Entanglement in quantum theory is a peculiar concept to scientists. With this concept we are forced to re-consider the cluster property which means that one event is irrelevant to another event when they are fully far away. In the recent works we showed that the quasi-degenerate states induce the violation of cluster property in antiferromagnets when the continuous symmetry breaks spontaneously. We expect that the violation of cluster property will be observed in other materials too, because the spontaneous symmetry breaking is found in many systems such as the high temperature superconductors and the superfluidity. In order to examine the cluster property for these materials, we studied a quantum nonlinear sigma model with U(1) symmetry in the previous work. There we showed that the model does have quasi-degenerate states. In this paper we study the quantum nonlinear sigma model with SU(2) symmetry. In our approach we first define the quantum system on the lattice and then adopt the representation where the kinetic term is diagonalized. Since we have no definition on the conjugate variable to the angle variable, we use the angular momentum operators instead for the kinetic term. In this representation we introduce the states with the fixed quantum numbers and carry out numerical calculations using quantum Monte Carlo methods and other methods. Through analytical and numerical studies, we conclude that the energy of the quasi-degenerate state is proportional to the squared total angular momentum as well as to the inverse of the lattice size.
Entanglement in quantum theory is a peculiar concept to scientists. With this concept we are forced to re-consider the cluster property which means that one event is irrelevant to another event when they are fully far away. In the recent works we showed that the quasi-degenerate states induce the violation of cluster property in antiferromagnets when the continuous symmetry breaks spontaneously. We expect that the violation of cluster property will be observed in other materials too, because the spontaneous symmetry breaking is found in many systems such as the high temperature superconductors and the superfluidity. In order to examine the cluster property for these materials, we studied a quantum nonlinear sigma model with U(1) symmetry in the previous work. There we showed that the model does have quasi-degenerate states. In this paper we study the quantum nonlinear sigma model with SU(2) symmetry. In our approach we first define the quantum system on the lattice and then adopt the representation where the kinetic term is diagonalized. Since we have no definition on the conjugate variable to the angle variable, we use the angular momentum operators instead for the kinetic term. In this representation we introduce the states with the fixed quantum numbers and carry out numerical calculations using quantum Monte Carlo methods and other methods. Through analytical and numerical studies, we conclude that the energy of the quasi-degenerate state is proportional to the squared total angular momentum as well as to the inverse of the lattice size.
Entanglement in quantum theory is a concept that has confused many scientists.This concept implies that the cluster property, which means no relations between sufficiently separated two events, is non-trivial.In the works for some quantum spin systems, which have been recently published by the author, extensive and quantitative examinations were made about the violation of cluster property in the correlation function of the spin operator.The previous study of these quantum antiferromagnets showed that this violation is induced by the degenerate states in the systems where the continuous symmetry spontaneously breaks.Since this breaking is found in many materials such as the high temperature superconductors and the superfluidity, it is an important question whether we can observe the violation of the cluster property in them.As a step to answer this question we study a quantum nonlinear sigma model with U(1) symmetry in this paper.It is well known that this model, which has been derived as an effective model of the quantum spin systems, can also be applied to investigations of many materials.Notifying that the existence of the degenerate states is essential for the violation, we made numerical calculations in addition to theoretical arguments to find these states in the nonlinear sigma model.Then, successfully finding the degenerate states in the model, we came to a conclusion that there is a chance to observe the violation of cluster property in many materials to which the nonlinear sigma model applies.
There are concepts that are accepted in our daily life, but are not trivial in physics.One of them is the cluster property that means there exist no relations between two events which are sufficiently separated.In a paper recently published by the author, it has been pointed out that this cluster property violates in the correlation function of the spin operator for the spin 1/2 XXZ antiferromagnet on the square lattice.In this paper, we investigate the spin 1/2 Heisenberg antiferromagnet on the square lattice, which has SU(2) symmetry.In order to study the cluster property, we need to calculate the ground state accurately.For this purpose, we employ the effective model based on the magnetization of the sub-lattices.Then we can define the quasi-degenerate states to calculate the ground state.Including two kinds of interactions which break SU(2) symmetry into the Hamiltonian, we obtain the ground state quantitatively.We find that two kinds of spin correlation functions due to degenerate states are not zero when the lattice size is large but finite.The magnitude of one of them is the same as the one previously found in the XXZ antiferromagnet, while another one is much larger when the additional interaction is strong.We then conclude that in Heisenberg antiferromagnet correlation functions violate the cluster property and the magnitude of the violation qualitatively differs from the one in the XXZ antiferromagnet.
The cluster property is one of fundamental properties in physics.This property means that there are no relations between two events that are sufficiently separated.Because the cluster property is directly connected with entanglement in quantum field theory and in many-body systems, theoretical and experimental progress on entanglement stimulates us to study this property deeply.In this paper we investigate the cluster property in the spin 1/2 XXZ antiferromagnet on the square lattice with an explicitly symmetry breaking interaction of strength g.In this model spontaneous symmetry breaking occurs when the lattice size N is infinitely large.On the other hand, we have to make g zero in order to obtain quantities in the XXZ model with no symmetry breaking interaction.Since some results depend on the sequence of limit operations -N → ∞ and 0 g → , it is difficult to draw a clear conclusion in these limits.Therefore we study the model with finite g on the finite lattice, whose size N is supposed to be 10 20 , for our quantitative calculations.Then we can obtain the concrete ground state.In order to study the cluster property we calculate the spin correlation function.It is known that the function due to Nambu-Goldstone mode (gapless mode), which is calculated using linear spin wave theory, satisfies this property.In this paper we show that almost degenerate states also induce the spin correlation.We assume that the spin correlation function in measurements is a sum of the function due to Nambu-Goldstone mode and one due to these degenerate states.Then we examine whether the additional correlation function violates the cluster property or not.Our conclusion is that this function is finite at any distance, which means the violation of the cluster property, and it is of order of ( )for the case of extremely small g, this violation is the fine effect.Therefore the correlation function due to the degenerate states can be observed only when it is larger than the spin correlation function due to Nambu-Goldstone mode.We show that g required for this condition depends on the distance between positions of two spin operators.
Spontaneous symmetry breaking is well known to be a macroscopic phenomenon in quantum physics in many body systems. The essential features of this phenomenon are the energy degeneracy and the existence of the local operator that can modify the degenerate states. Due to these properties the lowest energy state is not stable. In order to obtain the stable ground state we introduce the explicitly breaking interaction, by which it is difficult for the local operator to modify the ground state. In the superconducting qubit there exists the stable eigenstate with the definite charge number. We point out that in this system, the local operator can change this state to another state. As results the cluster property violates, which means that the events in the far distance are not irrelevant. In this paper we examine the cluster property in the superconducting qubit. Then we propose a method by cavity quantum electrodynamics with two cavities in order to observe the violation.
We study antiferromagnetic systems with enhanced “nearly-SU(N)” symmetry, which can be realized in systems of ultracold spinor atoms in optical lattices. Examples of N = 3 (for S = 1 bosons) and N = 4 (for S=32 fermions) are considered. Near the SU(N) point, the low-energy physics can be described by the CPN−1 model with an additional symmetry-breaking term lowering the symmetry down to SU(2) and favoring the Néel ordering. We show that the effective theory of such systems can be cast in the form of a nonlinear sigma model with the SO(3) matrix-valued field, which is typically obtained for frustrated magnets with non-collinear order. Further, we show that those systems possess a peculiar effect of topological binding: for a system with the underlying spin S, lowering of the symmetry from SU(2S + 1) to SU(2) leads to binding of topological unit-charge excitations of the CP2S model (skyrmions for space dimension d = 2, instantons for d = 1, and hedgehogs for d = 3) into 2S-multiplets.
There are some concepts that are accepted in our daily life but are not trivial in physics.One of them is the cluster property that means there exist no relations between two events which are sufficiently separated.In the works recently published by the author, the extensive and quantitative examination has been made about the violation of cluster property in the correlation function of the spin operator for the quantum spin system.These works have shown that, when we include the symmetry breaking interaction, the effect by the violation is proportional to the inverse of the system size.Therefore this effect is tinny since the system size is quite large.In order to find the effect due to the violation even when the size is large, we propose a new system where additional spins couple with the spin system on the square lattice, where the coupling constant between these systems being assumed to be small.Applying the perturbation theory, we obtain the effective Hamiltonian for the additional system.This Hamiltonian includes Curie-Weiss model that is induced by the violation of the cluster property.Then we find that this effective Hamiltonian has the factor which is the inverse of the system size.Since Curie-Weiss model, which is known to be exactly soluble, has to contain this factor so that the thermodynamical properties are well-defined, the essential factor for the Hamiltonian is determined by the coupling and the strength of the symmetry breaking interaction.Our conclusion is, therefore, that it is possible to observe the effect by the violation of the cluster property at the inverse temperature whose order is given by these parameters.