Download Quantum spin

Subject Area: Algebra and Mathematical Physics
Title: Quantum Spin-Chains (Level 5 project)
Supervisor: Dr Christian Korff ([email protected])
Prerequisites: algebra, representation theory, interest in physical applications
WHAT? Quantum spin-chains are particular examples of exactly solvable or "quantum
integrable" systems in 1+1 spacetime dimensions. Picture a ring of atoms (in order to
have periodic boundary conditions) each of which possesses a quantum "degree of
freedom", called a "spin", which can point in two directions, up or down. "Quantum"
means that we allow for all complex linear superpositions of the different possible spin
configurations of the ring, this set forms the physical state space.
The dynamics of the system, i.e. how a particular state evolves in time, is governed via
the famous Schrödinger equation which involves the Hamiltonian, an operator over the
state space which encodes the microscopic interaction between the quantum spins. A
much studied example is the Heisenberg spin-chain
H  J ∑g x  xn  xn1  g y  yn  yn1  g z  zn  zn1 
n
Here σx,y,z are the Pauli-matrices (2 by 2 complex matrices) and the lower indices
indicate on which atom in the ring the matrices act. Obviously, the Heisenberg chain only
involves nearest neighbour interaction. The Pauli matrices "rotate" each spin in different
directions and depending on the coupling constants gx,y,z in front of each term certain
spin-arrangements are particularly favourable in the sense that they possess a minimal
energy. In order to compute these energies and the associated stationary states one has
to solve the eigenvalue problem of the above Hamiltonian. Before we turn to the
techniques and mathematical structures which arise in this context, there are first some
comments in order which explain the long-standing interest in these systems.
WHY? Many-particle systems (quantum or classical) are usually quite difficult to solve
and except for a few exceptions one faces often insurmountable difficulties in the
computations of physically relevant quantities. Take for instance, correlation functions
which encode the probability to find two spins in the ring separated by a distance x to be
aligned (either at a certain time or in a quantum statistical ensemble). Quite generally,
these quantities can only be computed numerically (running into difficulties with growing
numbers of atoms) or perturbatively (works only for small values of the coupling
constants, where the system is almost "free", i.e. close to a non-interacting ring of
atoms).
Both approaches are often in danger to miss out on essential physical effects. The
importance of "exactly solvable" or "quantum integrable" systems lies in the fact that they
provide non-trivial interacting systems where exact solutions can be obtained (for all
values of the coupling constant). These models hold therefore valuable insight into
transport phenomena, magnetic and electric properties, they are discussed as toy
models for quantum computers and even appear as formal constructs in computations
related to string theory. But they are more than purely theoretical constructs: the
Heisenberg spin-chain nowadays can even be experimentally realized in condensed
matter systems and the correlation functions can be measured in the laboratory.
In addition to this physical motivation to study quantum spin-chains there exists also
significant mathematical interest. The methods used to solve these physical models,
which we describe in a moment, have given rise to several algebraic structures which
have turned into research areas in pure mathematics, most famously quantum groups or
algebras. Unfortunately, physicists and mathematicians often speak a different language
and over the years the research area has evolved in different directions. Ideally one
wants to harness the progress made by pure mathematicians on the mathematical
structures and apply it to solve the physical problems like the computation of correlation
functions, that's where mathematical physics comes in. It tries to bridge the gap between
abstract mathematical theories and concrete physical applications keeping in mind that
both have mutually benefited from each other over the past years and continue to do so
until today.
HOW & WHO? Let us now turn to the mathematical aspects involved in solving the
aforementioned eigenvalue problem of the Heisenberg spin-chain. It is worthwhile to
introduce them by giving a rough overview over the historical development.
Historically, Bethe's 1931 work on the isotropic case (gx = gy = gz), known as the XXX
model, had a major impact and was the starting point for many of the subsequent
developments in this area. He made an "ansatz" for the stationary states of the XXX
spin-chain to be a superposition of plane waves (in quantum mechanics particles or
spins are described by so-called wave functions) whose momenta/wave vectors have to
satisfy an intricate set of non-linear equations, called Bethe's equations. In the literature
his approach is nowadays referred to as "coordinate Bethe ansatz" and has been
applied to numerous other quantum integrable systems. It is the combinatorics and the
algebraic aspects behind Bethe's ansatz which are of mathematical importance.
Another milestone in the history of integrable systems is Onsager's 1944 solution of the
planar Ising model. Quite a few modern mathematical concepts have their origin in
methods going back to his work; for instance his 1944 paper contains the star-triangle
relation (a precursor of the famous Yang-Baxter equation) and an infinite-dimensional
algebra, today called Onsager's algebra which is a special quotient of the sl2 loop
algebra (the latter had not been invented yet by mathematicians). The planar Ising
model, however, is a model in classical statistical mechanics and physically it is
unexpected that this model should have anything to do with a quantum spin-chain. The
relation is purely mathematical (the statistical transfer matrix shares a set of common
eigenvectors with the quantum spin-chain Hamiltonian) and was made apparent through
Baxter's seminal works in the 1970's, who took many of Onsager's techniques to the
next level by generalizing them as well as adding numerous new ideas to the subject
area.
The next major step in the mathematical development was the introduction of the
"quantum inverse scattering method" (QISM) by the Faddeev-school. Roughly, the latter
method introduces a spectrum generating algebra (i.e. operators which "generate" the
eigenvectors of the Hamiltonian by successive action on a pseudo-vacuum state).
Central to this approach is the aforementioned Yang-Baxter equation and Baxter's idea
of commuting transfer matrices. The QISM laid the foundation for many algebraic
structures, the following table gives an overview for Heisenberg spin-chains:
Model
Couplings
Algebra
XXX
gx = g y = gz
sl2 Yangian
XXZ
gx = g y
affine quantum algebra of sl2
XYZ
all independent
elliptic algebra of sl2
Today this continues to remain an active international research field with many open
problems. The exact, analytic computation of correlation functions is still an important
problem under investigation, so are aspects of the eigenvalue problem and symmetries
of the Hamiltonian. If you want to know more refer to the review articles and books below
and references therein.
Some suggested literature:
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B. M. McCoy, The Baxter Revolution, J. Stat. Phys. 102 (2001) 375; cond-mat/0001256
L.D. Faddeev, How the algebraic Bethe ansatz works for an integrable model, hepth/9605187
M. Fowler, Quantum integrable systems in one dimension, Physica D 86 (1995) 189
M. Jimbo and T. Miwa, Algebraic analysis of solvable lattice models, CBMS vol 85, AMS
(1995)
R. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press (1982)