This thesis is strongly based on Refs. [28–30], but it also contains a broader discus-sion about some technical aspects and methods. It is organized as follows:
• In chapter 2 we introduce the three spin-1/2 models that will be investigated:
the XXZ chain, the isotropically dimerized chain and the two-leg ladder. Their
1.1 Outline of the thesis
corresponding Hamiltonians and their phase diagrams are discussed. In the case of the ladder we show how the model can be mapped onto a purely 1D chain in some specific regimes, and other subsequent approximations which allow a direct analytical solution for the quantities we are interested in. Our results for the dimerized chain and for the ladder are computed with explicit reference to existing compounds, which are presented in the last part of the chapter.
• In chapter 3 we provide a description of two theoretical techniques, one nu-merical and one analytical, which are particularly suited for the treatment of low dimensional physics: DMRG (with its MPS reformulation) and the TLL theory. Starting from the generic MPS representation of a quantum state, we discuss how the search for a ground state of a given Hamiltonian is performed and how dynamics and finite temperature can be introduced in this framework.
We then combine these two ingredients to tackle the computation of dynam-ical correlation functions at finite temperature. In the second part we present the TLL theory focusing on its applications to the case of the spin chain, and we propose a new method to determine numerically the TLL parameters.
• Chapter 4 is dedicated to the numerical computation of the NMR spin-lattice relaxation rate (often called1/T1) for the chain models. After having defined the quantity and discussed its connection with spin-spin correlation functions, we show how MPS results for finite temperature dynamics can offer very good description of it, also in regimes where analytical treatments are particularly difficult.
• In chapter 5 we present the MPS results for dynamical correlation functions at finite temperature of a dimerized spin-1/2 chain at different magnetic fields and temperatures. We discuss the main features of the spectra through differ-ent quantum phases, and try to interpret some of these structures in the strong dimerization limit or using the TLL theory. We also investigate how the tem-perature acts on the spectra and in particular on the dispersion of a single triplet excitation.
• In chapter 6 we deal with the computation of the so called Grüneisen param-eter for the ladder model. After having defined this quantity and highlighted its importance for the detection and the investigation of quantum phase transi-tions, we compute it analytically close to criticality using the strong coupling expansion and an effective free fermion theory. Our results are compared to experimental measurements and DMRG simulations.
• In chapter 7 we give a general overview on our results and discuss further perspectives.
1. INTRODUCTION
• In appendix A we give additional technical details about some calculations and approximations performed in chapter 4.
• In appendix B we present the extrapolation method adopted in chapter 4 and how we associate an errobar to the extrapolated results.
• In appendix C we benchmark our MPS results for dynamical correlations at finite temperature against exact results available for the XX model.
• In appendix D we describe, in the case of the dimerized chain in the gapped phases and in the strong dimerization limit, how some computed spectra can be interpreted by simply considering the dynamics of a single excitation.
CHAPTER 2
Spin-1/2 Chains and Ladders
In this chapter we present in detail the three spin-1/2 models which we will study in this work: we start from the XXZ chain, then we move to the isotropically dimerized chain, and finally we consider the two-leg ladder model. For all of them we remind briefly their main physical features and for the last one we introduce the spin chain mapping and the boson representation, which provides a simpler interpretation of the physics of the system in specific limits. Finally we present more in detail some interesting experimental realizations of the dimerized chain and of the two-leg ladder to which our calculations refer directly.
2.1 Homogeneous spin-1/2 chains
We start by considering the very well known XXZ Hamiltonian. The chain is sub-jected to a magnetic fieldhapplied along thezdirection. The Hamiltonian is given by
H=JX
j
1
2 Sj+Sj+1− +h.c.
+ ∆SjzSj+1z
−hX
j
Sjz, (2.1) whereSjα = 12σjαis a spin operator for a spin1/2on sitej,α = x, y, zdenotes its direction, andσαj the Pauli matrices. Sj± = Sxj ±iSjy are the spin rising and lowering operators. The parameterJ gives the spin coupling strength,∆is dimen-sionless and measures the anisotropy. Thegfactor, the Bohr magneton and~have been absorbed intohandJ, which both have here the dimensions of an energy. For
2. SPIN-1/2 CHAINS AND LADDERS
Figure 2.1: Schematic phase diagram at zero temperature of the XXZ model as a func-tion of the magnetic fieldhand of the anisotropy∆. The phase denoted with ’Ferro’
is theferromagnetic one in which spins are polarized along thez direction. XY de-notes a massless phase characterized by dominant in-plane antiferromagnetic correla-tions which is a TLL [7]. ’Néel’ denotes an antiferromagnetically ordered Ising phase alongz. Dashed lines separate the different phases. The behavior of the boundary as a function ofharound the point∆ = 1reflects the Berezinski-Kosterlitz-Thouless (BKT) behavior of the gap at the transition. The figure is taken from Ref. [29] and inspired by Fig. 1.5 in [2].
the isotropic case∆ = 1, the model corresponds to the Heisenberg (or XXX) Hamil-tonian, isotropic in the three directions. For∆ = 0we have the XX model which can be mapped via a Jordan-Wigner transformation [7] onto a free-fermion model with a fixed chemical potential and possesses exact solution. The phase diagram at zero temperature of the XXZ model as a function of the magnetic field and of the anisotropy is given in Fig. 2.1. The boundary between the XY and ferromagnetic phases is described by the relationh=J(1 + ∆). The boundary between the XY and Néel phases is given by the triplet gap, which is a function of∆[2]. In this work we will limit ourselves to the case0≤∆≤1.