University of Amsterdam - UvA/FNWI

73 Pages·2016·1.77 MB·English

University of Amsterdam
MSc Physics
Theoretical Physics
Master Thesis
Using tensor network renormalization to simulate the 2D
classical Ising model at criticality
by
Schelto Crone
10285288
July 2016
60 ECTS
Supervisor: Examiner:
dr. P. Corboz prof. dr. J. de Boer
Institute of Theoretical Physics Amsterdam

Abstract
Recently a new coarse-graining algorithm for tensor networks has been introduced: the ten-
sor network renormalization algorithm (TNR). By employing the techniques of entanglement
renormalization it can properly coarse-grain a 2D classical system around and at its critical
point and reproduce the correct fixed point structure. In this thesis this method is implemented
and its capabilities have been tested in context of the 2D classical Ising model. It is shown
that this algorithm can accurately reproduce the trivial and the critical fixed points of the
model. This allows the algorithm to properly coarse-grain the system at criticality, resulting in a
scale invariant system. The results have been compared with previous coarse-graining schemes,
namely the tensor renormalization group and the corner transfer matrix method, in terms of the
accuracy and the computational cost. It has been found that, although the TNR can simulate a
system at its critical point, it has a huge computational cost which prohibits simulations for large
bond dimensions. Still, the qualitative features produced by the algorithm allow for accurate
simulations at criticality.
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Populaire Nederlandse samenvatting
Een veel-deeltjes systeem is een model bestaande uit velen elementaire kwantumdeeltjes die een
interactie hebben met elkaar. Het beschrijven van deze veel-deeltjes systeem met een sterke
interactie term is in de meeste gevallen niet exact mogelijk: er zijn maar enkele oplossingen
bekend. Veel interessante fenomenen, bijvoorbeeld hoge temperatuur supergeleiding of het
fractionele kwantum Hall effect zouden alleen hierdoor kunnen worden verklaart. Hierom
zijn numerieke methoden van een essentieel belang om informatie te verkrijgen over dit soort
modellen.
In de jaren 90’ is door S. White een methode ontwikkeld om dit effici¨ent te doen voor een
dimensionale kwantumsystemen, de DMRG methode. Deze methode probeert op een effici¨ente
manier de meest relevante toestanden van het systeem te vinden. Deze methode heeft een
revolutie binnen het numeriek simuleren teweeggebracht door zijn zeer hoge nauwkeurigheid.
Het is een van de eerste methoden die een goed werkende numerieke renormalizatie groep
transformatie implementeert. Later bleek dat de nauwkeurigheid van dit algoritme te verklaren
valt met behulp van een netwerk van tensoren.
Het algoritme is gemaakt voor een dimensionale kwantumsystemen die niet op een kritiek
punt zitten. Een kritiek punt kan worden beschouwd als het punt waar het systeem zich op een
faseovergang bevindt. Om de systemen op een kritiek punt te simuleren is een andere aanpak
nodig. Dit is later opgelost met de MERA algoritme, welke door alle korte correlaties in het
systeem te verwijderen dit kritieke punt wel kan simuleren.
De methode van DMRG is ook toegepast op een twee dimensionaal klassiek systeem, alleen
die methodes zijn ook niet nauwkeurig het kritieke punt. Recentelijk is een nieuwe methode
voorgesteld,TNR,welkewelinstaatzoumoetenzijnomhetklassiekesysteemtesimulerenopzijn
kritieke punt. In deze these is deze nieuwe methode onderzocht en vergeleken met de gevestigde
methodes. De methoden zijn vergeleken door te kijken naar de kwalitatieve eigenschappen, de
nauwkeurigheid en de benodigde rekenkracht. Er is gevonden dat dit algoritme het systeem echt
op het kritieke punt kan simuleren, waardoor het nieuwe soort simulaties mogelijk maakt. Echter,
de nauwkeurigheid van het algoritme weegt helaas niet op tegen de veel grotere rekenkracht
nodig om het fatsoenlijk te kunnen simuleren. Dit zou waarschijnlijk kunnen worden verbeterd,
maar hiervoor is meer onderzoek nodig.
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Contents
1 Introduction 2
2 Theoretical background 4
2.1 Truncating the singular values: White’s rule . . . . . . . . . . . . . . . . . . . . . 4
2.2 Entanglement renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Intuition from the renormalization group . . . . . . . . . . . . . . . . . . . . . . . 11
3 Tensor network methods 14
3.1 Tensor networks in graphical notation . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2 Permuting and reshaping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.3 Contracting tensors efficiently . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.4 The singular value decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.5 Tensor network states and the area law . . . . . . . . . . . . . . . . . . . . . . . 18
4 The 2D Ising model 22
4.1 Introduction to the 2D Ising model . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.2 Tensor network representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.3 Calculation of observables from the tensor network representation. . . . . . . . . 27
4.4 1D quantum 2D classical correspondence . . . . . . . . . . . . . . . . . . . . . . . 28
5 Algorithms 29
5.1 The corner transfer matrix (CTM) . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.2 The tensor renormalization group (TRG) . . . . . . . . . . . . . . . . . . . . . . 34
5.3 Tensor network renormalization (TNR) . . . . . . . . . . . . . . . . . . . . . . . 37
5.4 The symmetrization step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.4.1 Finding the isometries and disentangler . . . . . . . . . . . . . . . . . . . 40
5.4.2 Details of the implementation . . . . . . . . . . . . . . . . . . . . . . . . . 46
6 Results 49
6.1 Free energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6.2 Spontaneous magnetization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
6.3 The two-point correlation function . . . . . . . . . . . . . . . . . . . . . . . . . . 51
6.4 Qualitative comparison between TNR and TRG. . . . . . . . . . . . . . . . . . . 52
7 Discussion 56
7.1 The computational cost of the algorithms . . . . . . . . . . . . . . . . . . . . . . 56
7.2 Quantitative comparison of the algorithms . . . . . . . . . . . . . . . . . . . . . . 57
7.3 Qualitative review of TNR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
8 Conclusion 61
A Additional figures 65
B Correspondence between a 1D quantum 2D classical Ising model 68
1 Introduction
The capability to produce accurate simulations of strongly correlated many-body systems is a
key challenge in condensed matter physics. In these systems the strong interactions between
the particles give rise to exotic collective behaviour like for example the fractional quantum
hall effect or high-temperature superconductivity. However, the models that describe these
systems are only in few cases analytically solvable and good numerical simulations can give
good insight in the behaviour and different phases. But numerical calculations are still hard
to perform exact for large systems due to the exponential scaling of the Hilbert space, which
prevents any exact diagonalization like approach. Therefore proper approximate methods are
necessary to solve these models. Some systems can be accurately simulated with Monte Carlo
methods, howeverthenegativesignproblempreventsthesesimulationsforfermionicorfrustrated
models[29]. A method that does not suffer from this sign problem is White’s density matrix
renormalization group (DMRG)[35]. The DMRG allows simulations of 1D quantum systems
and, due to its accuracy, this method it has become the state of the art method of simulating
these systems[25, 23]. Later it has been shown that the DMRG has an underlying tensor
network variational ansatz called the matrix products states (MPS)[31, 32], which allow this
to be rewritten as a tensor network. The approach of DMRG is to truncate the Hilbert space
in order to find a fixed number of most relevant states. This idea is in the same sprit as the
renormalization group, as originally introduced by Kadanoff and Wilson[14, 36, 37]. One can
view it as a renormalization group transformation in the Hilbert space, resulting in a more
effective Hilbert space. From quantum information it was shown that the ground state of local
gapped Hamiltonians possess are only locally entangled [1], which is most times called an area
law for the entanglement. This means that only a small portion of the Hilbert space needs
to be considered for accurate simulations of the system. A representation of the ground state
which automatically possesses this property of local entanglement are the MPS [31]. The DMRG
algorithm tries (and succeeds) to employ this local entanglement to find these relevant states,
resulting in very accurate simulations[32, 23].
These methods however fail at criticality, because then the system no longer follows a perfect
area law. The systems do not possesses the gapped Hamiltonian with only local entanglement,
but have entanglement at each length scale[34, 1]. Although it had been shown that the 1D
systems can still accurately be simulated by an MPS[31] there is much improvement to be
made here. This problem was eventually fixed by Vidal[33] with his multi-scale entanglement
renormalization ansatz (MERA). This is an ansatz that can accurately simulate 1D (and perhaps
higher) quantum systems at criticality. By inserting disentanglers this ansatz fully removes all
the short-range entanglement, allowing systems which follow this logarithmic correction to the
area law to be properly simulated. The MERA can therefore simulate 1D quantum systems at
criticality.
All the above mentioned methods are made to simulate 1D quantum system, however the
same methods can be applied to 2D classical systems. The partition function of a 2D classical
systemhasadirectcorrespondencewiththepartitionfunctionofa1Dquantumsystem, therefore
the same methods can be applied to a 2D classical system[17]. The first algorithms which employ
this are the corner transfer matrix (CTM)[21] and the tensor renormalization group (TRG) [17].
Although both work well away from criticality, at criticality they suffer from the same problem
as the DMRG method [17]. Many variations of TRG[39, 40, 11] have been given, but all suffer
from this failure at criticality. Recently, in the same spirit as the MERA, the tensor network
renormalization (TNR) algorithm has been introduced [5, 2]. It coarse-grains classical systems
with the employment of the entanglement renormalization. One can view this as a variation on
the TRG algorithm with added disentangler, and a clear connection with the MERA algorithm
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has been shown [7]. Using this a proper coarse-graining of a classical system at criticality can be
made. In this thesis the recently proposed TNR algorithm is investigated. The theory behind
TNR suggests that it should have some qualitative properties: it should be able to properly
coarse-grain the system and produce the right fixed point structure. Therefore it should also
be able to coarse-grain the correct critical fixed point and accurately calculate quantities at
criticality. In the thesis these qualitative and quantitative behaviour of TNR is investigated.
Then TNR will be compared with some of the previous coarse-graining algorithms, namely the
previously mentioned TRG[17] and CTM[21] algorithms. Both the accuracy and computational
cost are investigated to conclude which algorithm performs best in which situation. All the
calculations will be performed on the tensor network representation of the partition function
of the two-dimensional classical Ising model, although they can easily be adapted for other
networks.
This thesis is organised in the following way: in section 2 the main theoretical foundation of
all the algorithms is presented. A way of truncating the Hilbert space effectively is shown and
the principles of entanglement renormalization are presented. Both are motivated in context
of a 1D quantum spin-1 chain, although they are also be applicable to the 2D classical Ising
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model. Then a short overview of principles of tensor networks and why they are relevant in the
context of many body physics is given in section 3. In section 4 the 2D classical Ising model is
introduced with its exact solution. Here also a way of rewriting the partition function of the 2D
classical Ising model as a tensor network is presented, and the correspondence between a 2D
classical Ising model and a 1D quantum Ising model is used to argue why the same methods
can be used in this context. After these sections the algorithms are explained in more detail,
they are all reviewed in section 5. Finally in section 6 the results are presented and they are
discussed in 7. Then the conclusions of this discussion are given in section 8.
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2 Theoretical background
In this section the main theoretical background of the algorithms is presented. First the method
of truncating a system to its largest singular values, as first introduced by Steven White in his
DMRG algorithm[35], will be discussed. This method can be seen as the basis for the CTM and
TRG algorithms. Then the problem of accumulation of short-range entanglement is introduced,
which can be solved by the method of entanglement renormalization[33]. The techniques will be
presented in context of the 1D quantum spin chain, however they can also be applied (as will be
the case with the rest of the thesis) on 2D classical models. Why this is allowed will be argued
later in context of the 2D classical Ising model. Last, a short qualitative review on some of the
basic principles of the renormalization group will be given.
2.1 Truncating the singular values: White’s rule
Themainapproachforallthealgorithmsistoapplyarenormalizationgroup(furtherabbreviated
as RG) like transformation on the configuration space (in the 2D classical system). This results
in a reduction of the degrees in freedom while still describing the relevant physics. The RG flow
through this space allow model to be represented faithfully while at the same time removing
the exponential scaling. Still, the question remains how this transformation can be performed
in a way that the system is still represented properly. The first who invented a way of doing
this efficiently for 1D quantum spin chains was S. White with his DMRG method[35]. The
justification of this method can be found in the ideas of quantum information and the finite
scaling of the entanglement entropy. In the DMRG a RG transformation is applied on the
Hilbert space, but the same principle can be applied to the configuration space off a classical
system. Therefore an outline of the ideas is presented below. However, because they are best
motivated by looking at the 1D spin chain, this is the context in which it is presented below.
The techniques can best be explained by looking at a 1D quantum spin-1 chain with a
2
gapped Hamiltonian (one can imagine for example a Ising transverse Hamiltonian, however the
specific Hamiltonian does not matter for the following story). The 1D quantum spin-1 chain has
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on each lattice site a Hilbert space given as H = { | ↓(cid:105), | ↑(cid:105)}, assuming this is the eigenstate
1
2
of the local Hamiltonian, so therefore an N particle spin-1 chain has a Hilbert space which is
2
given as
H = H ⊗H ⊗...⊗H = (H )⊗N. (1)
N 1 1 1 1
2 2 2 2
Here one can see that the size of the Hilbert space for this spin chain grows exponentially
(|H | = 2N). Due to this exponential scaling exact diagonalization of a Hamiltonian H in this
N
basis is not efficient. To solve this a block of lattice sites (2 or more) is, in accordance with the
RG group ideas, coarse-grained into an effective site. In this blocking step the total degrees
of freedom of the system is reduced - the new Hilbert space should have a smaller dimension
than the original Hilbert spaces combined. The question of when and how this is possible and
will be addressed later. Doing this RG transformation multiple times results in an ’effective’
Hilbert space with such a small dimension that it has become computationally tractable. The
Hamiltonian in each step also should be updated to an effective Hamiltonian H(cid:48) that works on
this truncated Hilbert space. Combining these two an RG-like flow arises which usually can be
written as
(H,H) → (H(cid:48),H(cid:48)) → (H(cid:48)(cid:48),H(cid:48)(cid:48)) → ... → (Heff,Heff) (2)
where |H| ≥ |H(cid:48)|.
The coarse-graining of the Hilbert space can be performed by introducing isometries. These
are operators which act on two (or more) lattice sites (more specific, the Hilbert space associated
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Figure 1: Here the idea of coarse-graining the lattice with isometries is shown pictorially. The
isometries truncate the Hilbert space of the original spin lattice H into a new, effective Hilbert
1
2
space H(cid:48), which should describe the same physics.
with this lattice site) and coarse-grain them into a single site. An isometry v, which acts on
two lattice sites with vector space V (In the first step of our lattice this would be H ) and
1
2
coarse-grains them into a single lattice site with vector space V(cid:48), can be written as a map
v : V⊗V → V(cid:48) (3)
If the vector space V has dimension n, then the vector space V(cid:48) has the dimension m ≤ n2.
When m = n2 this coarse-graining is exact and V(cid:48) contains the same degrees of freedom as the
earlier lattice. However when m < n2, the isometry truncates the vector spaces. This naturally
leads to the property of the isometry that v†v = I, but vv† ≈ I. There is a simple reasoning for
this, v† leads V(cid:48) into a larger or equal vector space V⊗V, therefore if this is again truncated into
V(cid:48) the total information can be recovered and this operation is essentially an identity operation.
The other way (vv†), however, starts with a truncation (n2 → m) and then is brought back to a
vector space of n2. This can only be achieved without loss of information when m = n2. However,
if m < n2 this is not exactly an identity. Still is can be seen as a good approximation of the
identity, because the isometry is constructed in such a way that the most relevant information is
kept. When the lattice is uniform the same isometry can be used at each lattice site, which can
speed up computational time. An example of this truncation for our original spin-1 lattice can
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be seen in figure 1.
The question still remains how to perform this truncation in such a way that it is efficient.
For a random state vector in the Hilbert space one would not expect that it would be possible
to find a general scheme such that this can be done. A priori nothing is known of its behaviour,
therefore the full Hilbert space is needed to properly describe such a vector. Luckily, in most
cases one is only interested in the ground state of the system (or low lying excitations). When
looking for the ground state of the system, it turns out that it is less entangled than a random
state which allows this truncation. This is a very important result from quantum information
theory and first used by White[35] in his DMRG paper. He suggested that the most efficient
truncation can be found by looking at the reduced density matrix of the ground state. Below
a review of his argument is given. One can see the additional structure by investigating the
entanglement in the system. To make this more concrete consider a lattice L of spins and block
B of size L. A wave function on this lattice can in general always be written as
(cid:88)
|Ψ(cid:105) = Ψ |i(cid:105) |j(cid:105) (4)
ij B L−B
ij
where |i(cid:105) ∈ H⊗L and |j(cid:105) ∈ H⊗N−L and Ψ is a matrix with the respective weights of each
ij
possible state vector. This is a very general statement for every wave function, which with the
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help of the singular value decomposition (SVD, see section 3.4) one can rewrite this as
(cid:88)
|Ψ(cid:105) = Ψ |i(cid:105) |j(cid:105)
ij B L−B
ij
= (cid:88)((cid:88)U s V† )|i(cid:105)|j(cid:105)
in nn nj
ij n
r (5)
= (cid:88)((cid:88)U |i(cid:105))s ((cid:88)V† |j(cid:105))
in nn nj
n i j
r
(cid:88)
= s |n(cid:105) |n(cid:105) .
nn i j
n
This process is called the Schmidt decomposition and it shows that the substructure of the
wave function allows it to always be written in a diagonal form. The index of summation goes
to r, which is due to the SVD defined to be r ≤ min(|H |,|H |. If this would be applied to our L
i j
spin-1 lattice sites r would be equal to r = |H⊗H| = 2L. The singular values squared sum by
2
construction (the wave function is normalized) to unity, which allow them to be interpreted as
the square root of the probability of a certain state[25, 19]. The reduced density matrix (tracing
out the whole lattice except the block B) is given as
ρB = Tr |Ψ(cid:105)(cid:104)Ψ|
env
r
(cid:88)
= s s |n(cid:105) (cid:104)m| Tr(|n(cid:105) (cid:104)m| )
nn mm i i j j
n,m
(cid:88)r (6)
= s s |n(cid:105) (cid:104)m| δ
nn mm i i mn
n,m
r r
(cid:88) (cid:88)
= s2 |n(cid:105) (cid:104)n| = P |n(cid:105) (cid:104)n|
nn i i i i i
n n
where s2 can be seen as a probability measure: it gives the probability of encountering the state
nn
|n(cid:105) (cid:104)n| . A reduced density matrix can be interpreted as a measure of how much one would
i i
learn about our states when the rest of the lattice is measured. It can also be viewed as a way
of seeing the entanglement between the block and the lattice. This can be better quantified in
terms of the entanglement entropy. The von-Neumann entanglement entropy is defined as
r
(cid:88)
S[ρs1s2] = −Tr[ρs1s2logρs1s2] = − P log(P ). (7)
i i
i=1
The entanglement entropy has again this very nice intuitive interpretation. It calculates how
much qubits one would need to describe the entangled state, or in a similar definition it gives a
measure of the number of states (which scales exponential with the entropy) one would at least
need to represent this entanglement structure. This second behaviour is the reason why it is
interesting to investigate, because it gives insight in how much states are needed to represent
this state faithfully.
This can be made more concrete by looking at the behaviour of this measure. When
r = 1 → S = 0, which means that there is no entanglement between the two sites and the
rest of the lattice. This means that the wave function can be written as a product state
|Ψ(cid:105) = |i(cid:105) |j(cid:105) which can easily be seen from equation 6. When r > 1, the block is
s1s2 L−s1s2
entangled with the rest of the lattice (as one would expect), with a maximum entanglement
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