Quantum Spin Chains And Luttinger Liquids With Junctions : Analytical And Numerical Studies
We present in this thesis a series of studies on the physical properties of some one dimensional systems. In particular we study the low energy properties of various spin chains and a junction of Luttinger wires. For spin chains we speciﬁcally look at the role of perturbations like frustrating interactions and dimerisation in a nearest neighbour chain and the formation of magnetisation plateaus in two kinds of models; one purely theoretical and the other motivated by experiments. In our second subject of interest we study using a renormalisation group analysis the eﬀect of spin dependent scattering at a junction of Luttinger wires. We look at the physical eﬀects caused by the interplay of electronic interactions in the wires and the scattering processes at the junction. The thesis begins with an introductory chapter which gives a brief glimpse of the ideas and techniques used in the speciﬁc problems that we have worked on. Our work on these problems is then described in detail in chapters 25. We now present a brief summary of each of those chapters. In the second chapter we look at the ground state phase diagram of the mixed-spin sawtooth chain, i.e a system where the spins along the baseline are allowed to be diﬀerent from the spins on the vertices. The spins S1 along the baseline interact with a coupling strength J1(> 0). The coupling of the spins on the vertex (S2) to the baseline spins has a strength J2. We study the phase diagram as a function of J2/J1 . The model exhibits a rich variety of phases which we study using spinwave theory, exact diagonalisation and a semi-numerical perturbation theory leading to an eﬀective Hamiltonian. The spinwave theory predicts a transition from a spiral state to a ferrimagnetic state at J2S2/2J1S1 = 1 as J2/J1 is increased. The spectrum has two branches one of which is gapless and dispersionless (at the linear order) in the spiral phase. This arises because of the inﬁnite degeneracy of classical ground states in that phase. Numerically, we study the system using exact diagonalisation of up to 12 unit cells and S1 = 1 and S2 =1/2. We look at the variation of ground state energy, gap to the lowest excitations, and the relevant spin correlation functions in the model. This unearths a richer phase diagram than the spinwave calculation. Apart from revealing a possibility of the presence of more than one kind of spiral phases, numerical results tell us about a very interesting phase for small J2. The spin correlation function (for the spin1/2s) in this region have a property that the nextnearest-neighbour correlations are much larger than the nearest neighbour correlations. We call this phase the NNNAFM (nextnearest neighbour antiferromagnet) phase and provide an understanding of this phase by deriving an eﬀective Hamiltonian between the spin1/2s. We also show the existence of macroscopic magnetisation jumps in the model when one looks at the system close to saturation ﬁelds. The third chapter is concerned with the formation of magnetisation plateaus in two diﬀerent spin models. We show how in one model the plateaus arise because of the competition between two coupling constants, and in the other because of purely geometrical eﬀects. In the ﬁrst problem we propose  a class of spin Hamiltonians which include as special cases several known systems. The class of models is deﬁned on a bipartite lattice in arbitrary dimensions and for any spin. The simplest manifestation of such models in one dimension corresponds to a ladder system with diagonal couplings (which are of the same strength as the leg couplings). The physical properties of the model are determined by the combined eﬀects of the competition between the ”rung” coupling (J’ )and the ”leg/diagonal” coupling (J ) and the magnetic ﬁeld. We show that our model can be solved exactly in a substantial region of the parameter space (J’ > 2J ) and we demonstrate the existence of magnetisation plateaus in the solvable regime. Also, by making reasonable assumptions about the spectrum in the region where we cannot solve the model exactly, we prove the existence of ﬁrst order phase transitions on a plateau where the sublattice magnetisations change abruptly. We numerically investigate the ladder system mentioned above (for spin1) to conﬁrm all our analytical predictions and present a phase diagram in the J’/J - B plane, quite a few of whose features we expect to be generically valid for all higher spins. In the second problem concerning plateaus (also discussed in chapter 3) we study the properties of a compound synthesised experimentally . The essential feature of the structure of this compound which gives rise to its physical properties is the presence of two kinds of spin1/2 objects alternating with each other on a helix. One kind has an axis of anisotropy at an inclination to the helical axis (which essentially makes it an Ising spin) whereas the other is an isotropic spin1/2 object. These two spin1/2 objects interact with each other but not with their own kind. Experimentally, it was observed that in a magnetic ﬁeld this material exhibits magnetisation plateaus one of which is at 1/3rd of the saturation magnetisation value. These plateaus appear when the ﬁeld is along the direction of the helical axis but disappear when the ﬁeld is perpendicular to that axis. The model being used for the material prior to our work could not explain the existence of these plateaus. In our work we propose a simple modiﬁcation in the model Hamiltonian which is able to qualitatively explain the presence of the plateaus. We show that the existence of the plateaus can be explained using a periodic variation of the angles of inclination of the easy axes of the anisotropic spins. The experimental temperature and the ﬁelds are much lower than the magnetic coupling strength. Because of this quite a lot of the properties of the system can be studied analytically using transfer matrix methods for an eﬀective theory involving only the anisotropic spins. Apart from the plateaus we study using this modiﬁed model other physical quantities like the speciﬁc heat, susceptibility and the entropy. We demonstrate the existence of ﬁnite entropy per spin at low temperatures for some values of the magnetic ﬁeld. In chapter 4 we investigate the longstanding problem of locating the gapless points of a dimerised spin chain as the strength of dimerisation is varied. It is known that generalising Haldane’s ﬁeld theoretic analysis to dimerised spin chains correctly predicts the number of the gapless points but not the exact locations (which have determined numerically for a few low values of spins). We investigate the problem of locating those points using a dimerised spin chain Hamiltonian with a ”twisted” boundary condition . For a periodic chain, this ”twist” consists simply of a local rotation about the zaxis which renders the xx and yy terms on one bond negative. Such a boundary condition has been used earlier for numerical work whereby one can ﬁnd the gapless points by studying the crossing points of ground states of ﬁnite chains (with the above twist) in diﬀerent parity sectors (parity sectors are deﬁned by the reﬂection symmetry about the twisted bond). We study the twisted Hamiltonian using two analytical methods. The modiﬁed boundary condition reduces the degeneracy of classical ground states of the chain and we get only two N´eel states as classical ground states. We use this property to identify the gapless points as points where the tunneling amplitude between these two ground states goes to zero. While one of our calculations just reproduces the results of previous ﬁeld theoretic treatments, our second analytical treatment gives a direct expression for the gapless points as roots of a polynomial equation in the dimerisation parameter. This approach is found to be more accurate. We compare the two methods with the numerical method mentioned above and present results for various spin values. In the ﬁnal chapter we present a study of the physics of a junction of Luttinger wires (quantum wires) with both scalar and spin scattering at the junction (,). Earlier studies have investigated special cases of this system. The systems studied were two wire junctions with either a fully transmitting scattering matrix or one corresponding to disconnected wires. We extend the study to a junction of N wires with an arbitrary scattering matrix and a spin impurity at the junction. We study the RG ﬂows of the Kondo coupling of the impurity spin to the electrons treating the electronic interactions and the Kondo coupling perturbatively. We analyse the various ﬁxed points for the speciﬁc case of three wires. We ﬁnd a general tendency to ﬂow towards strong coupling when all the matrix elements of the Kondo coupling are positive at small length scales. We analyse one of the strong coupling ﬁxed points, namely that of the maximally transmitting scattering matrix, using a 1/J perturbation theory and we ﬁnd at large length scales a ﬁxed point of disconnected wires with a vanishing Kondo coupling. In this way we obtain a picture of the RG at both short and long length scales. Also, we analyse all the ﬁxed points using lattice models to gain an understanding of the RG ﬂows in terms of speciﬁc couplings on the lattice. Finally, we use to bosonisation to study one particular case of scattering (the disconnected wires) in the presence of strong interactions and ﬁnd that suﬃciently strong interactions can stabilise a multichannel ﬁxed point which is unstable in the weak interaction limit.