Phy5670

Welcome to the Quantum Many Body Physics PHY5670 Fall2010

PHY5670 is a one semester graduate level course. Its aim is to introduce basic concepts, and logical framework, of this vast and developing discipline: broken symmetry and adiabatic continuity. Theoretical techniques, such as coherent state path integrals and diagrammatic perturbation expansions, will be used to emphasize these deeper underlying concepts, as well as to provide practical means of calculations. Few illustrative physical systems and quantum many-body models will also be studied.

The key component of the course is the collaborative student contribution to the course Wiki-textbook. Each team of students is responsible for BOTH writing the assigned chapter AND editing chapters of others.

Team assignments: Fall 2010 student teams

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Outline of the course:

Conceptual basis of many body physics

Broken symmetry

Our experience shows us, then, that as matter cools down it usually no longer retains the full symmetry of the basic laws of quantum mechanics which it undoubtedly obeys; our task here is to understand that the questions we must ask are "Why", "In what sense", and "What are the consequences?" P.W. Anderson (Basic Notions of Condensed Matter Physics)

The laws that govern the universe present different kind of symmetries as translational and rotational symmetries, which tell us about the homogenity and isotropy of the space. Also we can talk about inversion and time-reversal symmetry. We can expect that basic symmetries are compatible with the matter at high temperatures, but not necessarly when it cools down as a crystal where the atoms are in fixed positions and the translational symmetry is no loger valid.

Following this idea we can talk about a broken symmetry when we pass from a liquid to a solid, or also when we pass from paramagnet to a ferromagnet. Obiously in a change of phase the system can no longer be described with the same equation of state and a new description is necessary.

We need to remark that boken symmetry is not the only way to have a change in the qualitatived behavior of the matter, but these cases are relatively rare. For example when a water pass from gas to liquid water, there is not broken symmetry.

"Why" broken symmetry?

Under surprisingly general circumstances the lowest energy state of a system does not have the total symmetry group of its Hamiltonian, and so in the absence of thermal fluctuations the system assumes an unsymmetrical state. P.W. Anderson (Basic Notions of Condensed Matter Physics)

The majority of real physical systems, with interaction between the particles, tend to exhibit the phenomenon of a lowest-energy state's not having the full symmetry of space. As example we can talk about the "Wigner solid" (see next section), which present a lattice array of electrons with coulomb interactions when the coulomb forces dominate at very low density.

The essential phenomenon is that the lowest state of a potential energy of interaction between particles must occur either a unique relative configuration of all particles or, in artificial cases, perhaps for a highly restricted subset. It is clear that in any sitution where energy dominates kinetic energy and entropy a system of particles obeying a simple potential will take up a regular lattice structure.

As anoter examples we can talk about broken time-reversal invariance in systems as magnetic ordering, ferro and antiferromagnetic systems. In general these phenomena also break an at least approximate spin rotation symmetry. Liquid crystals exhibit another type: broken local rotationl symmetry. Superconductivity is a broken gauge symmetry.

In general we can say that the lowest-energy state of a system does not have the total symmetry group of its Hamiltonian, and so in the absence of thermal fluctuation the system assumes an unsymmetrical state.

Examples: Wigner crystallization and He-3

At low density, electrons in the jellium model crystalize into a bcc Wigner crystal. At high density (and sufficiently finite temperature) they behave as Fermi gas/liquid with all the invariance properties of the ordinary Boltzman gas, which this system is when $T>>E_{F}$.

If we compress a high enough density Fermi liquid with short range repulsive interactions, approximately as $^{3}He$, it will form a regular solid.

The melting curve of He3 at low temperature from E. R.Grilly, J. Low Temp. Phys., 11:33–52, 1973. Note that for a range of pressures, the solid He3 melts upon cooling(!). The reason behind such anomaly has its origin in solid having a greater paramagnetic entropy than (Fermi) liquid.

The essential phenomenon is either case is that the lowest state of a potential energy of interaction between the particles -- for example, a pair interaction

$V_{tot}=\sum _{i<j}V(|r_{i}-r_{j}|)$ -- must occur for either a unique relative configuration of all particles $\{r_{1},r_{2},\ldots r_{N}\}$ and all translations and rotations,

or, in artificial cases, perhaps for a highly restricted subset of all configurations. P.W. Anderson (Basic Notions of Condensed Matter Physics)

It is then clear that in any situation where the potential energy dominates kinetic energy and entropy, as in the two cases mentioned, a system of particles obeying a simple potential will take up a regular lattice structure. P.W. Anderson (Basic Notions of Condensed Matter Physics)

The less symmetric state tends to be one lower in temperature, simply because the more symmetric one is usually a distribution of thermal fluctuations among all the available values of the order parameter . But this order of phases is not a general rule; He3, for instance, violates it because the solid has a greater nuclear paramagnetic entropy than liquid, and at low temperatures the melting curve has a negative slope. So in this temperature regime, the solid is the high-temperature phase and the liquid is the low-temperature phase. P.W. Anderson (Basic Notions of Condensed Matter Physics)

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"What are the consequences?" of broken symmetry

• discreteness of phase transitions, and the resulting failure of continuation: disjointness of physical phases
• development of collective excitations
• generalized rigidity
• defect structures: dissipation and topological considerations

Discreteness and Disjointness

First theorem "It is impossible to change symmetry gradually. A given symmetry element is either there or it is not; there is no way for it to grouw imperceptibly" (Landau and Lifshitz, 1958)

Let's analyze the phase diagram of the water (see figure above). As we can see in the diagram we can go from vapor to liquid in a "smooth path" just going around the critical point, actually this means that these two phases doesn't present a broken symmetry. On the other hand, it's impossible to go from liquid to solid smoothly (or in opposite way). The liquid-gas transition is typical of a symmetry-nonbreaking transition. There is no possibility that the fluid and gas can be in equilibrium at the same density except at a point on the boiling curve.

In the cases of true broken symmetry, the unsymmetrical state is normally characterized by and "order parameter". By Landau's definition this is simply any parameter that is zero in the symmetric state and has a nonzero average uniquely specifying state when the symmetry is broken; it is an additional variable necessary to specify the microscopic state in the lower symmetry state. Thus by broken symmetry a new variable is created. For example, in a nonmagnetic material the order parameter is the magnetization M, which in the absence of a magnetic field is zero by time-reversal symmetry and the state is specified by the usual intesive variables P and T.

It is possible to predict that $F=-T\ln {<e^{-\beta H}>}$ in the new system is a new mathematical function from than in the old. For instace, call the new variable $\psi$. Then in general we have

$F=F(V,T,\psi )$

We calculate $F=F(V,T)$ by appending to this the equilibium condition

${\frac {\partial F}{\partial \psi }}=f=0$

where f is the generalized force variable (like the magnetic field H) corresponding to $\psi$. Above the critical temperature Tc, this is satisfied by symmetry; below, it is nontrivial. Then from the las two equations the result is a new function $F'(V,T)$, which is not analytic at Tc. This is Landau's essential insight.

In simplest terms, it is quite clear that the First Theorem requires that F have a boundary of singularities between the two regimes of symmetry, and that therefore analytic continuation between the two is not possible.

Collective excitations

At low enough temperatures the matter doesn't obey the initial symmetry, which the Hamiltonian of the system do. From this fact the most important and interesting properties of matter appear:

1. Affectation of the spectrum of elemetary excitations and fluctuations.

2. Causes a new static properties call "generalized rigidity".

In this sectio we will talk about the first consequence. The new state can be either an eienstate of H or not.

The original Hamiltonian has a group of symmetries G, and only when this group is a continuous one we can see dynamic effects at low temperature. G may be described by a set of generators $L_{i}$, i.e., the group elements A belonging to G

$AHA^{-1}=H$

may be written as

$A=e^{i\theta L_{i}}$

where $L_{i}$ is related to infinitesimal transformations. For an infinitesimal transformation $\epsilon$

$A=1+i\epsilon L_{i}$

and continuous groups may in general be built up by products of infintesimal operators. The spin rotaion group, for instance, has the generators $S_{x},S_{y},S_{z}.$

The only unsymmetrical operators which commute with H are the various group generators $L_{i}$. These are therefore conserved quantities; and if the order parameter can be taken as one of them, it is conserved. For example in the case of ferromagnets the order parameter may be taken as the total spin in some fixed direction z. That may well be the only example of a simple, conserved order parameter.

The "dimensionality of the order parameter" plays a great role in critical fluctuation theory. When the violated symmetry is only that of a point group (a discrete symmetry) we speak of a "one dimensional order parameter". An example is the Ising model for ferromagnetism, where there is no question of collective modes, because there are no continuous degrees of freedom to support them.

Even though H does not obey full rotation group symmetry, the underlying dynamical object do: they are spins, a consequence of the rotation group properties of the fundamental particles, only coupled together and to a lattice in such a way as not to exhibit their full symmetry. Nonetheless, the group generators play a role in their excitation spectrum, and it is easy to deduce from that the underlying objects.

Spin waves

Phonons

Near a second-order Tc, no matter what the dimensioality and symmetry, there will be the finite temperature equivalent of zero-frequency modes, namely critical fluctuations with divergent amplitude as $T\rightarrow Tc$. One imagines the coarse-grained average of the order parameter to be specified as a (necessarily very slowly varying) function of position: $\mathbf {M} =\mathbf {M} (\mathbf {r} )$, and then asks for the free energy as a functional of this functio, i.e., with $\mathbf {M} (\mathbf {r} )$ constrained. It is clear that if an analytic expression exists near Tc for such an F, it must be approximated as

$F=\alpha (T-Tc)M^{2}+A(\nabla M)^{2}+BM^{4}+...$

in close analogy to the observatio about the energy at T=0 and also to the Landau free energy of a ferroelectric. The (T-Tc) term is the simplest way to ensure that M will grow from zero aboe Tc to a finite value below it. Then the probability of a fiben fluctuation M(r) will be

$P\propto e^{-{\frac {F(M)}{T}}}$

which leads to fluctuatio amplitudes

$\langle M^{2}(k)\rangle ={\frac {\langle M^{2}e^{-F/T}\rangle }{\langle e^{-F/T}\rangle }}\propto {\frac {1}{\alpha (T-Tc)+Ak^{2}}}$

which in turn diverges as $k\rightarrow 0$ and $T\rightarrow Tc$. This divergence in the fluctuation amplitudes in turn implies that the relevant frequencies approach zero in some manner.

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