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Entanglement in first excited states of some many-body quantum spin systems: indication of quantum phase transition

George Biswas, Anindya Biswas

Published Sep 27, 2019• 0 citations
OpenAlex

Abstract

We compute concurrence, a measure of bipartite entanglement, of the first excited state of the $1$-D Heisenberg frustrated $J_1$-$J_2$ spin-chain and observe a sudden change in the entanglement of the eigen state near the coupling strength $\alpha=J_2/J_1\approx0.241$, where a quantum phase transition from spin-fluid phase to dimer phase has been previously reported. We numerically observe this phenomena for spin-chain with $8$ sites to $16$ sites, and the value of $\alpha$ at which the change in entanglement is observed asymptotically tends to a value $\alpha_c\approx0.24116$. We have calculated the finite-size scaling exponents for spin chains with even and odd spins. It may be noted that bipartite as well as multipartite entanglement measures applied on the ground state of the system, fail to detect any quantum phase transition from the gapless to the gapped phase in the $1$-D Heisenberg frustrated $J_1$-$J_2$ spin-chain. Furthermore, we measure bipartite entanglement of first excited states for other spin models like $2$-D Heisenberg $J_1$-$J_2$ model and Shastry-Sutherland model and detect quantum phase transitions.

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We compute concurrence, a measure of bipartite entanglement, of the first excited state of the 111-D Heisenberg frustrated J1J_1J1​-J2J_2J2​ spin-chain and observe a sudden change in the entanglement of the eigen state near the coupling strength α=J2/J1≈0.241\alpha=J_2/J_1\approx0.241α=J2​/J1​≈0.241, where a quantum phase transition from spin-fluid phase to dimer phase has been previously reported. We numerically observe this phenomena for spin-chain with 888 sites to 161616 sites, and the value of α\alphaα at which the change in entanglement is observed asymptotically tends to a value αc≈0.24116\alpha_c\approx0.24116αc​≈0.24116. We have calculated the finite-size scaling exponents for spin chains with even and odd spins. It may be noted that bipartite as well as multipartite entanglement measures applied on the ground state of the system, fail to detect any quantum phase transition from the gapless to the gapped phase in the 111-D Heisenberg frustrated J1J_1J1​-J2J_2J2​ spin-chain. Furthermore, we measure bipartite entanglement of first excited states for other spin models like 222-D Heisenberg J1J_1J1​-J2J_2J2​ model and Shastry-Sutherland model and detect quantum phase transitions.
Abstract We compute concurrence, a measure of bipartite entanglement, of the first excited state of the 1-D Heisenberg frustrated J 1 - J 2 spin-chain and observe a sudden change in the entanglement of the eigen state near the coupling strength α = J 2 / J 1 ≈ 0.241, where a quantum phase transition from spin-fluid phase to dimer phase has been previously reported. We numerically observe this phenomena for spin-chain with 8 sites to 16 sites, and the value of α at which the change in entanglement is observed, asymptotically tends to a value α c ≈ 0.24116. We have calculated the finite-size scaling exponents for spin chains with even and odd spins. It may be noted that bipartite as well as multipartite entanglement measures applied on the ground state of the system, fail to detect any quantum phase transition from the gapless to the gapped phase in the 1-D Heisenberg frustrated J 1 - J 2 spin-chain. Furthermore, we measure bipartite entanglement of first excited states for other spin models like 2-D Heisenberg J 1 - J 2 model and Shastry-Sutherland model and find similar indications of quantum phase transitions.
We study here different regions in phase diagrams of the spin-1/2, spin-1 and spin-3/2 one-dimensional antiferromagnetic Heisenberg systems with frustration (next-nearest-neighbor interaction J2) and dimerization (δ). In particular, we analyze the behaviors of the bipartite entanglement entropy and fidelity at the gapless to gapped phase transitions and across the lines separating different phases in the J2-δ plane. All the calculations in this work are based on numerical exact diagonalizations of finite systems.
Abstract We show that the nearest neighbour entanglement in a mixture of ground and first excited states—a subjacent state—of the J 1 − J 2 Heisenberg quantum spin chain can be used as an order parameter to detect the phase transition of the chain from a gapless spin fluid to a gapped dimer phase. We study the effectiveness of the order parameter for varying relative mixing probabilities between the ground and first excited states in the subjacent state for different system sizes, and extrapolate the results to the thermodynamic limit. We observe that the nearest neighbour concurrence can play a role of a good order parameter even if the system is in the ground state, but with a small finite probability of leaking into the first excited state. Moreover, we apply the order parameter of the subjacent state to investigate the response to separate introductions of anisotropy and of glassy disorder on the phase diagram of the model, and analyse the corresponding finite-size scale exponents and the emergent tricritical point in the former case. The anisotropic J 1 − J 2 chain has a richer phase diagram which is also clearly visible by using the same order parameter.
We investigate the behavior of genuine multiparticle entanglement, as quantified by the generalized geometric measure, in gapless-to-gapped quantum transitions of one- and two-dimensional quantum spin models. The investigations are performed in the exactly solvable one-dimensional XYXYXY models, as well as two-dimensional frustrated J1−−J2{J}_{1}--{J}_{2}J1​−−J2​ models, including the Shastry-Sutherland model. The generalized geometric measure shows nonmonotonic features near such transitions in the frustrated quantum systems. We also compare the features of the generalized geometric measure near the quantum critical points with the same for measures of bipartite quantum correlations. The multipartite quantum correlation measure turns out to be a better indicator of quantum critical points than the bipartite measures, especially for two-dimensional models.
We study Heisenberg spin-1/2 and spin-1 chains with alternating ferromagnetic (J1FJ_1^FJ1F​) and antiferromagnetic (J1AJ_1^AJ1A​) nearest-neighbor interactions and ferromagnetic next-nearest-neighbor interaction (J2FJ_2^FJ2F​). In this model frustration is induced by the non-zero J2FJ_2^FJ2F​ and the effect of dimerization is realized by changing the strength of J1AJ_1^AJ1A​ for a fixed value of J1FJ_1^FJ1F​. The model with site spin sss behaves like a Haldane spin chain with site spin 2sss in the limit of vanishing J2FJ_2^FJ2F​ and large J1F/J1AJ_1^F/J_1^AJ1F​/J1A​. We show that the exact ground state of the model can be found along a line in the parameter space. The phase diagram in the space of J1A−J2FJ_1^A-J_2^FJ1A​−J2F​ is determined using numerical techniques complemented by some analytical calculations. A number of quantities, including the ground state spin, energy gap, structure factor, entanglement entropy, and zero temperature magnetization, are studied to understand the complete phase diagram. An interesting and application-wise important feature of this model is that it exhibits a macroscopic magnetization jump in the presence of a magnetic field; we study this using an effective Hamiltonian.
We extend the definition of the recently introduced valence-bond entanglement entropy to arbitrary SU(2) wave functions of S=1/2S=1/2S=1/2 spin systems. Thanks to a reformulation of this entanglement measure in terms of a projection, we are able to compute it with various numerical techniques for frustrated spin models. We provide extensive numerical data for the one-dimensional ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ spin chain where we are able to locate the quantum phase transition by using the scaling of this entropy with the block size. We also systematically compare with the scaling of the von Neumann entanglement entropy. We finally underline that the valence-bond entropy definition does depend on the choice of bipartition so that, for frustrated models, a ``good'' bipartition should be chosen, for instance, according to the Marshall sign.