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Finite-Temperature Scaling of Magnetic Susceptibility in the XY Spin Chain

H. T. Quan

Published Jun 15, 2008• 0 citations
OpenAlex

Abstract

We study the magnetic susceptibility of 1D quantum XY model, and show that when the temperature approaches zero, the magnetic susceptibility (MS) exhibits finite-temperature scaling behavior. This scaling behavior of MS in 1D quantum XY model, due to the quantum-classical map, can be regarded as a witness of macroscopic quantum entanglement, and can be experimentally tested. Furthermore, the universality in the critical properties of MS in quantum XY model is verified. Our study also reveals the close relation between MS and geometric phase in some quantum phase transition systems, which has been studied recently.

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We study the magnetic susceptibility of 1D quantum XY model, and show that when the temperature approaches zero, the magnetic susceptibility (MS) exhibits finite-temperature scaling behavior. This scaling behavior of MS in 1D quantum XY model, due to the quantum-classical map, can be regarded as a witness of macroscopic quantum entanglement, and can be experimentally tested. Furthermore, the universality in the critical properties of MS in quantum XY model is verified. Our study also reveals the close relation between MS and geometric phase in some quantum phase transition systems, which has been studied recently.
We study the magnetic susceptibility of 1D quantum XY model, and show that when the temperature approaches zero, the magnetic susceptibility exhibits the finite-temperature scaling behavior. This scaling behavior of the magnetic susceptibility in 1D quantum XY model, due to the quantum-classical mapping, can be easily experimentally tested. Furthermore, the universality in the critical properties of the magnetic susceptibility in quantum XY model is verified. Our study also reveals the close relation between the magnetic susceptibility and the geometric phase in some spin systems, where the quantum phase transitions are driven by an external magnetic field.
We show that the geometric phase of the ground state in the XY model obeys scaling behavior in the vicinity of a quantum phase transition. In particular we find that the geometric phase is nonanalytical and its derivative with respect to the field strength diverges at the critical magnetic field. Furthermore, the universality in the critical properties of the geometric phase in a family of models is verified. In addition, since the quantum phase transition occurs at a level crossing or avoided level crossing and these level structures can be captured by the Berry curvature, the established relation between the geometric phase and quantum phase transitions is not a specific property of the XY model, but a very general result of many-body systems.
A measure of how sensitive the entanglement entropy is in a quantum system, has been proposed and its information geometric origin is discussed. It has been demonstrated for two exactly solvable spin systems, that thermodynamic criticality is directly \textit{indicated} by finite size scaling of the global maxima and turning points of the susceptibility of entanglement entropy through numerical analysis - obtaining power laws. Analytically we have proved those power laws for ∣ λc(N)−λc∞∣| \ λ_c(N)-λ_c^{\infty}|∣ λc​(N)−λc∞​∣ as N→∞N\to \inftyN→∞ in the cases of finite 1D transverse field ising model (TFIM) (λ=hλ=hλ=h) and XY chain (λ=γλ=γλ=γ). The integer power law appearing for XY model has been verified using perturbation theory in O(1N)\mathcal{O}(\frac{1}{N})O(N1​) and the fractional power law appearing in the case of TFIM, is verified by an exact approach involving Chebyshev polynomials, hypergeometric functions and complete elliptic integrals. Furthermore a set of potential applications of this quantity under quantum dynamics and also for non-integrable systems, are briefly discussed. The simplicity of this setup for understanding quantum criticality is emphasized as it takes in only the reduced density matrix of appropriate rank.
Abstract We investigate the quantum coherence of an XY spin chain with Dzyaloshinskii–Moriya interaction using a quantum renormalization group-based method, and show that the relative entropy of coherence increases with Dzyaloshinskii–Moriya interaction strength at zero temperature, as well as at finite temperature. We also study the quantum criticality of the system by making use of the relative entropy of coherence and violation of Bell inequality, and find that the first derivatives of the renormalized quantum coherence exhibit singularity near the critical point of the quantum phase transition. Finally, we explore the finite-size scaling behaviors of the derivatives of the quantum coherence at the critical point of the quantum phase transition, and obtain several universal finite-size scaling laws.
We study the macroscopic entanglement properties of a low dimensional quantum spin system by investigating its magnetic properties at low temperatures and high magnetic fields. The tempera- ture and magnetic field dependence of entanglement from the susceptibility and magnetization data comparing the experimental extraction with theoretical estimates are given. Extraction of entan- glement has been made possible through the macroscopic witness operators magnetic susceptibility. The protocol followed in doing so has been outlined in some recent work. Various plots of entan- glement comparing the experimental extraction with theoretical estimates are given. Quantitative comparison between concurrence and entanglement witness is given for both the theoretical and experimental results. Theory and experiments match over a wide range of temperature and field. The spin system studied is a chain, which exhibits dimerisation and yields fascinating entanglement properties when the temperature and magnetic field is varied. These spin systems exhibit quantum phase transition (QPT) at low temperatures, when the magnetic field is swept through a critical value. We show explicitly for the first time, using tools used in quantum information processing (QIP), that quantum phase transition (QPT) can be captured experimentally using canonically conjugate observables. Macroscopically, quantum complementarity relation clearly delineates entangled states from separable states across the QPT. We have estimated the partial information sharing in this system from our magnetization and susceptibility data. The complementarity relation has been experimentally verified to hold in this system
We use finite-size methods to estimate the critical properties of s=1 xy chains with uniaxial single-ion anisotropy of strength D. Strong evidence is presented for a phase transition, belonging to the universality class of the 1D quantum 0(2) model, with the weak coupling phase gapless, an essential zero in the gap at the critical coupling D*, and critical exponents z(D*)≊1, η(D*)≊1/4. Like the quantum 0(2) system, one of the models turns out to be a 1D quantum analog of the classical and isotropic 2D xy model.