Seminar on Statistical Physics

University of Heidelberg
Lecturer: Prof. Dr. Georg Wolschin

Topological Phase Transitions

Melda Akyazi
Summer Semester 2020

Abstract
Considering the two-dimensional XY model the Berezinskii-Kosterlitz-Thouless
transition is introduced. As the Hohenberg-Mermin-Wagner theorem forbids phase
transitions resulting from a transition from a disordered to an ordered state with
rising temperature, a new state called quasi-ordered state is defined and the
crossover to the disordered high-temperature phase is shown to be due to the
unbinding of pairs of vortices and antivortices.
Contents
1. Motivation 3

2. Two-Dimensional XY Model 3
2.1. High Temperature Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2. Low Temperature Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3. Vortices 6
3.1. Topological Charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.2. Energy Cost of Vortices and Vortex-Antivortex Pairs . . . . . . . . . . . 8

4. Experimental Realisation 10

5. Summary and Outlook 11

A. Appendix 12
A.1. Gaussian Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
A.2. Logarithmic Decrease of Gaussian Variables in Two Dimensions . . . . . 13

References 15

2
1. Motivation
Before the 1970s, when topological phase transitions were found, the conventional type
of phase transitions was based on the idea of symmetry breaking since the critical point
was thought of as the crossover from an ordered to a disordered state. Therefore, if
there was no long-range order (LRO) in the low-temperature phase, there should be no
phase transition at all. Furthermore, it was expected and rigorously shown in 1966 by
Mermin and Wagner in the case of the XY and Heisenberg model that there is no LRO
at nonzero temperatures in one and two dimensions [11]. In the same year, Hohenberg
showed rigorously that there can be no superfluidity and superconductivity at nonzero
temperatures in one and two dimensions [4]. Nevertheless, numerical results and se-
ries expansions seemed to be in a contradiction to these results (cf. [9] and references
therein), therefore, it was clear that under the assumption that the Hohenberg-Mermin-
Wagner theorem was true the investigation of some other kind of phase transition was
missing. In the year 1971 Berezinskii first came up with the idea that there has to be
another kind of phase transition due to topological defects [1] and two years later Koster-
litz and Thouless had the same idea and expanded the theory [9]. In the following year
Kosterlitz used renormalisation group techniques to investigate this problem in some
more detail [7]. Since then topological defects became very important and had a wide
impact on a large number of systems due to the concept of universality classes which
allows to describe systems having the same dimension and nature of order parameter by
the same effective theory. This made it possible to use this kind of theory to describe,
for instance, two-dimensional superfluids.

In the following we will investigate therefore the simplest model showing a topolog-
ical phase transition, namely the two-dimensional XY model. The difference in the
spin-spin correlations at high and low temperatures will lead us to the definition of a
new kind of order and a new kind of phase transition due to topological defects.

2. Two-Dimensional XY Model
We begin our investigations with a regular two-dimensional square lattice Λ with lattice
spacing a and assign to each site x ∈ Λ a two-component unit vector sx , interpreted as
a classical spin:

sx = (cos(θx ), sin(θx′ )), θx ∈ [0, 2π],

where the spin is constrained to rotate in the (XY-) plane of the lattice and θx is the
angle the spin of site x makes with some arbitrary axis. This classical spin model is
therefore called XY model or rotator model and has a continuous symmetry. It is a
special case of the n-vector model for n = 2.

3
The Hamiltonian of this system is given by
X
H = −J sx · sx′
⟨x,x′ ⟩
X
= −J cos(θx − θx′ ),
⟨x,x′ ⟩

where ⟨x, x′ ⟩ denotes nearest neighbour pairs. In the last line we used a trigonometric
identity. In the following we consider only the ferromagnetic case, therefore J is a pos-
itive interaction term.

This system has two important invariances (cf. [5]): first, we have a global invariance
which causes that a rotation of every spin by the very same amount, say α,
θx → θx + α
leaves the Hamiltonian unchanged. In addition, there is a gauge invariance as the
Hamiltonian is invariant under
θx → θx + 2π
for any site x. The latter invariance is important for us as it is responsible for the
occurrence of vortices.

As we are interested in the possibility of a phase transition at finite temperature, we

want to know how spin-spin correlations behave at low and high temperatures. We ex-
pect no LRO at low temperatures and thus we suspect that the correlation function of
the spins fall off for largely separated lattice sites at all temperatures. Nevertheless it
is insightful to examine whether there is a change in correlations at all and, as we will
see, there is indeed a difference.

2.1. High Temperature Phase

Although it should be clear that at high temperatures there can be no LRO due to
thermal fluctuations and therefore spins on lattice sites which are far apart from each
other should not be correlated, we will show for the sake of completeness a way to see
this in the case of the two-dimensional XY model immediately by looking at the spin-
spin correlation function. We follow here [6] and consider the spins at the lattice sites 0
and r. Defining K := kBJ T for kB being the Boltzmann constant, we see that for a lattice
with N lattice sites
⟨s0 · sr ⟩ = ⟨cos(θ0 − θr )⟩
Z  X YN
1 2π dθi
= cos(θ0 − θr ) exp K cos(θx − θx′ )
Z 0 ′
2π
⟨x,x ⟩ i=1
N Z 2π Yh
1 Y
i dθi
= cos(θ0 − θr ) 1 + K cos(θx − θx′ ) + O K 2 (1)
Z 0 2π
i=1 ⟨x,x′ ⟩

4
for Z being the partition function. In the last line, we used that we are allowed to
expand the exponential function around zero at high temperatures. Neglecting higher
order terms in the expansion we see that each bond on the lattice, i.e., each pair of
nearest neighbours, contributes either a factor one or K cos(θx − θx′ ). The product over
nearest neighbours leads to a large sum with the first term being equal to one and the
others depending on K or higher orders of K. We enumerate the lattice sites and see
immediately that, for example,
Z 2π
dθ1
cos(θ1 − θ2 ) = 0.
0 2π
By using this we do not only see that terms depending on K and with endpoints within
the lattice sites 0 and r should vanish, but also paths of such internal points as, for
example,
Z 2π Z 2π
dθ2 dϕ
cos(θ1 − θ2 ) cos(θ2 − θ3 ) = cos(θ1 − θ3 − ϕ) cos(ϕ)
0 | {z } 2π 0 2π
:=ϕ
Z 2π
dϕ
= cos(θ1 − θ3 ) cos2 (ϕ)
2π
Z0 2π
dϕ
+ sin(θ1 − θ3 ) sin(ϕ) cos ϕ
2π
|0 {z }
=0
1
= cos(θ1 − θ3 ).
2
We used in the second step that according to trigonometric identities cos(θ1 − θ3 − ϕ) =
cos(θ1 − θ3 ) cos(ϕ) + sin(θ1 − θ3 ) sin(ϕ). But comparing this result with (1) we see, how
non-vanishing terms resulting from the product over nearest neighbours have to look
like: we need a term which depends on cos(θ0 − θr ) such that it adds up to a quadratic
term after being multiplicated with the cosine in (1). Integrating over a quadratic cosine
would not vanish, therefore the only integrals which are nonzero are those connecting
paths between the sites 0 and r such as
Z 2π
dθ0 dθr 1
cos2 (θ0 − θr ) 2
= .
0 (2π) 2
Hence each bond along the path contributes K/2 and as we are in the high temperature
phase, i.e. K ≪ 1, the leading term will be the shortest path |r|. Therefore to lowest
order we have
 K |r|
⟨s0 · sr ⟩ ≈ = e−|r|/ξ with ξ ≈ (ln(2/K))−1 .
2
Hence we see that spins at lattice sites of far distance are not correlated and that the
correlation falls off very fast as the correlation function tends exponentially to zero for
r → ∞, just as expected.

5
2.2. Low Temperature Phase
At low temperatures we have a quite different situation. It is important to bear in mind
that we are only considering the ferromagnetic case of this model, thus the ground state
is the fully aligned state θx = θ for all x, where θ ∈ [0, 2π] is a fixed value. Assuming
low temperatures there will be only slowly varying configurations, i.e. adjacent angles
nearly equal. Therefore we expand the cosine up to terms quadratic in the angles
1
cos(θx − θx′ ) ≈ 1 − (θx − θx′ )2 .
2
Furthermore, as the range of θx is otherwise inconvenient, we extend the range to go
from minus infinity to infinity resulting now in a completely Gaussian problem which
can be solved. Therefore this is called Gaussian or spin wave approximation to the XY
model and is reliable for all dimensions if the temperature is low enough. We go on by
considering the spin-spin correlation function again:

⟨s0 · sr ⟩ = ⟨cos(θ0 − θr )⟩
D E
= ℜ exp(i(θ0 − θr )
  
= ℜ exp −⟨(θ0 − θr )2 ⟩/2
 |r| − k4πJ
BT

≈ ,
a
where we used in the penultimate step that we have a set of Gaussian distributed vari-
ables with zero mean because of symmetry such that we are able to use the results of
section A.1, especially equation (2). In the last step, we used that Gaussian fluctuations
grow logarithmically in two dimensions, a proof can be found in the appendix, section
A.2, especially equation (5). Now we see that the decay of correlations is algebraic
rather than exponential implicating the possibility of a phase transition. We say that
this low-temperature phase has quasi-LRO or topological-LRO in contrast to the totally
disordered, high-temperature phase.

3. Vortices
The results from the low-temperature phase showed an algebraic behaviour of spin-
spin correlations as one would expect according to the theory of conventional phase
transitions from the spin-spin correlation at a critical point. Therefore the resulting
temperature-dependent exponent is reminiscent of a critical exponent with the impor-
tant difference that in the spin wave approximation the system seems to be at a critical
point for all temperatures, which is clearly unphysical. Hence, as Berezinskii, Kosterlitz,
and Thouless argued [1], [9], there has to exist another set of excitations. The disorder-
ing at low temperatures is, according to them, caused by topological defects, or more

6
explicit vortices in the case of the XY model, that can not be regarded as simple defor-
mations of the ground state. Those vortex excitations are responsible for the change in
the correlations as the critical temperature for the phase transitions is the one at which
vortex-antivortex pairs unbind, whereas at higher temperatures there is a plasma of free
vortices and antivortices.

Before continuing we consider the Hamiltonian using the spin wave approximation and
defining the ground state energy E0 = −JNnn with Nnn being the number of nearest
neighbours
1 X
H − E0 ≈ J (θx − θx′ )2
2
⟨x,x′ ⟩
Z
continuum limit
→ J (∇θr )2 d2 r.

For the continuum limit we assumed that the lattice spacing a is infinitesimal, then one
gets according to [13]:
1 X
H − E0 ≈ J (θx − θx′ )2
2 ′ ⟨x,x ⟩

1 Xh θi − θR 2  θi − θT 2  θi − θL 2  θi − θB 2 i
N
= + + +
2a2 i=1 a a a a
X
N
≈a 2
|∇θi |
i=1

where we enumerated each of the n lattice sites and where θR , θT , θL , θB represent some
θj being the right, top, left, or bottom nearest neighbour of θi . The first two and the last
two terms in the large bracket represent the one-sided gradient at the site i. Finally we
want to replace the sum by an integral and have to divide by a2 (size of an elementary
cell in the lattice):

X
N Z
a 2
|∇θi | → J (∇θr )d2 r.
i=1

In some literature, such as in [6], the continuum limit is taken in a different way such that
there is an additional factor of one half. This leads to slightly different results but does
not change the nature of topological phase transitons. The way we took the continuum
limit leads to the same results as in the original paper of Kosterlitz and Thouless [9].

3.1. Topological Charge

We follow here [6] and [12] and recall that we already mentioned that the XY model has
a gauge invariance and therefore the angle θ describing the orientation of a spin is defined

7
 Figure 1: Vortex and antivortex in the two-dimensional XY model [6].

up to an integer multiple of 2π. As a result we are able to construct spin configurations

for which going around a closed path the angle rotates by 2πq if we define q ∈ Z to be the
topological charge or winding number. For instance, in figure (1) there are elementary
defects which can not be destroyed by small fluctuations of the neighbouring spins and
going around a closed path around these causes a change in the orientation of spin by
2π respectively −2π, thus we call the left one a vortex and the right one an antivortex.
In contrast to that integrating along a closed path if there is no topological defect leads
to a topological charge equal to zero. In the continuum limit, we expect
I
1 1 dθ 1
q= ∇θ · dℓ = (2πr) = |∇θ|2πr
2π 2π dr 2π
q
⇒ |∇θ| = ,
r
as θ has a spherical symmetry and for r being a radial coordinate. This approximation
clearly fails close to the center of the vortex due to the lattice structure which becomes
more important.

Using this result we can now calculate the energy cost of a single vortex. We neglect the
contributions from the core region as this will only lead to a constant which will anyway
cancel out when we are considering the change in the free energy.

3.2. Energy Cost of Vortices and Vortex-Antivortex Pairs

We now want to check whether it is favourable that the system makes these vortex-
excitations. Therefore we calculate the cost to create a single vortex of topological
charge q according to [6] and [12]. Although we expect to have contributions not only
from the relatively uniform distortions away from the center but also from the core
region, we use without loss of generality a circle of radius r0 , a cutoff of the order of the
lattice spacing, to distinguish the two and neglect the contribution from the core as we

8
are interested in the overall behaviour:
Z
Ev = J (∇θr )2 d2 r
r0
Z L  2
q
= 2πJ rdr
r0 r
L
= 2πJq 2 ln
r0
for L being the linear dimension of the system. Thus we see, that the energy di-
verges in the thermodynamic limit. Now we can conclude that single vortices can
not spontaneously formate. The configurational entropy of a single vortex is given
by S = 2kB ln(L/r0 )) as there are L2 /r0 2 possibilities to locate a vortex in a domain of
area L2 . Therefore the change in free energy due to the formation of a vortex is just
  L
∆F = 2πJq 2 − 2kB T ln .
r0
Not only the entropy but also the energy grows as ln L. Hence, at low temperatures,
the energy term dominates and configurations with single vortices vanish, whereas at
high temperatures the entropy dominates and is large enough to favour the spontaneous
formation of vortices. Therefore we can consider the case of vortices with a topological
charge equal q = ±1 being clearly the energetically ”cheapest” case and get for the
critical temperature:
πJ
TBKT = .
kB
It is important to bear in mind that this estimate for the critical temperature is an
upper bound because at lower temperatures the formation of vortex-antivortex pairs is
favourable as we will show next (cf. [9]). We consider a vortex-antivortex pair separated
by a distance d and investigate the distortions far away from the dipole center, r ≫ d.
This case can be considered similar to the case of a dipole in an electric field, therefore
we expect now that |∇θ| ∝ rd2 . Calculating the energy leads to
Z L
d2
Ev,a ∝ 2πJ dr
a r3

which is converging in the thermodynamic limit, thus causing a finite energy. Looking at
these results, it becomes clear that at low temperatures there can be vortex-antivortex
pairs and at high temperatures above TBKT free vortices and antivortices. As a result
TBKT marks the critical temperature at which, starting at the low-temperature phase,
the unbinding of vortex-antivortex pairs takes place.

9
 Figure 3: Proliferation of free
Figure 2: Sketch to the experimental setup [3]. vortices [3].

4. Experimental Realisation
As mentioned above the Berezinskii-Kosterlitz-Thouless (BKT) transition had a wide
impact on future research in general and therefore was also reason to more than 500
experimental articles touching this theory [8]. Due to the concepts of universality the
BKT theory is applicable to a wide variety of two-dimensional phenomena. One example
which we will consider in the following are superfluids. The condensate wave function
of a superfluid is given by

ψ(r) = |ψ(r)|eiθ(r) ,

(cf. [8]). Hence the important thermal fluctuations are only in the phase θ such that the
order parameter can be described by a single angle just as in the case of the XY model.
Now we can conclude that the critical behaviour of two-dimensional superfluids can be
described by the same effective theory as in the case of the two-dimensional XY model
and we expect a BKT transition.

There is a noteworthy experiment providing direct experimental evidence for the mi-
croscopic mechanism underlying the BKT theory, which was carried out by Hadzibabic
et. al. in 2006 [3]. Their work differs from previous ones by revealing the binding
and unbinding of vortex-antivortex pairs by using a trapped quantum degenerate gas of
rubidium 87 Rb atoms. According to the theory a uniform two-dimensional fluid of iden-
tical bosons, in contrast to the three-dimensional case, can not undergo Bose-Einstein
condensation as shown by Hohenberg [4], but become superfluid below a finite critical
temperature TBKT due to the pairing of vortices and antivortices. Therefore first a

10
quantum degenerate cloud of 87 Rb atoms is trapped in a two-dimensional optical lattice
such that the three-dimensional gas is split into two independent clouds and compressed
into the two-dimensional regime with about 105 atoms per plane, see figure (2a). These
Bose-Einstein condensates are allowed to equilibrate independently and then the trap-
ping potentials are turned off. The two clouds expand predominantly perpendicular
to the planes, thus forming a three-dimensional matter wave interference pattern as
they overlap which allows to get information about the correlation function and the
microstructure. A jump between a quasi-LRO phase and a disordered phase could be
identified as the waviness of the interference fringes due to phase fluctuations in the two
planes increased for temperatures beyond TBKT = (290±40)nK, see figure (2c) and (2d).
Moreover, it was occasionally possible to observe a sudden onset of vortex proliferation
with increasing temperature as in figure (3), in (a) the change in the interference pattern
is attributed to the presence of a free vortex and in (b) to several ones. The critical
temperature at which those vortices disappeared due to the pairing up of vortices and
antivortices coincides with the loss of quasi-LRO indicating conclusive evidence for the
observation of the BKT transition.

5. Summary and Outlook

The Berezinskii-Kosterlitz-Thouless transition solved the problem of a lack of explana-
tion for phase transitions in two-dimensional systems which could not be explained by
the conventional type of phase transitions due to the destruction of LRO with rising
temperature. The theory of such topological phase transitions opened up many new
areas of physics. Until 2013 the work of Kosterlitz and Thouless has been mentioned
by nearly 2200 papers [10] and due to the wide impact their theory had, they got the
Nobel Prize for physics in 2016. We already saw applications to magnetic and superfluid
systems, another application is the theory of melting of two-dimensional crystals, al-
though the situation there is much more complicated and therefore ”their physical ideas
are correct but incomplete” ([8], p.21).

We considered the simplest model showing this new kind of phase transition in de-
tail, namely the XY model, and were able to see that the behaviour of the spin-spin
correlation function implicates a phase transition at some nonzero temperature. In
the low temperature phase there is an algebraic fall off of correlations rather than an
exponential one as in the case of a disordered, high-temperature phase, therefore the low-
temperature phase can be identified to be quasi-ordered rather than disordered. The idea
of Berezinskii, Kosterlitz and Thouless to explain the crossover by an unbinding process
of vortices [1], [9], makes it accessible to consider the energy costs of creating single vor-
tices and vortex-antivortex pairs. This leads to the derivation of a critical temperature
which defines the temperature at which single vortices become energetically favourable.
These heuristic arguments made it possible to get an insight into the processes behind
topological phase transitions such as the experimental realisation of Hadzibabic et. al.

11
[3] which proved in a direct way the existence of those unbindings. Nevertheless one
could argue that the spin wave approximation might be too inaccurate and therefore
the concluded critical temperature a fallacy. Notwithstanding it was possible to verify
the nature of this critical temperature by renormalisation group techniques, cf. [8] and
references therein. Furthermore, it is also already shown in a rigorously manner by
Fröhlich and Spencer that the Berezinskii-Kosterlitz-Thouless transition exists [2]. In
contrast to that, it remains an unsolved problem to show that the Heisenberg model in
two dimensions has no phase transition at nonzero temperatures, neither a conventional
one nor a topological as expected from non-rigorous theoretical physics literature [14].
From the experimental point of view there are also still many improvements to be done.
For example, in the above mentioned direct experiment it was expected to reach a pure,
fully coherent Bose-Einstein condensate at extremely low temperatures which could not
be achieved due to residual heating.

A. Appendix
A.1. Gaussian Distributions
If X with components X1 , ..., XN is a set of Gaussian distributed variables, then the
expectation value of an exponential formed from a linear combination of these variables
is an exponential of a quadratic form in the coefficients, i.e., the generating function
⟨exp(iq · X)⟩ for q ∈ RN being an n-component vector is given by
 q
⟨exp(iq · X)⟩ = exp iq · ⟨X⟩ − q · G · (2)
2
where G is related to the correlation matrix. To motivate this result one can look at
the case of a single random variable X with variance σ 2 and considering the generating
function one obtains
Z ∞ !
1 (x − ⟨X⟩)2
⟨exp(iqX)⟩ = √ exp − exp(iqx)dx
2πσ 2 −∞ 2σ 2
Z ∞ !
1 −(x − (⟨X⟩ + iqσ 2 ))2  q2σ2 
=√ exp exp iq⟨X⟩ − dx
2πσ 2 −∞ 2σ 2 2
 q2σ2 
= exp iq⟨X⟩ − .
2
Therefore (2) is just the case for having a set of Gaussian variables. We will not go
into greater detail, some of these steps and more about Gaussian distributions can be
found in [5] and one can get the same result as in (2) by considering cumulants, see for
example [6]. Using cumulants it becomes clear that q · G · q = ⟨X 2 ⟩ − ⟨X⟩2 for sets of
Gaussian distributed variables.

12
A.2. Logarithmic Decrease of Gaussian Variables in Two Dimensions
We want to calculate ⟨ei(θ0 −θr ) ⟩ and we do this according to [6]. The probability of a
particular configuration is given by
Z !
J
P(θr ) ∝ exp − (∇θ(r))2 dr
kB T

and in terms of Fourier components,

!
J X 2
P(θk ) ∝ exp − k |∇θ(k)|2 . (3)
kB T k

Each mode θk is an independent random variable with Gaussian distribution of zero

mean, and with
kB T δk,−k′
⟨θk θk′ ⟩ = (4)
2Jk 2
as we can read off from (3) the variance of these Gaussian fluctuations to be equal to
kB T
2Jk2
if k = −k′ , and vanishing else. Using this we can calculate the correlations in the
phase θr in real space. Clearly, ⟨θr ⟩ = 0 by symmetry, while with (4)
1 X ik·r+ik′ ·r′
⟨θr θr′ ⟩ = e ⟨θk θ−k′ ⟩
a2 k,k′
1 X kB T eik·(r−r )
′

= 2
a k 2Jk 2
Z ′
kB T eik·(r−r )
→ dk
(2π)2 2Jk 2
Z
kB T C2 (r − r′ ) eik·r
=− with C2 (r) := − dk.
2J (2πk)2

The function C2 is just the Coulomb potential due to a unit charge at the origin in a
two-dimensional space, since it is the solution to
Z 2 ik·r
k e
∇ C2 (r) =
2
(2πk)2
= δ 2 (k).

Using Gauss’ theorem one gets

Z I
2 2
d x∇ C2 = dS · ∇C2 ,

13
and for a spherically symmetric solution, ∇C2 = dC2
ê ,
dx x
therefore

dC2
1 = 2π|r|
dr
dC2 1
⇒ =
dr 2π|r|
ln(|r|/a)
⇒ C2 = ,
2π
where we introduced a short-distance cutoff a which is in the order of the lattice spacing.
As by symmetry the mean values are vanishing, we get

1 ln(|r|/a)kB T
⟨(θ0 − θr )2 ⟩ = ⟨θ02 ⟩ − ⟨θ0 θr ⟩ = (5)
2 4πJ

14
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