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Thesis

Reference

Numerical study of thermal effects in low dimensional quantum spin systems

COIRA, Emanuele

Abstract

This thesis is composed by three projects in which we investigate thermal effects in low dimensional quantum spin systems. Thanks to recent developments in DMRG/MPS techniques, it is possible to simulate for such systems the dynamics at finite temperature. This allows us in the first two parts to study the temperature dependence of the NMR spin-lattice relaxation rate and of the INS spectra. Numerical calculations are performed for a wide range of temperatures. This represents a bridge between analytical results available for very low and very high temperatures. Interesting deviations from bosonization results valid at low energy, are seen at temperatures comparable to the energy scale of the system. In the third part we study the temperature and magnetic field dependence of the Grüneisen parameter, a valuable quantity for the detection and investigation of quantum phase transitions. Using an effective theory valid close to criticality, we show deviations from the critical behavior.

COIRA, Emanuele. Numerical study of thermal effects in low dimensional quantum spin systems . Thèse de doctorat : Univ. Genève, 2016, no. Sc. 5011

URN : urn:nbn:ch:unige-905301

DOI : 10.13097/archive-ouverte/unige:90530

Available at:

http://archive-ouverte.unige.ch/unige:90530

Disclaimer: layout of this document may differ from the published version.

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UNIVERSITÉ DE GENÈVE Section de Physique

Departement of Quantum Matter Physics

FACULTÉ DES SCIENCES Professeur T. Giamarchi

UNIVERSITÄT BONN H.I.S.K.P.

Professeur C. Kollath

Numerical study of thermal effects in low dimensional quantum spin

systems

THÈSE

présentée à la Faculté des Sciences de l’Université de Genève pour obtenir le grade de docteur ès Sciences, mention Physique

par

Emanuele Coira

de Côme (Italie)

Thèse n5011

GENÈVE

Atelier d’impression numérique ReproMail 2016

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Acknowledgements

This PhD thesis represents for me a great reward after an unforgettable five-year period of extremely intense life, both professionally and per- sonally. I am very grateful to all the people I met along this path. Luck- ily, I have always found in them friendship, helping hands and reassur- ing words. Although it is impossible in practice to mention all of them, I would like in particular to thank:

My two supervisors T. Giamarchi and C. Kollath for their constant help and support. They represent an example not only as world-leading sci- entists but also for the kindness and the empathy they show in everyday’s life. I also thank them for the great effort they put in proofreading my papers and my thesis.

P. Barmettler, B. Sciolla and C. Berthod for their technical support and for their help in solving my programming issues.

My present and past group mates and colleagues for their friendship and for the fruitful discussion I had with them: D. Poletti, P. Bouillot, E.

Agoritsas, A. Tokuno, T. Ewart, A. Kantian, S. Uchino, A. M. Novello, B. Wehinger, S. Furuya, N. Burgermeister, P. Grisins, C. Bardyn, N. A.

Kamar, S. Takayoshi and M. Filippone.

L. Foini and N. Kestin for their kindness, friendship and for the nice atmosphere they created in the office we shared in the last two years.

The Swiss National Science Foundation under MaNEP and Division II, and the financial and academic support of the University of Geneva. In particular the department of quantum matter physics (DQMP) and the department of theoretical physics (DPT), with their staff and secretaries.

The University of Bonn and the HISKP, the DFG and the ERC.

To conclude I would also like to thank my family, Ilaria and my close friends for their fundamental presence, patience and love. I will never forget your daily support and sacrifices, and this thesis is dedicated es- pecially to you.

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Résumé en français

Dans cette thèse, composée de trois projets, nous étudions les effets thermiques dans les systèmes quantiques à basse dimension pour des quantités physiques statiques et dynamiques. Ces systèmes sont soumis à un champ magnétique pour pouvoir explorer differentes phases quantiques.

Dans les deux premières parties, nous calculons numériquement les fonctions de corrélation dynamique, à température finie, pour des chaînes de spin-1/2. En partic- ulier, on considère les modèles XXX et XXZ, et la chaîne isotropiquement dimérisée.

Nous explorons les differentes phases quantiques de ces modèles en faisant varier l’intensité du champ magnétique. La méthode numérique choisie dans ce travail est le DMRG (density matrix renormalization group) dans sa reformulation en MPS (matrix product states). Cette méthode a été récemment étendue au calcul de la dynamique à température finie. Nous considérons deux quantités interessantes qui sont directement liées à ces fonctions de corrélation: le taux de relaxation longitu- dinal en résonance magnétique nucléaire (NMR)1/T1, et les spectres de diffusion inélastique de neutrons (INS). Dans le premier cas, nous étudions le taux NMR1/T1

en fonction de la température et nous montrons des déviations interessantes entre les résultats numériques et les prédictions analytiques valides à basse température, obtenues par bosonization et par la théorie du liquide de Tomonaga-Luttinger (TLL).

Dans le deuxième cas, nous analysons les structures observées dans les spectres INS pour le modèle dimerisé et pour différentes phases quantiques. Nous interprétons certains résultats en utilisant la théorie TLL ou une approximation des liens forts indépendants. Dans ce cas, nous montrons également comment la température agit sur les spectres INS en redistribuant l’intensité et en élargissant les structures. Nous proposons aussi une méthode pour le calcul numerique des paramètres TLLuetK.

Dans la troisième partie nous calculons analytiquement en fonction de la tem- pérature et du champ magnétique pour des modèles d’échelles de spin-1/2 le paramètre

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de Grüneisen. Cette quantité permet efficacement d’identifier et d’étudier les tran- sitions de phase quantiques. En utilisant une théorie effective, valide près du point critique et basée sur l’approximation de fort couplage de l’échelle, nous étudions en detail le paramètre de Grüneisen en fonction de la température et les déviations entre nos calculs et les résultats au point critique.

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Abstract

This thesis is composed by three projects, all of them focusing on the investigation of thermal effects on dynamical or static physical quantities in low dimensional quan- tum spin systems. These systems are all subjected to a magnetic field in order to drive the system through different quantum phases.

In the first two parts we compute numerically dynamical correlation functions at finite temperature in spin-1/2 chains. In particular, we consider the XXX and XXZ models, and the isotropically dimerized chain. We explore different quantum phases by varying the value of the applied magnetic field. The numerical tool we choose is the density matrix renormalization group (DMRG) in its matrix product states (MPS) reformulation. This method has been recently extended to the computation of the dynamics at finite temperature. We consider two interesting quantities directly related to these correlations: the nuclear magnetic resonance (NMR) relaxation rate 1/T1, and the inelastic neutron scattering (INS) spectra. In the first case we study the temperature dependence of the NMR rate1/T1and we show interesting deviations at intermediate temperatures from the behavior expected at low temperatures from bosonization and Tomonaga-Luttinger liquid (TLL) theory. In the second case we study the structures seen in the INS spectra for the dimerized model and for different quantum phases. We interpret some of these results using TLL theory or the single strong-bond picture. Also in this case we show how temperature acts on these spectra by redistributing the intensity and broadening the structures. We also propose a method to compute numerically the TLL parametersuandK.

In the third part we focus on the analytical computation of the so called Grüneisen parameter for spin-1/2 two-leg ladders as a function of the temperature and of the magnetic field. This quantity represents a valuable tool for the detection and the investigation of quantum phase transitions. Using an effective theory, valid close to criticality and based on the strong bond expansion, we study in detail the temperature

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dependence and the deviations from the critical behavior.

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Contents

1 Introduction 1

1.1 Outline of the thesis . . . 2

2 Spin-1/2 Chains and Ladders 5 2.1 Homogeneous spin-1/2 chains . . . 5

2.2 Inhomogeneous spin-1/2 chains: the isotropically dimerized case . . 6

2.3 Spin-1/2 two-leg ladders . . . 9

2.4 Spin chain mapping and boson representation for the ladder model . 10 2.4.1 Beyond spin chain mapping: treatment aroundhc1 . . . 11

2.4.2 Beyond spin chain mapping: treatment aroundhc2 . . . 12

2.5 Experimental realizations . . . 13

3 Methods 15 3.1 DMRG & MPS . . . 16

3.1.1 Basic ideas of DMRG & MPS . . . 16

3.1.2 Ground state search . . . 19

3.1.3 Time evolution . . . 21

3.1.4 Finite temperature . . . 22

3.1.5 Dynamical correlations at finite temperature . . . 23

3.2 Tomonaga-Luttinger Liquid (TLL) . . . 25

3.2.1 Bosonization of the spin-1/2 chain . . . 26

3.2.2 Evaluation of TLL parameters . . . 27

4 NMR Relaxation Rate 31 4.1 Spin-lattice relaxation rate1/T1 . . . 31

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CONTENTS

4.2 What we compute, why and how . . . 33

4.3 Results for1/T1as a function of the temperature . . . 35

4.3.1 Heisenberg model . . . 36

4.3.2 XXZ model . . . 40

4.3.3 Dimerized model . . . 41

4.4 Summary & conclusions of the chapter . . . 43

5 Dynamical Correlations at Finite Temperature of a Dimerized Spin-1/2 Chain 45 5.1 What we compute: definitions . . . 46

5.2 Results for dynamical correlations at finite temperature . . . 47

5.2.1 Isotropic gapped systemh= 0 . . . 48

5.2.2 Gapped regime at finiteh. . . 50

5.2.3 Tomonaga-Luttinger liquid regime . . . 52

5.2.4 High field gapped phase . . . 53

5.3 Band-narrowing effects . . . 56

5.4 Summary & conclusions of the chapter . . . 58

6 Grüneisen Parameter and Quantum Phase Transitions in Spin-1/2 Lad- ders 63 6.1 The Grüneisen parameter: definition and computation . . . 63

6.1.1 Computation ofΓmagvia spin-chain mapping . . . 64

6.1.2 Strong coupling expansion & refined results . . . 67

6.2 Analytical results against experiments and DMRG . . . 69

6.3 Summary & conclusions of the chapter . . . 71

7 General Conclusions & Perspectives 75 A Technical Aspects of Chapter 4 77 A.1 Equality between the time integrals of onsiteS+−andS−+correla- tions . . . 77

A.2 Derivation of the result in Eq. (4.11) . . . 78

B Extrapolation Method for Spin-Lattice Relaxation Rate 81 C Consistency Test Using the XX Model 83 D Dynamics of a Single Excitation in the Strong Dimerization Limit 87 D.1 h= 0 . . . 87

D.2 S+−correlations forh > hc2. . . 88

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CHAPTER 1

Introduction

Quantum spin systems are an extremely fascinating subject of study because of their rich physics. Depending on their microscopic characteristics (dimensionality, geom- etry, local spinS, type of interaction) and on the external environment (temperature, applied magnetic field, pressure), one can detect many interesting phenomena. Vari- ous combinations of these ingredients have been investigated theoretically [1, 2] for decades, leading to several important results and observations. Among them one can cite for example the discovery of the spinon nature of the excitation spectrum of the spin-1/2 antiferromagnetic chain [3], the different nature of integer and half-integer spin chains (which are gapped or gapless, respectively) [4, 5], or the connections between high temperature supeconductivity and quantum magnetism in a 2D square lattice [6].

More specifically, low-dimensional quantum magnets like spin chains or ladders are particularly intriguing because quantum fluctuations are extreme and no ordered state is usually possible. In one dimension, the interaction between the excitations leads to various exotic states, ranging from gapped phases to phases possessing quasi-long-range magnetic order known as Tomonaga-Luttinger liquids (TLL) [7].

Thanks to recent advancements in chemistry, it is now possible to grow pure, single crystalline samples of low-dimensional spin systems, characterised by small cou- pling constants (in the order of∼ 10K) [8, 9]. On the one hand, this allows one to experimentally manipulate the sample and to drive it through different quantum phases, for example via the application of a magnetic field [10, 11] or by exerting pressure on it [12]. On the other hand, the energy scale of the exchange coupling gets closer to temperatures accessible by experiments, so that thermal effects start to

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1. INTRODUCTION

play a fundamental role in the description of the system.

In order to provide a complete picture of the different types of order that can be present, the nature of the different phases and also what happens at the quantum crit- ical points (QCP), a set of probes sensitive to the correlation functions of the system, which can be evaluated both experimentally and theoretically, is needed. On the ex- perimental side, fortunately, a set of suitable probes such as neutron scattering [13], electronic spin resonance [14], Raman scattering [15] and nuclear magnetic reso- nance (NMR) [16, 17] exists. On the theoretical side, it is important to have reliable numerical and analytical methods to access the desired quantities and to allow for a direct comparison with experiments. Field theory methods such as bosonization [7]

provide a very good description of the correlation functions of such systems [10] for temperatures very low compared to the characteristic energy scale. Via Bethe ansatz it is possible to compute for integrable models the neutron spectra at zero temper- ature [18–20]. Density matrix renormalization group (DMRG) techniques [21] has provided a quantitative description of the thermodynamics, and of dynamical prop- erties at zero temperature of low-dimensional spin systems [10, 11, 22]. However, a direct method of computation representing a bridge between the very low temper- ature regime and the high temperature one was missing. Recent developments in DMRG/MPS techniques [23, 24] have opened the path for the computation of real time dynamics at finite temperature [25] and here we will make intensive use of this new feature. This technique offers some advantages when compared to other nu- merical methods. It can access directly the real time dynamics avoiding the delicate problem of the analytic continuation from imaginary time which characterizes quan- tum Monte-Carlo (QMC) [26], and it is capable to treat systems of much bigger size if compared for instance to exact diagonalization methods [27].

In this thesis we show how MPS methods can be used for a quantitative computa- tion of dynamical correlation functions in one dimensional spin-1/2 systems at finite temperature, especially in regimes where the temperature plays a crucial role. By putting together the analytical results in the low temperature limit, and the numerical results at finite T, one can now get a full description of quantities like the NMR re- laxation time or the spectra obtained via inelastic neutron scattering measurements.

We also perform an analytical study of the thermal effects close to criticality in spin- 1/2 two-leg ladder systems. A more detailed outline of the thesis is given in the following.

1.1 Outline of the thesis

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

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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.

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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.

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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

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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.

2.2 Inhomogeneous spin-1/2 chains: the isotropically dimerized case

The second model we consider in this work is the dimerized Heisenberg chain sub- jected to a magnetic fieldhalong thezdirection, which is described by the Hamil- tonian

H=X

j

J+ (−1)jδJ

Sj·Sj+1−hX

j

Sjz. (2.2)

whereSj denotes the vector of the spin at sitej. The spin coupling strength is alternated with valuesJs=J+δJ(strong bonds) andJw=J−δJ (weak bonds).

Also in this case thegfactor, the Bohr magneton and~have been absorbed intohand

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2.2 Inhomogeneous spin-1/2 chains: the isotropically dimerized case

J-δJ J+δJ J-δJ

j+1 j+2

j-1 j

Figure 2.2: Pictorial representation of the dimerized Heisenberg chain: a spin 1/2 is located on each black square, the strength of the coupling (between nearest neighbor spins only) is alternated with valuesJs = J +δJ (solid lines) andJw = J−δJ (dashed lines).

(J±δJ), which all have here the dimensions of an energy. A pictorial representation of this system is given in Fig. 2.2.

The phase diagram of this model as a function of the magnetic field and at low temperature is reported in Fig. 2.3, lower part. At zero magnetic field (h = 0) the model has a non-trivial spin-0 ground state (quantum disordered phase) with a gap of orderJsto the first excitation which is a band of spin-1 excitations (triplons) [7].

The role of the magnetic field alongzis to progressively reduce this gap, up to the first critical fieldhc1when the gap closes. Ath > hc1a quantum critical phase arises with gapless excitations (TLL phase). If we further increase the magnetic field, the spins of the chain progressively polarize and above the second critical magnetic field hc2the ground state is fully polarized with a gapped spectrum [7].

A very intuitive explanation of this behavior can be provided by considering the limit of strong dimerization. As we will see in the end of this chapter, we will stick to a choice of parameters that will put us exactly in that limit. WhenJ ∼δJ ←→

JsJwthe strong bonds are essentially decoupled from each other. In this single strong bond picture, each of them has four eigentstates: the singlet state

|si= |↑↓i − |↓↑i

√2 (2.3)

with energyEs=−3Js/4, total spinS= 0andz−projectionSz= 0, and the three triplet states

t+

=|↑↑i, t0

= |↑↓i+|↓↑i

√2 , t

=|↓↓i (2.4)

withS = 1; Sz = 1,0,−1, and energiesE+ = Js/4−h, E0 = Js/4,E = Js/4+h, respectively. The ground state is|sibelow the critical value of the magnetic fieldhc1and|t+iabove. The dependence of the energies onhis shown in Fig. 2.3.

A small but finiteJwdelocalizes triplets and creates bands of excitations with a finite bandwidth for each triplet branch. This leads, as anticipated, to three distinct phases in the dimerized system of Eq. (2.2) depending on the magnetic field:

1. Quantum disordered phase, characterized by a spin-singlet ground state and a gapped spectrum. This phase appears for magnetic fields ranging from 0 to

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2. SPIN-1/2 CHAINS AND LADDERS

Figure 2.3: Sketch of the energy spectrum of excitations for the dimerized chain under the application of a magnetic fieldhin the limit of large dimerization (J ≈δJ). The diagram can be very well understood in the single strong bond picture. Ath= 0there is an energy gap of the order ofJsbetween the singlet (ground state) and the three triplet states. The magnetic fieldhsplits the triplets and brings down the excitation energy of the state

t+

. Due to the presence of the weak bonds the triplets can be delocalized and, thus, have a dispersion in energy of the order ofJw(the boundaries of which are represented by the dotted lines). At sufficiently high magnetic field, the energy of the lowest triplon band is close to the energy of the singlet state. This leads in the extended system to a quantum critical phase forh > hc1with gapless excitations. The system remains in this phase up to the pointh=hc2for which the triplon band is totally filled and a fully polarized phase arises. Picture adapted from Fig. 2 in Ref. [10]

.

hc1. The magnetic field progressively splits the triplets energy, bringing down the|t+ione, and closes the gap.

2. TLL phase, characterized by a gapless excitation spectrum. It occurs between the critical fieldshc1andhc2. The magnetization per site increases from 0 to 0.5 forhrunning fromhc1tohc2. The low energy physics can be described by the TLL theory.

3. Fully polarized phase, characterized by a fully polarized ground state and a gapped excitation spectrum. This phase appears abovehc2.

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2.3 Spin-1/2 two-leg ladders

𝑺

𝑗−1,1

𝑺

𝑗,1

𝑺

𝑗+1,1

𝑺

𝑗−1,2

𝑺

𝑗,2

𝑺

𝑗+1,2

J

J

J

Figure 2.4: Pictorial representation of the two-leg spin-1/2 ladder: a spin 1/2 is located on each black square, the strength of the coupling (between nearest neighbor spins only) isJon the rungs andJwon the legs.Jis considerably bigger thanJkin this work.

2.3 Spin-1/2 two-leg ladders

Finally we consider also the spin-1/2 two-leg ladder model, in a magnetic fieldh along thezdirection, described by the Hamiltonian

H =H+Hk, (2.5)

where

H=JX

j

Sj,1·Sj,2−hX

j

Szj,1+Sj,2z

, (2.6)

and

Hk=JkX

j 2

X

l=1

Sj,l·Sj+1,l. (2.7)

HereSl,kis the spin-1/2 (vector) located on thel-th leg and on thej-th rung.J is the coupling along the rung,Jk is the coupling along the legs. Again theg factor, the Bohr magneton and~have been absorbed intoh,JandJkwhich all have here the dimensions of an energy. A sketch of the model is given in Fig. 2.4. This model shares many features with the dimerized chain presented in the previous section and for the phase diagram we can easily refer to Fig. 2.3 with the mappingJ ←→ Js

andJk←→Jw, andJJk. Therefore we have again

1. AQuantum disordered phase, characterized by a spin-singlet ground state on each rung and a spectrum with a gap of the orderJ. This phase appears for magnetic fields ranging from 0 tohc1.

2. A TLL phase, characterized by a gapless excitation spectrum, occurring be- tween the two critical fields hc1 and hc2. The low energy physics can be described by the TLL theory.

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2. SPIN-1/2 CHAINS AND LADDERS

3. AFully polarized phase, characterized by full polarization and a gapped exci- tation spectrum, occurring abovehc2.

2.4 Spin chain mapping and boson representation for the ladder model

WhenJJk, the ladder problem can be reduced to a simpler spin chain problem, especially in the vicinity of the two critical fieldshc1andhc2. In these regions, in the single strong rung picture, states|siand|t+iare close to each other in energy, while the other two

t0

and|tirepresent highly excited states. The essence of the spin-chain mapping[31–34] is to project out

t0

and|tibands from the Hilbert space of the model, since they are almost completely unoccupied. One can thus map each strong bond (rung) to a pseudo-spin 1/2 according to the relations

|↓i˜ =|si= 1

√2(|↑↓i − ↓↑i),

|↑i˜ = t+

=|↑↑i. (2.8)

The transformation that expresses the original spin operatorsSi(i = 1,2) of each rung in terms of the new pseudo-spin 1/2 operatorsS˜is given by:

S1,2± =:∓ 1

√2

±, S1,2z =:1 4

1 + 2 ˜Sz

. (2.9)

The ladder Hamiltonian in Eqs. (2.5-2.7) becomes after this transformation an XXZ chain Hamiltonian, with anisotropy1/2:

XXZ =JkX

j

1 2

j+j+1 + ˜Sjj+1+ +1

2 S˜jzj+1z

h−J−Jk 2

X

j

jz+L

−J 4 +Jk

8 −h 2

. (2.10)

L is the length of the ladder. This new Hamiltonian can be solved analytically using Bethe ansatz, see for instance Refs. [18,19]. Close to the first critical field and at low temperature, we are in a regime of extreme diluteness of excitations (very few triplets

“+” in a sea of singlets). One can get an idea of the physics in this limit adopting an effective theory which consists in considering the excitations as non interacting. We then apply to Hamiltonian (2.10) a Jordan-Wigner transformation given by

(S˜j+= (−1)jcje−iπPl<jclcl,

jz=cjcj12, (2.11)

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2.4 Spin chain mapping and boson representation for the ladder model

and we then neglect the term quadratic in density. This leads to the following free fermion Hamiltonian

Hf f =−Jk 2

X

j

cj+1cj+cjcj+1

−(h−J)X

j

cjcj−3

4LJ, (2.12) which can be exactly solved for all quantities of interest.

This spin chain mapping represent a part of a more general strong coupling ex- pansion of the model. In order to go beyond the previous results, it becomes neces- sary to say more about this approach. As anticipated, the four-dimensional Hilbert space on each rung is spanned by the four states|si,|t+i,

t0

and|ti, and we in- troduce the four corresponding operatorssj,tj,+,tj,0andtj,− creating on the rung ja singlet or a triplet. By imposing an hard-core constraint on each rungj(only one of the four state can exist on a single rung)

sjsj+tj,+tj,++tj,0tj,0+tj,−tj,−= 1, (2.13) we can rewrite the Hamiltonian of the ladder model in terms of boson operators.H

is quadratic in these operators, whileHkis quartic and it has a much more complex structure [35]. This procedure is the starting point also for the simple spin chain mapping discussed just above. To go back to Eq. (2.10) what we need to do is to neglect completely the possibility of having higher excitations

t0

and|ti. This can be achieved by suppressing all terms containing operatorstj,0,tj,−,tj,0andtj,−, and by simplifying the hard-core constraint Eq. (2.13) as

sjsj+tj,+tj,+= 1. (2.14) If we now collect the remaining term and make an additional transformation back to the spin operators of the form

(S˜lz=tl,+tl,+12,

l+=tl,+sl, (2.15)

the structure of Eq. (2.10) is restored.

2.4.1 Beyond spin chain mapping: treatment around h

c1

Let’s try to refine a little bit the picture given in Eq. 2.12 and let’s consider the vicinity ofhc1. We start as always from a landscape where only few triplets|t+iare present in a sea of singlets|si, but we want to add very few excitations

t0

and|ti created by temperature. Following this idea we rearrange the ladder Hamiltonian written in terms of rung operators by re-expressingsjsj in terms of the densities of the other triplets (exploiting the hard-core constraint), by setting elsewheresj

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2. SPIN-1/2 CHAINS AND LADDERS

sj ≈1, and by neglecting all terms cubic or quartic in triplet operators. After some algebra we are left with an Hamiltonian containing three species of non-interacting fermions (i.e.the three triplets) which we write in a more compact form as

H1= X

l=+,0,−

X

j

Jk 2

cl,jcl,j+1+h.c.

−˜hlcl,jcl,j

, (2.16)

with the evident associationcl,j =tl,jfor alljand forl= +,0,−. Also,

˜h+ =h−J,

˜h0=−J, (2.17)

˜h =−h−J.

According to our expectations, the magnetic field for the triplets t0

is fixed and prevents the creation of these excitations at zero temperature. The same happens for the remaining triplets|tisince we are in the vicinity of the first critical field.

This Hamiltonian can be solved exactly for all the quantities we are interested in, for values of the field around the first critical one.

2.4.2 Beyond spin chain mapping: treatment around h

c2

Close tohc2, the scene is completely dominated by triplets|t+iand there are only few singlets. As before, we take into account the possibility of having in addition very few higher excitations

t0

and|tithermally activated. This time we rearrange the ladder Hamiltonian in terms of rung operators by re-expressingt+,jt+,jin terms of the densities of the other triplets and of the singlets (exploiting again the hard-core constraint), by setting elsewheret+,j ≈t+,j ≈1, and by neglecting all terms cubic or quartic insj,t0,j andt−,j operators. After some algebra we are left once more with an Hamiltonian containing three species of non-interacting fermions (i.e. the

“0” and “-” triplets, and singlets) of the form:

H2 = Jk 2

X

j

h

sjsj+1+h.c.i

− J+Jk−h X

j

sjsj+

+Jk 2

X

j

htj,0tj+1,0+h.c.i

− Jk−h X

j

tj,0tj,0

−2 Jk−h X

j

tj,−tj,−+ J

4 +Jk 2 −h

L. (2.18)

This Hamiltonian can be solved exactly for all the quantities we are interested in, for values of the field around the second critical one.

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2.5 Experimental realizations

Copper Nitrogen

Hydrogen Oxygen

Figure 2.5: Chemical structure of the copper nitrate

Cu(NO3)2·2.5D2O . Red spheres are Cu2+ions which are the spin carriers. Solid, red thick lines represent the strong couplingJs, Solid, red thin lines represent the weak couplingJw, while dashed green lines stand for the intra-chain interaction, which is weak and therefore not consid- ered in this work. The sketch has been realised according to the crystal structure detailed in Ref. [39].

2.5 Experimental realizations

Results that will be presented in this work for the dimerized chain and for the ladder model are computed within choices of parameters referring to existing compounds.

As for the dimerized chain we explicitly refer to the quantum magnet copper nitrate[Cu(NO3)2·2.5D2O]experimentally investigated via neutron scattering in Refs. [36–38]. This compound shows a dimerized structure with coupling constants Js = J +δJ = 5.28kBK and Jw = J −δJ = 1.474kBK, and critical fields experimentally measuredhc1 = 4.39kBK andhc2 = 6.73kBK. A sketch of the atomic structure of the compound is shown in Fig. 2.5.

As for the ladder model we consider two organic compounds belonging to the so called “Hpip” family, namely[(C5H12N)2CuBr4], which is often called BPCB, and its chlorine counterpart[(C5H12N)2CuCl4]. The first of these materials was pre- sented originally in Ref. [40] and it has been intensively investigated in the following years both theoretically and experimentally, see for instance Ref. [10] and references therein. The atomic structure of the bromine compound is sketched in Fig. (2.6).

It is characterized by coupling constantsJ = 12.92kBK andJk = 3.33kBK, and critical fieldshc1 = 9.69kBK andhc2 = 19.95kBK. However, the values of the couplings of this compound make it difficult to access experimentally the upper critical field. More recently a new member of this family of compounds have been

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2. SPIN-1/2 CHAINS AND LADDERS

Figure 2.6: Chemical structure of BPCB compound. Blue spheres are Cu2+ions which are the spin carriers. Solid thick blue lines and dashed thick blue lines stand for the interaction pathJandJk, respectively. Picture adapted from Fig.1(a) in Ref. [41].

synthetized, by replacing Br atoms with Cl ones [42]. This substitution induces a re- duction of the exchange parameters and therefore of the critical fields, making both of them more easily accessible in experiments. For the chlorine compound we thus haveJ = 3.42kBK andJk = 1.34kBK, and critical fieldshc1= 2.39kBK and hc2= 6.06kBK.

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CHAPTER 3

Methods

In this chapter we present the methods that will be used in this work for the the- oretical investigation of the spin systems already introduced. In particular we will focus on the DMRG, especially in its more recentmatrix product states(MPS) re- formulation, and on the TLL theory: these techniques are particularly suited for the treatment of one-dimensional or quasi-one-dimensional systems.

We start from a general overview on DMRG, which represents a very power- ful tool to describe strongly correlated quantum lattice systems, especially in one dimension. This techinque was originally introduced by S. R. White in the begin- ning of the 90’s [43, 44]. Along the years many extensions and generalizations have been proposed and successfully implemented, offering the possibility to access static properties at finite temperature [45–47] and dynamical properties at zero tempera- ture [48–50]. These two results have opened the path to the computation of dynam- ical correlations at finite temperature. In this work we will make extensive use of these recent algorithms, following what is proposed in Refs. [23, 24].

Finally we introduce an analytical low-energy description for the gapless regime of our spin systems, the TLL theory [7]. This theory has proven to be extremely suc- cessful in describing quantitatively the low-energy physics of many one-dimensional systems. It will be used in this work to benchmark some of the numerical results and to help in giving a physical interpretation of them. We will see the importance in this theory of two parameters, the TLL parametersuandK, and we propose a method to compute them numerically via MPS.

The combination of numerical and analytical results (MPS+TLL theory) allows today to obtain a complete, detailed picture of the model for all energy and temper-

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3. METHODS

ature regimes.

3.1 DMRG & MPS

In this section we first give a short review of the basic ideas of the method, focusing on its application to one-dimensional, finite systems at zero temperature. Then we discuss the implementation of the real time evolution (useful for the simulation of the dynamics) and then of the imaginary time evolution (allowing for the computation of thermodynamic quantities). Finally, by combining these two procedures as explained in Refs. [23, 24], we describe how to compute finite temperature dynamics. We present the algorithm we use in this work and we also discuss how to get momentum- frequency correlations from our results.

3.1.1 Basic ideas of DMRG & MPS

The most difficult problem while dealing with numerics for quantum many-body system is the exponential growth of the Hilbert space of the system with its size.

If we take for instance a spin-1/2 chain ofLsites, the corresponding Hilbert space would be of size2L. Luckily, the first assumption on which DMRG relies is that one can always find a reduced Hilbert space, often extremely small if compared to the full one, which contains the relevant physics and which can be parametrized efficiently by the MPS formalism. In fact, as one would guess from the name of the formalism, quantum many-body states can be represented as products of matrices.

Also the operators we use to measure the relevant quantities can be re-interpreted in this matrix language, in this case we speak aboutmatrix product operators(MPO).

The ground state search is then performed, as we will see, as a variational procedure in the MPS space.

Before entering into details, let us introduce one of the most useful results of linear algebra: the so calledsingular value decomposition(SVD). SVD guarantees that for any arbitrary rectangular matrixM of dimensionsm×nthere exists a de- composition such that

M =U SV. (3.1)

S is diagonal with non-negative elementsSaa = sa calledsingular valuesand it has dimensions(min(m, n)×min(m, n)). Singular values are organized such that sa ≤ sa0 ifa > a0. The numberrof singular values which are non-zero gives the rank of M.Uhas dimensionsm×min(m, n)andUU =I, whileVhas dimensions min(m, n)×nandV V=I. It is particularly important for the following to stress that it is possible to optimally approximate a matrix M of rankrby a matrix M’ of rankr0 < rproperly chosen. This approximation consists in setting to zero all but the firstr0singular values of S to zero (and to shrink the column dimension of U and the row dimension ofVaccordingly).

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3.1 DMRG & MPS

The basic idea of the DMRG techinque consists in splitting the Hilbert space in two blocks which we callleft(L) andright(R). Any pure state on a bipartite lattice LRcan be written as

|Ψi=X

α,β

Cαβ

φLα φRβ

, (3.2)

where φL,R

are in the set of the orthonormal basis of the two blocks, of dimensions NLandNR, in which we have divided the system. The coefficientsCαβcan be read as the elements of a matrixC. As previously said, the size of each block basis grows exponentially with the lattice size. In order to make the computation accessible one needs at some point to reduce the size of the bases keeping onlyN < NL, NRstates to approximate each of them.

The SVD procedure represents the basis of a very useful representation of quan- tum states of a bipartite systemLR, theSchmidt decomposition. It also provides a powerful optimization of the truncated bases. Let us now reconsider the state|Ψiin Eq. (3.2) and let’s perform an SVD of the matrixC. We get

|Ψi=X

α,β

min(NL,NR)

X

r=1

UαrSrrVβr

φLα φRβ

=

=

min(NL,NR)

X

r=1

X

α

Uαr φLα

! sr

 X

β

Vβr φRβ

=

=

min(NL,NR)

X

r=1

sr

rL rR

, (3.3)

wheresr=Srrare the elements on the diagonal of the matrixS. If this last sum is performed over only theN ≤min(NL, NR)positive and nonzero singular values, what we get is theSchmidt decompositionof our quantum state. This decomposition is related to the reduced density matrices by

ρL=TrR|Ψi hΨ|=X

r

s2r

rL rL

, (3.4)

ρR=TrL|Ψi hΨ|=X

r

s2r

rR rR

, (3.5)

where rL,R

are eigenvectors ofρL,R, ands2rare the eigenvalues. An approximate description of the state|Ψican be obtained by truncating its Schmidt decomposition retaining only the states

rL and

rR

corresponding to the D largest valuessr:

0i ≈

D

X

r=1

sr

rL rR

, (3.6)

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3. METHODS

withs1 ≥ s2 ≥ s3 ≥ · · · ≥ 0. If normalization is desired, the retained singular values need to be rescaled. The accuracy of the approximation can be evaluated by thetruncation errorεDdefined as the difference in norm between the exact state and the approximated one using the optimized basis:

D= X

r>N

s2r=k|Ψi − |Ψ0ik2 (3.7) For a givenDthis error is minimized by the truncation procedure adopted in Eq. (3.6) and it clearly depends on the distribution of thesr.

Let’s go back now to our chain ofLsites with d-dimensional local state spaces {σi}on each sitei = 1, . . . , L. It can be shown that any representative state of a spin-1/2 chain (and more in general of any quantum state on a lattice) can be written as:

|Ψi= X

σ1,...,σL

cσ1...σL1. . . σLi. (3.8) Let’s see now how it can be represented exactly in a matrix product form. Additional details can be found in Ref. [21], here we follow the same procedure described there.

We start by reshaping the vectorcσ1...σL with dimensiondL into a matrix M with dimensions d×dL−1

:

cσ1...σL −→ Mσ1,(σ2...σL) (3.9) An SVD of M gives

Mσ1,(σ2...σL)=

r1

X

`1

Uσ1,`1S`1,`1(V)`1,(σ2...σL)=

r1

X

`1

Uσ1,`1c`1σ2...σL, (3.10) where we have reshaped back into a vector the product between matrices S and V, andr1 ≤ d. Now we can decompose the matrix U into a collection ofdrow vectorsLσ1with elementsLσ`1

1, and reshape at the same timec`1σ2...σL into a matrix M(`1σ2),(σ3...σL)of dimensions(r1d×dL−2). We then get

cσ1...σL =

r1

X

`1

Lσ`1

1M(`1σ2),(σ3...σL). (3.11) Now this new matrixM can undergo a new SVD following the same idea as before to give

cσ1...σL =

r1

X

`1 r2

X

`2

Lσ`1

1Lσ`2

1,`2M(`2σ3),(σ4...σL), (3.12) whereM(`2σ3),(σ4...σL)has now dimensions(r2d×dL−3)andr2 ≤r1d≤d2. If we continue to SVD until the end of the chain, what we get is

cσ1...σL = X

`1,...`L−1

Lσ`1

1Lσ`2

1`2. . .Lσ`L−2L−1,`L−1Lσ`L−1L =Lσ1Lσ2. . .LσL−1LσL, (3.13)

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3.1 DMRG & MPS

where the last compact form implies the matrix product induced by the sums over the`iindexes. The quantum state can be now represented in the matrix product state form as:

|Ψi= X

σ1...σL

Lσ1Lσ2. . .LσL−1LσL1. . . σLi. (3.14) These matricesLare such thatP

σiLσiLσ=I and they are calledleft-normalized, and the corresponding MPSleft-canonical. The MPS formalism is quite versatile, therefore one can easily get for exampleright-canonicalMPS, if one starts the series of SVDs from the right, or evenmixed-canonicalif one starts from both ends of the chain. In this last case the decomposition gives:

|Ψi= X

σ1...σL

Lσ1. . .LσlSRσl+1. . .RσL1. . . σLi, (3.15)

with singular values at the bond(l+ 1, l). Within this notation the Schmidt decom- position presented above into two blocks (left block from 1 tol, right block from l+ 1toL) is easy to reintroduce. If we reconsider Eq. (3.3) one immediately notes the parallelism

rR

= X

σ1...σl

(Lσ1. . .Lσl)|σ1. . . σli (3.16) rL

= X

σl+1...σL

(Rσl+1. . .RσL)|σl+1. . . σLi (3.17)

From a simple dimensional analysis it turns out immediately that in practical calcu- lations such decompositions cannot be performed exactly given the limited computa- tional resources. Luckily it turns out that in most cases the state can be nevertheless described accurately using matrices of reduced dimensions. If we consider for in- stance the mixed-canonical representation we have just introduced, it is possible to cut the spectra of the reduced density operators and to optimally approximate (in the 2-norm) the state following the same philosophy of Eq. (3.6). This argument can be generalized from the approximation incurred by a single truncation to that caused by L−1truncations, one at each bond. If we make no assumptions on the normalization of the matrices, we can write a general MPS for open boundary conditions as:

|Ψi= X

σ1,...,σL

Mσ1Mσ2. . .MσL−1MσL1. . . σLi. (3.18)

3.1.2 Ground state search

Let us now move to the presentation of the standard algorithm which allows one to compute the ground state|Ψ0ifor a given HamiltonianHˆ (expressed as an MPO)

(33)

3. METHODS

𝜎1

𝑀𝑖𝑛.

𝜎2… ℳ𝜎𝐿−1𝜎𝐿 𝐿𝑎𝑛𝑐𝑧𝑜𝑠1𝜎1𝜎2… ℳ𝜎𝐿−1𝜎𝐿

↙ 𝑆𝑉𝐷 ↙

1𝜎1𝜎2

𝑀𝑖𝑛.

… ℳ𝜎𝐿−1𝜎𝐿 𝐿𝑎𝑛𝑐𝑧𝑜𝑠1𝜎11𝜎2… ℳ𝜎𝐿−1𝜎𝐿

↙ 𝑆𝑉𝐷 ↙

( … 𝑡𝑜 𝑡ℎ𝑒 𝑟𝑖𝑔ℎ𝑡 𝑒𝑛𝑑 𝑜𝑓 𝑡ℎ𝑒 𝑐ℎ𝑎𝑖𝑛 … ) 1𝜎11𝜎2… ℳ1𝜎𝐿−1𝜎𝐿

𝑀𝑖𝑛.

𝐿𝑎𝑛𝑐𝑧𝑜𝑠1𝜎11𝜎2…ℳ1𝜎𝐿−11𝜎𝐿 1𝜎11𝜎2…ℳ1𝜎𝐿−12𝜎𝐿 𝐿𝑎𝑛𝑐𝑧𝑜𝑠1𝜎11𝜎2… ℳ1𝜎𝐿−11𝜎𝐿

𝑀𝑖𝑛.

↘ 𝑆𝑉𝐷 ↘

1𝜎11𝜎2…ℳ2𝜎𝐿−12𝜎𝐿 𝐿𝑎𝑛𝑐𝑧𝑜𝑠1𝜎11𝜎21𝜎𝐿−1

𝑀𝑖𝑛.

2𝜎𝐿

↘ 𝑆𝑉𝐷 ↘

( … 𝑡𝑜 𝑡ℎ𝑒 𝑙𝑒𝑓𝑡 𝑒𝑛𝑑 𝑜𝑓 𝑡ℎ𝑒 𝑐ℎ𝑎𝑖𝑛 … ) 1)

2)

3) 4)

5)

6) Repeat points from 1) to 5) using the updated matrices.

Sweeping procedure

Figure 3.1: Pictorial representation of the finite size DMRG algorithm: starting from the initial MPS, we optimize the first matrix in order to minimize the energy (point 1), we then move to the next site and do the same thing (point 2), and so on until we reach the end of the chain (point 3). Then the direction is reversed (points 4-5) and the procedure continued until one reaches the other end of the chain. This sweeping procedure stops when convergence onto the ground state is reached. In the figure indexes1,2are there to distinguish explicitly the original matrix from the optimized one.

with open boundary conditions. In order to find the optimal approximation to it, we have to find the optimal MPS|Ψi=|Ψ0iof some dimension D that minimizes

E= hΨ|Hˆ|Ψi

hΨ|Ψi . (3.19)

The most efficient way of doing it is a variational search in the MPS space. The algorithm starts from some initial guess|Ψi. It minimizes the energy on the first site by solving the standard eigenvalue problem (exploiting Lanczos algorithm1) for the representative matrixMσ1, taking its initial value as a starting point. Then it moves to the second site, usually via SVD to mantain the desired normalization structure,

1The Lanzos algorithm [51] is an adaptation of the power method, particularly suited for numerical computation, for finding the extremal eigenvalues of a matrix A (in our case the minimal). The power method guarantees that, starting from a random vector|ψiand applying A several times to it such that

|ψin+1=A|ψin, in the large n limitxn/kxnkapproaches the normed eigenvector corresponding to the desired eigenvalue.

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