# Geometric and bond frustration in transverse field Ising clusters

Abhiraj Jalagekar  Amador Valley High School, 1155 Santa Rita Rd, Pleasanton,
CA 94566

(August 31, 2024)

###### Abstract

The Ising model, often seen as the paradigmatic spin model, has been heavily
studied for its mathematical description of ferromagnetism in statistical
mechanics. We explore a quantum version of this model, the transverse field
Ising model, and investigate how the quantum property of magnetization
fluctuates with geometrical frustration, a phenomenon arising from the
geometry of a spin system’s ground state. We introduce a general measure for
frustration and then implement a computational model built in Python that
translates spin lattices into adjacency matrices, solves the spectrum of the
transverse field Ising model, calculates the expected magnetization of the
ground state, plots the variable over randomly generated spin systems with
controllable degrees of geometrical frustration, and compares the evolution of
these plots over an increasing transverse field. Our finite-size studies
exhibit an expected decrease in magnetization with increasing frustration and
suggest the possibility of robust phases in the large N limit over finite
ranges of frustration, which may be precursors to spin liquids or spin
glasses.

##  I Introduction

The classical Ising model, often seen as the paradigmatic spin model [1, 2,
3], is a model of ferromagnetism in statistical mechanics. The transverse
field Ising model is a quantum version of the classical model describing
qubits with nearest-neighbor couplings in a transverse field, and can be
mapped exactly to the classical Ising model in one higher dimension [4]. While
this model is paradigmatic in the field of quantum condensed matter theory, it
has only been exactly solved in up to two dimensions for the quantum case (up
to three dimensions for the classical case) [5, 6]. With the advent of quantum
simulators with the ability to coherently manipulate tens of qubits and couple
them arbitrarily [7], the study of the quantum Ising model in various
dimensions, on graphs of arbitrary connectivities, and with bond strengths of
arbitrary strengths, is worth revisiting.

Beyond the classical Ising model’s original application in the context of
magnetism [8, 9], the model’s simplicity has given it a breadth of application
in a diversity of settings, as seen through the comprehensive list found in
the detailed article by Reinhart and Cuevas [10]: a toy model of matter in
quantum gravity models [11], lattice gas models [12], the Potts model in knot
theory [13], Hopfield’s proposal in artificial neural networks [14, 15], the
size of canopy trees in ecology [16], a model for flocks of birds [17],
viruses as quasi-species [18, 19], and protein folding [20, 21, 22, 23], just
to name a few. No matter the application, it is clearly a model that provides
great value to an extensive range of scientific insight.

Geometrical Frustration in Spin Systems

In this work, we will be interested in the effects of geometrical frustration
in spin systems. First introduced in the context of magnetic systems by Gerard
Toulouse in 1977 [24, 25], the term frustration refers to a phenomenon where
qubits in spin systems become frustrated due to being forced to be both spin
up and down at the same time, thus struggling to pick one spin orientation.
More specifically, geometrical frustration is frustration that stems from the
lattice geometry of spin configuration. Due to conflicting couplings, a qubit
may face competing interactions that frustrate it. For example, one coupling
may favor the qubit being spin up, whereas another coupling may favor the
qubit being spin down. Thus, frustration can arise within the spin system
depending on how the couplings are arranged geometrically. One simple example
of geometrical frustration would be a triangular arrangement of spins with
three magnetic ions residing on the three vertices, as studied initially by G.
H. Wannier in 1950 [26]. The couplings are antiferromagnetic, meaning that the
spins between any two vertices must be antiparallel. Considering the first two
vertices, we quickly see that one must be spin up and the other spin down for
the antiparallel condition to be satisfied. However, the third ion struggles
to achieve the lowest energy level along with the other two [26, 27] because
it must be antiparallel with both the first two ions. The ion cannot be
antiparallel with a spin up ion at the same time that it must be with a spin
down ion, and thus, frustration arises within the spin system. This idea of
frustration in the triangular lattice can be extended to other more complex
lattices. From here on out, we shall reference geometrical frustration as
simply frustration for convenience.

Graph Theory and Adjacency Matrices

Qubit lattice and coupling geometry can be described by graph theory through
what are known as general dimensional graphs. These graphs consist of vertices
that represent qubits and edges between vertices that represent couplings
between qubits. These edges may be labeled with real numbers where a positive
number would represent a ferromagnetic coupling of some magnitude, a negative
number would represent an antiferromagnetic coupling of some magnitude, and
zero would represent no coupling. These general dimensional graphs can be
represented in the language of linear algebra through what are known as
adjacency matrices [28]. An $N$ qubit system would be represented by an
$N\times N$ adjacency matrix, where the entry in the $i$th row and $j$th
column would store the value of the coupling between the $i$th and $j$th
qubits. For example, if the 2nd and 3rd qubits had an antiferromagnetic
coupling of -100, the entry in the 2nd row and 3rd column of the corresponding
adjacency matrix would read -100. Observe that the entry in the 3rd row and
2nd column would also read -100 as the coupling exists equally between both
qubits. Extending this logic to the rest of the couplings, we see that
adjacency matrices, at least for our purposes, must be symmetric [29, 30].
Additionally, the coupling between a qubit and itself is defined to be 0,
making the diagonal of all adjacency matrices full of 0’s. To summarize, the
conditions for an adjacency matrix, representing spin systems, are first that
it must be symmetric and second that its diagonal must contain only 0’s.
Adjacency matrices make it easier for computers to interpret general
dimensional graphs and perform relevant calculations.

Outline

The outline of the paper is as follows. In section II, we elaborate upon our
research model in greater depth. We describe the primary equations in our
calculations, the constructed measures for our variables of interest, and the
methods used for answering our research question. We also analyze a simple
case of a three qubit system to better illustrate the ideas introduced
throughout section II. In section III, we thoroughly investigate our results,
the plots generated from our coded simulations, and we answer our research
question while also highlighting some interesting observations. In section IV,
we conclude with a discussion of our research in a broader context and address
potential focuses for future studies covering this topic.

##  II Model

Overview

Our starting point is the Hamiltonian

$H=-\sum_{i,j}J_{ij}Z_{i}Z_{j}-g\sum_{i}X_{i}$ (1)

which describes a frustrated transverse field Ising model on $N$ qubits
$(i=1,\ldots,N)$ with bond strengths $J_{ij}$. When $J_{ij}=J$, this describes
the familiar transverse field Ising model. When $J_{ij}$ varies in sign, we
say the model is frustrated.

In this work, we define a measure for frustration through what we call the $s$
parameter, and we study how magnetization over the transverse field Ising
model changes for spin systems with different levels of frustration. To do so,
we calculate and plot the dependent variable of magnetization against the
parameter $s$ (essentially representing frustration) over randomly generated
adjacency matrices with varied values of $s$.

We define the $s$ parameter to be the fraction of couplings that are negative
(antiferromagnetic) in a spin system, or in the context of adjacency matrices,
the fraction of negative entries. This parameter ranges from 0, all positive
couplings, to 1, all negative couplings. Note that the denominator of this
fraction, the total number of entries, excludes the diagonal of 0’s which
don’t represent any couplings between distinct qubits, and simply considers
the entries above and below the diagonal. The frustration of spin systems is
quite approximately measured by this parameter as after all, it is negative
couplings that give rise to contradictory conditions for spin alignment and
ultimately frustrate qubits. Consider a system of only positive couplings; we
quickly see that all the spins must be aligned and that it is completely
unfrustrated. Now consider a system of only negative couplings, and we can
imagine how various complications could arise from qubits’ spins having to be
antiparallel. Therefore, for general purposes, the $s$ parameter is a strong
measure of frustration in spin systems.

Methods

Our goal is to study magnetization vs. frustration, so we produce
magnetization vs. $s$ plots to do so. In our code, we control the $s$ value of
adjacency matrices by randomly multiplying the also random entries of the
matrix by -1 until we have our desired $s$ value. For example, if there were 7
couplings and our $s$ value was 3/7, we would randomly pick 3 couplings whose
magnitudes we would negate. We can perform this process for any $s$ value we
desire; however, there are important restrictions to be noted. Due to the
number of couplings being discrete, $s$ is discrete as well and increases from
0 in increments of 1/${N}\choose{2}$ for an $N\times N$ adjacency matrix. The
number ${N}\choose{2}$ is the total number of possible couplings in an $N$
qubit system; thus, increasing the number of negative couplings by 1 would
correspond to increasing s, the fraction of negative couplings, by
1/${N}\choose{2}$. For our plots, we generate a certain number of adjacency
matrices per each possible $s$ value, $i$/${N}\choose{2}$ where $i$ ranges
from $0$ to ${N}\choose{2}$.

Next, we calculate the magnetizations of the adjacency matrices by first
calculating the Hamiltonians of the spin system according to Eq.(1). We do so
using Python scientific computing packages like NumPy and SciPy that are
capable of performing linear algebra operations such as matrix multiplication
and tensor products. We then diagonalize the matrix representing the
Hamiltonian to find the smallest eigenvalue algebraically. The eigenvector
corresponding to this eigenvalue represents the ground state of our spin
system. We then calculate the expectation value of the magnetization of the
ground state according to the formula

$\overline{M}=\frac{1}{N_{\text{couplings}}}\sum_{i,j}\langle
Z_{i}Z_{j}\rangle_{\text{gnd}}$ (2)

where $N_{\text{couplings}}$ is the total number of nonzero couplings.

For the final step of answering our research question, we produce plots of
magnetization vs. $s$ over numerous qubit graphs with $s$ ranging from 0 to 1.
There are several parameters we set when producing a plot. First are the lower
and upper bounds for the magnitudes of the entries in our adjacency matrices,
which we call $(j,k)$ with $j$ being the lower bound and $k$ being the upper
bound. For example, a matrix with $(j,k)=(3,5)$ would contain entries of real
numbers with magnitudes ranging from 3 to 5, inclusive. Recall that their
signs are determined randomly by the given $s$ value. The next parameter is
simply the number of qubits $N$ in the spin system we model; this parameter
plays a role when creating the $N\times N$ adjacency matrices and is simply
ubiquitous in our formulae. Note that as $N$ scales up and the spin system
grows larger, calculations take increasingly longer to be performed.
Throughout the majority of our paper, we stick with $N=8$, which gives us a
sufficiently large enough system to gain insight into the question we wish to
answer. The last parameter is the strength of the transverse field (an
external magnetic field), or the parameter $g$ as seen in the formula for the
Hamiltonian Eq.(1). Setting $g$ equal to 0 is the equivalent of turning off
the effects of the transverse field. In this work, we initially take a look at
results with the transverse field turned off, but later, we analyze
interesting trends in the evolution of plots with transverse fields of
increasing strengths.

Simple Case: Three Qubits

To better introduce the ideas described earlier, we examine a simple case of
our model, a three qubit system with the transverse field turned off, and we
solve for the ground state and magnetization. Let the three coupling strengths
between the qubits be $c_{1}$, $c_{2}$, and $c_{3}$, and without loss of
generality, let $c_{1}\leq c_{2}\leq c_{3}$. Thus, our corresponding adjacency
matrix is

$J=\left[\begin{array}[]{ccc}0&c_{1}&c_{2}\\\ c_{1}&0&c_{3}\\\
c_{2}&c_{3}&0\end{array}\right].$

Using our initial formula for the Hamiltonian Eq.(1), we get the $8\times 8$
matrix

$\begin{gathered}H=\text{diag}[(-c_{1}-c_{2}-c_{3}),(-c_{1}+c_{2}+c_{3}),\\\
(c_{1}-c_{2}+c_{3}),(c_{1}+c_{2}-c_{3}),(c_{1}+c_{2}-c_{3}),\\\
(c_{1}-c_{2}+c_{3}),(-c_{1}+c_{2}+c_{3}),(-c_{1}-c_{2}-c_{3})]\end{gathered}$

where diag[ ] represents a diagonal matrix whose entries from left to right
and top to bottom on the diagonal are the values inside the brackets.
Consequently, given that $c_{1}\leq c_{2}\leq c_{3}$, the ground state energy
algebraically is one of two options:

$\begin{gathered}{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}-----}\lambda_{\text{gnd,1}}=(-c_{1}-c_{2}-c_{3})\\\
{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}-----}\lambda_{\text{gnd,2}}=(c_{1}+c_{2}-c_{3}).\end{gathered}$

depending on whether $-c_{1}-c_{2}\leq c_{1}+c_{2}$. The corresponding ground
states are

$\displaystyle|\lambda\rangle_{\text{gnd,1}}$
$\displaystyle=\alpha\hat{e}_{1}+\beta\hat{e}_{8}$
$\displaystyle|\lambda\rangle_{\text{gnd,2}}$
$\displaystyle=\gamma\hat{e}_{3}+\delta\hat{e}_{4}$

where $|\alpha|^{2}+|\beta|^{2}=|\gamma|^{2}+|\delta|^{2}=1$ and
$[\hat{e}_{j}]_{k}=\delta_{jk}$. Thus, we observe that the system has
classical frustration but lacks quantum frustration without the transverse
field due to the differing bond strengths $c_{1},c_{2},c_{3}$.

Now, we proceed with solving for the $n$ possible expectation values of the
ground state’s magnetization. Using its formula from earlier Eq.(2), we get

$\begin{gathered}{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}---}\overline{M_{n}}=\frac{1}{3}\sum_{i,j}\langle
Z_{i}Z_{j}\rangle_{\text{gnd,$n$}}\\\
{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}--}=\frac{1}{3}(|\lambda\rangle_{\text{gnd,$n$}})^{\dagger}(\sum_{i,j}Z_{i}Z_{j})(|\lambda\rangle_{\text{gnd,$n$}}).\end{gathered}$

Performing similar calculations to those of the Hamiltonian except without the
coupling strengths, we have

$\sum_{i,j}Z_{i}Z_{j}=\text{diag}[3,-1,-1,-1,-1,-1,-1,3].$

Substituting this back into the magnetization calculation and multiplying with
the ground state vector and its adjoint, we get $\overline{M_{1}}=1$ and
$\overline{M_{2}}=-\frac{1}{3}$. Recall the condition we had established for
choosing between the two ground states: $-c_{1}-c_{2}\leq c_{1}+c_{2}$. This
inequality can be rewritten in simpler terms for the purpose of determining
the expected magnetization for a three qubit system’s ground state:

$\begin{gathered}{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}-----}c_{1}+c_{2}\geq
0\text{ }\text{ }\Longrightarrow\text{ }\text{ }\overline{M}=1\\\
{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}-----}c_{1}+c_{2}<0\text{
}\text{ }\Longrightarrow\text{ }\text{
}\overline{M}=-\frac{1}{3}.\end{gathered}$

Figure 1: Expected magnetization vs. $s$ plot with transverse field turned off
for 3 qubits.

Our results for magnetization are confirmed by our code through the plot it
generates for three qubits, as seen above in Fig. 1. For each of the ten
systems plotted at each of the four $s$ values, the magnetization is either 1
or $-\frac{1}{3}$, just as we calculated earlier. Furthermore, for matrices
with lower $s$ values which indicate mostly positive couplings and would thus
satisfy the inequality $c_{1}+c_{2}\geq 0$, our plot shows magnetizations of
1. Likewise, for matrices with higher $s$ values which indicate mostly
negative couplings and would thus satisfy the inequality $c_{1}+c_{2}<0$, our
plot shows magnetizations of $-\frac{1}{3}$.

##  III Results

Figure 2: Evolution of expected magnetization vs. $s$ plots with transverse
field strengths increasing from $g$ = 0 to $g$ = 10 for 8 qubits.

We now proceed to an in-depth investigation of the magnetization of the
frustrated transverse field Ising model described in Eq.(1) as a function of
the transverse field and the degree of frustration in the bonds $J_{ij}$. This
is plotted in Fig. 2 for various values of transverse field strength
increasing from $g$ = 0 to $g$ = 10. Each plot captures an eight-qubit cluster
with $(j,k)=(1,1.1)$ and $s$ values ranging from 0 to 1 in increments of
1/${8}\choose{2}$ $\approx 0.036.$ At each $s$ value, 10 random adjacency
matrices are generated, and the corresponding magnetizations and $s$ values
are computed and plotted. Observe that it is very possible for these points to
overlap, as seen abundantly in the $g$ = 0 plot where some $s$ values give the
false impression of having only one point.

Let us make a primary, elementary observation about the behavior of these
plots. In general for all $g$, we see that as frustration increases,
magnetization tends to decrease. This is expected because an increase in
frustration corresponds to a greater fraction of antiferromagnetic bonds,
which favor spins anti-aligning, leading to a decrease in the magnetization.

Next, let us turn our attention to the case of $g$ = 0 plot. In this case, $H$
is diagonal in the $Z$ basis, and the frustration can be considered a
classical effect because the energy eigenstates are always product states in
the $Z$ basis and do not have any entanglement between different qubits. It is
remarkable that this classical frustration still has pronounced discrete
effects – magnetization jumps – which are possible analogs of quantum phase
transitions in more complicated frustrated systems. In particular, for these
eight qubit clusters, the expected magnetization jumps from $1$ to $0.5$ at a
critical value of

$s_{\text{crit}}\approx 0.2.$

From numerical observations, this corresponds to the typical lowest value at
which a single row of the adjacency matrix $J_{ij}$ (or column) contains a
majority of negative entries, and this results in an order 1 fraction of the
cluster having mutually antiferromagnetic couplings. Finally, a sequence of
“classical” phase transitions (i.e. phase transitions between different
product states) ensues from tuning $s$ from $0$ to $1$ at critical values
$s\approx 0.25,0.3,0.35$.

Next, let us review the familiar case of $s=0$ and varying $g$. In this case,
the familiar results of the transverse Ising model tell us that the
magnetization drops from 1 to 0 with a critical point at $g=1$, which is
consistent with our results.

Finally, when $s>0$ and $g>0$, we observe that the transverse field smears the
discrete magnetization jumps which come from the classical frustration. When
$g$ is order 1, the states are already far from product states and highly
entangled, and we can regard the finite $s$ transition from a polarized to an
unpolarized state as a true quantum phase transition or crossover (which has
its origin in the classical frustration phase transitions). For instance, when
$g=2$, we observe a smooth drop from a polarized to an unpolarized state with
a “critical value” of approximately $s\approx 0.4$.

##  IV Discussion

In conclusion, we have studied finite-size clusters of a variant of the
transverse field Ising model, where we have allowed the bond strengths to vary
in magnitude and (crucially) sign, allowing for arbitrary degrees of both
geometric and bond frustration. We have found that the magnetization of the
resulting clusters jumps discretely as we tune the frustration, measured here
in a coarse-grained fashion as the fraction $s$ of negative bonds, from $0$ to
$1$. These magnetization jumps are due to an order $1$ fraction of the graph
belonging to an antiferromagnetic sub-graph. With the addition of the
transverse field, these transitions are broadened and smoothed.

Although this study has been done with a finite system size ($N=8$), we
include a few predictions about the behavior of magnetization with frustration
in larger systems sizes $(N\to\infty$). In this case, we expect that
frustration is tunable as a continuous parameter (as the fraction of negative
bonds) and the crossovers at finite $g$ from polarized to unpolarized states
may be sharpened into true phase transitions. The classical frustration case
($g=0$) may be amenable analytically. Whether these expectations are borne out
remains an interesting open question.

Finally, the study of Ising clusters is a cartoon model of the transverse
field Ising model in arbitrary dimensions because, for instance, the
coordination number of a cubic lattice in $d$ dimensions grows as $2^{d}$, and
this behavior is mimicked by an all-to-all coupled graph. In future studies,
it will be interesting to understand which features of the general $d$
dimensional transverse field Ising model are well-captured by randomly coupled
all-to-all Ising clusters.

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