Remember that U is the potential experienced by a boson and V
¯
characterize the interaction
between two bosons. eqn (4.10.2) tells that how the ground state energy density depends on those
parameters. Because the ground state is the state at zero temperature, how ground state energy
depends on those parameters will tell us if there are quantum phase transitions
4
or not. As we
change the parameters that characterize a system, if the ground state energy changes smoothly,
we will say that there is no quantum phase transition. If the ground state energy encounter a
singularity, the singularity will represent a quantum phase transition. From eqn (4.10.2) we see
¯
that there is a quantum phase transition at U = 0 if V > 0. Since it is the second order derivative
∂
2
u/∂U
2
that has a discontinuity, the phase transition is a second order phase transition. The
phase for U > 0 contains no bosons since it cost energy have a boson. The phase for U < 0 has a
non-zero boson density n =
−U/V
¯
. This is because for negative U the system can lower its energy
by having more bosons. But if there are too many bosons, there will be large cost of interaction
energy due to the repulsive interaction between the bosons. So the density n =
−U/V
¯
is a balance
between the potential energy due to U and the interaction energy due to V
¯
. Later we will see that
the phase with non-zero φ is a superfluid phase. It is also called the boson condensed phase.
We would like to remark that the eqn (4.10.2) is really the ground state energy density of
the classical field theory. It is an approximation of the real ground state energy density of the
interaction boson system. So we are not sure if the real ground state energy density contains a
singularity or not. Even if the singularity does exist, we are not sure if it is the same type as
described by eqn (4.10.2). A more careful study indicates that the real ground state energy density
of the interaction boson system does have a singularity that is of the same type as in eqn (4.10.2)
if the dimensions of the space is 2 or above. In 1D, the real ground state energy density has a
singularity at U = 0 but the form of the singularity is different from that in eqn (4.10.2).
4.11 Continuous phase transition and symmetry
• Two mechanisms for phase transitions.
• A continuous phase transition is a symmetry breaking transition.
• The concept of order parameter.
We know that ground state energy (4.10.2) is obtained by minimizing the energy functional
(4.9.2). U and V
¯
are the parameters in the energy functional. Can we have a more general and
a deeper understanding when the minimum of an energy functional has a singular dependence on
the parameters in the energy functional?
Let us consider a simpler question: when the minimum of an energy function has a singular
dependence on the parameters in the energy function? To be concrete, let us consider a real function
parameterized by a, b, c, d (where d > 0):
E
abcd
(x) = ax + bx
2
+ cx
3
+ dx
4
(4.11.1)
Let E
0
(a, b, c, d) be the minimum of E
abcd
(x). How can the minimum E
0
(a, b, c, d) to have a singular
dependence on a, b, c, d, knowing that the function E
abcd
(x) itself has no singularity.
One mechanism for generating singularity in E
0
(a, b, c, d) is through the “minima switching” as
shown in Fig. 4.9. When E
abcd
(x) has multiple local minima, a singularity in the global minimum
E
0
(a, b, c, d) is generated when the global minimum switch from a local minimum to another. The
singularities generated by “minimum-switching” always correspond to first order phase transitions
since the first order derivative of of the ground state energy E
0
is discontinuous at the singularities.
4
A quantum phase transition, by definition, is a phase transition at zero temperature.
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