Ruben Serral Gracia- Spin Bosons and Spin Glasses

8/3/2019 Ruben Serral Gracia- Spin Bosons and Spin Glasses

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Spin Bosons and Spin Glasses

ACADEMISCH PROEFSCHRIFT

ter verkrijging van de graad van doctor

aan de Universiteit van Amsterdamop gezag van de Rector Magnificus

prof. mr. P.F. van der Heijdenten overstaan van een door het college voor promoties ingestelde

commissie in het openbaar te verdedigen in de Aula der Universiteitop donderdag 25 november 2004, te 14:00 uur.

doorRuben Serral Gracia

geboren te Xerta, Spanje


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Promotor: prof. dr. B. NienhuisCopromotor: dr. Th. M. Nieuwenhuizen

Promotiecommissie: prof. dr. F. A. Baisprof. R. Baliandr. J.-S. Cauxprof. dr. L. F. Cugliandolodr. L. Leuzzidr. L. G. Suttorpprof. dr. C. J. M. Schoutens

Faculteit der Natuurwetenschappen, Wiskunde en Informatica

This thesis is part of the research programme of the Sticht-ing voor Fundamenteel Onderzoek der Materie (FOM), whichis financially supported by the Nederlandse Organisatie voorWetenschappelijk Onderzoek (NWO).


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Paranimfen:Jordi Busquets Blanco

Daniele Moroni

This thesis is based on the following papers:

R. Serral Gracia and Th. M. NieuwenhuizenQuantum spherical spin modelsPhys. Rev. E 69 056119 (2004) [cond-mat/0304150]

R. Serral Gracia, Th. M. Nieuwenhuizen and I. V. LernerConcentration dependence of the transition temperature in metallic spin glassesEur. Phys. Lett. 66 (3) 419-422 (2004) [cond-mat/0311491]

A. E. Allahverdyan, R. Serral Gracia and Th. M. NieuwenhuizenWork extraction and cooling in the spin boson modelIn preparation

A. E. Allahverdyan, R. Serral Gracia and Th. M. NieuwenhuizenCooling spins via bosonic bath

In preparationR. Serral Gracia, L. Leuzzi and F. RitortNumerical simulation on the hypercubic cell spin glassIn preparation


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A la meua famlia,


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Contents

Introduction 11

I Spherical Model 15

1 Phase transitions and critical phenomena 17

1.1 Path Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.1.1 Bosonic coherent state representation for a single oscillator . 22

2 Quantum spherical model 25

2.1 Classical spherical model . . . . . . . . . . . . . . . . . . . . . . . . . 272.1.1 Vector spherical spins . . . . . . . . . . . . . . . . . . . . . . 272.1.2 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.1.3 Spherical constraint on the length of the total spin . . . . . . 282.1.4 Spherical constraint on the number of spin quanta . . . . . . 282.1.5 Comparison of the two constraints . . . . . . . . . . . . . . . 29

2.2 Ferromagnetic Hamiltonians with creation and annihilation operators 302.2.1 Coherent state representation for spherical spins . . . . . . . 302.2.2 General solution . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.2.3 Ferromagnetic couplings with transversal field in d dimensions 332.3 Hamiltonians involving spins but not their momenta . . . . . . . . . 402.3.1 Ferromagnetic couplings in the presence of a transversal field 42

2.4 Generalization and mapping from Heisenberg spins . . . . . . . . . . 462.5 Summary and discussion . . . . . . . . . . . . . . . . . . . . . . . . . 49

II Spin-Boson Model 53

3 Introduction 55

3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.1.1 Work Extraction . . . . . . . . . . . . . . . . . . . . . . . . . 563.1.2 Cooling of spins . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.2 Open quantum systems and spin boson model . . . . . . . . . . . . . 57


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8 Contents

4 Solvable model 61

4.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.1.1 Heisenberg equations and their exact solution . . . . . . . . . 634.1.2 Factorized initial conditions . . . . . . . . . . . . . . . . . . . 644.1.3 Correlated initial conditions . . . . . . . . . . . . . . . . . . . 66

4.2 Pulsed Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.2.1 Parametrization of pulses . . . . . . . . . . . . . . . . . . . . 70

4.3 Setup of pulsing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.4 Realizations of the model . . . . . . . . . . . . . . . . . . . . . . . . 71

4.4.1 Why do we insist on the exact solvability of the model . . . . 724.5 The spectral density . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.5.1 Ohmic spectrum of the bath . . . . . . . . . . . . . . . . . . 734.5.2 Other types of environment . . . . . . . . . . . . . . . . . . . 74

4.A Correlated versus factorized initial conditions . . . . . . . . . . . . . 764.B Quantum noise generated by ohmic bath . . . . . . . . . . . . . . . 78

5 Work extraction 81

5.1 Definition of work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.2 General restrictions on work extraction . . . . . . . . . . . . . . . . . 82

5.2.1 Efficiency of work-extraction . . . . . . . . . . . . . . . . . . 845.3 Work-extraction via two pulses . . . . . . . . . . . . . . . . . . . . . 85

5.3.1 Formulas for work . . . . . . . . . . . . . . . . . . . . . . . . 855.3.2 Work extraction for T > TS . . . . . . . . . . . . . . . . . . . 875.3.3 Work extraction for T < TS . . . . . . . . . . . . . . . . . . . 895.3.4 Efficiency of work extraction . . . . . . . . . . . . . . . . . . 90

5.4 Work-extraction via spin-echo pulses . . . . . . . . . . . . . . . . . . 915.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.5.1 Lasing without inversion . . . . . . . . . . . . . . . . . . . . . 955.5.2 Quantum heat engine . . . . . . . . . . . . . . . . . . . . . . 96

5.A Derivation of Eqs. (5.9, 5.10) . . . . . . . . . . . . . . . . . . . . . . 97

6 Cooling of spins 996.1 Cooling via pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 996.2 Cooling using spin-echo pulses . . . . . . . . . . . . . . . . . . . . . . 1016.3 1/f spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1026.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

III Spin Glasses 105

7 Introduction to spin glasses 107

7.1 Theory and mean field models . . . . . . . . . . . . . . . . . . . . . . 109

8 Transition temperature in metallic spin glasses 113

8.1 Concentration dependence of the transition temperature . . . . . . . 113


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Contents 9

9 The Ising Spin-Glass on a Hypercubic Cell 117

9.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

9.2 Simulation and Data Analysis . . . . . . . . . . . . . . . . . . . . . . 1189.2.1 Exchange Monte Carlo . . . . . . . . . . . . . . . . . . . . . . 1189.2.2 Multispin coding . . . . . . . . . . . . . . . . . . . . . . . . . 1229.2.3 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

9.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1299.3.1 Phase Transition . . . . . . . . . . . . . . . . . . . . . . . . . 1299.3.2 Low temperature behaviour . . . . . . . . . . . . . . . . . . . 1319.3.3 Evidence for Ultrametricity . . . . . . . . . . . . . . . . . . . 135

9.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

Bibliography 137

Samenvatting 147

Acknowledgements 151


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Introduction

The last centuries showed the birth of, among others, statistical mechanics andquantum mechanics; theories which are the basic building bricks of todays physics.By now we are familiar with concepts like phase transitions, critical phenomena, orquantum coherence; and with quantities like the partition function or the densitymatrix. It is during the last two centuries that physicists realized that a completedescription of reality is intractable from a mathematical point of view. Not onlythis, they realized that it was unnecessary, as is the case of statistical mechanics, orthat it was impossible, as is the case for quantum mechanics. Indeed, the answerto problems in physics changed from a given deterministic solution to a solution interms of averages, or a set of probabilities for each of the possible events. Commonly,

full solutions, even in terms of averages, were not possible since the mathematicsat disposal was not sufficient to tackle the forthcoming problems. Approximationswere called for.

Presently theoretical physics is mainly based on approximations. We can onlyfind approximate solutions, or at most, solutions valid in a restricted range of theparameters involved in the problem. In spite of that, these solutions can give usmost of the needed information and understanding. Another successful approach isto use simplified models which, however, keep the key features of more complicatedand more realistic ones. From these toy models, we can infer the behavior ofrealistic models, we can test general ideas like, for example, critical phenomena and

universality. This approach is, in fact, followed in the first part of this thesis.This theoretical development was followed, or even sometimes pushed, by ex-

perimental development. For instance, the range of temperatures of contemporaryexperiments is much larger than only a few decades ago. Quantum effects are atthe moment seen in many laboratories in the world. Very low temperatures, ap-proaching T = 0, are now accessible and controllable. Understanding the physicsat such low temperatures is important and new problems and worries arose fromthat. New phases of matter were discovered, such as Bose-Einstein condensates.New phase transitions were found, which brought a better understanding of criticalphenomena. In addition, a new type of transitions was proposed, quantum phase

transitions. Because they occur strictly at T = 0, they are rarely observed di-rectly, but their presence can be inferred indirectly since many effects in very lowtemperature physics arise from lying in the vicinity of a quantum critical point.

One of the most interesting features discovered in the last century, was the exis-


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12 Introduction

tence of spins. We describe spins in many different ways depending on the purposeand the situation. For instance, spins are responsible for magnetic effects. Many ap-

proximations were needed to understand their behavior. Often, in a magnet, detailson the dynamics of each of the spins are not of leading importance. Furthermore,the crystalline structure can break the symmetries at disposal, leading then, in thecase of spins 1/2, to Ising or XY spins. The interactions between each pair of spinsthroughout the lattice can sometimes be treated as classical, though their originis purely quantum mechanical. In spite of all these approximations, the problemstypically remain unsolvable. To overcome that, several approximations and gener-alizations in the algebra describing spins have been performed, during many yearsof study. A very successful generalization is the so-called spherical spin approxima-tion, to which the first part of this thesis is devoted. A quantum spherical model

is presented and solved there in order to study quantum phase transitions. This isone of the simplest models showing non-trivial critical phenomena. We will showhow spherical spins can arise as a limit of Heisenberg spins and that leads to ananalytically tractable problem. With this model we can analytically study criticalphenomena of a quantum phase transition, i.e., we can calculate exactly the in-volved critical exponents, the critical amplitudes and the relation with the classicalcounterpart.

As technology improved, new ways to study spins appeared. Nowadays, forinstance, it is common to deal with SQUIDs (superconducting quantum interference

device) which can be understood as a two level system, thus analogous to a spin1/2. In this situation, with only one SQUID, no spin-spin interaction is present butthe dynamics play an important role. Furthermore, these types of systems cannotbe isolated from the external world around them, therefore the interaction with theenvironment must be taken into account. Thus we jump from a static approach,i.e. in equilibrium, like the one we often have in magnetism, towards a dynamicalone. Many other systems can be approached in this manner. In nuclear magneticresonance (NMR), one deals with an ensemble of spins that can, sometimes, beconsidered as non-interacting. There, by using very strong magnetic pulses, it ispossible to tune the dynamics of the spin up to the point to be able to perform

controlled rotations on the Bloch sphere. Experimentally the dynamics is analyzedin free induction decays (FID). The environment also plays an important role here.The dynamics of the system is biased by the surroundings of the sample, bringingdecoherence and damping. This is the subject of part II (chapters 3 to 6) of thisthesis. In this part, an exactly solvable limit of the spin-boson model is presentedand solved. The model is a description of the dynamics of an ensemble of spinson which NMR pulses are applied. As in the previous part, the approximationsare performed when posing the problem, and not half way as is most frequentlydone. This reduces its applicability but in exchange, it leads to an analyticallysolvable model. The first objective of this part is to use this model to study new

mechanisms of work extraction, for instance lasing, in such microscopic systems.These techniques are expected to have a number of advantages over the standardones. The second objective is to cool spins, i.e. to obtain purer spin states, sincethe efficiency of, e.g., NMR studies depends strongly on the initial polarization of


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13

the spins. Enhancing that quantity enhances the output signal in the experiment.The problem of cooling is well-known. However, some important objectives are not

yet met with the existing methods. This led us to propose a new method of cooling,where the bath is involved in an essential manner.

Advances in the fields of chemistry and physics of materials allowed precision cre-ating new materials. For instance, creating alloys of different metals at a given pre-cise concentration of their constituents. Examples of that are for instance Au1cFec,a transition metal embedded in a noble metal, which for certain concentrations c isa spin glass. The discovery of spin glasses boosted an at that time young branch ofstatistical mechanics, the study of disordered systems. Indeed, spin glasses can beunderstood as disordered magnets which at low temperatures condense in a frozendisordered configuration instead of a periodic one.

Part III (chapters 7 to 9) is focused on spin glasses. In this part our aim is moremodest than to start from a simplified Hamiltonian and evolve analytically till theend. In chapter 8 departing from the fact that at finite temperature the range ofthe RKKY interaction is cut-off, we give a description of the dependence of thetransition temperature on the concentration of magnetic impurities in metallic spinglasses.

In chapter 9, finally, another way to tackle intractable problems is considered,namely computer simulations. With the help of computers, simulations of unsolvablemodels can be carried out and important results can be found. This brings a wholenew dimension to physics, since the problems are no longer the unsolvability but, forinstance, the too small size of the simulated systems that has to be chosen, because acomputer has finite memory and limited speed. In chapter 9 a numerical simulationof a spin glass on a hypercubic cell is presented. With this simulation we study howthis finite size and finite connectivity model approaches the well known infinite size,infinitely connected mean field model of the spin glasses.


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Part I

Spherical Model


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1

Phase transitions and criticalphenomena

In this chapter we introduce fundamental concepts of phase transitions and criti-cal phenomena. There are many good text books on critical phenomena where thereader can find an extensive treatment of the problem. In order to write this chap-ter the author has mainly benefited from Refs. [Ma82, Bel91, Sac99, NO98, ID89,BGZJ76] to which the reader is refered for further reading.

When describing a piece of material, we are mainly interested in quantities suchas, the energy, the mass, the total magnetic moment, or the corresponding intensivecounterparts, the energy density, mass density, magnetization. These are refered toas mechanical variables [Ma82]. In most cases they are properly defined once wecharacterize the environment of the material of interest, say, we know the externalfields applied to it, the temperature, magnetic field, pressure, etc. These externalfields define the parameter space, and the response or state of the system in each ofthe points of this parameter space define the phase diagram. At some points or lines

of the different external field variables, the system undergoes a phase transition. Atthis stage, one of the mechanical variables, or a combination of them, will define anorder parameter. The order parameter is a variable which is zero in one phase andnon-zero in the other. At the transition it starts to grow. Most studies in criticalphenomena are devoted to find the order parameter or to understand how it evolves.Examples of order parameters are for a ferromagnet: the magnetization, for a liquidgas transition: the relative density with respect to the one at the critical point, forsuperconductors: the complex field density amplitude of Cooper pair bosons.

Since in this thesis we study the critical phenomena of magnetic systems, weproceed using them as the example for defining the critical exponents.

The different critical exponents are defined as:

: Specific heat divergence [BGZJ76]:


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18 I.1. Phase transitions and critical phenomena

At zero field h = 0, the specific heat diverges at Tc as a power law

C =

A (T Tc)

+ B for T > Tc,A(Tc T) + B for T < Tc.

(1.1)

: Order parameter m as a function of temperature T:

For zero external magnetic field, h = 0, the magnetization is a decreasingfunction of T and vanishes at Tc

m (Tc T) for T < Tc. (1.2): Magnetic susceptibility divergence:

The magnetic field susceptibility = mhT

diverges at h = 0 at the transitiontemperature

=

C(T Tc) for T > Tc,C(Tc T) for T < Tc.

(1.3)

: Order parameter m as a function of the external magnetic field h:

At the transition temperature, Tc, the magnetic field follows a power law with

the external magnetic field, vanishing for vanishing field

m |h|1/ for T = Tc. (1.4): Spatial dependence of the correlation function Gij

Near the transition temperature Tc the correlation function

Gij = SiSj SiSj, (1.5)can be expressed for long enough distances rij = ri rj

G(r) =g(r/)

rD+2, G(q) q2+. (1.6)

where G is the correlation function in Fourier space, D corresponds to thedimension of the system, g is a function that at r decays exponentiallyg(r/) exp(r/), and is the correlation length.

: Divergence of the correlation length

At the transition temperature T = Tc and for vanishing field h = 0, thecorrelation length diverges

=

D (T Tc) for T > Tc,D(Tc T) for T < Tc.

(1.7)


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19

z: Divergence of the correlation time

At the transtition temperature T = Tc and for vanishing field h = 0, thecorrelation time diverges

z |T Tc|z ; |T Tc|z , (1.8)

where corresponds to the characteristic energy fluctuation.

Each system has its own critical exponents. However, although being very dif-ferent, many systems share the same critical exponents. Such systems are said tobelong to the same universality class. At the critical point and in nearby surround-ings, the scaling limit can be taken. The scaling limit of an observable is definedas its value when all corrections involving the ratio of small to large lengths areneglected. In the case of magnetic spins on a lattice, the small length would be thelattice spacing a, while the large ones would be the correlation length and thesystem size. The assertion of universality is that the results of the scaling limit arenot sensitive to the precise microscopic model used. This can be seen as a formalconsequence of the physically reasonable requirement that correlations at the scaleof large should not depend upon the details of the interactions on the scale of thelattice spacing, a. In this situation, we can map our Hamiltonian onto a continuumfield theory that represents the universality class. Typically, only essential quali-tative features, such as the symmetry of the order parameter, the dimensionalityof the space, and constraints placed by conservation laws, survive the continuumlimit, and the structure of the quantum field theory is severely constrained by theserestrictions.

The critical exponents depend strongly on the 0 1/2 1 3 0 1/2

Table 1.1: The mean fieldcritical exponents

spatial dimension of the system. In next chapter

we will solve anallytically the critical phenomena ofthe quantum spherical model for arbitrary spatialdimension. We will see there that below a certaindimension the phase transition does not occur any-more, which is a general phenomenon. The dimension where this happens is calledlower critical dimension. Qualitatively this can be understood because cooperativeeffects are not strong enough to overcome the disorder created by temperature. Onthe contrary, above a certain dimension, the cooperative effects are so strong thatfluctuations can be ignored and the mean field approximation turns out to be valid.This dimension is called upper crticial dimension. The system then has the mean

field critical exponents, see table 1.1. Between the upper and the lower criticaldimension, there are the hyperscaling relations, i.e. the critical exponents are func-tions of the space dimension and relations among them can be found in terms of thisdimension d. Some examples can be


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=

2 , (1.9) = 2 d, (1.10) =

1

2(d 2 + ), (1.11)

=d + 2 d 2 + . (1.12)

So far in these definitions we were considering classical critical points. Classi-cal critical points occur at finite temperature. In such situations, all fluctuations

have a thermal origin. No matter how quantum the system behaves, for instance thetransition to a superconductor phase, as long as the transition occurs at a finite tem-perature, thermal fluctuations are responsible for the transition and the transitionis classical. On the contrary, quantum phase transitions occur at zero temperature,T = 0, [Sac99, Voj00a, Voj00b]. Classical systems at zero temperature usually freezeinto a fluctuationless ground state. In contrast, quantum systems have fluctuationsdriven by the Heisenberg uncertainty principle even in the ground state. These areresponsible for the quantum phase transitions since in such situations, temperatureand the corresponding thermal fluctuations cannot play a role, and the only possiblesource of fluctuations is quantum mechanical. A simple model that exhibits a quan-

tum phase transition is the Ising model with transversal field. This model reproducesthe main ideas of quantum phase transitions. We will reproduce it qualitatively herenot only for pedagogical reasons, but also because it was a source of inspiration forthe work described in the following chapter.

The Ising model is one of the simplest models to study magnetism. It consists ofpairwise interactions among the Ising spins. Ising spins can only take 2 values, +1and 1. Physically, they describe strongly anisotropic magnetic moments on whichthe typical 3 directions available to the microscopic magnetic moment, only one isavailable leaving as the only degree of freedom the orientation, whether positiveor negative. A physical realization of such model are the low-lying excitaitons of

the insulator LiHoF4 which consist of fluctuations of the Ho ions between two spinstates that are aligned parallel or antiparallel to a particular crystaline axis [Sac99].Dipolar interactions among these ions at low temperatures align all spins parallelforming thus a ferromagnet. We will simplify these interactions to only nearestneighbours. For this study of quantum phase transitions the relevant situation ishaving a transversal field, say in the x-direction. The Hamiltonian reads

H = Jij

zi zj g

i

xi , (1.13)

where J is the coupling constant, positive for ferromagnetic couplings, i.e. when thespins tend to aling with respect to each other, g > 0 is a coupling proportional to theexternal magnetic field, and ai are the Pauli matrices (a = x,y,z) that correspondto the spin on the lattice site i with z =diag(1, 1). The transversal field induces


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I.1.1 Path Integrals 21

quantum tunneling between the two states of each Ho ion, and a sufficiently strongtunneling rate can eventually destroy the long range order.

In the limit of zero temperature and zero external field, T 0, g 0, thesystem finds itself in its ground state: all spins point parallel or antiparallel to thez direction. Conversely, at zero temperature but in the limit of very large magnetictransversal field, T 0, g the ground state consists of having all spins pointingin the x direction. Then by tuning the transversal field g, one is able to go from aphase where all spins are ordered in the z direction towards one where the spins arepointing mostly to the x direction. In between these two limiting situations theremust be a critical value of the external field g = gc where the transition takes placesince the system cannot analytically go from one situation to the other.

In this quantum critical point, all

Paramagnet

T

Tc g

ggc

T

Figure 1.1: Phase diagram exhibiting aquantum phase transition at T = 0, g = gc

definitions of critical exponents madeabove must be reformulated. In this sit-uation, the temperature plays the rolethe transversal magnetic field was play-ing in the classical critical point. Figure1.1 a typical phase diagram is sketched.One can see how temperature pulls thesystem away from the critical point inthe same manner the transversal mag-

netic field g does in the classical crit-ical point. The role played before bytemperature is now played by the con-trol parameter of quantum fluctuations,which drive the phase transition. In this case it is the transversal magnetic field g.Then, in all critical exponent definitions above, temperature has to be changed bytransversal field, T g and Tc gc. On the contrary, since the longitudinalmagnetic field h always helps the transition in the same way, its role is unchanged.

Using renormalization group arguments, Hertz [Her76] proved that the criticalexponents in the quantum critical point are related with their counterparts in the

classical critical point. The relation is set by the dynamical critical exponent z. Thequantum critical point behaves as the classical one for dimensions D = d + z. So theupper and lower critical dimensions are shifted down in the critical point by z andthe hyperscaling relations in between these two dimensions are found by changing dby D.

1.1 Path Integrals

In this section we give a brief introduction on how to compute the partition function

of a quantum system using Feynman path integrals [FH65]. To find the partitionfunction, using one technique or another, is usually the starting point of any cal-culation in statistical mechanics. Here, we will only focus on finding the partitionfunction of a system of bosons in their coherent state representation. This tech-


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nique will be exploited and extended in the following chapter. The reader is referedto e.g. [NO98] or [Sto] for a review of path integrals and coherent states and to

e.g. [KS85] for a complete study of coherent states. These books served as sourceand inspiartion for the elaboration of this section.

1.1.1 Bosonic coherent state representation for a single oscil-lator

Fock space is the Hilbert space of states labeled by the number of oscillator quanta.Coherent states are defined as the eigenstates of the anihilator operator a. Then itcan be proven that for a system with many particles

| = eP

a | 0 =

n

(a)n

n!

| 0 , (1.14)

is a coherent state, where | 0 is the vacuum representation in Focks space, and stands for an index of a mode of the system. Indeed, because of the identitya(a

)n = n(a

)n1 + (a)

n a, it holds that a | = | . The scalarproduct of two coherent states gives

| = eP

. (1.15)

A crucial property of the coherent states is that they form an overcomplete set ofstates. Any vector in Fock space can then be expanded in terms of coherent states.This is expressed by the closure relation [NO98]

d()d()

eP

| | = 1, (1.16)

where the measure in the integral comes from gaussian integration with complexvariables ( stands for imaginary part and for real part and these parts areintegrated from

to

) and the exponential term is due to the fact that coherent

states are not normalized. Let us check that eq. (1.16) is indeed a representation ofthe identity of Fock space. We insert it in the left hand side of eq. (1.15) and we get

| 1 | =

d()d()

eP

| | =

d()d()

eP

(

) = e

P

,

(1.17)

which indeed is the right hand side of eq. (1.15).The partition function of any quantum system Z = tr

eH(a

,a)

can be com-

puted by the Trotter approach. The exponential has the same form as a time evo-lution operator in imaginary time; thus it is possible to create a path integral over


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I.1.1 Path Integrals 23

closed paths. The procedure is to split the exponential in a product of M equalterms. Between each pair of them a representation of the identity, eq. (1.16), is

inserted. The partition sum then has the following shape

Z = tr

eH(a

,a)M

= tr

eH(a,a) 1 eH(a

,a)1 1 eH(a,a)

, (1.18)

where = /M and each 1 is the above identity operator. Each of these identities isgiven an index; they represent the steps the system passes through in a discretizedpath. By using the identity defined in Eq. (1.16) only the following matrix elementis needed in the calculation:

j | eH(a,a) | j1 . (1.19)Provided the Hamiltonian is normal ordered, the outcome is [NO98]

j | eH(a,a) | j1 j | 1 H(a, a) | j1 =e

j j1(1 H(j , j1)) = e

j j1H(

j ,j1) + O(2).

(1.20)

Correction terms can be neglected in the limit M [NO98]. Each identitybrings an integral at each time step. These integrals cover any path between its

initial and its final state. The trace will finally tie the ends giving a closed path.The partition function finally reads

Z =

()=(0)

D(()())

exp

=0

d

() d()

d+ H((), ( d))

,

(1.21)

where the subindex of the integral reflects the trace structure of the partition functionsince it gives a closed path integral; stands for the imaginary time step, so () =i; d is the imaginary time difference between steps, so ( d) = i1; and

d()

d=

() ( d)d

=i i1

/M. (1.22)

Despite the fact that the nomenclature used in these formulas suggests a continu-ous time, it should always be understood as being discrete. The limit M shouldalways be taken at the end of the calculations, otherwise some indeterminacies mayarise. Continuous notation is used nevertheless because it is more compact.


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2

Quantum spherical model

In this chapter we are going to study the quantum spherical model. This model is aperfect ground up to understand the concepts of the previous chapter and generallythe ideas of critical phenomena, since it is an exactly solvable model that reproducesnon-trivial critical phenomena. The fact of being quantum mechanical allows us tostudy quantum phase transitions analytically. A definite expression is found for thedifferent critical exponents in terms of the dimensionality of the system.

Originally the spherical model, in its classical version, was introduced by Kac.After being introduced in 1947 to Onsagers rather intricate solution of the 2d-Isingmodel, he desired to formulate a simpler spin model. As a first step he took thespins to be continuous Gaussian variables, nowadays called the Gaussian model.This had unphysical behavior at low temperatures which led Kac to consider thespherical model. The spherical model has continuous spins that are restricted by thespherical constraint

Ni=1 S

2i = N, which represents the hypersphere intersecting all

vertices of the hypercube sustained by the Ising spins, Si = 1. In the end, thespherical model is formally the same as the Ising model, i.e. the Hamiltonian is the

same, with a global constraint instead of a local one: the sum of the square of thelength of the spins is constrained rather than each of them. At that time the saddlepoint method, needed in the solution, was not widely known, and here Berlin camein, leading the celebrated joint publication on the spherical model in 1952 [BK52].Kacs personal reminiscence of this history is presented in Ref. [Kac64].

The spherical model for a ferromagnet has been considered in great detail inliterature. Actually, the paramagnetic to ferromagnetic transition is similar to anideal Bose-Einstein condensation. As already pointed out, the critical behavior canbe solved exactly. Critical exponents and scaling functions can be derived. In par-ticular, the model with short range interactions exhibits dlc = 2 as the lower critical

dimension; for d 2 no stable ferromagnetic phase occurs. Likewise, duc = 4 is theupper critical dimension; for d > 4 critical exponents take their mean-field values,see previous chapter. These analytic results have been used to test approximationsand general ideas of phase transitions for a wide range of interactions, short and


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26 I.2. Quantum spherical model

long range. For a review on the classical spherical model see Ref. [Joy72].As said, the spherical model was introduced for its mathematical simplicity.

However, Stanley [Sta68] proved that the free energy of a model of arbitrary spindimension , incorporating thus the Ising model (for spin dimension = 1), thex y model (spin dimension = 2) and Heisenberg model ( = 3), approaches thatof the spherical model in the limit of infinite spin dimensionality . Hence,it gives a geometrical interpretation to the spherical model. Since various criticalproperties where proven to be monotonic functions of the spin dimensionality , thecritical properties of the Heisenberg model appeared to be bounded on one side bythose of the Ising model and on the other by those of the spherical model.

The spherical model for antiferromagnets was studied by Knops. The sphericalconstraint imposes < S2i >= 1 for ferromagnets. However, this does not work forantiferromagnets because of the lack of translational invariance. To recover this,Knops added a second constraint; more generally, one constraint has to be addedfor each translationally invariant set, which in the case of antiferromagnets meanseach of the two sublattices. He found that the two constraints reduce to a uniqueone provided the staggered external field is zero. The fact that the spherical spinsare scalars makes it impossible to define an order parameter that can be identifiedwith the spontaneous staggered magnetization. To solve that and get the properorder parameter Knops used a vector version of the spherical model [Kno73a]. Healso generalized Stanleys arguments to non-translational interactions [Kno73b].

The spherical model has also been applied to disordered systems. Though, inview of Knops finding, perhaps an infinite number of spherical constraints shouldbe used, typically no analog of the staggered external field is applied, and one mayexpect that all constraints collapse into a single one. Therefore spherical spin glassmodels may still give insight in the physics of the problem which would be moredifficult to study e.g. with Ising spins. In the case of pair couplings the exact solutionexhibits no breaking of replica symmetry and the replica trick need not be used[KTJ76]. The family of p-spin spin glasses (p-spin models) [CS92] has been shownto exhibit one step replica symmetry breaking by studying the spherical version.For spin glasses with random pair and quartet interactions ({p = 2} + {p = 4},p = 2 + 4), Nieuwenhuizen showed that an exact solution exists, exposing the fullreplica symmetry breaking scenario. The simplicity of spherical models thus maygive insight in difficult problems for which otherwise no exact solution is available.For an early review on the use of the spherical model in disordered systems, seeRef. [KKPS92].

So far the discussion has been classical. The classicality can be understood inparticular because the entropy diverges at low temperature as ln T, just as for aclassical ideal gas. Different quantum versions of the spherical model have been pro-posed. In this chapter we will discuss the two main approaches and the differencesbetween them. The spins have to become quantum mechanical operators and their

adjoint operators also have to be included in the formalism. The spherical constraintcan then be kept the same though in operators language, as it was started by Ober-mair [Obe72], or, conversely, one can constrain both the total spin length as beforeplus its global kinetic energy, as developed by Nieuwenhuizen [Nie95a, Nie95b]. The


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I.2.1 Classical spherical model 27

fact of having the dynamics inside the spherical constraint allows one to considerHamiltonians without an explicit a kinetic part. Both approaches, though in differ-

ent setups, exhibit a quantum phase transition [Voj96, SN04]. Obermairs model,belongs to a universality class, i.e. O(n) non-linear sigma model for large n, whileNieuwenhuizens can, depending how the dynamics is considered, belong to the sameuniversality class or to another, i.e. SU(n) Heisenberg ferromagnet in the limit forlarge n.

2.1 Classical spherical model

The spherical constraint was conceived as a relaxation of the Ising constraint. In-

deed, Ising spins, Si si,z = 12, obviously satisfy it. Adjusting the coefficientsfrom the original version it may be written as

1

2

Ni=1

S2i = N , (2.1)

with = 2/8 having dimension (Js)2. So the spins are treated as real variablesconstraint by Eq. (2.1). The Berlin-Kac spherical model is defined by the partitionsum

Z =

DS eH (

1

2

Ni=1

S2i N ) =

DS

ii

d

2ieH

12PN

i=1 S2i+N (2.2)

where

DS =i

dSi. (2.3)

2.1.1 Vector spherical spinsFor vector spins the generalization of Eq. (2.1) in the case of m spin dimensionsreads

1

2

Ni=1

ma=1

(Sai )2

= Nm. (2.4)

It is worth mentioning that the spin dimensionality in eq. (2.4) is not related tothe approach of Stanley, who started with vector spins and ended up with scalar

spherical spins. We only introduce vector spherical spins to avoid the restrictionscalar spins have. We benefit from the fact that the vector character allows to studythe behavior in a transverse field. A similar step allowed Knops to define a properorder parameter for the antiferromagnetic spherical model [Kno73a].


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2.1.2 Quantization

It is natural to consider the Si analogous to position variables of harmonic oscillators.In quantum mechanics they become hermitian operators Si with the dimension of, Js. The conjugate momentum operator ai is dimensionless and postulated tosatisfy the commutation relation

[Sai , bj ] = ii,ja,b. (2.5)

As for harmonic oscillators, this allows to define creation and annihilation operators

a i =1

2Sai

i2

ai , ai =

1

2Sai +

i2

ai (2.6)

satisfying the commutation relation

[ai , b j ] = i,ja,b. (2.7)

2.1.3 Spherical constraint on the length of the total spin

There is some freedom to choose the spherical constraint, which amounts to describ-ing different physical situations. The standard quantum constraint considered inliterature is just the quantized version of the mean of eq. (2.4),

Constraint 1 :1

2i,a

< (Sai )2 >= Nm, (2.8)

where < ... > denotes the quantum expectation value. Obermair took as the quan-tum Hamiltonian the classical H(S) with spins replaced by operators, and addedthe kinetic term that one expects for physical rotors,

H(S, ) =1

2gi

2i + H(S), (2.9)

where g1 is the rotors moment of inertia. An effective Hamiltonian which includes

the constraint can be derived with a Lagrange multiplier. One ends up with

Htot =1

2gi

2i + H(S) + (t)[1

2

i,a

(Sai )2 N m], (2.10)

where is the Lagrange multiplier that enforces the constraint. In equilibrium itsvalue is given by the equation of the spherical constraint F = 0.

2.1.4 Spherical constraint on the number of spin quanta

The constraint can restrict additionally to spin length also the global dynamics ofthe system. This is possible by using

Constraint 2 :i,a

< a i ai >=

i,a

< nai >= Nm

2. (2.11)


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I.2.1 Classical spherical model 29

A constraint analogous to this one was introduced by Nieuwenhuizen [Nie95a,Nie95b]. The kinetic energy is also constrained here since this constraint includes

the momenta as well. This can be seen by writing it in the form

Constraint 2 :1

2

i,a

(Sa 2i + 2a 2i ) = Nm( +2

2). (2.12)

For a Hamiltonian H(S, ) that may, but need not, depend explicitly on the mo-menta, the effective spherical Hamiltonian is

Htot = H(S, ) +1

2i,a(S

ai )

2 + 2(ai )2

N m( +2

2). (2.13)

Now, the situation where the Hamiltonian does not depend explicitly on themomenta (no kinetic term), H(S, ) H(S), still leads to sensible dynamics,since the constraint already depends on the momenta. Different constraints describedifferent physics. However, at high temperatures one expects the differences tobecome small.

In the remaining of this paper we will simplify the notation by taking units inwhich = 1.

2.1.5 Comparison of the two constraintsThe main difference between the two constraints is obviously the presence or ab-sence of momenta. In the second case eq. (2.11), the spherical constraint can carryall the dynamics of the model. On the contrary, using the first constraint eq. (2.8),a kinetic term, with an external parameter g, has to be added to the Hamiltonian[Obe72]. This parameter determines the strength of quantum fluctuations; the clas-sical model can be recovered for g = 0. This fact makes models with the firstconstraint describe quantum rotors, as was pointed out in Ref. [Voj96]. The firstconstraint, eq. (2.8), brings actions which are invariant under orthogonal transforma-

tions. Conversely, using the second constraint, eq. (2.11), the choice of Hamiltoniancan bring symmetry under unitary transformations or orthogonal ones dependingon the question whether the Hamiltonian contains momenta or not. Hamiltonianswith unitary transformation symmetry yield free energies analogous to the large Nlimit of the generalization of SU(2) Heisenberg spins to SU(N). Hamiltonians withorthogonal transformation symmetry share the critical phenomena with the large

N limit of O(N) non-linear sigma model and describe therefore quantum rotors asoccurs by using the first constraint, eq. (2.8).

Each of the symmetries belongs to different universality classes in the quantumregime, yet classical critical phenomena are always the same as in the classical model,

consistent with the expectation that quantum effects do not lead to qualitativechanges at finite temperatures. We will see that the dynamical critical exponent zis different in both symmetries, causing the difference in critical exponents at thequantum critical point as was pointed out in Ref. [Her76].


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2.2 Ferromagnetic Hamiltonians with creation and

annihilation operatorsWe want to study the Hamiltonian

H(, ) = i=j

Jiji j

i

i(i + i)

2

= 12

i=j

Jij(SiSj + ij) i

iSi,

(2.14)

where in the second equality we inserted Eq. (2.6). The iSij cancelled since weassumed symmetric couplings, Jij = Jji . Obviously, the momentum operators dooccur in this expression. The couplings Jij can in principle express any kind of inter-action, ferromagnetic, antiferromagnetic, spin glass... The i represent an externalfield, that can be constant, variable, random... Later on, we will focus on ferromag-netic couplings in the presence of constant magnetic field. This Hamiltonian withoutthe external magnetic field is symmetric under unitary transformations, a fact thatwill determine the critical behavior.

The first step to get the partition function is to diagonalize the couplings,

i() =

()ei ,

() =i

i()ei ,

(2.15)

where ei is the normalized eigenvector of the coupling matrix Jij.

2.2.1 Coherent state representation for spherical spins

In this section we explain, following Ref. [NR98], how to add the spherical constraintto a quantum Hamiltonian using the path integral formalism for models with thesecond constraint Eq. (2.11). In second quantization the spins are given a bosonicalgebra. Then we can use the formalism introduced in section 1.1. We can deal withspherical spins using almost the same approach described there. The operator aiis identified with ai , where the index a denotes the spin vector direction, and thecorresponding fields i are denoted as

ai .

In order to impose this constraint, Eq. (2.11), in the path integral formalism,the identity definition Eq. (1.16) is modified to adopt to the spherical case, in a wayinspired by Ref. [Nie95a, Nie95b, NR98]: one restricts the path integral to states

which exactly satisfy the constraint by employing the truncated identity

1 1spherical C

ia

d(ai )d(ai )

e | | (n Nm), (2.16)


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I.2.2 Ferromagnetic Hamiltonians with creation and annihilation operators 31

where the number operator n,

n =i,a

a i

ai , (2.17)

counts the total number of spin quanta. We insert

(n N m) =

d

2ei(nNm) =

ii

d

2ie(nNm) (2.18)

where = i is imaginary. (Strictly speaking, we should insert a Kronecker- func-tion, rather than the Dirac-, but for large N this amounts to the same.) Repeatingthe same procedure with this new identity we get

Z =

()=(0)

DDD exp(A), (2.19)

with the action

A =

=0

d

() d()

d

+ ()(()

(

d)

N m) + H((), (

d))(2.20)

and integration measures defined as

DD =

ia

d(ai ())d(ai ())

, (2.21a)

D = C

ii

d()

2i, (2.21b)

so () is the Lagrange multiplier introduced to impose the spherical constraint andthe prefactor C(M) is added to ensure, if needed, a proper normalization. Details onthis factor were given in Ref. [NR98].

It should be noted that the particle number operator in the definition of theidentity eq. (2.16) will be surrounded, as is the case for the Hamiltonian, by spinoperators on different timesteps; therefore its creation and annihilation operatorswill also be projected on different timesteps, a i

ai a i ()ai ( d). In

Refs. [Nie95a, Nie95b, NR98] the spherical constraint was slightly different fromthe one presented here but equivalent up to an additive constant.

It is worth remarking that, up to now, we imposed the spherical constraint

strictly, no thermal average has been performed. In the following section will beintegrated over by the method of steepest descent. This approximation allows theparticle number to fluctuate and therefore the satisfiability of the constraint in theend remains only in average.


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2.2.2 General solution

We may write the partition function sum as a continuum expression,

Z =

D DD

exp

da,

a

()da()

d+ () (a

()a( d) N m)

Ja()a( d) 1

2 (

a() + a( d))

.

(2.22)

In discrete notation, the action of Eq. (2.19) reads

A =j

1

a,

a ,j

a,j a ,ja,j1

+ (j)

a,

a ,ja,j1 N m

a,

Ja ,j

a,j1

a,

(a ,j +

a,j1)

2

,

(2.23)

where = d is the imaginary time step, j the time index and = iie

i is the

field in the basis of eigenvectors of Jij. Collecting all terms we have

Z =

D,a

j

da ,jd

a,j

2i

exp

ij

a ,iBija,j +

j

(a ,j +

a,j1)

2

ePj Nm(j),

(2.24)

where Bij = ij (1 + J (j))i,j+1; here the prime stands for the fact that1,M+1 1 due to the trace structure of the partition function. We can now integrateover the spins

Z =

D exp

,a

m ln det Bij +

222

ij

Bij1 + m

j

(j)

. (2.25)

As usual, in thermodynamics, the saddle point value of one-time quantities like() can be taken independent of . We will employ this simplification throughoutthe rest of this chapter. The determinant and the matrix inversion can then be

performed [NO98]. Integrating over by the saddle point method we obtain

F = m + 1N

,a

ln(1 a) M

222(1 a)

, (2.26)


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where a = 1 ( J). Sending M we finally get

F = m + mN

ln(1 e(J))

2

2( J)

=

m( + 12

) + m

dJ(J)

ln

2sinh

2( J)

2

2( J)

,

(2.27)

where in the last equality we have assumed that the couplings satisfy 1N J = 0.

Otherwise, we would get an additional term 1/2mJ. The saddle point equationreads

+ 1 =1

N

1

1 e(J) +2

2( J)2

=

dJ(J)

1

1 e(J) +2

2( J)2

.

(2.28)

The sums over the different eigenvalues of the coupling matrix have been changedinto integrals. Each J has a weight in this integral given by (J). The actual form

for this weight function will depend on the type of couplings. A set of weightfunctions for ferromagnets in different cubic lattices can be found in Ref. [Joy72],and for spin glasses with long range interactions in Refs. [NR98, KTJ76].

At large temperatures these equations reduce to

F = m + m

dJ(J)

ln ( J)

2

2( J)

, (2.29)

+ 1 =

dJ(J) T

J +2

2( J)2

. (2.30)

Apart from a factor two, these are exactly the equations of the classical sphericalmodel, see e.g. [Joy72]. This factor two arises because the momenta double thedegrees of freedom, see e.g. [Nie95a, Nie95b]. Near the phase transition they arealready approximate, but the transition stays within the classical universality class.

2.2.3 Ferromagnetic couplings with transversal field in d di-mensions

In this section we will use the results given in the previous one for the concrete caseof ferromagnetic couplings with uniform transversal field. The Hamiltonian in thiscase differs from the one before Eq. (2.14) in the fact that the couplings only act


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in the z-direction while the external field only acts in the x-direction (we restrictourselves therefore to m = 2). The free energy reads

F = (2 + 1) +

ddk

(2)dln

2sinh

2( J(k))

+ ln

2sinh

2

2

2

(2.31)

and the saddle point equation

2( + 1) = ddk

(2)d1

1

e(J(k))+

1

1

e+

2

22, (2.32)

where we have applied the changes J J(k) anddJ(J) =

ddk

(2)d. (2.33)

We choose J(k) J0 J|k|x for |k| 0. In the case of short range couplings,for instance, one has x = 2 since J(k) =

Jcos ki J(0) 12J|k|2. A long range

coupling that decays as J(r) 1/r at large r gives x = d.As in the theory of Bose-Einstein condensation, the saddle point equation fixes

the dependence of on temperature. There should be a solution at any T. In

order to have a real free energy, cannot be smaller than the maximum value forJ(k). Therefore, we should investigate the convergence of the integral in the limit J0. If the integral diverges, must go to infinity before reaches J0 in orderto satisfy the saddle point equation, so there exists a for all temperatures and nophase transition occurs. If the integral converges, however, there will be a rangeof temperatures in which the saddle point as it stands cannot hold. This indicatesthat we have overlooked a macroscopic occupation of the ground state, as occurs inBose-Einstein condensation. The relevant integral behaves as

d

dk

(2)d 11 e(J0J(k)) d(2)d0

dkkd1 11 eJkx 0

dkkd1x (2.34)

where d is the hypersurface of a sphere in d dimensions. At k = 0, this integralconverges for d > x, hence there will be a phase transition for dimensions largerthan x.

At low temperatures, may get stuck at J0 and the saddle point equation as itis in Eq. (2.32) is no longer valid. This is because, as in Bose-Einstein condensationcalculations, the ground state is not properly included in the integral. It shouldbe taken out of the sum before this one is converted to an integral. This causesa change in the free energy by a factor (

J0)q, where q =

1

Nz

k=0

zk=0

is the

ground state occupation, and the saddle point equation becomes

2( + 1) =

ddk

(2)d1

1 e(J(k)) +1

1 e +2

22+ q, (2.35)


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q can be evaluated from the saddle point equation ( J0)q = 0. Thus when = J0 the occupation of the ground state can take non-zero values that can be

determined using Eq. (2.35). Hence the ground state occupation is macroscopic inthe ordered phase.

A transversal field will lower the transition temperature. Above a certain valuec, the transition does not exist anymore, thus T = 0, = c is a quantum criticalpoint, see chapter 1 or for a complete study over quantum phase transitions, see e.g.[Sac99]. We will now first study the classical critical point, at = 0.

Finite temperature phase transition

For the dimensions where the phase transition exists, the critical temperature is

found by solving the equation

2( + 1) =

ddk

(2)d1

1 ec(J0J(k)) +1

1 ecJ0 . (2.36)

The dependence of the chemical potential (the Lagrange multiplier introducedin Eq. (2.18) is equivalent to the chemical potential and both terminologies will beused) on temperature near the transition is the first thing needed. To get it, weexpand the saddle point equation around the critical point T = Tc+ , = J0 + .The integral gives, up to first order in and

ddk

(2)d1

1 e(J(k))

ddk

(2)d

1

1 ec(J0J(k))

+J0 J(k)

4T2c sinh2J0J(k)

2Tc

14Tc sinh

2J0J(k)

2Tc

.

(2.37)

The coefficient of is an integral that diverges for d 2x. This means thatfor these dimensions the leading term in the expansion of Eq. (2.37) has a powersmaller than one. For dimensions d > 2x we will have which will lead tothe mean field exponents, thus duc = 2x is therefore the upper critical dimension.To study the system near the critical point we subtract Eq. (2.36) from the saddlepoint equation, a procedure that will cancel the zeroth order term in the expansionin and , giving finally

ad 2x.


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ad>2x = Tc ddk

(2)d1

sinh2J0J(k)

2Tc

+

1

sinh2J02Tc

where

=

ddk

(2)dJ0 J(k)

sinh2J0J(k)

2Tc

+ J0sinh2

J02Tc

1

(2.39)

is a finite, positive number.The internal energy of the system reads

U = (2 + 1) +

ddk(2)d

J(k)2

coth

( J(k))2

+

2coth

2

2

2.

(2.40)

The specific heat close to the transition from the paramagnetic side can be writtenas

C

C0 +

xad2x C1 for d > 2x,

(2.41)

where

C0 =1

4T2

ddk

(2)d[ J(k)]2

sinh2J(k)2T

+ 24T2 sinh2

2T

, (2.42)

C1 = 2 1 +

ddk

(2)d

1

2coth

J(k)

2T

J(k)

4Tsinh2J(k)2T

+

1

2 coth

2T 4Tsinh2 2T +

2

22 ,

(2.43)

where ad are the prefactors in Eq. (2.38) for the corresponding dimension. In theordered phase is stuck in its minimum value ( = J0) for any temperature. Hence,C = C0( = J0) in the ordered phase. The critical exponent , see section 1 fordefintion of the critical exponents, is the expected one: = d2xdx for x < d < 2x,and the mean field value = 0 holds for d > 2x, which describes a jump in thespecific heat.

Adding a small longitudinal field h, the free energy reads

F = (2 + 1) + ddk(2)d

ln

2sinh

2

( J(k))+ ln

2sinh

2

2

2 h

2

2( J0)(2.44)


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and the saddle point equation becomes

2( + 1) =

ddk(2)d

11 e(J(k)) +

11 e +

222

+ h22( J0)2 (2.45)

By differentiating the free energy with respect to h can be seen that the magne-tization is

Mz =h

J0 . (2.46)

In the limit h 0, it is proportional to the square root of the occupation ofthe ground state, since by comparing Eq. (2.45) with Eq. (2.35) one finds q =

1Nz k=0zk=0 = M

2

z

2 . The factor 12 appears because it is actually the real part ofthe spin field that is macroscopically occupied. This factor comes in from the trans-formation Eq. (2.6). From Eq. (2.45), we can approach the transition by sending thelongitudinal field to zero at the critical temperature. The saddle point equation nowaccounts for the dependence of the chemical potential on the field. The calculationis similar, yielding finally

h 2dTc

(2)dJd/xx sin(dx)x

12

d+x2x

for x < d < 2x,

h

2dTc

(2)dJ2xln

12

32 for d = 2x, (2.47)

h

ddk

(2)d1

2Tc sinh2J0J(k)

2Tc

+1

2Tc sinh2J02Tc

12

3/2

for d > 2x.

Inserting these expressions in Eq. (2.46), the critical exponent is found to be = d+xdx for dimensions x < d < 2x and the mean field value = 3 is recovered for

d > 2x. From the magnetization, the susceptibility follows as 1 . Thereforewe find = xdx for x < d < 2x and = 1 for d > 2x. In the ordered phase, theexpansion of the saddle point equation, Eq. (2.45), for T near the transition yields

M2z 1

2T2c

ddk

(2)dJ0

J(k)

sinh2J0J(k)

2Tc

+

J0

sinh2J02Tc

. (2.48)

Since M2z for all dimensions where the phase transition exists, one has = 12 .


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For other critical exponents the correlation function is needed. It can be com-puted adding the right source term to the Hamiltonian, (g(q) + g(r))and differentiating

G(, q|, r) =T[z (q)z(r)]

=,2

g(q)g(r)

Z(g, g)

Z0

g=g=0

(2.49)

where T stands for the time ordered product. Z(g, g) is the partition function ofthe Hamiltonian including the source terms and Z0 is the partition function withoutthem. This procedure is carefully explained in Ref. [NO98] giving the result

G(k, ) = G(k, |k, 0) = e(J(k)) {( )(1 + nk) + ( + )nk} (2.50)where is a Heaviside step function and is a positive infinitesimal that indicatesthat the second term is the relevant one at = 0. Furthermore,

nk =1

e 1 (2.51)is the boson occupation probability, with = ( J(k)). Fourier transforming thislast result to Matsubara frequencies we get

G(k, n) =1

J(k) ink0 1

J|k|x + + in . (2.52)

So when we approach the critical point, we can see from this equation thatx , then using Eq. (2.38) we find that = 1dx for dimensions x < d < 2xand = 1x for d > 2x. = 2 x due to the fact that the couplings depend on kx,and z = x because in the denominator n appears as a linear term. Both and zare valid for any dimension.

This finally gives all the critical exponents of this finite temperature phase tran-

sition, which are exactly the same as in the classical model. This is expected fromrenormalization group arguments [Her76]. The critical behavior is controlled by aclassical fixed point, therefore quantum dynamics does not play a qualitatively newrole. Hence, the results are the same as in the classical spherical model [Joy72]or other models with different quantum dynamics considered at finite temperatures[Voj96].

T=0 quantum phase transition

In this section we analyze the behavior of the system at T = 0. As it can be seen from

Eq. (2.32), when the transversal field increases, the temperature of the transitiondecreases till it reaches zero. This defines a quantum critical point Tc = 0 at = c.In order to study it, an analogous procedure as before should be followed. At T = 0everything happens to be rather simple. The free energy reduces to


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F = 2 2

2 . (2.53)

The saddle point equation turns out to be

2 =2

22in the paramagnetic phase,

2 =2

2J20+ q in the ferromagnetic phase. (2.54)

where q = 12M2z is the occupation of the ground state for small transversal fields.

Since the temperature vanishes, quantum fluctuations, controlled by , give rise tothe phase transition. Therefore, the parameter that should be used to control thetransition is the transversal field and not the temperature. Then the proper analogof the specific heat will be proportional to the second derivative of the free energywith respect to the source of fluctuations, the transversal field.

C 2F

2= 1

+

2

. (2.55)

As before, we must know the dependence of ( = J0 + ) on the distance tothe critical point () in the paramagnetic phase. The upper critical dimension willbe duc = x. Between those two dimensions, 0 < d < x, the analysis of the saddlepoint equation, Eq. (2.32), yield that the product goes to a finite, strictlypositive value for T 0. This leads to a scaling form x/d. On the T = 0line, the analog of the specific heat goes continuously from the paramagnetic valueC 1/J0 + c/J20 (xd)/d to the simple ferromagnetic value C = 1/J0. Thisimplies that = (d x)/d. Adding a longitudinal field we find the dependence h2x/(d+2x) bringing = (d + 2x)/d and = x/d. Subtracting the saddlepoint equation near the transition in the ferromagnetic phase from the one at the

transition, we get q = (2 2c)/2J20 , which in the lowest order gives M2z andtherefore, as always, = 12 . Equation (2.52) can be used here once it is transformedto real frequencies, in = + i. Then we find = 1/d, = 2 x and z = x.

For dimensions d > x, from eqs. (2.55,2.54), it can be seen that the analog ofthe specific heat has a jump discontinuity, implying = 0.

C = 0 in the paramagnetic phase,

C =1J0

in the ferromagnetic phase. (2.56)

where the minus sign comes from the fact that the T = 0 free energy, Eq. (2.53), isnegative. Adding a small longitudinal field as before, we find the critical exponent = 3, since Mz h and h

23 at c. For the susceptibility, we find = 1


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since 1 1. As before we find = 12 and since we find = 1x , = 2

x and z = x. Hence we find always the mean field values. This occurs because

the quantum critical point of a d-dimensional model shares the critical exponents ofa classical critical point of a (d+z)-dimensional model, as it was shown by generalrenormalization group arguments [Her76].

Summary of the critical exponents

Finally all the critical exponents together read

Exponent Classical Quantum Classical (d > 2x)(x = dlc < d < duc = 2x) (d < duc = x) Quantum (d > x)

d2xdx dxd 0 12

12

12

xdxxd 1

d+xdxd+2xd 3

1dx1d

1x

2 x 2 x 2 xz x x x

Table 2.1: Critical exponents for the Hamiltonian with momenta

We can see how the third column corresponds to the table 1.1 for the normalnearest neighbors interaction where x = 2.

2.3 Hamiltonians involving spins but not their mo-menta

In this section we extend the analysis of section 2.2 to a Hamiltonian which onlydepends on the spin operators S and not on the momenta . When going from the

classical to the quantum model, we have to keep in mind that the Hamiltonian mustbe Hermitian. To be precise, the Hamiltonian we will deal with is

H = 12

ij

JijSiSj i

iSi (2.57)

with real valued Jij and i and where S = (+)/

2 is the real part of the former

spin field. Hence, the Hamiltonian does not involve momenta, but the sphericalconstraint does, see Eq. (2.11). This changes the symmetry of the problem frominvariance under unitary transformations to orthogonal ones. In terms of boson

creation and annihilation operators the coupling term for symmetric interactions,Jij = Jji , is proportional to Jij(2

i j + ij +

i

j), where we can notice the

symmetry of the problem. We will see that this model reproduces the O(N) quantumrotor model.


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I.2.3 Hamiltonians involving spins but not their momenta 41

We can get the partition function in many ways. A similar procedure usingdiscrete imaginary time path integrals can be done as before. This gives us many

problems due to the fact that creation and annihilation operators are projected ondifferent time steps which is a lengthy and tedious procedure. However, the formof the Hamiltonian makes it suitable to apply a Bogoliubov transformation (fordetails see e.g. [Aue94]). Due to that, we get the same Hamiltonian as before butwith different coefficients. In order to do so, we must add the spherical constraintdirectly to the Hamiltonian. The procedure is as follows: first the couplings matrixis diagonalized as done before in Eq. (2.15) and then the Ss are shifted to absorbthe field term, i.e. S S J . This finally gives

H =

( J2 ) J4 ( + )

2

2( J)

Nm. (2.58)Performing the Bogoliubov transformation it turns into

H =

( J)

22( J)

Nm, (2.59)

where the canonical transformation consists of an imaginary rotation

= cosh + sinh ;

= cosh

+ sinh (2.60)

where are real and even in . The condition to get 2.59 from 2.58 is that

tanh2 =J

2

J2 . (2.61)

Finally Eq. (2.59) is a Hamiltonian analogous to Eq. (2.14). So it can be diago-nalized as explained, giving finally

F = m( + 12

)

+ m

dJ(J)

ln

2sinh

2

( J)

2

2( J)

(2.62)

where we have put back the factor m standing for the number of components of thevector spin. The saddle point equation is obtained as

+1

2=

dJ(J)

2 J

4

( J)

coth

2

( J) +

2

2( J)2

. (2.63)

At large temperatures these equations reduce to

F = m( + 12

) + m

dJ(J)

ln

( J) 2

2( J)

, (2.64)


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+1

2= dJ

(J

) T

2+

T

2( J)+

2

2( J)2 . (2.65)These equations are very similar to the standard ones of the classical spherical

model (up to a factor 2), see Eq. (2.29), but they are only identical where theyshould be, namely at large T, where also T is very large, see also Ref. [Joy72].

2.3.1 Ferromagnetic couplings in the presence of a transversalfield

Analyzing the phase transition of the Hamiltonian that does not contain momenta is

analogous to the previous case. We begin again by choosing the coupling term in thez direction and the external field in the x and we assume ferromagnetic couplings.The saddle point equation gives a phase transition via a macroscopic occupation ofthe ground state, which in the present case is a bit more complicated. The criticalexponents are different, due to the fact that the symmetries of the system havechanged. The free energy reads

F = (2 + 1) + ddk

(2)dln 2sinh

2( J(k))+ ln

2sinh

2

2

2,

(2.66)

and the saddle point equation

4 + 2 =

ddk

(2)d2 J(k)

2

( J(k)) coth

2

( J(k))

+ coth

2

+

2

2.

(2.67)

We now analyze this model in detail.

Finite temperature phase transition

Following the same procedure as before we can find that the transition exists ford > x and that the upper critical dimension is d = 2x. The critical temperature isthe solution of

4 + 2 = ddk

(2)d

2J0 J(k)2

J0(J0 J(k))coth

2J0(J0 J(k))+ coth

J0

2 .(2.68)

The dependence of the chemical potential in the temperature near the classicalcritical point reads


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ad 2xad>2x =

ddk

(2)d2 J(k)

2( J(k))coth

2( J(k))+ coth

2 =J0

where

=

ddk

(2)d2J0 J(k)

2sinh2

J0(J0J(k))

2Tc

+ J0sinh2

J02Tc

1

. (2.70)

The internal energy of the system reads

U = (2 + 1) +

ddk

(2)d

( J(k))

2coth

2

( J(k))

+

2coth

2

2

2.

(2.71)

The specific heat has the same expression as in Eq. (2.41) where ad now corre-spond to the prefactors of Eq. (2.69) and with coefficients

C0 =1

4T2

ddk

(2)d( J(k))

sinh2

(J(k))

2T

+ 24T2 sinh2

2T

, (2.72)

C1 = (2 + 1) +

ddk

(2)d

2 J(k)

4

( J(k)) coth

2

( J(k))

2 J(k)8T sinh2

(J(k))

2T

+1

2coth

2T

4T sinh2 2T.

(2.73)

This is analogous to the previous model and gives the same exponent, = d2xdxfor x < d < 2x and = 0 for d > 2x. Adding a small magnetic field longitudinal to


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the couplings, the free energy becomes

F = (2 + 1) +

ddk

(2)dln

2sinh

2

( J(k))

+ ln

2sinh

2

2

2 h

2

2( J0) ,(2.74)

therefore the magnetization is Mz =h

J0, which is as before the square root of the

occupation of the ground state q = 1N < Sz(|k| = 0)2 >= M2z . The saddle pointequation is now

2(2 + 1) =

ddk

(2)d2 J(k)

2

( J(k)) coth

2

( J(k))

+ coth

2

+

2

2+

h2

( J0)2 ,(2.75)

With all these and following the algebra of the previous section one finds thesame critical exponents for the magnetization for the same dimensions since we arein the classical critical point.

The time ordered correlation function TSzk()S

zk(0) differs from the previous

one, Eq. (2.50) since in this case the S are not the variables that diagonalize theHamiltonian in Eq. (2.59). We must write S in terms of and then compute thecorrelations. This brings

G(k, |k, 0) = J(k)4

( J(k))

nk cosh

( J(k))

+ e||(J(k)),

(2.76)

where

nk =1

e 1 (2.77)

is just like before in Eq. 2.51 but with =

( J(k)). The correlation functionin frequency space reads

G(k,in) =J(k)

2[2

n ( J(k))]. (2.78)

When approaching the critical point we find that x as before and weget the same critical exponent = 1dx for dimensions x < d < 2x and =

1x

for d > 2x. Since couplings appear in the same way as before we also get the


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same critical exponent, = 2 x for all dimensions.. The difference appears in thedynamical critical exponent. Here n appears squared, therefore z = x/2. Here we

see how the model reproduces the critical exponents of the rotor model as in Ref.[Voj96] bringing thus a different behavior at the T = 0 quantum critical point fromthe model of section 2.2.3.

T=0 quantum phase transition

In this case the T = 0 phase transition the dynamical critical exponent z = x2 issmaller than z = x of the previous section. The free energy reads

F = (2 + 1) + ddk

(2)d( J(k))2 + 2

2

2 (2.79)

and the saddle point is set by

4 + 1 =

ddk

(2)d2 J(k)

2

( J(k)) +2

2. (2.80)

We find that the transition exists for dimensions larger than d > x2 and d =3x2

is the upper critical dimension. The chemical potential depends on the source offluctuations = c as

ad 3x2

ad=3x/2 =J202

22cJ30

ddk

(2)d

1

J0(J0 J(k)) 2J0 J(k)

4 (J0(J0 J(k)))32

where the s on the right hand side of the first equality are Eulers Gamma functions.The specific heat (see eq. (2.55) ) coming from the disordered region will behave as

C

1J0

4cad< 3x

2J20(2dx)

2d+3x2dx for x2 < d 3x

2 1

for d > 3x2 ,(2.82)

where ad is the prefactor in Eq. (2.81) for the proper dimension. Coming from theordered region, conversely, C 1J0 . Therefore = 2d3x2dx for x2 < d < 3x2 and = 0 for d > 3x2 .


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The dependence of a small longitudinal field on the chemical potential, in casethe transversal field is at its critical value, reads

h a 12d< 3x

2

2d+3x4x for

x

2< d 3x

2

12

32 for d >

3x

2.

From these equations and the ones for the magnetization and the susceptibility

we can find that = 2d+3x2dx and = 2x2dx for x2 < d < 3x2 , while = 3 and = 1for d > 3x2 . As before =

12 for every dimension. For the correlation function the

calculation is the same as in the finite temperature case, projected into real time,the exponents are = 22dx for dimensions

x2 < d 2x)(x = dlc < d < duc = 2x) (

x2 < d

3x2 )

d2xdx2d3x2dx 0

1212

12

xdx2x

2dx 1

d+xdx2d+3x2dx 3

1dx2

2dx1x

2 x 2 x 2 xz x2

x2

x2

Table 2.2: Critical exponents for the Hamiltonian without momenta

2.4 Generalization and mapping from Heisenbergspins

In this section we generalize the two preceding Hamiltonians and we map the Heisen-berg model onto the spherical model. In a more compact way, we can write the

former Hamiltonians in absence of external field as

H = ij

Aij

i j +

Bij2

i

j + ij

. (2.84)


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I.2.4 Generalization and mapping from Heisenberg spins 47

If the matrices Aij and Bij can be diagonalized simultaneously, the techniquesfrom previous sections can be used. The free energy reads

F = m( + 12

) +m

N

A

2+ ln

2sinh

2

( A)2 B2

(2.85)

where satisfies the saddle point equation

+1

2=

1

N A

2( A)2

B2

coth

2( A)2 B2

. (2.86)

The coefficient Bij in Eq. (2.84) is the responsible for a change in the symmetriesof the problem. IfBij is zero, the action is symmetric under unitary transformationswhile if it is non-zero the symmetry is reduced to orthogonal.

The mapping from Heisenberg spins comes as follows. The Hamiltonian can bewritten in terms of Schwinger bosons [Aue94]. The Schwinger boson transformationfor SU(2) spins reads

S+ = a1a2, S = a1a

2, S

z =1

2(a1a1 a2a2). (2.87)

This can be generalized to SU(N) spins and expand around the large-N limit[AA88]. In a path integral formalism for ferromagnetic interactions

H = 12

ij

JijSi Sj 12N

ij,mn

Jijajmaima

inajn, (2.88)

where i, j represent lattice sites and m, n represent the boson flavor. The Hilbertspace spanned by Schwinger bosons is much larger than the one given by Heisenbergspins. The constraint needed to restrict it to the physical Hilbert space is that

the number of Schwinger bosons at each site has to be kept fixedNm nm = NS.

This is inserted into the formalism in the same way as we have done it with thespherical constraint, a Lagrange multiplier i() appears. The biquadratic termscan be decoupled by a Hubbard-Stratonovich transformation (see e.g. [NO98]). Inthe case of a ferromagnet, the transformation at each time step and for each flavorin the path integral reads

exp

2Nij

Jijajaia

i aj

i,j

dQij exp

N2

i,j

QijJijQji 2

ij

QijJijajai

(2.89)


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where a field Qij() has been generated. In the mean field approximation, one putsQij() = Q and i() = . Hence, one gets up to a non interesting constant

HFMBMF (N) =i,m

aimaim Qij,m

Jijajmaim +

NQ2

2

ij

Jij NN S (2.90)

where we have already added the Schwinger boson constrainti,m a

imaim =NNS.

The free energy per particle reads

F =NNk

ln(1 e(QJ(k)) + NQ2

2J(k = 0) SN (2.91)

and the saddle point equations are

1

N

k

nk = S, (2.92)

1

N

k

J(k)nk = QJ(k = 0), (2.93)

where nk is the boson occupation number Eq. (2.51) with = QJ(k). Sub-tracting the two saddle point equations we can see that for large S and small Twe can approximate Q S recovering then the spherical model Eq. (2.27,2.28) forzero external field or Eq. (2.85,2.86) for Bij = 0. From this approach we thus seethat the free energy of a SU(N) Heisenberg ferromagnet for large N is formally thesame as the quantum spherical model proposed in Eq. (2.14) in the thermodynamiclimit, so when the radius of the hypersphere that defines the model (N in eq. (2.11)) is also very large. Thus the large N limit is analogous to Stanleys large spindimensionality limit.

In the case of an SU(N) antiferromagnet the procedure is more or less the samebut the symmetries are different. The lattice is divided in two sublattices A, B.

In one of the sublattices a spin rotation is performed that allows us to write theHamiltonian in the form [AA88]

H =1

2

ij

JijSi Sj 12N

ij,mn

Jijaima

imajnajn. (2.94)

Performing a Hubbard-Stratonovich transformation as before, the Hamiltonianwith the Schwinger boson constraint in the mean field approximation finally reads

H

AFMB

MF (N) =i,m a

imaim Q

2ij,mJij(a

ima

jm + aimajm)

+N Q2

2

ij

Jij NNS.(2.95)


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I.2.5 Summary and discussion 49

It is important to stress that here the SU(N) symmetry has been reduced to aresidual O(

N). The free energy per particle reads

F =NN

k

ln

2sinh

2

2 Q2J2(k)

N

S+1

2

+

NQ22

J(k = 0)

(2.96)

and the saddle point equations read

1

Nk

2 Q2J2(k)

nk +

1

2

= S+1

2 , (2.97)

1

N

k

J2(k)Q2 Q2J2(k)

nk +

1

2

= QJ(k = 0), (2.98)

where nk is Eq. (2.51) for =

2 Q2J2(k). Subtracting the first equation times from the second times Q we get

1

Nk

2 Q2J2(k)nk 1

2 = S+ 1

2 Q2J(k = 0). (2.99)

The first term is proportional to T, so for very small temperatures and very largeS, near the transition where QJ(k = 0), we can approximate Q S+ 12 . Theneqs. (2.96,2.97) are analogous to eqs. (2.85,2.86) for Aij = 0. This will have thesame critical behavior as the model in section 2.3 due to the fact that it comes fromthe term coth[

J(k)]/ J(k) which also appears here due to the equality

2nk + 1 = coth[

( + QJ(k))( QJ(k))].

2.5 Summary and discussionIn this chapter we have discussed a way of working with quantum spherical spinmodels using path integrals and coherent states. Some examples of the use of thisformalism are given, eqs. (2.14,2.57), and their critical phenomena are studied.We propose a comparison with SU(N) Heisenberg models that gives a geometricalinterpretation to the quantum spherical spins. The spherical constraint we use, fixesthe number of spin quanta , Eq. (2.11); in other words, it fixes both the averagelength square of the spin operator, S2, and the one of its conjugate momentum, 2.The usual version of the quantum spherical model, on the contrary, involves only the

spin part S. The presence of momenta in the spherical constraint allows to considerHamiltonians that have no kinetic term, since it can be induced by the constraint,a fact that can change the symmetries of the problem, and due to that, the criticalbehavior.


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The Hamiltonian in Eq. (2.14) yields an action invariant under unitary transfor-mations. It brings formally the same free energy as a SU(

N) Heisenberg ferromagnet

in the limit of largeN. The other Hamiltonian studied, Eq. (2.57), brings an actioninvariant under orthogonal transformations; it gives the same critical behavior asan SU(N) Heisenberg antiferromagnet in the limit of large N, which is, in its turn,analogous to an O(N) nonlinear -model or quantum rotor model [Voj96, Sac99].The main difference between these models lies in the dynamical critical exponentz which brings a different behavior at the quantum critical point. Classical criticalphenomena are, as expected, the same in both models and equal to those of theclassical spherical model.

In the formulation of the model, the strict spherical constraint has been usedwhere fluctuations on the particle number are not allowed. The constraint is added to

the action via a Lagrange multiplier. The strict approach has to be abandoned whenwe integrate this Lagrange multiplier since we cannot perform the int

Ruben Serral Gracia- Spin Bosons and Spin Glasses

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