Spin Chains Thesis Examples

Abstract

Spin systems are discussed into details in this thesis. It majorly focuses spin systems in quantum physics whereby elementary particles are majorly considered. The paper elaborates various properties of spin systems like how spin algebra can be applied in spin systems. The
Heisenberg Hamiltonian is also discussed, though with a great focus on the Heisenberg for integer and the half-integer, that is, where s= 1/2 and s=1. It furthers probes into the gap and gapless spin systems are looked into in the first chapter. Later, dimerization and the physical properties of spin chain, spin ladder, are looked into. Spin ladders are amazing, hence they are not left out while looking into the spin systems. Even and odd spin ladders are also included to help understand the spin ladders into details.
Chapter three features frustration and perturbation of spin chain and ladder and the quantum phase transitions in the systems. The paper has experimental considerations of both the spin chain and spin ladders. Finally, a strong conclusion ends the paper with relevant citations.

Chapter 1

Introduction
The interactions in particle systems between electrons can cause changes in the physical properties of the material. These changes include superconductivity and magnetism. Condensed matter Physicists have had their theoretical and experimental goals in understanding these properties of particle systems.
Low-dimensional magnetism is one of the most fundamental areas in condensed matter physics, at the simplest level, there is the uniform ID spin chain. It can be demonstrated by different behaviours for half-integer and integer spins with antiferromagnetic (AFM) coupling. For integer-spin chains, a spin-singlet ground state is observed with a gap in the magnetic excitation spectrum, as was first predicted by Haldane in 1983 and is therefore referred to as the Haldane gap. The uniform AFM chain with half-integer spin, on the other hand, behaves quite differently, possessing a gapless magnetic excitation spectrum.
Elementary particles in quantum physics usually have an intrinsic property known as spin. The ordinary matter is constituted by particles known as fermions. The elementary fermions usually have half-integer spin.

A system with one-dimensional line of spins forms a spin chain. When the individual spins are parallel or antiparallel to in a certain direction, they form an Ising spins. Alternatively, the individual spins may be a certain point in a fixed plane to form XY spins or they may be in any directions as the Heisenberg spins. Crystals usually experience spin chains (Doikou, 1999).
1.2 Spin Algebra
Usually an electron becomes a fermion having an intrinsic angular momentum with a spin quantum number S=1/2 in quantum mechanics. A solid material can have an individual atom obtaining a total magnetic momentum from the electrons configuration. Several particle systems with strong correlations have their collective behaviour understood by only thinking in terms of the physics of these localized magnetic moments while neglecting the remaining electronic degrees of freedom. The spin operators Six,Siy and Siz can be conform the following commutation relations (with ℏ = 1)
Siα,Sjβ=iδijϵαβγSiγ

While the anti-commutation relations,

Siα,Sjβ=12 δijδαβ
Spin operators commute on the different site and anti-commute on the same site. The state vectors existing in the spin-1/2 physics have two possible states of the vectors, “spin-up” and “spin-down”.
spin-up ↑ =10 , spin-down ↓ =01
The components of a single particle spin operators Siα=12 σiα where α=x,y,z are defined in terms of Pauli matrices σ1=0110 , σ2=0-ii0 , and σ3=1001

The components of the total spin operators, S, in the N particle system are the sum of single spin operators.

Sα=i=1NSiα σiα

Spin systems

The basic interactions for a collection of magnetic ions in a lattice can be represented by the equation H=Ji,jSi .Sj
Where J is the exchange interaction between the spins Si and Sj at lattice sites i and j respectively. The equation above forms the Heisenberg Hamiltonian (Dubbers and Stöckmann, 2013).

The Heisenberg Hamiltonian

Quantum spin operators can be used to describe phenomena like quantum magnetism which Heisenberg Hamiltonian forms the fundamental model. The nearest-neighbour interactions of localised quantum spins can be described by the Hamiltonian, H=Ji,jSi .Sj
Whereby Si is the spin operator at lattice site i,, i,j denotes the nearest-neighbour sites and J is the exchange coupling constant that provides the energy scale in the problem. The nearest neighbour interaction is the only thing taken into consideration in a regular Heisenberg Hamiltonian to have Jij=j=constant. The models become antiferromagnetic orderings with the antiparallel position of spins aligned along the same axis when J>0. The model becomes ferromagnetic orderings when J<0 with all spins aligned along the same axis (Greiter, 2011).
Spin algebra can be discussed with the help of the Heisenberg spin chain. The Hamiltonian of the system for the nearest neighbour interaction between the sites I and i+1 is expressed as H= i=1N{ Jxy SixSi+1x+SiySi+1y+JzSizSi+1z}
Where N is the number of sites in the chain. The long range orders in the isotropic chain with uniform antiferromagnetic couplings where Jxy = Jz is prevented due to the strong quantum fluctuations.

The arrows represent spins at each site i and the line represents the exchange coupling J.

When Jz =0 the model become the XY model and can be solved using the Jordan-Wigner transformation. The XY model of Heisenberg chain is defined byHXY=JijSixSjx+SiySjy

They act on the Sz eigen-states

s+↓=↑ s+↑=0
s-↑=↓ s-↓=0

Then we rewrite the Hamiltonian as

Hxy=J2i,j(si+sj-+si-sj+)
The Hamiltonian can thus be rewritten as Hxy=J2i,j(si+sj-+si-sj+)
They obey the Fermi commutation si-,si+=(si+si-+si-si+)=1 when the site is the same. Otherwise they obey the Bose commutation rules when they are at different sites. si-,si+=si-,si-=si+,si+=0 i≠j

And at different sites they obey the Bose commutational rules in that:

si-,si+=si-,si-=si+,si+=0 i≠j
The raising and lowering operators do not preserve their mixed set of commutation relations under a canonical transformation. For this reason, in 1928 Jordan and Wigner wrote the spin operators in terms of fermonic operators. The Jordan-Wigner transformation is used to change spin models into fermion models [9]. So, the JW transformation defines a new set of operators by multiplying the raising and lowering operators by a phase factor, which ensures that the commutation relations are preserved (Greiter, 2011).

After applying (JWT) the Hamiltonian can be write as

Hxy= J2 ici+1+ci+ci+ci+1
1.5 Review of Jordan-Wigner Transformation (JWT)

The JW transformation reads as [9].

si-=exp(-iπj=1i-1nj)ci
si+= ci+exp(iπj=1i-1nj)
siz=ni-12 ni=cici+
The operators ci and ci+ , are fermion creation and annihilation operators. In the JW transformation, the up and down spins correspond to the presence and absence of one fermion, ni=cici+ is the occupation- number operator.

One can check that the JW fermion operators obey the canonical anti-commutation relations

ci,cj+=δij , ci,cj=0 , and ci+,cj+=0

So we can write the operator at the same site

And for the different sites as
Now, we can rewrite the terms in the XY-Hamiltonian as:
si-si+1+=ciexpeiπj=1i-1nj-j=1injci+1+=-cici+1+=ci+1+ci

And

si+1-si+=ci+1expeiπj=1inj-j=1i-1njci+=-ci+1ci+=ci+ci+1
Hxy= J2 ici+1+ci+ci+ci+1
The JW fermion system is half filled, ci+ci=1/2 , because the ground state of the X Y model in ID has no net magnetization siz=ci+ci-12=0 (Kim et al., 2007).
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1.6 Haldane chains
Haldane conjectured that something different would happen for integer spin chains (i.e. chains of spins with S = 1, 2, 3,), that would be a gap in the excitation spectrum which occurs because of nonlinear quantum fluctuations in the ground state. A one-dimensional chain of integer spins is therefore known as a Haldane chain and the gap in the excitation spectrum is known as a Haldane gap. This fundamental difference between half-integer and integer spin chains is related to the difference between fermions and bosons under exchange; this different exchange symmetry has a topological origin and has a dramatic effect on the nature of the excitations. So, He conjectured that the spin-1 system is gapped whereas spin-1/2 system shows the gapless behaviour [7].
1.7 Dimerized spin system
The dimerized spin system is one of a gapped system in which dimerization the exchange coupling J along the chain by the parameter δ. where δ is the dimerization parameter. The coupling of a Heisenberg chain with dimerization has stronger and weaker bonds:
J→J(1+((-1)iδ)

Where i is the number of the site.

The arrows represent the spin at each lattice site, and thick and thin lines represent the exchange couplings J (1 ± δ) (Le Bellac, 2006).

For the structurally dimerized spin-1/2 antiferromagnetic Heisenberg chain, the Hamiltonian becomes

H=Ji1+(-1)iδ Si.Si+1
This Hamiltonian has no magnetic long range order (LRO). More detail about spin systems and Heisenberg spin systems is provided in Ref. [6].

References

Doikou, A. (1999). Integrable quantum spin chains.
Doikou, A. (1999). Integrable quantum spin chains.
Dubbers, D. and Stöckmann, H. (2013). Quantum physics. Berlin: Springer.
Greiter, M. (2011). Mapping of parent Hamiltonians. Heidelberg: Springer.
Jacak, J., Gonczarek, R. and Jacak, L. (2012). Application of Braid Groups in 2D Hall System Physics. Singapore: World Scientific Publishing Company.
Jung, P. and Rosch, A. (2007). Spin conductivity in almost integrable spin chains. Physical Review B, 76(24).
Kawamoto, H. (1989). Theoretical Study of Quantum Coherence in Spin-Boson System. Progress of Theoretical Physics, 82(6), pp.1044-1056.
Kim, W., Covaci, L., Dogan, F. and Marsiglio, F. (2007). Quantum mechanics of spin transfer in coupled electron-spin chains. Europhys. Lett., 79(6), p.67004.
Kruczenski, M. (2004). Spin Chains and String Theory. Phys. Rev. Lett., 93(16).
Le Bellac, M. (2006). Quantum physics. Cambridge: Cambridge University.
Nakahara, M. and Sasaki, Y. (2013). Quantum information and quantum computing. Singapore: World Scientific.
Shaikh Yasin, S. (2008). Electron spin resonance in low-dimensional spin chains and metals. [S.l.: s.n.].
Weiss, U. (2008). Quantum dissipative systems. Singapore: World Scientific.