Session B13: Focus Session: Low-Dimensional and Molecular Magnetism - 1D Magnetism/single-chain magnets - Theory

Ferromagnetic versus helical order in edge sharing CuO$_2$ chains - a computational study

The magnetic ground state of edge sharing CuO$_2$ spin 1/2 Heisenberg chains with nearest neighbor exchange $J_1$ and second neighbor exchange $J_2$ depends delicately on structural details of the crystal structure, like Cu-O-Cu bond angles, Cu-O distances and the position of the cations. Without taking into account a renormalization by the interchain coupling, a critical ratio $\alpha=-J_2/J_1$ separates a ferromagnetic from a helical ground state (FM for $\alpha < 1/4$, helical for $\alpha > 1/4$). Here, we present a density functional based band structure study that investigates the different influences of various structural parameters for Li$_2$ CuO$_2$ as example compound. We find that the ferromagnetic and antiferromagnetic contributions develop rather differently for the same structural changes. Therefore, the key parameter $\alpha$ for the ground state is especially sensitive for small structural changes that might be induced by temperature or pressure variation.   

Quantum criticality of dipolar spin chains

We show that a one-dimensional chain of Heisenberg spins, interacting with long-range dipolar forces in a magnetic field perpendicular to the chain, exhibits a quantum critical point belonging to the two-dimensional Ising universality class. Within linear spin-wave theory (corresponding to the so-called Gaussian approximation) the long-wavelength magnon dispersion is characterized by a logarithmic singularity in the magnon velocity for vanishing momenta, due to the long range nature of dipolar interactions in one-dimension. However, in the vicinity of the critical point this logarithmic correction is renormalized to zero by the effects of quantum fluctuations, signaling the reemergence of scale invariance, in accordance with the Ising critical scenario. The quantum critical regime where linear spin-wave theory breaks down is studied using two independent non-perturbative methods, namely the density-matrix renormalization group (DMRG) and the functional renormalization group (FRG). The Ginzburg regime where non-Gaussian fluctuations are important is found to be rather narrow on the ordered side of the transition, and very broad on the disordered side.   

Spin transport in spin chains: Possible applications for spintronics

One of the main issues in modern electronics is that as devices get ever smaller the removal of waste energy generated by Joule heating becomes problematic. A possible solution to this problem is offered by spintronics in non-itinerant systems. Here, we theoretically propose the spin-equivalent in such systems of several different components that are used in traditional electronics: the resistance, the diode, and the capacitance. The system we consider here consists of an antiferromagnetic spin chain with anisotropic exchange interaction, connected to two three-dimensional spin reservoirs. We use inhomogeneous Luttinger liquid theory to describe the system, and non-equilibrium methods to calculate the relevant transport properties. \begin{thebibliography}{100} \bibitem{1} K. A. van Hoogdalem and D. Loss, Phys. Rev. B {\bf 84}, 024402 (2011). \bibitem{2} K. A. van Hoogdalem and D. Loss, in preparation. \end{thebibliography}   

Fractionalization of electron's spin and orbital degrees of freedom in 1D

We show that electron's spin and orbital degrees of freedom can fractionalize in 1D antiferromagnets: although the orbital excitations are inherently coupled to spinons in antiferromagnetic Mott insulators, in 1D they separate into a {\it pure} orbiton and a single spinon. This is similar to the spin-charge separation in 1D but corresponds to an exotic regime where spinons are faster than holons [1]. The resulting large dispersion of the pure orbiton can be detected in e.g. quasi-1D cuprates [2]. [1] K. Wohlfeld, M. Daghofer, S. Nishimoto, G. Khaliullin, and J. van den Brink, Phys. Rev. Lett. {\bf 107}, 147201 (2011). [2] J. Schlappa {\it et al.}, to be published (2011).   

Accurate determination of the Gaussian transition in spin-1 chains with single-ion anisotropy

The Gaussian transition in the spin-one Heisenberg chain with single-ion anisotropy is extremely difficult to treat, both analytically and numerically. We introduce an improved DMRG procedure with strict error control, which we use to access very large systems. By considering the bulk entropy, we determine the Gaussian transition point to 4-digit accuracy, $D_{c}/J = 0.96845(8)$, resolving a long-standing debate in quantum magnetism. With this value, we obtain high-precision data for the critical behavior of quantities including the ground-state energy, gap, and transverse string-order parameter, and for the critical exponent, $\nu = 1.472(2)$. Applying our improved technique at $J_{z} = 0.5$ highlights essential differences in critical behavior along the Gaussian transition line.   

Generalized Fidelity Susceptibilities as Applied to the $J_1-J_2$ Heisenberg Chain

In this talk slightly generalized quantum fidelity susceptibilities for the antiferromagnetic Heisenberg $J_1-J_2$ chain will be introduced. The differential change in these fidelities differ from the typical fidelity in that they are measured with respect to a term other than the one used for driving the system towards a quantum phase transition. We study three fidelity susceptibilities; $\chi_{\rho}$, $\chi_D$ and $\chi_{AF}$, which are related to the spin stiffness, the dimer order and antiferromagnetic order, respectively. I will discuss how these quantities can accurately identify the quantum critical point at $J_2$=0.241167$J_1$ in this model. This phase transition, being in the Berezinskii-Kosterlitz-Thouless universality class, is controlled by a marginal operator and is therefore particularly difficult to observe. In addition more recent work on the anisotropic Heisenberg triangular model will be discussed.   

Excitation structure of frustrated spin chains with dimerization and the description by the effective field theory

Heisenberg antiferromagnetic chain with alternating exchange interaction is an important model, which describes magnetic properties of real materials. Field theoretical approach is a powerful tool to investigate such kind of one-dimensional quantum magnets, and it is known that this lattice model is related with corresponding sine-Gordon effective field theory through the bosonization technique. We investigate the excitation spectrum and the correspondence between $S=\frac{1}{2}$ and 1 frustrated chain with dimerization and their effective field theories by both analytical and numerical methods, focusing on the mass ratio $r$ of second breather to soliton. In the result, the $S=\frac{1}{2}$ and 1 cases are understood in a unified way. $r$ becomes $\sqrt{3}$, the value predicted from sine-Gordon model by the introduction of next-nearest neighbor coupling $J_2=J_{2{\rm c}}$ where the marginal term in effective field theory vanishes. The universality class of transition is Tomonaga-Luttinger liquid and first order for $J_2 [Preview Abstract]

Statistically interacting particles with shapes

Ising spin $s=\frac{1}{2},1,\frac{3}{2}$ chains with nearest and next-nearest neighbor coupling are interpreted as systems of floating particles. The particles are classified into species according to structure and into categories according to function. Species are distinguished by motifs consisting of several consecutive spins that interlink by sharing one or two sites. The four categories include compacts, hosts, tags, and hybrids. All particles from one set are excited from a selected Ising product state serving as pseudo-vacuum. Compacts and hosts float in segments of pseudo-vacuum. Tags are located inside hosts. Hybrids are tags with hosting capability. All particles are free of binding energies but subject to a generalized Pauli principle. In the Ising context, all particle energies are functions of the Hamiltonian parameters. However, the exact statistical mechanical analysis can be performed for particles with arbitrary energies. The entropy is a function of the particle populations from each species. Applications to jamming of granular matter in narrow channels and to DNA overstretching are in the works.   

Fisher zeros and correlation functions in Ising models

Phase transitions take place at singular points of the free energy. These correspond to zeros of the partition function when one tuning parameter is extended into the complex plane (so called Lee-Yang or Fisher zeros). It has been known since the 1960s that transition temperatures and critical exponents can be calculated from the distributions of these partition function zeros. We use this technique to calculate the spin-spin correlation function for the 1d Ising model and notice that it forms a spiral with a wavevector dependent on the position of the complex temperature on the contour of zeros. To extend this we will look at results from the 2d Ising model as well as the Ising model in a transverse field.   

Integrability in anyonic quantum spin chains via a composite height model

Recently, properties of collective states of interacting non-abelian anyons have attracted a considerable attention. We study an extension of the `golden chain model', a model of interacting Fibonacci anyons, where two- and three-body interactions are competing. Upon fine-tuning the interaction, the model is integrable. This provides an additional integrable point of the model, on top of the integrable point, when the three-body interaction is absent. To solve the model, we construct a new, integrable height model, in the spirit of the restricted solid-on-solid model solved by Andrews, Baxter and Forrester. The model is solved by means of the corner transfer matrix method. We find a connection between local height probabilities and characters of a conformal field theory governing the critical properties at the integrable point. In the anitferromagnetic regime, the criticality is described by the $Z_{k}$ parafermion conformal field theory, while the $su(2)_{1} \times su(2)_{1} \times su(2)_{k-2}/su(2)_{k}$ coset conformal field theory describes the ferromagnetic regime.   

Out of equilibrium energy dynamics in low dimensional quantum magnets

We investigate the real-time dynamics of the energy density in spin-1/2 $XXZ$ chains using two types of quenches resulting in initial states which feature an inhomogeneous distribution of local energies [1]. The first involves quenching bonds in the center of the chain from antiferromagnetic to ferromagnetic exchange interactions. The second quench involves an inhomogeneous magnetic field, inducing both, an inhomogeneous magnetization profile [2] and local energy density. The simulations are carried out using the adaptive time-dependent density matrix renormalization group algorithm. We analyze the time-dependence of the spatial variance of the bond energies and the local energy currents which both yield necessary criteria for ballistic or diffusive energy dynamics. For both setups, our results are consistent with ballistic behavior, both in the massless and the massive phase. For the massless regime, we compare our numerical results to bosonization and the non-interacting limit finding very good agreement. The velocity of the energy wave-packets can be understood as the average velocity of excitations induced by the quench. \\[4pt] [1] Langer et al. Phys. Rev. B in press; arXiv:1107.4136\\[0pt] [2] Langer et al. Phys. Rev. B 79, 214409 (2009)   

Dissipative phases in the one-dimensional Kondo-Heisenberg model

Atomic-sized magnetic structures built on clean metallic surfaces are currently under intense investigation {[}1{]}. Besides their potential uses in quantum information storage and processing, these systems allow to ask fundamental questions in condensed matter physics. In particular, the interplay between the Kondo effect (i.e., the screening of the atomic magnetic moment by conduction electrons) and Heisenberg exchange interactions between magnetic impurities has been recently investigated with scanning tunneling microscopy (STM) {[}2{]}. Inspired by the above developments, we study an one-dimensional chain of S=1/2 Kondo impurities coupled by anisotropic Heisenberg-Ising exchange and embedded in a two-dimensional metallic substrate. Remarkably, in the case of easy-plane exchange, we find a novel quantum phase exhibiting long-range order at zero temperature. We discuss implications of the existence of this phase for possible experiments. References: {[}1{]} R. Wiesendanger, RMP 81, 1495 (2009). {[}2{]} P. Wahl et al, PRL 98, 056601 (2007) and references therein.   

String solutions and Mott physics in quasi-one-dimensional antiferromagnets

Mott insulators are caused by repulsive interactions between electrons, whereas band insulators are due to full filling of a single-particle band. In the one-dimension (1D) Hubbard model, the upper Hubbard band (UHB) has been identified with $k$-$\Lambda$ string solutions [1]. Similarly, in the 1D spin-1/2 antiferromagnetic Heisenberg model, the high-energy magnetic excitations in a magnetic field have been identified primarily with 2-string solutions [2]. We can intuitively understand the correspondence between the high-energy states and the UHB, by mapping the Heisenberg model to the hard-core boson model with repulsive interactions. Furthermore, noting that the high-energy states persist in anisotropic triangular antiferromagnets [3], we can interpret the high-energy magnetic excitations observed in quasi-1D antiferromagnets such as Cs$_2$CuCl$_4$ and CuCl$_2$$\cdot$2N(C$_5$D$_5$) in a magnetic field in the context of the Mott physics: the high-energy magnetic excitations, whose origin can be traced back to the string solutions, are due to repulsive interactions between hard-core bosons (down-spins) mapped from the Heisenberg model. [1] M.K., Phys. Rev. Lett. 105, 106402 (2010). [2] M.K., Phys. Rev. Lett. 102, 037203 (2009). [3] M.K., Phys. Rev. Lett. 103, 197203 (2009).   

A family of spin-S chain representations of SU(2) level k Wess-Zumino-Witten models

We investigate a family of spin-$S$ chain Hamiltonians recently introduced by one of us [M. Greiter, Mapping of Parent Hamiltonians, Springer Tracts in Modern Phyiscs, Vol 244 (Springer, Berlin, 2011)]. For $S=1/2$, it corresponds to the Haldane--Shastry model. For general spin $S$, we numerically show that the low--energy theory of these spin chains is described by the SU(2)$_{k}$ Wess--Zumino--Witten model with coupling $k=2S$. In particular, we investigate the $S=1$ model whose ground state is given by a Pfaffian for even number of sites $N$. We reconcile aspects of the spectrum of the Hamiltonian for arbitrary $N$ with trial states obtained by Schwinger projection of two Haldane--Shastry chains.   

Inhomogeneous Spin Chains and Luttinger Liquids

We consider a one-dimensional spin chain with inhomogeneous coupling, which can also be modeled as an inhomogeneous Luttinger liquid. The Luttinger liquid paradigm has proved a very successful theoretical tool for investigating one-dimensional wires. However, there remain open questions about what happens when such a system becomes inhomogeneous. The mapping between the spin chain and the Luttinger liquid allows us to use both numerics and field theory to analyze the problem. Of particular interest is the case where the Luttinger liquid is attached to external leads, as is necessary for example when measuring the conductance of the wire. In this paper we use an abrupt shift in the parameters of the Luttinger liquid to model these connections and see how this affects its behavior. In particular we analyze the relevant back-scattering perturbations at the connections, and identify a case where this relevant operator can be tuned to zero within an otherwise still inhomogeneous system. This of course has consequences not only for transport in the Luttinger liquid system but also for the magnetic susceptibility of the spin chain.