifractal is a highly anisotropic structure where planar fractals are stacked to form a 3dimensional lattice. The localization lengths along fractal structure for the Anderson model defined on a bifractal are calculated. The critical disorder and the critical exponent of the localization lengths are obtained from the finite size scaling behavior. The numerical results are in a good agreement with previous results which have been obtained from the localization lengths along the perpendicular direction. This suggests that the anisotropy of the embedding lattice structure is irrelevant to the critical properties of the localization.
We study a class of super-rough growth models whose structure factor satisfies the Family-Vicsek ... more We study a class of super-rough growth models whose structure factor satisfies the Family-Vicsek scaling. We demonstrate that a macroscopic background spontaneously develops in the local surface profile, which dominates the scaling of the local surface width and the height-difference. The shape of the macroscopic background takes a form of a finite-order polynomial whose order is decided from the value of the global roughness exponent. Once the macroscopic background is subtracted, the width of the resulting local surface profile satisfies the Family-Vicsek scaling. We show that this feature is universal to all super-rough growth models, and we also discuss the difference between the macroscopic background formation and the pattern formation in other models.
The Bethe ansatz equation is solved to obtain analytically the leading finite-size correction of ... more The Bethe ansatz equation is solved to obtain analytically the leading finite-size correction of the spectra of the asymmetric XXZ chain and the accompanying isotropic 6-vertex model near the antiferromagnetic phase boundary at zero vertical field. The energy gaps scale with size N as N^-1/2 and its amplitudes are obtained in terms of level-dependent scaling functions. Exactly on the phase boundary, the amplitudes are proportional to a sum of square-root of integers and an anomaly term. By summing over all low-lying levels, the partition functions are obtained explicitly. Similar analysis is performed also at the phase boundary of zero horizontal field in which case the energy gaps scale as N^-2. The partition functions for this case are found to be that of a nonrelativistic free fermion system. From symmetry of the lattice model under π /2 rotation, several identities between the partition functions are found. The N^-1/2 scaling at zero vertical field is interpreted as a feature ar...
We investigate the thermodynamic and critical properties of an interacting domain wall model whic... more We investigate the thermodynamic and critical properties of an interacting domain wall model which is derived from the triangular lattice antiferromagnetic Ising model with the anisotropic nearest and next nearest neighbor interactions. The model is equivalent to the general five-vertex model. Diagonalizing the transfer matrix exactly by the Bethe Ansatz method, we obtain the phase diagram displaying the commensurate and incommensurate (IC) phases separated by the Pokrovsky-Talapov transitions. The phase diagram exhibits commensurate phases where the domain wall density q is locked at the values of 0, 1/2 and 1. The IC phase is a critical state described by the Gaussian fixed point. The effective Gaussian coupling constant is obtained analytically and numerically for the IC phase using the finite size scaling predictions of the conformal field theory. It takes the value 1/2 in the noninteracting limit and also at the boundaries of q = 0 or 1 phase and the value 2 at the boundary of q = 1/2 phase, while it varies smoothly throughout the IC region.
We investigate the interacting domain-wall model derived from the triangularlattice antiferromagn... more We investigate the interacting domain-wall model derived from the triangularlattice antiferromagnetic Ising model with two next-nearest-neighbor interactions. The system has commensurate phases with a domain-wall density q = 2/3 as well as that of q = 0 when the interaction is repulsive. The q = 2/3 commensurate phase is separated from the incommensurate phase through the Kosterlitz-Thouless (KT) transition. The critical interaction strength and the nature of the KT phase transition are studied by the Monte Carlo simulations and numerical transfer-matrix calculations. For strongly attractive interaction, the system undergoes a first-order phase transition from the q = 0 commensurate phase to the incommensurate phase with q = 0. The incommensurate phase is a critical phase which is in the Gaussian model universality class. The effective Gaussian coupling constant is calculated as a function of interaction parameters from the finite-size scaling of the transfer matrix spectra .
We present a model for evolving population which maintains genetic polymorphism. By introducing r... more We present a model for evolving population which maintains genetic polymorphism. By introducing random mutation in the model population at a constant rate, we observe that the population does not become extinct but survives, keeping diversity in the gene pool under abrupt environmental changes. The model provides reasonable estimates for the proportions of polymorphic and heterozygous loci and for the mutation rate, as observed in nature.
By the use of duality and star-triangle transformations together with a uniqueness assumption, th... more By the use of duality and star-triangle transformations together with a uniqueness assumption, the exact transition temperature of the Potts model with q states per site is obtained for the triangular and honeycomb lattices.
In the reaction-diffusion process A + B → ∅ on random scale-free (SF) networks with the degree ex... more In the reaction-diffusion process A + B → ∅ on random scale-free (SF) networks with the degree exponent γ, the particle density decays with time in a power law with an exponent α when initial densities of each species are the same. The exponent α is α > 1 for 2 < γ < 3 and α = 1 for γ ≥ 3. Here, we examine the reaction process on fractal SF networks, finding that α < 1 even for 2 < γ < 3. This slowly decaying behavior originates from the segregation effect: Fractal SF networks contain local hubs, which are repulsive to each other. Those hubs attract particles and accelerate the reaction, and then create domains containing the same species of particles. It follows that the reaction takes place at the non-hub boundaries between those domains and thus the particle density decays slowly. Since many real SF networks are fractal, the segregation effect has to be taken into account in the reaction kinetics among heterogeneous particles.
Physica A: Statistical Mechanics and its Applications, 2005
We introduce a growing network model which generates both modular and hierarchical structure in a... more We introduce a growing network model which generates both modular and hierarchical structure in a self-organized way. To this end, we modify the Baraba´si-Albert model into the one evolving under the principles of division and independence as well as growth and preferential attachment (PA). A newly added vertex chooses one of the modules composed of existing vertices, and attaches edges to vertices belonging to that module following the PA rule. When the module size reaches a proper size, the module is divided into two, and a new module is created. The karate club network studied by Zachary is a simple version of the current model. We find that the model can reproduce both modular and hierarchical properties, characterized by the hierarchical clustering function of a vertex with degree k, CðkÞ; being in good agreement with empirical measurements for real-world networks.
The energy gap between the ground state and the first excited state of the one-dimensional attrac... more The energy gap between the ground state and the first excited state of the one-dimensional attractive Bose-Hubbard Hamiltonian is investigated in connection with directed polymers in random media. The excitation gap ∆ is obtained by exact diagonalization of the Hamiltonian in the twoand three-particle sectors and also by an exact Bethe Ansatz solution in the two-particle sector. The dynamic exponent z is found to be 2. However, in the intermediate range of the size L where U L ∼ O(1), U being the attractive interaction, the effective dynamic exponent shows an anomalous peak reaching high values of 2.4 and 2.7 for the two-and the three-particle sectors, respectively. The anomalous behavior is related to a change in the sign of the first excited-state energy. In the two-particle sector, we use the Bethe Ansatz solution to obtain the effective dynamic exponent as a function of the scaling variable U L/π. The continuum version, the attractive delta-function Bose-gas Hamiltonian, is integrable by the Bethe Ansatz with suitable quantum numbers, the distributions of which are not known in general. Quantum numbers are proposed for the first excited state and are confirmed numerically for an arbitrary number of particles.
Transfer matrix spectra can be used to establish a connection between two dimensional critical la... more Transfer matrix spectra can be used to establish a connection between two dimensional critical lattice models and conformal field theories. Affine-D models are a class of spin models with (L+3) state per site (L=3, 4, ...) generalizing the multicritical line of the magnetic hard square model. There are two continuous parameters characterizing the models; crossing parameter lambda with which critical exponents vary continuously and spectral parameter u which controls anisotropy of the interactions. Their transfer matrix spectra combined with finite-size scaling theory show that the models in the positive u regime belong to the universality class of the Ashkin-Teller model and are related to the c=1 conformal field theory by r={2L2(1-lambda/pi)}-1/2, where r is the orbifold radius. The models in the negative u regime are found to be related to those in the positive u regime by a certain transformation and also belong to the Ashkin-Teller universality class with the correspondence r={2L2lambda/pi}-1/2.
During the last 10 years since the publication of pioneering papers on small-world and scale-free... more During the last 10 years since the publication of pioneering papers on small-world and scale-free networks, more than 5,500 distinct researchers produced more than 4,000 research papers on complex networks, setting an unprecedented example in the history of science. Based on the dataset of published papers on complex networks during the years 1998-2007, here we study the complete evolution of the co-authorship network in network science. This dataset allows us to study the complete trail of social network evolution from the inception, in particular in the early transient stage, which has not been addressed empirically in previous studies. We find that distinct patterns in network topology emerge during the evolution: A fractal, tree-like giant cluster forms in the early stage through the cluster aggregation process, akin to the pattern near the percolation point, then followed by the entanglement process due to appearance of large-scale loops in later times. This evolution pattern is also observed in the co-authorship network on string theory. The giant cluster is found to be dynamic yet robust upon removal of obsolete inactive links, providing the core part underneath the further developed network. Finally, based on the empirical observations, we introduce a network evolution model, successfully reproducing the observed patterns.
Transfer matrix methods are used to locate accurate phase boundary of the triangular lattice anti... more Transfer matrix methods are used to locate accurate phase boundary of the triangular lattice antiferromagnetic Ising model in magnetic field. Universal quantities such as the central charge and the first few scaling dimensions are obtained along the phase boundary except near the zero field point where the crossover effect degrades convergence. Numerical results are fully consistent with the operator content of the 3-state Potts model indicating that whole phase boundary belongs to the 3-state Potts universality class.
Exact analyses are given for two three-dimensional lattice systems: A system of close-packed dime... more Exact analyses are given for two three-dimensional lattice systems: A system of close-packed dimers placed in layers of honeycomb lattices and a layered triangular-lattice interacting domain wall model, both with nontrivial interlayer interactions,. We show that both models are equivalent to a 5-vertex model on the square lattice with interlayer vertex-vertex interactions. Using the method of Bethe ansatz, a closed-form expression for the free energy is obtained and analyzed. We deduce the exact phase diagram and determine the nature of the phase transitions as a function of the strength of the interlayer interaction.
We investigate the localization of two interacting particles in one-dimensional random potential.... more We investigate the localization of two interacting particles in one-dimensional random potential. Our definition of the two-particle localization length, ξ, is the same as that of v. Oppen et al. [Phys. Rev. Lett. 76, 491 (1996)] and ξ's for chains of finite lengths are calculated numerically using the recursive Green's function method for several values of the strength of the disorder, W , and the strength of interaction, U. When U = 0, ξ approaches a value larger than half the single-particle localization length as the system size tends to infinity and behaves as ξ ∼ W −ν 0 for small W with ν0 = 2.1 ± 0.1. When U = 0, we use the finite size scaling ansatz and find the relation ξ ∼ W −ν with ν = 2.9 ± 0.2. Moreover, data show the scaling behavior ξ ∼ W −ν 0 g(b|U |/W ∆) with ∆ = 4.0 ± 0.5.
Bifractal is a highly anisotropic structure where planar fractals are stacked to form a 3dimensio... more Bifractal is a highly anisotropic structure where planar fractals are stacked to form a 3dimensional lattice. The localization lengths along fractal structure for the Anderson model defined on a bifractal are calculated. The critical disorder and the critical exponent of the localization lengths are obtained from the finite size scaling behavior. The numerical results are in a good agreement with previous results which have been obtained from the localization lengths along the perpendicular direction. This suggests that the anisotropy of the embedding lattice structure is irrelevant to the critical properties of the localization.
We study a class of super-rough growth models whose structure factor satisfies the Family-Vicsek ... more We study a class of super-rough growth models whose structure factor satisfies the Family-Vicsek scaling. We demonstrate that a macroscopic background spontaneously develops in the local surface profile, which dominates the scaling of the local surface width and the height-difference. The shape of the macroscopic background takes a form of a finite-order polynomial whose order is decided from the value of the global roughness exponent. Once the macroscopic background is subtracted, the width of the resulting local surface profile satisfies the Family-Vicsek scaling. We show that this feature is universal to all super-rough growth models, and we also discuss the difference between the macroscopic background formation and the pattern formation in other models.
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Papers by Doochul Kim