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

What is 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)

In a system of many particles there could be many different kind of interactions between them that dictated the behavior of the system when it pass through different regions of the parameters space. For example we can think in one system where the potential interaction is predominant compare to the kinetic energy, but obviously this is not necessarily true for all the possible configuration of the system (at least not in general for all the systems). Therefore, in each different system there could be present many kind of symmetries and depending in which configuration is the system will be the different presented symmetries.

Talking about the different symmetries probably the easy one to use as example would be the translational symmetry. Imagine that you have a system of particles with homogeneous properties in the space, then it doesn't matter if you do a translation in the system you should still see exactly the same properties. A crystal is obviously not this kind of system because it has fix positions for the particles, therefore if you translate the system an arbitrary distance probably the density (which is not homogeneous, you only have a probability to find particles in a lattice position rather than everywhere) will not be longer exactly the same than the previous one. In this sense we can say that when we have a crystallization in the system the translational symmetry is broke.

In a similar way than the translational symmetry, the rotational symmetry tell us about the isotropy of the system. As example we can cite a crystal system, where each particle has a magnetic dipole pointing randomly. If we apply a magnetic field to the system pointing in some constant direction the different dipoles will tray to be align to this external field. Therefore we will have a broke rotational symmetry because right now there is a preferred direction.

Also we can talk about inversion and time-reversal symmetry which are a little bit more complicated to understand, but at the same time are responsible of many phenomenons.

In a natural way we could expect that many symmetries were compatible with the matter at considerable high temperatures, but when the temperatures gets down these symmetries are no longer valid as we have pointed in the previous examples. We need to clarify that not only the change in temperature is responsible for the broken symmetries and the presence of the different phases, as we will see in the different examples that we will discuss here.

Therefore we can talk about a change of phase when there is a broken symmetry. For example when a system turns from liquid to solid, or also when pass from paramagnet to a ferromagnet; are because have been presented a broken symmetry and obviously the system in different phases cannot be treat with the same ideas and we need to distinguish the considerations that we can apply to the different phases.

A point that should be clarify carefully is that when there is a change of phase could be because have been a broken symmetry, but this is not necessarily true. However, these are rare cases, but one should have present at least one example which is the change of phase from liquid water to vapor. In this case there is not a broken symmetry and one way to see it is because the critical point that exist in the water's phase diagram give us a chance to modify the system in such a way that liquid and vapor could coexist. We will talk a little more about this example.

"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.

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)

Symmetry breaking, the thermodynamic limit and phase transitions

The concept of phase transitions and symmetry breaking are closely interlinked. In particular, every transition between states with different symmetry is also a phase transition (the reverse is not necessarily true).

The classification of phase transition distinguishes between first and second order transitions

First order phase transition: accompanied by specific heat, like the transition from liquid to gaseous water

Second order phase transition: some

thermodynamic limit: spontaneous symmetry breaking is strictly speaking possible only in systems with infinite number of particles. The reason is that

Examples

Wigner Crystals

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 $T>>E_{F}$.

He 3

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.

other examples

In the next chart we show and compare different systems that exhibit broken symmetry.

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Example	Disordered Phase	Ordered Phase	Order Parameter	Constant of motion?
Ferromagnetism	Paramagnet	Ferromagnet	Magnetization M	Yes
Antiferromagnetism	Paramagnet	Antiferromagnet	Sublattice magnetization $M_{sl}$	No
Superconductivity	Normal Metal	Superconductor	$<\Psi _{\sigma }^{*},\Psi _{-\sigma }^{*}>$	No
Crystal	Liquid	Solid	Density $\rho _{G}$	No
He 3	Normal liquid	Anisotropic superfluid	$d_{ij}=<\Psi \Psi >_{M_{L},M_{S}}$	No

Note: To see more details and another phenomenons you can check "P.W. Anderson(Basic Notions of Condensed Matter Physics), pg.68"

As we can imagine there are just few examples were the order parameter is a constant of motion. The Ferromagnetism is one them.

"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.

The fluctuation of the last equation do become of quasimacroscopic size and dimension at the critical point, as a consequence of the divergences predicted. These fluctuations may or may not correspond to real collective modes. In the case of a continuous symmetry group, the low frequency collective excitations, which are necessarily there at absolute zero, are also present through out the temperature range up to Tc and merge, more or less, with the critical fluctuatio spectrum near Tc.

Phonons in solids and the spin waves are examples of "collisionless" modes, essentially quantum excitations required by the symmetries, or the conservation laws in some cases, of the system. But at frequencies low compared to microscopic relaxation times, these may merge into "hydrodynamic" modes, which have the character of ordinary sound waves, compressional or other modes of vibration in which the matter remains in local thermodynamic equilibrium.

The nature of the low-temperature collisionless excitations is more interesting in the much more common case in which the order parameter is not a constant of the motion, and in which the low-temperature state does not belong to a representation of the symmetry group of the Hamiltonian. A familiar example is the crystalline state.

crystalline state.

The plasma modes as a collisionless excitation is universal in charged condensed systems. For instance, it occurs in two important cases: the Wigner lattice, where only the two transverse modes behave as phonons, with $w\propto q$, and the superconductor, where there are no $w\rightarrow 0$ modes. In all cases, the zero-point motions of these collective modes play an important role in the energy and wave function of the ground state itself. The zero-point motion is central to the problem of restoring the broken symmetry

Generalized "rigidity"

When we move one end of a ruler, the other moves the same distance, such action at a distance is not built into the laws of nature except in the case of the long-range forces such as gravity or electrostatics. It is strictly a consequence of the fact that the energy is minimized when symmetry is broken in the same way throughout the sample: the phase and angle variables want to be uniform, so that the orientation and position of the lattice is the same everywhere. Of course, in general they are not quite the same, since the lattice can deform elastically, but nonetheless the lattice transmits that force from one end to the other even in equilibrium and without having to flow constantly like a viscous liquid. To break down the rigidity completely, we must supply the condensation energy of a macroscopic piece of the sample, which is very large.

The generalization of this concept to all of the instances of broken symmetry is it call here generalized rigidity. For instance, permanent magnets are so because the magnetization cannot change a little at a time. Superconductivity is the phase rigidity of the electron pair fluid.

Defect structures

There is a general rule that the breakdown of the generalized rigidity property, along with the resulting dissipation, is a consequence of the formation and motion of defect structures which are usually macroscopic in size.

The twin models for such effects are the domain theory of ferromagnetism, due to Landau (1941), Block (1930) and others, and the dislocation theory of G. I. Taylor (1934), Burgers (1939, 1940), and others. These two cases demonstrate nicely the one general statement which will be justified by the topological theory: that a one-dimensional order parameter allows only two-dimensional ("wall") defects; and that a two-dimensional order parameter allows linear defects such as dislocations, but not necessarily always; while a three- or more- dimensional case may allow point defects.

Clearly in any broken-symmetry system we can imagine forces which would disorient the order parameter in one region relative to another. As a simple example could be that different parts of the sample may have grown with different order parameters and eventually meet in the middle. In what ways can the system respond?

The most obvious and simplest is a boundary or "domain wall" (see next figure): one simply has two or more regions. each locally homogeneous, separated by boundaries. In the case of the one-dimensional order parameter, there is only a discrete set of local equilibrium states (directions of M for a ferromagnet, for instance, or of P for ferroelectric), and there must be a two-dimensional boundary (for a 3D sample) shape of the boundary or other singularity is determined by competition between the forces of generalized rigidity. In the boundary case, characteristically the wall is $\approx$ one lattice constant thick-- essentially, there is no length parameter except the fundamental microscopic length. Thin boundaries cannot move continuously in space --they are located at a specific crystal plan and must overcome an activation energy to move to the next.

In general, the response to external forces tending to twist or reverse the magnetization must in the first instance come from the motion of these boundaries. To make a permanent magnet, one can proceed in two ways: either to pin the existing boundaries, i.e., to make a very impure material, or to remove them, i.e., to make i of very small particles. Thus the response properties of magnets are wholly conditioned by the defect structures. This is, in fact, the general case.

A second type of defect is a line defect. An example is the dislocation (see next figure). We can imagine following a closed path through a crystal along which the local phase varies gradually, corresponding to a small local strain, but then returns to the original phase changed by $2\pi$. There must be associated a line of singularity where the lattice structure is essentially destroyed.

The two basic types of dislocations are edge dislocations, where an extra plane of atoms has been interpolated ending at a line singularity; and screw dislocations, where the lattice has the topology of a screw.

The easiest way in which a crystal can continuously slide against itself is by the continuous flow of dislocations across a line between the two. Again, there are two ways to make a strong crystal: the easy way is to pin the dislocations, one version of which is called work hardening; the hard way is to eliminate them altogether as in a "whisker" crystal.

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