eigenstate phase transitions...phase transitions have been discovered and studied; they are...
TRANSCRIPT
-
Eigenstate Phase Transitions
Bo Zhao
A Dissertation
Presented to the Faculty
of Princeton University
in Candidacy for the Degree
of Doctor of Philosophy
Recommended for Acceptance
by the Department of
Physics
Adviser: Professor David A. Huse
September 2015
-
c© Copyright by Bo Zhao, 2015.
All rights reserved.
-
Abstract
Phase transitions are one of the most exciting physical phenomena ever discovered.
The understanding of phase transitions has long been of interest. Recently eigenstate
phase transitions have been discovered and studied; they are drastically different
from traditional thermal phase transitions. In eigenstate phase transitions, a sharp
change is exhibited in properties of the many-body eigenstates of the Hamiltonian of
a quantum system, but not the thermal equilibrium properties of the same system.
In this thesis, we study two different types of eigenstate phase transitions. The first
is the eigenstate phase transition within the ferromagnetic phase of an infinite-range
spin model. By studying the interplay of the eigenstate thermalization hypothesis
and Ising symmetry breaking, we find two eigenstate phase transitions within the
ferromagnetic phase: In the lowest-temperature phase the magnetization can macro-
scopically oscillate by quantum tunneling between up and down. The relaxation of
the magnetization is always overdamped in the remainder of the ferromagnetic phase,
which is further divided into phases where the system thermally activates itself over
the barrier between the up and down states, and where it quantum tunnels. The
second is the many-body localization phase transition. The eigenstates on one side
of the transition obey the eigenstate thermalization hypothesis; the eigenstates on
the other side are many-body localized, and thus thermal equilibrium need not be
achieved for an initial state even after evolving for an arbitrary long time. We study
this many-body localization phase transition in the strong disorder renormalization
group framework. After setting up a set of coarse-graining rules for a general one
dimensional chain, we get a simple “toy model” and obtain an almost purely ana-
lytical solution to the infinite-randomness critical fixed point renormalization group
equation. We also get an estimate of the correlation length critical exponent ν ≈ 2.5.
iii
-
Acknowledgements
No one can achieve a PhD without others’ help. I am very thankful to have this
opportunity of pursuing the PhD in physics department in Princeton. First of all,
the honor should be given to my advisor, Professor David Huse. It is he who led me
to the palace of physics. Before entering graduate school in Princeton, I was just a
student who had some ability in learning in courses and doing homework. One of the
most important lessons David has shown me is how to tackle a research topic where
there might be no preexisting theory. By doing research with David during these PhD
years , I have experienced both exciting and painful moments. But David has taught
me, through his own attitude, that when doing research, one should keep cautious,
positive and energetic. What typically happens during research is that one keeps
trying and failing until stages where a new method, model or direction are found. I
am very grateful to learn from David, in terms of both his strong physical intuition
and his encouraging personality.
Secondly, I would like to thank my colleagues who have helped me in my academic
life in my PhD years. In David’s group, I have had many meaningful discussions about
physics with Hyungwon Kim and Liangsheng Zhang. Both of them have helped me a
lot in digesting the background physics in our research projects. I would also like to
thank my colleagues, Aris Alexandradinata and Vedika Khemani who have also given
me lots of advice about many academic aspects.
Thirdly, PhD’s life is not all about research. I enjoyed these years in Jadwin with
all of my friends, including, but not limited to, Chaney Lin, Akshay Kumar, Bin Xu,
Yu Shen and many more graduate students who often appeared on the 4th floor in
Jadwin. Without them, my life in Jadwin would be much more lonely and much less
interesting. There are countless many friends who I enjoyed staying with, so I also
would like to thank all of them.
iv
-
Last, but definitely not the least, I must give special thanks to my family. The
most selfless love and support were consistently given by my parents. Without them,
this thesis would have never been completed.
v
-
To my parents.
vi
-
Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
1 Introduction 1
1.1 Quantum Thermalization . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Eigenstate Thermalization Hypothesis . . . . . . . . . . . . . . . . . 5
1.3 Many-body Localization . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Eigenstate Phase Transition . . . . . . . . . . . . . . . . . . . . . . . 10
1.5 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Three ‘Species’ of Schrödinger Cat States in an Infinite-range Spin
Model 12
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 Infinite Range Transverse Field Ising Model . . . . . . . . . . . . . . 16
2.2.1 Integrability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.2 Energy Level Degeneracy . . . . . . . . . . . . . . . . . . . . . 17
2.2.3 Discrete WKB Method . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4 Eigenstate Thermalization . . . . . . . . . . . . . . . . . . . . . . . . 24
vii
-
2.5 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.5.1 Thermal Activation and Quantum Tunneling . . . . . . . . . . 26
2.5.2 Paired States and Unpaired States . . . . . . . . . . . . . . . 31
2.6 Analytical Calculation of the Thermal Activation and Quantum Tun-
neling Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.7 Asymptotic Behaviors of Both Transitions Near Both QCP and
u = 0 ,Γ = 0 Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.7.1 Near QCP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.7.2 Near u = 0 ,Γ = 0 Point . . . . . . . . . . . . . . . . . . . . . 43
2.8 Numerical Evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3 Strong Disorder Renormalization Group Approach to Many-body
Localization Transition 54
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.2 RG Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.2.1 RG Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.2.2 RG Flow of Probability Distributions . . . . . . . . . . . . . . 67
3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.3.1 Fixed Point Distribution . . . . . . . . . . . . . . . . . . . . . 68
3.3.2 Critical Exponent . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.3.3 Numerical Evidence . . . . . . . . . . . . . . . . . . . . . . . . 91
3.4 Conclusion and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . 95
A Mathematical Proof of the Degeneracy Formula 97
B Estimation of the Magnitude of Disorder 103
B.1 Lower Bound of Disorder . . . . . . . . . . . . . . . . . . . . . . . . . 104
B.2 Upper Bound of Disorder . . . . . . . . . . . . . . . . . . . . . . . . . 107
viii
-
C Divergence of First Order Derivative of γ(s̄) at s̄ = s̄min 109
D Discussion on Convexity of Function γ(s̄) 112
Bibliography 116
ix
-
List of Tables
1.1 A list of some properties of the many-body localized phase contrasted
with properties of the thermal phases. Table from [19]. . . . . . . . . 9
x
-
List of Figures
2.1 The phase diagram of our model. u is the energy per spin and Γ is
the transverse field. The ground state (zero temperature) is indicated
by the black (solid) line, with the quantum critical point (QCP) indi-
cated. The green (dot-dashed) line is the thermodynamic phase tran-
sition (PT) between the paramagnetic and ferromagnetic (F) phases.
There are two dynamical (eigenstate) phase transitions within the fer-
romagnetic phases, indicated by the blue (dashed) lines. See the text
for discussions of the sharp distinctions between these three phases F1,
F2 and F3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Two “potential-energy curves” in our model. . . . . . . . . . . . . . . 20
2.3 Constant energy (E) line described by equation (2.25) by setting s̄y = 0
or just by equation (2.22). It determines the WKB turning point xt
when the total spin density s̄ is fixed. . . . . . . . . . . . . . . . . . . 27
2.4 A sketch of the entropy Σ(s̄) and the tunneling rate γ(s̄). . . . . . . 29
2.5 A sketch of the difference between thermal activation and quantum
tunneling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
xi
-
2.6 The level-spacing statistics using 100 realizations of H at N = 15 in
phase F1 within the even sector. δ is the ratio between the smaller
level spacing δ< to the larger level spacing δ> for three consecutive
eigenenergies in the even sector. f is the relative frequency for each
bin in this histogram. . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.7 Averages of log (D) in phases F1 and F3, respectively, where D is the
‘eigenstate distance’ defined in the text.. The energy density range we
used in F1 is from the first excited state in each sector up to uc − 0.02
where uc is the energy density at the phase boundary between F1 and
F2, whereas in F3 we used the phase’s full energy density range. N
is the total number of spins varying from 8 to 15. The exponential
decrease of D with increasing N indicates thermalization. The error
bars come from averaging over 100 realizations. . . . . . . . . . . . . 48
2.8 The mean ᾱn and the standard deviation ∆αn of the quantity αn de-
fined in Eq. (2.81). The number of realizations is 1600 for N = 11
(blue dash-dotted lines) and 100 for N = 15 (red dashed lines). The
green (solid) line gives the theoretical quantity α(u,Γ) defined in Eq.
(2.32) for the system size N → ∞. . . . . . . . . . . . . . . . . . . . 49
3.1 A sketch of typical RG moves. For example, (a) is to fuse two adjacent
thermal (T) blocks into one thermal block, called “TT move”; all others
are similar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.2 Q∗(η) up to η = 20 using composite Trapezoidal rule. (a) shows the
curve and (b) demonstrates it in a semi-log plot with a linear regression
fit. (c) plots the cumulative area under the curve. . . . . . . . . . . . 80
3.3 The eigenfunction f−(η) corresponding to eigenvalue 0.3995: (a) Linear
scale (b) Absolute value on log scale. . . . . . . . . . . . . . . . . . . 90
xii
-
3.4 The eigenfunction f−(η) using either the numerical integration directly
or the diagonalization. . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.5 The cumulative distribution function (Cdf) for both the T-blocks and
I-blocks as the total number of blocks N decreases from 107 to 103 and
the cutoff Λ grows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.6 The difference of cumulative distribution functions (Cdf) between the
T-blocks and I-blocks as the total number of blocks N decreases from
107 to 103 and the cutoff Λ grows. . . . . . . . . . . . . . . . . . . . . 94
D.1 Comparison between numerical and analytical result of the thermal ac-
tivation and quantum tunneling transition line; the dots are numerical
points and the line underneath the dots is the analytical result given
by equation (2.47). They fit perfectly well. . . . . . . . . . . . . . . . 113
D.2 Numerical results of ∂γ∂s̄
as a function of s̄ by sowing different pairs of
Γ and u randomly within the entire ferromagnetic phase region. . . . 114
xiii
-
Chapter 1
Introduction
Our understanding of the laws of nature has taken giant leaps in the past several
centuries, beginning from the time of Galileo. For mechanics, Newton first postulated
the classical equations of motion that every macroscopic object should obey, these
equations, we now call “Newton’s Laws”. These laws solve nearly every mechani-
cal phenomenon we observe in daily life! However, when we would like to further
understand the microscopic structures of thermal behaviors, we must consider many
interacting degrees of freedom, too many to explicitly solve for the dynamics, say
from Newton’s Laws or from Schrödinger equation for a quantum system. This gave
birth to the area of statistical mechanics.
Classical statistical mechanics is very successful at explaining equilibrium ther-
mal behavior in daily life, for example the concept of temperature. Thus, at least
phenomenologically speaking, we know the solution based on the essence of classical
equilibrium statistical mechanics: the ensemble theory. But the foundation of the
ensemble theory is to assume a probabilistic nature of so-called microstates. If we
want to get the equilibrium probability distributions in ensemble theory purely from
Newton’s Laws, either we assume that the initial state is itself an ensemble, or we get
the ensemble by including all long time states of the system (ergodicity). Fortunately,
1
-
people have successfully discovered a more fundamental theory, quantum mechanics,
which is itself a probabilistic theory. Quantum statistical mechanics is the product
of combining quantum mechanics and statistical mechanics. In quantum statistical
mechanics, ensemble theory naturally arises from the probabilistic nature of quantum
mechanics, so the equilibrium ensemble can emerge at any specific long time even if
the initial state is a pure state.
In quantum statistical mechanics, one of the most fundamental questions is
whether a closed quantum system equilibrates. This question is by no means as easy
as it appears. One of the difficulties is the apparent time reversal symmetry breaking
which arises from low entropy initial states. However, the quantum mechanical
formalism obeys the time reversal symmetry, through the unitary time evolution of
a quantum state. Thus, at least, we need to formulate a more rigorous definition of
equilibration called quantum thermalization.
In the remaining sections of this chapter, I will briefly review the concepts of
quantum statistical mechanics, including, but not limited to, quantum thermalization,
eigenstate thermalization hypothesis and many-body localization.
1.1 Quantum Thermalization
Consider a closed quantum system with a time independent Hamiltonian H. In
quantum mechanics, the system is described by a ket vector |ψ〉 in the Hilbert space.
But in quantum statistical mechanics, |ψ〉 is not enough. It only covers the pure
states, a subset of the quantum states described by the probability operator (also
known as density matrix) ρ. As a probability operator, ρ needs to satisfy the following
constraints:
ρ† = ρ , tr ρ = 1 , (1.1)
2
-
and also, the spectrum of ρ is in the interval [0, 1].
After describing the quantum state of the system, what we need next is the law
of time evolution. If we choose the Schrödinger picture, the time evolution of ρ(t) as
a function of time t is given by the unitary operator e−iHt/�:
ρ(t) = e−iHt/�ρ(0)eiHt/� , (1.2)
or the differential equation form:
i�d
dtρ(t) = [H, ρ(t)] . (1.3)
By establishing both the quantum state and the rule of time evolution, we are
able to formulate a definition for quantum thermalization. Roughly speaking, as
mentioned earlier, thermalization means that the system goes to equilibrium in the
limit of a long time evolution. But if we consider the full probability operator of an
entire closed quantum system, the time evolution is unitary; it will never reach equi-
librium, because such a state is, by definition, fully determined by initial conditions.
In other words, the unitary time evolution “remembers” all information about the
initial state. Instead, we consider thermalization in terms of subsystems. And the
idea is to use the rest of the full system to serve as a heat bath to thermalize the
selected subsystem.
When we are talking about thermalization, we need to send our system to the
thermodynamic limit where the total degrees of freedom N goes to infinity. In gen-
eral, the full system can have several extensive conserved quantities, for example,
total energy, particle number, total spin, etc. But to keep the discussion simple, we
assume the system only has one extensive conserved quantity, the total energy. So if
the system goes to thermal equilibrium, the equilibrium state has only one thermody-
namic parameter, the temperature T . In order to take the thermodynamic limit, we
3
-
need a sequence of both the initial states ρN(t = 0) and the Hamiltonian HN labeled
by N . In addition, we need to restrict to initial states ρN(0) where the uncertainty in
total energy is subextensive, so T has zero uncertainty for N → ∞. Now we consider
a subsystem S. The degrees of freedom in the full system F can be decomposed
into the degrees of freedom in the subsystem S and the remaining degrees of freedom
—called the “bath”— B:
F = S ⊕ B , (1.4)
for the sets of degrees of freedom. (It is a tensor product in Hilbert space.) Then
the thermodynamic limit corresponds to taking both F and B to infinity but keeping
S finite. We consider the probability operator ρS(t) on S (known as the “reduced”
density matrix)
ρS(t) = trB ρN(t) . (1.5)
Thermalization on the subsystem S means that by sending the system to the ther-
modynamic limit, the probability operator of the subsystem S goes to its thermal
equilibrium value indicated by the temperature T :
limN→∞t→∞
ρS(t) = ρeqS (T ) , (1.6)
where
ρeqS (T ) = lim
N→∞trB ρ
eqN (T ) (1.7)
4
-
and
ρeqN (T ) =
1
Ze−βH . (1.8)
The last expression is the standard quantum statistical mechanical formula describing
the equilibrium state in the canonical ensemble where Z is the partition function
Z = trF e−βH and β is the temperature parameter β = 1/kBT .
In other words, quantum thermalization of a subsystem S means that the subsys-
tem behaves as if the full system is exactly at the thermal equilibrium state. And
the rest of the full system serves as a heat bath. And if for all subsystems S, at long
time t the above thermalization condition is true for the same temperature T , and
for all initial states corresponding to that T , we say the system thermalizes for this
temperature. [10, 26, 27, 22, 19]
1.2 Eigenstate Thermalization Hypothesis
By defining quantum thermalization, we can further study the destiny of a given
closed quantum system H. From everyday thermal phenomena, one would expect
that if a system can equilibrate, then the system must thermalize from any initial
conditions. Then, it follows that all the exact many-body eigenstates of the full
system must be thermal because the probability operators, ρeigen = |ψ〉〈ψ|, do not
change with time, where |ψ〉 satisfies H|ψ〉 = E|ψ〉. This motivates the following
hypothesis called the Eigenstate Thermalization Hypothesis (ETH):[10, 26, 27, 22]
Every single eigenstate thermalizes.
By saying eigenstate, we are making no approximations. We mean the exact many-
body eigenstate of the full system. By saying thermalize, we mean all subsystems
thermalize for the temperature determined by the eigenenergy of the eigenstate.
5
-
ETH is a hypothesis. It is not true for systems that are many-body localized
(MBL) as I will discuss in the next section. Also, it is not true for integrable systems
which contain infinite number of local conservation laws. By saying local, we do
not necessarily mean local in real space, it can be local in for example momentum
space. But it should not be global because it is trivial that for any given full system
eigenstate |n〉, the projection operator |n〉〈n| is a conserved operator. And so is any
weighted sum over them. Thus there are infinite number of “global” conservation
laws for any given system. But these projection operators are global and presently
inaccessible to measurement.
ETH is a hypothesis in another sense, that even for those systems where ETH
appears to be true, for example everyday thermal phenomena, it is extremely hard to
prove, even numerically, because one needs to test the exact many-body eigenstates
which requires the exact diagonalization of the full system. In practice, this approach
is limited to N only up to about 20, due to the exponential growth of the dimension
of H. But that has not deferred people from finding numerical evidence for ETH.
Indeed, there has been plenty of research tackling the numerical test of ETH and
there is strong support that ETH is true for a large number of systems, for example
Rigol, Dunjko and Olshanii [22]; Pal and Huse [20]; Kim and Huse [16]; Beugeling,
Moessner, and Haque [6] and many more.
Thus it is worthwhile to mention some consequences based on ETH. One of them
is for the diagonal ensemble. We write the probability operator in the basis of the
exact many-body eigenstates of the full Hamiltonian:
ρ =∑
m,n
|m〉ρmn〈n| , (1.9)
then based on the time evolution of the probability operator, we have that the diagonal
matrix element ρnn stays constant and the off-diagonal matrix element ρmn, m = n,
6
-
evolves by multiplying a “simple” phase factor:
ρmn(t) = e−i(Em−En)t/�ρmn(0) , m = n , (1.10)
where Em and En are the eigenenergies of the corresponding eigenstates |m〉 and |n〉:
H|n〉 = En|n〉 . (1.11)
When t → ∞, the phase factors in the off-diagonal terms become essentially random
and when they contribute to the probability operator of any local subsystem, they
effectively go to their mean value 0. This procedure is called dephasing which causes
the full system to thermalize when ETH is true. After dephasing, the probability
operator becomes diagonal:
ρD =∑
n
|n〉ρnn〈n| , (1.12)
which is called the diagonal ensemble. It neglects all the off-diagonal terms, and the
diagonal terms are set by the initial condition. As being two special cases of the
diagonal ensemble, we also have the canonical ensemble as appears in the standard
equilibrium statistical mechanics
ρeq =1
Ze−βH =
1
Z
∑
n
|n〉e−βEn〈n| (1.13)
and the single many-body eigenstate “ensemble”
ρ(n) = |n〉〈n| (1.14)
as a limit of the standard microcanonical ensemble by sending the available energy
window ∆E → 0. When ETH is true, all different forms of the diagonal ensembles
7
-
including the two special cases above are equivalent for small subsystems, provided
the uncertainty in the total energy remains less than extensive.
1.3 Many-body Localization
As mentioned in the previous section, there exists a large class of systems, localized
systems, which do not obey ETH. The idea of localization first came from Anderson [1]
in 1958. In this section, I will focus on one subset of the localized systems: interacting
many-body localized (MBL) systems.
To briefly demonstrate the idea of many-body localization, I use the model of Pal
and Huse [20]. It is a one dimensional spin chain with spin-1/2 with Hamiltonian:
H =∑
i
hiσzi +
∑
i
Jσi · σi+1 , (1.15)
where σi are the Pauli operators for the spin-1/2 at ‘site’ i. The onsite magnetic
fields hi are static random variables with a probability distribution that is uniform in
[−h, h].
At J = 0, the many-body eigenstates are just the product states which are the
basis of the σz representation and the system is fully localized. For nonzero J , if
we assume J ≪ h, we can apply perturbation theory with zeroth order J = 0 to
construct the many-body eigenstates [5]. Because the local level spacing produced
by the J = 0 product states are comparable to h, it is in general much larger than
the interaction J . It implies that the eigenstates are very weakly hybridized. This
argument further implies that there is no DC spin transport or energy transport
and so the quantum thermalization is violated [5]. In addition to this perturbative
argument, Pal and Huse [20] demonstrated numerical evidence for the generic non-
perturbative case by doing the exact diagonalization up to 16 spins. They showed
8
-
that when h/J > hc/J ≈ 3.5, the system fails to behave thermal and goes into the
localized phase.
There are several properties that differ between thermalization and localization.
Some are summarized in Table 1.1 cited from [19]. Note that these differences are all
in dynamical properties or properties of the exact eigenstates. In fact, the two phases
do not differ at all in their static thermal equilibrium properties.
Thermal Phase Many-body Localized Phase
Memory of initial conditions ‘hid-den’ in global operators at longtimes
Some memory of local initial con-ditions preserved in local observ-ables at long times
ETH true ETH false
May have non-zero DC conduc-tivity
Zero DC conductivity
Continuous local spectrum Discrete local spectrum
Eigenstates with volume-law en-tanglement
Eigenstates with area-law entan-glement
Power-law in time spreading ofentanglement from non-entangledinitial condition
Logarithmic in time spreading ofentanglement from non-entangledinitial condition
Dephasing and dissipation Dephasing but no dissipation
Table 1.1: A list of some properties of the many-body localized phase contrasted withproperties of the thermal phases. Table from [19].
I close this section by mentioning the definition of temperature in the MBL phase.
As we know in the standard statistical mechanics, temperature is a crucial concept
in the equilibrium state. But in the MBL phase, the system fails to obey the ETH
hence the system cannot reach the equilibrium state. So temperature is ill-defined in
the MBL phase. One way to define temperature is to say that it is the corresponding
temperature if the system could thermalize. This implies a brand new type of phase
transition called the eigenstate phase transition and it will be discussed in the next
section.
9
-
1.4 Eigenstate Phase Transition
The eigenstate phase transition, as formulated e.g. in [13], is a brand new type
of phase transition compared with traditional thermal phase transition. In thermal
phase transition, we see a sharp change in terms of the thermal equilibrium when we
go from one phase to another. For example, from water to ice, the sharp change can
be observed directly by solving the minimal value of the free energy in the standard
statistical mechanical way and the minimal point of the free energy just corresponds to
the thermal equilibrium. However, in eigenstate phase transitions, the sharp change is
in properties of the many-body eigenstates but not the thermal equilibrium because
thermal equilibrium is an average over lots of eigenstates and this average washes
out the sharp change in terms of single many-body eigenstates. Thus, eigenstate
phase transition is invisible to the equilibrium statistical mechanics. For example,
as I mentioned in the previous section, the phase transition between the thermal
phase and the many-body localized phase is an eigenstate phase transition. In the
MBL phase, the system cannot even evolve to its thermal equilibrium value and some
information is still stored locally for a long time. In fact, if one would use the standard
equilibrium statistical mechanics to solve the MBL system, one could still get some
result and there would be no sharp change between the two phases in this sense.
The traditional equilibrium statistical mechanics however, does not correctly give the
long-time behavior in the MBL phase.
Because the eigenstate phase transition is a brand new type of phase transition,
it by itself has become a topic of research. Also, the usual equilibrium statistical
mechanics may break down when applied to the eigenstate phase transition, so the
properties of the critical point in the eigenstate phase transition, for example the
MBL transition, need to be reestablished and many questions about it are still open
[19]. In addition, eigenstate phase transition is not limited to the MBL transition.
Even in the regime where ETH is true on both sides, there remain eigenstate phase
10
-
transitions, at least in the limit of long-ranged interaction [32]. These are one of the
main topics in this thesis.
1.5 Thesis Outline
In this thesis, I will discuss two examples of eigenstate phase transitions as men-
tioned in the previous section. In Chapter 2, we explore a transverse-field Ising
model that exhibits both spontaneous symmetry-breaking and eigenstate thermaliza-
tion. Within its ferromagnetic phase, the exact eigenstates of the Hamiltonian of any
large but finite-sized system are all Schrödinger cat states: superpositions of states
with ‘up’ and ‘down’ spontaneous magnetization. This model exhibits two eigenstate
phase transitions within its ferromagnetic phase: In the lowest-temperature phase the
magnetization can macroscopically oscillate by quantum tunneling between up and
down. The relaxation of the magnetization is always overdamped in the remainder
of the ferromagnetic phase, which is further divided into phases where the system
thermally activates itself over the barrier between the up and down states, and where
it quantum tunnels.
In Chapter 3, we study the many-body localization transition in one dimensional
systems via the strong disorder renormalization group approach. In this framework,
we impose a set of rules for coarse-graining the system. The result from this set
of rules turns out to be a beautifully simple “toy” renormalization group. We can
almost solve for the critical fixed point distribution analytically. In addition, we also
get an estimate of the correlation length critical exponent ν ≈ 2.5 by both solv-
ing the analytical equations numerically and directly simulating the coarse-graining
procedure.
11
-
Chapter 2
Three ‘Species’ of Schrödinger Cat
States in an Infinite-range Spin
Model
2.1 Introduction
The dynamical properties of isolated many-body quantum systems have long been
of interest, due to their role in the fundamentals of quantum statistical mechanics.
More recently, experiments approximating this ideal of isolated many-body quantum
systems have become feasible in systems of trapped atoms [17, 8] and ions [7], and
as a consequence this topic has received renewed attention. It appears that a broad
class of such systems obey the Eigenstate Thermalization Hypothesis (ETH) [10, 26,
27, 22, 19]. The ETH asserts that each exact many-body eigenstate of a system’s
Hamiltonian is, all by itself, a proper microcanonical ensemble in the thermodynamic
limit, in which any small subsystem is thermally equilibrated, with the remainder of
the system acting as a reservoir. In the present chapter we present some interesting
12
-
results for an infinite-range transverse-field Ising model that obeys the ETH and also
has spontaneous symmetry-breaking.
Quantum many-body systems with static randomness may fail to obey the ETH
due to many-body Anderson localization stopping thermalization [1, 5, 20]. The
interesting interplay of many-body localization and discrete symmetry-breaking was
recently explored in Refs. [13, 21, 29, 18]; the present chapter instead explores an
example of the interplay of the ETH and Ising symmetry breaking.
We start with the infinite-range transverse-field Ising model:
H0 = −1
N
N∑
1=i
-
unchanged and can be used in our analysis. This randomness is too weak to produce
any localization. We have chosen to put the randomness on the interactions, since we
have found in exact diagonalizations that this produces much better thermalization
at numerically accessible system sizes as compared to, e.g., only making the local
transverse fields random.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7�0.35
�0.3
�0.25
�0.2
�0.15
�0.1
�0.05
0
Γ
u
Para
F3F1
Ground State
Thermodynamic PT
Eigenstate PT
QCP
F2
Figure 2.1: The phase diagram of our model. u is the energy per spin and Γ is thetransverse field. The ground state (zero temperature) is indicated by the black (solid)line, with the quantum critical point (QCP) indicated. The green (dot-dashed) line isthe thermodynamic phase transition (PT) between the paramagnetic and ferromag-netic (F) phases. There are two dynamical (eigenstate) phase transitions within theferromagnetic phases, indicated by the blue (dashed) lines. See the text for discussionsof the sharp distinctions between these three phases F1, F2 and F3.
We now briefly summarize our results, before deriving and discussing them in
more detail below. The phase diagram of this spin model as a function of the energy
14
-
per spin u = 〈H〉/N and the transverse field Γ is shown in Fig 2.1. There are the
usual two thermodynamic phases of a ferromagnet: the paramagnetic phase (Para)
at high energy and/or high |Γ|, and the ferromagnetic phase (F) when both |Γ| and
the energy are low enough. In the ferromagnetic phase, for any finite N , we can
ask about the dynamics of the system’s order parameter. There are three regimes
of behavior that are sharply distinguished from one another in the thermodynamic
limit N → ∞: At the highest energies within region F3 of the ferromagnetic phase
the system is a sufficiently large thermal reservoir for itself so that the most probable
path by which it flips from ‘up’ to ‘down’ magnetization under unitary time evolution
is by thermally activating itself over the free energy barrier between the two ordered
states. At lower energies (F1 and F2) the barrier is higher and wider and as a
result the reservoir is inadequate, so the system quantum tunnels through the barrier
when it flips the Ising order parameter. At the lowest energies in region F1 one
can in principle prepare a state that is a linear combination of two Schrödinger cat
eigenstates of H that will coherently oscillate via macroscopic quantum tunneling
between up and down magnetizations. In the intermediate energy regime (F2) the
magnetization dynamics due to quantum tunneling is always overdamped.
Throughout the ferromagnetic phase, the exact eigenstates of H for any finite N
are Schrödinger cat states that are superpositions of up and down magnetized states,
and the properties of these cats differ in the three regimes of the ferromagnetic phase
that are indicated in Fig 2.1. Thus the two phase transitions between these three
dynamically distinct ferromagnetic phases are not only dynamical phase transitions
but also ‘eigenstate phase transitions’ [13], while the equilibrium thermodynamic
properties are perfectly analytic through these two phase transitions.
We consider the infinite-range model not only because this allows a controlled
calculation of this novel physics within the ferromagnetic phases, but also because
finite-range, finite-dimensional models obeying ETH do not show these features. In
15
-
the latter models the free energy needed to flip the magnetization by making a domain
wall and sweeping it across the system is sub-extensive, while at any nonzero tem-
perature the system is a reservoir of extensive size, so a large system will always flip
via the thermal process without macroscopic quantum tunneling through the energy
barrier; phases F1 and F2 thus do not exist for such models. One can also consider
intermediate cases of transverse-field Ising models with interactions that fall off as a
power of the distance between spins. When this power is small enough, the resulting
free energy barrier to flip the magnetization in the ferromagnetic phase is extensive,
and we thus expect phases F1 and F2 to also occur in those models, although we do
not see a way to simply calculate the locations of the phase boundaries as we can for
the infinite-range model.
2.2 Infinite Range Transverse Field Ising Model
2.2.1 Integrability
First we examine the the unperturbed Hamiltonian H0, which has the same ther-
modynamics as our full model H. The Hamiltonian H0 commutes with all permuta-
tions of the spins, as does the total spin operator:
S ≡N∑
i=1
si . (2.3)
The magnitude S2 of the total spin squared also commutes withH0, so we can choose a
set of eigenstates of H0 that are also eigenstates of S2. This unperturbed Hamiltonian
16
-
only depends on the total spin and thus can be written as
H0 = −1
N
N∑
1=i
-
is the total spin value, then it is shown1 that
fN(S) = CN2−S
N − CN2−S−1
N . (2.5)
Then the entropy per spin Σ(S/N) defined by
Σ(S
N) ≡ 1
Nln fN(S) (2.6)
is given by2
Σ(S
N) = −
[
(1
2− S
N) ln(
1
2− S
N) + (
1
2+
S
N) ln(
1
2+
S
N)]
, (2.7)
due to all the different ways one can add together N spin-1/2’s to get total spin S
depends only on the ratio S/N .
For each value of S the spectrum of H0 has (2S+1) eigenenergies. These eigenen-
ergies and the corresponding eigenstates can be approximated for large S using a
discrete version of the WKB method [9], as we discuss below.
2.2.3 Discrete WKB Method
After solving the degeneracy of the energy level, the remaining task we need to
deal with is one copy of the block diagonal Hamiltonian with the total spin S. The
subspace is 2S + 1 dimensional and the Hamiltonian in the subspace is
H0 = −1
2NS2z − ΓSx
= − 12N
S2z −Γ
2(S+ + S−) , (2.8)
1Please see Appendix A for the mathematical proof.2Please see the end of Appendix A for the proof.
18
-
where S+, S− are raising and lowering operators. In (S2, Sz) representation, we have
S2z |S,m〉 = m2|S,m〉
S+|S,m〉 =√
(S −m)(S +m+ 1)|S,m+ 1〉
S−|S,m〉 =√
(S +m)(S −m+ 1)|S,m− 1〉 .
(2.9)
If we set the eigenstate wave function as
|ψ〉 =S∑
m=−SCm|S,m〉 , (2.10)
we arrive at the discrete version of the Schrödinger equation:
− 12N
m2Cm −Γ
2
√
(S +m)(S −m+ 1)Cm−1 −Γ
2
√
(S −m)(S +m+ 1)Cm+1 = E · Cm .
(2.11)
By applying the discrete WKB method reviewed by P.A.Braun[9], we define
wm =1
2Nm2
pm =Γ
2
√
(S +m)(S −m+ 1) ,(2.12)
then the Schrödinger equation reduces to the standard form in [9]:3
pmCm−1 + (wm + E)Cm + pm+1Cm+1 = 0 . (2.13)
By following the steps mentioned in [9], we introduce “momentum” ϕ = −i ∂∂m
, then
H0 = −(wm + pme−iϕ + pm+1eiϕ)
≃ −(wm + 2pm+ 12
cosϕ) . (2.14)
3In the original paper, the coefficient before Cm is “wm − E”, what we only need to do is toreplace all E’s into −E.
19
-
From now on, we will simplify the notation by denoting w ≡ wm and p ≡ pm+ 12
=
Γ
2
√
(S + 12)2 −m2. Then we introduce the “potential-energy curve”:4
U+(m) ≡ w + 2|p|
U−(m) ≡ w − 2|p| .(2.15)
The result given by [9] about U+ and U− is that the classical accessible energy region
is confined by U+ and U−:
−U+(m) ≤ E ≤ − U−(m) . (2.16)
In our model, the relation among U+, U− and E is demonstrated by Fig. 2.2. From
-U -
-U +
E
m_tm
U
Figure 2.2: Two “potential-energy curves” in our model.
Fig. 2.2, the classical accessible region is between the two curves, so if the energy is
specified as shown in Fig. 2.2, it would generically have the quantum tunneling effect
between the left and right well.
4In the original paper, the definition was U± ≡ w± 2p by assuming p > 0. Actually when p < 0,the situation is the same as p > 0 if we define U± ≡ w ± 2|p|.
20
-
Based on the connection relation associated by U+ turning point mt, we have the
WKB wave function[9]:
Cm>mt =A√vm
cos(
∫ m
mt
arccosB dm− π4
)
Cm
-
Then the turning point equation (2.20) reduces to:
1
2x2t + |Γ|
√
s̄2 − x2t + u = 0 ; (2.22)
⇒ xt =√2 ·
√
−u− Γ2 −√
Γ2(2u+ s̄2 + Γ2) . (2.23)
Meanwhile, the scaled logarithm of the tunneling rate γ reduces to:
γ =1
N
∫ mt
−mtarccosh
−E − 12N
m2
|Γ|√S2 −m2
dm
= 2
∫ xt
0
arccosh−u− 1
2x2
|Γ|√s̄2 − x2
dx . (2.24)
2.3 Thermodynamics
For the thermodynamics (but not the dynamics) of this system in the limit of large
N we can treat the components of S classically and ignore their nonzero commutators
when obtaining the extensive thermodynamic properties (energies, entropies, magne-
tizations). Then we obtain the energy density
u =E
N=
H0N
= −12s̄2z − Γs̄x , (2.25)
where
s̄x = ±√
s̄2 − s̄2y − s̄2z . (2.26)
Note that this equation is the same as the turning point equation (2.22) by setting
s̄y = 0 and s̄z = xt and choosing the sign of s̄x as the same as the sign of Γ. It
means that we can also use the semiclassical point of view to determine the WKB
22
-
turning point xt and further obtain the tunneling rate γ(s̄). The ground state is given
by minimizing the energy density u as a function of s̄, s̄y and s̄z. Here we choose
the sign of s̄x as the same as the sign of Γ so that the term −Γs̄x gives a negative
contribution to u. Since s̄ ≤ 1/2, it is easy to see that when s̄ = 1/2 and s̄y = 0,
u will be minimized in terms of them. By plugging these two conditions into u, we
have
u = −12s̄2z − |Γ|
√
1
4− s̄2z . (2.27)
By setting the first order derivative of s̄z equals zero, we have s̄z = 0 or s̄z =√
14− Γ2.
After evaluating the second order derivative, it becomes clear that when 14− Γ2 < 0,
s̄z = 0 is the minimal point; when14− Γ2 > 0, s̄z =
√
14− Γ2 minimizes u:
when |Γ| < 12, s̄z =
√
14− Γ2, umin = −18 − 12Γ2
when |Γ| > 12, s̄z = 0, umin = −12 |Γ|
. (2.28)
Thus, the ground state of H0 always has the maximum value of S = N/2. For
|Γ| ≥ 1/2, the ground state is paramagnetic with the spins polarized along the x-
direction: s̄y = s̄z = 0, s̄x =12sign(Γ) and u = 〈H0〉/N = −|Γ|/2. For |Γ| < 1/2, the
two nearly-degenerate ground states are ferromagnetic, with s̄z = ±(1/2)√1− 4Γ2,
s̄x = Γ, s̄y = 0 and u = −(1 + 4Γ2)/8. So the system has ground state quantum
phase transitions at Γ = ±1/2 between ferromagnetic phase (s̄z = 0, |Γ| < 1/2) and
paramagnetic phase (s̄z = 0, |Γ| > 1/2). The corresponding quantum critical points
are at Γc = ±1/2 , uc = −1/4.
For the excited states, the phase transition between ferromagnetic and paramag-
netic phase also exists. Since we are interested here in eigenstates, which are at a
given energy E = Nu, we will do the statistical mechanics in the microcanonical en-
semble. For a given transverse field Γ and energy E, the equilibrium (most probable)
23
-
state of the system is the one that maximizes the entropy, which means minimizing
the total spin S. To minimize S for a given E, clearly we set Sy = 0, since Sy does not
appear in the Hamiltonian. In the paramagnetic phase the total spin points along
the x-direction and the equilibrium value of the total spin is thus s̄eq = |u/Γ|. In
the ferromagnetic phase the system can go to higher entropy (lower total spin) for
a given u by making s̄z = 0. Some algebra shows that in the ferromagnetic phase,
which is |Γ| < 1/2 and −(1 + 4Γ2)/8 ≤ u < −Γ2, the equilibrium is at s̄x = Γ,
s̄z = ±√
2(−u− Γ2) and s̄eq =√−2u− Γ2. The line of critical points separating the
para- and ferromagnetic phases is u = −Γ2, for |Γ| < 1/2, as indicated in Fig 2.1; this
critical line ends at the quantum critical points at |Γ| = 1/2, u = −1/4. Note that
in this whole ferromagnetic phase regime, the sign of s̄x is the same as the sign of Γ.
Since all the quantities related to Γ, except s̄x, are even functions of Γ (or functions
of |Γ|), from now on we assume Γ > 0 and use Γ and |Γ| interchangeably. So it also
implies that s̄x ≥ 0.
2.4 Eigenstate Thermalization
Due to its full symmetry under all permutations of the N spins, the Hamiltonian
H0 is integrable, with all the good quantum numbers associated with this permutation
symmetry, including the magnitude S of the total spin. We want to study a more
generic system, so we add to the Hamiltonian the small term H1 (see Eq. (2.2))
to break the permutation symmetry, lift all degeneracies, and make the eigenstates
thermal. The only symmetry that remains in our full H is the Ising (Z2) symmetry
under a global rotation of all spins by angle π about their x axes.
For a given E and Γ, the eigenstates of H0 have total spin ranging from the
minimum and equilibrium value Seq up to the maximum value of S = N/2. To make
these in to thermal eigenstates we need H1 to perturb the system enough so that
24
-
the eigenstates of the full H are linear combinations of all these total spin values,
weighted as at thermal equilibrium. At first order, the perturbation we are adding,
H1, flips at most two spins, so it can change the total spin by at most ±2. The
spectrum of H0 at each value of total spin S contains (2S + 1) eigenenergies spread
over a range of energy that is of order S. Thus the level-spacing in the spectrum of
H0 at a given S remains of order one in the limit of large N . This is reflected in the
dynamics under H0, which is spin precession about the mean field, and the mean field
is of order one, so the rate of precession is also of order one.
For the eigenstates of H to strongly and thermally mix the different values of S
we thus need the matrix elements of H1 between states at different S to be large
compared to the (order-one) level spacing of H0 in the large N limit. This is why we
require that the exponent p in the definition of H1 satisfies p < 1, since this is the
condition for these matrix elements to diverge in the large N limit.5 This should be
sufficient to make all the eigenstates of H thermal in that limit. We have not yet
found a way to actually prove that this is sufficient to make all the eigenstates of H
satisfy the ETH, but below we provide some numerical evidence for this from exact
diagonalization of finite-size systems.
2.5 Dynamics
In addition to making sure that H1 is strong enough to thermalize the system, we
also want it to be weak enough so that we can use the well-understood dynamics and
thermodynamics of H0 in our analysis. By restricting the exponent p to be greater
than 1/2, in the large N limit the effective field that each spin is precessing about
is the mean field from H0, with only a small correction from H1 that vanishes as
N → ∞.6 This small correction is enough to thermalize the system for 1/2 < p < 1,5For the details of the order analysis of the lower bound of disorder, see Appendix B.1.6For the details of the order analysis of the upper bound of disorder, see Appendix B.2.
25
-
which is the range where the perturbation due to H1 on a single spin’s dynamics
vanishes for N → ∞, while the perturbation to the dynamics of the full many-body
system diverges. In this regime, the system’s primary dynamics is the S-conserving
dynamics due to H0, which is spin precession at a rate of order one, and, assuming
we are in the ferromagnetic phase, ‘attempts’ at rate of order one to tunnel through
the energy barrier between total Sz up and down. At a rate that is slower by a
power of N , the dynamics due to H1 allow ‘hopping’ between different values of the
total spin S, and thus thermalization to the equilibrium probability distribution of
S dictated by the entropy Σ. And at a rate that is slower still, exponentially slow
in N , the system succeeds in crossing the free energy barrier between up and down
magnetizations. It is the separation between these three time scales that allows us to
systematically understand the dynamics of this system for large N .
Since our full Hamiltonian H has Ising symmetry under a global spin flip, and
the randomness in H1 means there are no exact degeneracies in the spectrum of H,
for finite N any eigenstate of H is either even or odd under this Ising symmetry
(with probability one). In the ferromagnetic phase, this means the exact eigenstates
of H are all Schrödinger cat states that are either even or odd linear combinations
of states with total Sz up and down. These two equal (in magnitude) and opposite
(in sign) values of Sz are extensive, thus ‘macroscopically’ different, which is why it
is appropriate to call these eigenstates ‘Schrödinger cats’.
2.5.1 Thermal Activation and Quantum Tunneling
Next we examine the rate at which this system, in its ferromagnetic phase, will flip
from the up state with positive total Sz to the down state with negative Sz under the
unitary time evolution due to its Hamiltonian H. In Section 2.2.3, we noticed that
the system has an energy level splitting due to double-well quantum tunneling which
is given by ∆E ∼ e−γ(s̄)N where γ(s̄) is the scaled logarithm of the tunneling rate
26
-
as a function of the total spin density s̄ = S/N . By drawing the classically constant
energy line shown in Fig. 2.3 described by equation (2.25) by setting s̄y = 0 or (2.22)
E
s_0
x_t
s_min
s_z
s
Figure 2.3: Constant energy (E) line described by equation (2.25) by setting s̄y = 0or just by equation (2.22). It determines the WKB turning point xt when the totalspin density s̄ is fixed.
in the ferromagnetic phase region, we obtain a double-well shaped curve which has
a local maximum at s̄z = 0 where s̄ = s̄0 = −u/Γ and two local minima at |s̄z| = 0
where s̄ = s̄min = Seq/N .
There are two steps to this process: First the system gets ‘excited’ from its usual
(high entropy) total spin Seq ‘up’ to a larger total spin S with lower entropy, with a
probability ∼ exp {−N(Σ(Seq/N)− Σ(S/N))} given by the resulting decrease of the
entropy. As S is increased, the energy barrier, whose top is at energy E = −ΓS,
decreases. For E ≥ −ΓN/2, one way the system can flip is to increase S enough
so that E ≥ −ΓS and then it will simply cross over the top of the barrier without
quantum tunneling. In the higher-energy part (F3) of the ferromagnetic phase, in the
limit of large N this is the dominant process that flips the magnetization: the system
‘thermally activates’ itself (via its unitary time-evolution) to a low-entropy, high-
total-spin state where the energy barrier can be crossed without quantum tunneling.
27
-
The ‘height’ of the entropy barrier it must cross to do this is extensive:
N∆Σ = N(Σ(√−2u− Γ2)− Σ(−u/Γ)) , (2.29)
where ∆Σ is the reduction in entropy per site needed to go over the barrier.
If the system does not or can not go over the energy barrier by increasing the total
spin S, then in order to flip the magnetization it must quantum tunnel through the
barrier. For large N , this tunneling probability can be estimated using a version of the
WKB method [9] I have explained in Section 2.2.3. As a small summary of Section
2.2.3, the total Sz serves as the ‘position’, while the operator −ΓSx serves as the
‘kinetic energy’. What we need to calculate is the probability of the system tunneling
between positive and negative total Sz for a given total spin S = Ns̄ satisfying
Seq ≤ S < −E/Γ. This probability behaves as ∼ exp (−Nγ), with γ(u, s̄,Γ) being
an intensive quantity. If we define a scaled ‘position’ x = Sz/N , the WKB ‘turning
points’ adjacent to the barrier are at x = ±xt with
xt =√2 ·
√
−u− Γ2 −√
Γ2(2u+ s̄2 + Γ2) . (2.30)
Then the intensive factor in the exponent of the tunneling probability is given by the
WKB tunneling integral
γ = 2
∫ xt
0
arcosh−u− 1
2x2
Γ√s̄2 − x2
dx . (2.31)
Since the probabilities of being ‘excited’ to total spin S and of quantum tunneling
through the barrier with total spin S are both exponentially small in N , in the limit of
large N the dominant process by which the magnetization flips is given by a standard
‘saddle point’ condition. The total spin S = Ns̄ at which the system tunnels is the
value that maximizes the product of these two probabilities and thus minimizes the
28
-
quantity
α(u,Γ) = mins̄{Σ(s̄min =√−2u− Γ2)− Σ(s̄) + γ(u, s̄,Γ)} . (2.32)
This means in order to tunnel through the barrier, the system will finally choose one
particular value of the total spin s̄ to maximize {Σ(s̄)− Σ(s̄min)− γ(u, s̄,Γ)}. Then
the distinction between thermal and quantum tunneling becomes clear: if the total
spin is chosen to be in the interval (s̄min, s̄0) which is below the top of the barrier, it is
quantum tunneling; if the total spin is chosen to be bigger than or equal to s̄0 = −u/Γ
which is at the top of the barrier, it is thermal activation over the barrier.
The sketch of Σ(s̄) and γ(s̄) is shown in Fig. 2.4, where dγ(s̄)ds̄
diverges7 at s̄ = s̄min;
�(s)
Γ(s)
0 s_min s_0 1/2s
�, Γ
Figure 2.4: A sketch of the entropy Σ(s̄) and the tunneling rate γ(s̄).
γ(s̄) becomes zero at s̄ = s̄0 since it is at the top of the barrier. From Fig. 2.4, if
we assume γ(s̄) is a convex8 function of s̄, then the distinction between the thermal
activation and quantum tunneling would be shown straightforward in Fig. 2.5 where
in thermal activation, Σ − γ reaches its maximum at s̄ = s̄0; in quantum tunneling,
Σ− γ reaches its maximum in the middle between s̄min and s̄0. Then, the condition7Please see Appendix C for proof of the divergence.8For detailed discussion of this issue, see Appendix D.
29
-
s_0s_mins
��Γ
(a) Thermal activation
s_min s_0s
��Γ
(b) Quantum tunneling
Figure 2.5: A sketch of the difference between thermal activation and quantum tun-neling.
reduces to:
if ∂∂s̄(Σ− γ)
∣
∣
∣
s̄=s̄0−0> 0 ⇒ Thermal Activation;
if ∂∂s̄(Σ− γ)
∣
∣
∣
s̄=s̄0−0< 0 ⇒ Quantum Tunneling.
(2.33)
We have located the saddle point numerically at many points within the ferro-
magnetic phase and it appears to always be unique, without any discontinuities as
the parameters u and Γ are varied. Some straightforward analysis as I will discuss in
Section 2.6 in details shows that in the higher-energy part (F3) of the ferromagnetic
phase where
πΓ√−u− Γ2
≥ ln Γ− 2uΓ+ 2u
(2.34)
the saddle point is ‘thermal’: the ‘entropy cost’ of going to higher S is less than the
‘tunneling cost’, and the system goes over the barrier without any quantum tunneling.
We call the eigenstates in this regime ‘thermal cats’, since these Schrödinger cat states
flip by thermally activating themselves over the barrier. In regions F1 and F2 this
inequality is instead false, and the system quantum tunnels through the barrier at a
value of S satisfying Seq ≤ S < −E/Γ, so the eigenstates are instead ‘quantum cats’.
30
-
The location of the dynamical phase transition between phases F2 and F3 is given
by converting the above inequality (2.88) to an equality.
The asymptotic behavior of this transition curve will be discussed in details in
Section 2.7 and the result is as follows. Near the quantum critical points (∆Γ =
1/2 − |Γ| ≪ 1), this transition line becomes exponentially adjacent to (and above)
the straight line u = −|Γ|/2: u = −1/4 + (∆Γ/2) + O(exp(−1/√∆Γ)); near Γ = 0,
it behaves as a power law: −u ≈ (|Γ|√
π/4)4/3.
2.5.2 Paired States and Unpaired States
Within the lower-energy phases (F1 and F2) of ‘quantum cats’, there is a second
dynamical phase transition within the ferromagnetic phase. This is also a ‘spectral
phase transition’ [13] in the level-spacing statistics of the eigenenergies. If one starts
with a state (not an eigenstate) that is magnetized up, the rate at which the system
crosses the barrier to down is ∼ exp (−Nα(u,Γ)), and as a result the uncertainty of
the energy of this initial ‘up’ state must be at least this large, by the time-energy
uncertainty relation. Compare this minimum energy uncertainty to the typical many-
body level spacing ∼ exp (−NΣ(√−2u− Γ2)) of the eigenstates of H at that energy.
There is clearly a sharp change at the phase transition line, which is the line where
α(u,Γ) = Σ(√−2u− Γ2). In another word,
if maxs̄
(Σ− γ) < 0 ⇒ Paired States;
if maxs̄
(Σ− γ) > 0 ⇒ Unpaired States.(2.35)
This transition line between phases F1 and F2 is shown in Fig 2.1. The location of
this transition was obtained numerically, since we do not have a simple closed-form
expression for α(u,Γ).
The asymptotic behavior of this transition curve will also be discussed in details
in Section 2.7 and the result is as follows. Near the quantum critical points (∆Γ =
31
-
1/2− |Γ| ≪ 1), this transition line also becomes exponentially adjacent to (but now
below) the straight line u = −|Γ|/2: u = −1/4 + (∆Γ/2) − O(exp(−1/√∆Γ)); near
Γ = 0, it is logarithmically tangent to the u-axis: −u ∼ 1/(ln |Γ|)2.
2.6 Analytical Calculation of the Thermal Activa-
tion and Quantum Tunneling Transition
The transition between thermal activation and quantum tunneling is illustrated
by condition (2.33). Thus, in order to find the transition, we need to calculate the
derivatives ∂Σ∂s̄
and ∂γ∂s̄
at s̄ = s̄0 − 0. The first one is straightforward from equation
(2.7):
∂Σ
∂s̄= − ln 1 + 2s̄
1− 2s̄ . (2.36)
Therefore we need to focus on solving ∂γ∂s̄
at s̄ = s̄0 − 0.
When s̄ = s̄0, by definition we have the turning point xt = 0 (It implies γ(s̄0) = 0.).
Plugging xt = 0 into the turning point equation (2.22) gives us:
s̄0 =−uΓ
. (2.37)
Now we perturb xt a little away from 0 by setting xt = ε ≪ 1. Then we have:
1
2ε2 + Γ
√s̄2 − ε2 + u = 0
Γ√s̄2 − ε2 = −u− 1
2ε2
s̄2 − ε2 = 1Γ2
(−u− 12ε2)2
s̄2 =u2
Γ2+ ε2 +
uε2
Γ2+O(ε4) .
32
-
Then,
⇒ s̄ =√
u2
Γ2+ ε2 +
uε2
Γ2+O(ε4)
=−uΓ
√
1 +u+ Γ2
u2ε2 +O(ε4)
=−uΓ
[
1 +u+ Γ2
2u2ε2 +O(ε4)
]
= −uΓ− u+ Γ
2
2uΓε2 +O(ε4) .
Thus the infinitesimal change of s̄ is:
∆s̄ = −u+ Γ2
2uΓε2 +O(ε4) < 0 . (2.38)
Now it is time to see ∆γ:
∆γ = γ(xt = ε) = 2
∫ ε
0
arccosh−u− 1
2x2
Γ√s̄2 − x2
dx . (2.39)
Since in this integral x runs from 0 to ε, we may do the rescaling to let x ≡ εb where
b runs from 0 to 1. After that, we have:
B =−u− 1
2ε2b2
Γ√s̄2 − ε2b2
=−u− 1
2ε2b2
Γ
√
u2
Γ2+ (1 + u
Γ2)ε2 − ε2b2 +O(ε4)
=−u− 1
2ε2b2
√
u2 + (u+ Γ2 − b2Γ2)ε2 +O(ε4)
=−u− 1
2ε2b2
(−u)√
1 + u+Γ2−b2Γ2u2
ε2 +O(ε4)
=1 + b
2
2uε2
1 + u+Γ2−b2Γ22u2
ε2 +O(ε4)
= 1 +( b2
2u− u+ Γ
2 − b2Γ22u2
)
ε2 +O(ε4) . (2.40)
33
-
Since arccoshB = ln(B +√B2 − 1), then if B = 1 +∆ where ∆ ≪ 1, we have
arccosh(1 +∆) = ln(1 +∆+√2∆+∆2)
= ln[1 +∆+√2∆+O(∆
√∆)]
= ∆+√2∆+O(∆
√∆)− 1
2[∆+
√2∆+O(∆
√∆)]2 +O(∆
√∆)
= ∆+√2∆− 1
2· 2∆+O(∆
√∆)
=√2∆+O(∆
√∆) . (2.41)
So if ∆ =(
b2
2u− u+Γ2−b2Γ2
2u2
)
ε2 +O(ε4), then
arccosh(1 +∆) =
√
b2
u− u+ Γ
2 − b2Γ2u2
ε+O(ε3) . (2.42)
⇒ γ(xt = ε) = 2∫ 1
0
√
b2
u− u+ Γ
2 − b2Γ2u2
ε · ε db+O(ε4)
= 2ε2∫ 1
0
1
−u√
ub2 − (Γ2 + u) + b2Γ2 db+O(ε4)
= 2ε2∫ 1
0
√−u− Γ2−u
√1− b2 db+O(ε4) . (2.43)
If we change the variable b = sin θ, then∫ 1
0
√1− b2 db =
∫ π
2
0cos θ ·cos θ dθ = π
4. Thus
we arrive at:
γ(xt = ε) = 2 ·
√−u− Γ2−u ·
π
4· ε2 +O(ε4) , (2.44)
34
-
∂γ
∂s̄
∣
∣
∣
s̄=s̄0−0= lim
ε→0
γ(xt = ε)− 0∆s̄
=2 ·
√−u−Γ2−u ·
π4
−u+Γ22uΓ
= − πΓ√−u− Γ2
. (2.45)
In addition,
∂Σ
∂s̄
∣
∣
∣
s̄=s̄0= − ln 1 + 2s̄0
1− 2s̄0= − ln Γ− 2u
Γ+ 2u. (2.46)
Therefore the boundary between thermal activation and quantum tunneling is given
by:
∂γ
∂s̄
∣
∣
∣
s̄=s̄0−0=
∂Σ
∂s̄
∣
∣
∣
s̄=s̄0
⇒ πΓ√−u− Γ2
= lnΓ− 2uΓ+ 2u
. (2.47)
To be more specific, we arrive at the result:
if πΓ√−u−Γ2 > lnΓ−2uΓ+2u
⇒ ∂∂s̄(Σ− γ)
∣
∣
∣
s̄=s̄0−0> 0 ⇒ Thermal Activation;
if πΓ√−u−Γ2 < lnΓ−2uΓ+2u
⇒ ∂∂s̄(Σ− γ)
∣
∣
∣
s̄=s̄0−0< 0 ⇒ Quantum Tunneling.
(2.48)
2.7 Asymptotic Behaviors of Both Transitions
Near Both QCP and u = 0 ,Γ = 0 Point
Up to now, we have already obtained the mathematical condition of both tran-
sition lines. To further explore the properties of both lines, it is better to also have
the asymptotic behaviors near the edges of the ferromagnetic regime demonstrated
35
-
by Fig. 2.1. One edge is the Quantum Critical Point(QCP); the other is the origin
of the phase diagram: u = 0 ,Γ = 0 point. We shall study the asymptotic behaviors
one by one.
2.7.1 Near QCP
The QCP is at Γc = 1/2 , uc = −1/4. We perturb the system a little away from
the QCP into the ferromagnetic region to let Γ = 12−∆Γ where ∆Γ ≪ 1. Then the
goal is to see what asymptotic behavior the energy density u should have in terms of
∆Γ. Let us discuss these two transitions one by one.
Thermal Activation and Quantum Tunneling Transition
In the thermal activation and quantum tunneling transition case, the transition
line must be placed above the straight line u = −Γ/2. If not, the top of the barrier
s̄0 =−uΓ
in Fig. 2.3 would be greater than 1/2 which is the upper bound of the
available total spin density s̄. Then there would be always a quantum tunneling
effect and no thermal activation can be achieved. So the transition line must be
placed in the regime where s̄0 =−uΓ
≤ 12or u ≥ −Γ/2. Also we know that u ≤ −Γ2
since it is in ferromagnetic region. Note that u = −Γ2 has a first order derivative −1
at Γc = 1/2. Therefore we set9
u = −14+ (
1
2+ δ) ·∆Γ , (2.49)
where 0 ≤ δ ≤ 1/2. Plugging this definition into the transition line equation:
πΓ√−u− Γ2
= lnΓ− 2uΓ+ 2u
, (2.50)
9This is only a redefinition of a variable but not a Taylor expansion since the essential singularityoccurs in general. And the same concern applies to all cases afterwards.
36
-
we then arrive at:
L.H.S. =π(1
2−∆Γ)
√
(12− δ)∆Γ−∆Γ2
� O( 1√
∆Γ
)
;
R.H.S. = ln1− 2(1 + δ)∆Γ
2δ∆Γ∼ O
(
− ln(δ∆Γ))
. (2.51)
Thus, if δ ∼ O(1), then the right hand side would be O(
− ln(∆Γ))
which is much
smaller than O(
1√∆Γ
)
. So in order to let the right hand side big enough, we need
δ ≪ 1. Under this condition, the left hand side is exactly O(
1√∆Γ
)
. By setting the
left hand side to be equal to the right hand side, we finally get:
1√∆Γ
∼ − ln(δ∆Γ)
⇒ δ ∼ 1∆Γ
e− 1√
∆Γ . (2.52)
Note that the form e− 1√
∆Γ behaves essentially singular at ∆Γ = 0 point and does
not have a proper Taylor expansion in terms of ∆Γ; in fact, it is much smaller than
any finite power of ∆Γ, so all the Taylor coefficients would be zero. It implies the
transition line is extremely near (and above) the straight line u = −Γ/2. So finally,
the asymptotic behavior of the thermal activation and quantum tunneling transition
is:
Γ =1
2−∆Γ
u = −14+
1
2∆Γ+O
(
e− 1√
∆Γ
)
,
(2.53)
where ∆Γ ≪ 1.
37
-
Paired and Unpaired States Transition
In the paired and unpaired states transition case, on the contrary, the transition
line must be placed below the straight line u = −Γ/2. If not, then s̄0 = −uΓ would be
an available candidate for taking the maximal value of Σ(s̄) − γ(s̄). By noting that
γ(s̄0) = 0, we would have
maxs̄
(Σ− γ) ≥ Σ(s̄0)− γ(s̄0)
= Σ(s̄0) > 0 , (2.54)
then it would be always in unpaired states region. So the transition line must be
placed in the regime where u ≤ − Γ/2. Also we know that u ≥ − 18− 1
2Γ2 since the
right hand side is the ground state energy. Note that at Γc = 1/2, u = −18 − 12Γ2 has
a first order derivative −1/2 which is the same as that of the straight line u = −Γ/2.
Since the transition line must be placed between these two bounds, we obtain the
first order derivative of the transition line must also be −1/2. Therefore we set
u = −14+
1
2∆Γ− λ1∆Γ2 , (2.55)
where 0 ≤ λ1 ≤ 1/2.
In order to solve maxs̄
(Σ − γ), we need to first solve the upper and lower bound
of the available s̄. The upper bound is 1/2 since in this case s̄0 ≥ 1/2. The lower
bound is exactly s̄min indicated in Fig. 2.3. By solving s̄ in the turning point equation
(2.22), we arrive at:
s̄2 =u2
Γ2+ (1 +
u
Γ2)x2t +
1
4Γ2x4t , (2.56)
38
-
then the minimal available s̄2 is exactly
s̄2min = −2u− Γ2 (2.57)
at (xt)2min = −2u − 2Γ2. By plugging the definition of u and Γ in terms of ∆Γ and
λ1 into s̄2min, we have:
s̄2min = −2u− Γ2 =1
4− (1− 2λ1)∆Γ2 . (2.58)
Then we define
s̄2 ≡ 14− (1− 2λ1) · α1 ·∆Γ2 , (2.59)
where 0 ≤ α1 ≤ 1. Then the task converts into finding the maximal value of Σ − γ
in terms of α1.
Since ∆Γ ≪ 1 and λ1 ,α1 � O(1), we have
s̄ =1
2− (1− 2λ1)α1∆Γ2 +O
(
[
(1− 2λ1)α1∆Γ2]2)
(2.60)
⇒ Σ(s̄) = −[
(1
2− s̄) ln(1
2− s̄) + (1
2− s̄) ln(1
2− s̄)
]
= −[
(1− 2λ1)α1∆Γ2 ln(
(1− 2λ1)α1∆Γ2)
+O(
(1− 2λ1)α1∆Γ2)]
� −∆Γ2 ln(∆Γ2) . (2.61)
Then for
γ(s̄) = 2
∫ xt
0
arccosh−u− 1
2x2
Γ√s̄2 − x2
dx
= 2
∫ 1
0
(
arccosh−u− 1
2x2t b
2
Γ√
s̄2 − x2t b2)
xt db , (2.62)
39
-
we have:
x2t = 2 ·(
− u− Γ2 − Γ√2u+ s̄2 + Γ2
)
= 2 ·[1
4− 1
2∆Γ+ λ1∆Γ
2 − (12−∆Γ)2
− (12−∆Γ)
√
1
4− (1− 2λ1)α1∆Γ2 −
1
4+ (1− 2λ1)∆Γ2
]
= 2 ·[(1
2− 1
2
√
(1− 2λ1)(1− α1))
∆Γ+(
λ1 − 1 +√
(1− 2λ1)(1− α1))
∆Γ2]
≡ 2 · (a1∆Γ+ a2∆Γ2) , (2.63)
where a1 ≡ 12 − 12√
(1− 2λ1)(1− α1) , a2 ≡ λ1 − 1 +√
(1− 2λ1)(1− α1) . Then, we
have:
−u− 12x2t b
2
Γ√
s̄2 − x2t b2
=14− 1
2∆Γ+ λ1∆Γ
2 − (a1∆Γ+ a2∆Γ2)b2
(12−∆Γ)
√
14− (1− 2λ1)α1∆Γ2 − 2(a1∆Γ+ a2∆Γ2)b2
≈[1
2+
λ1∆Γ2 − a1b2∆Γ− a2b2∆Γ2
12−∆Γ
]
· 2[
1 + 2(1− 2λ1)α1∆Γ2 + 4a1b2∆Γ2 + 4a2b2∆Γ2]
= 1 +2(λ1∆Γ
2 − a1b2∆Γ− a2b2∆Γ2)12−∆Γ + 2(1− 2λ1)α1∆Γ
2 + 4a1b2∆Γ+ 4a2b
2∆Γ
2
= 1 +1
12−∆Γ
[
2λ1∆Γ2 − 2a1b2∆Γ− 2a2b2∆Γ2 + (1− 2∆Γ)(1− 2λ1)α1∆Γ2
+ 2a1b2∆Γ− 4a1b2∆Γ2 + 2a2b2∆Γ2 − 4a2b2∆Γ3
]
≈ 1 + 2∆Γ2[
2λ1 + (1− 2λ1)α1 − 4a1b2 − 4a2b2∆Γ]
, (2.64)
where all the terms I have omitted are definitely much smaller than at least one of
the present terms in the last line given ∆Γ ≪ 1 ,λ1 � O(1) ,α1 � O(1) .
Next comes the key point: in order to have Σ and γ in same order at some
value of α1, both α1 and λ1 have to be much smaller than 1. Let me prove it by
contradiction. If the conclusion is not true, then at least one of α1 or λ1 would be
40
-
O(1). Then, we would have both a1 ∼ O(1) and a2 ∼ O(1). So xt ∼√∆Γ and
−u− 12x2t b
2
Γ
√s̄2−x2t b2
∼ 1 + O(∆Γ2). Since we know arccosh(1 + ∆) =√2∆ + O(∆
√∆) when
∆ ≪ 1, it means that:
γ(s̄) = 2
∫ 1
0
(
arccosh−u− 1
2x2t b
2
Γ√
s̄2 − x2t b2)
xt db
∼√∆Γ2 ·
√∆Γ
= ∆Γ3
2 . (2.65)
However, from equation (2.61), we have Σ � ∆Γ2 ln(∆Γ2). So we would arrive at
Σ(s̄) ≪ γ(s̄) which is not the condition of the transition line. Thus, by contradiction,
we conclude both α1 ≪ 1 and λ1 ≪ 1 have to be true in order to satisfy the transition
condition.
By knowing α1 ≪ 1 and λ1 ≪ 1, all quantities are further simplified. First, we
have:
a1 ≈1
2λ1 +
1
4α1 , a2 = λ1 − 2a1 ≈ −
1
2α1 , (2.66)
which are both linear in λ1 and α1. Then, we get:
xt ≈√
2a1∆Γ ≈√
(λ1 +1
2α1)∆Γ , (2.67)
and
−u− 12x2t b
2
Γ√
s̄2 − x2t b2≈ 1 + 2(2λ1 + α1 − 4a1b2)∆Γ2
≈ 1 + 2(2λ1 + α1)(1− b2)∆Γ2 . (2.68)
41
-
⇒ γ(s̄) ≈ 2∫ 1
0
√
2 · 2(2λ1 + α1)(1− b2)∆Γ2√
(λ1 +1
2α1)∆Γ db
= 2√2(2λ1 + α1)∆Γ
3/2
∫ 1
0
√1− b2 db
=
√2
2π(2λ1 + α1)∆Γ
3/2 . (2.69)
In addition, from equation (2.61) we have:
Σ(s̄) ≈ − (1− 2λ1)α1∆Γ2 ln(
(1− 2λ1)α1∆Γ2)
≈ α1∆Γ2 ln(
α1∆Γ2)
. (2.70)
In order to let Σ and γ be in same order, we finally arrive at:
(2λ1 + α1)∆Γ3/2 ∼ α1∆Γ2 ln
(
α1∆Γ2)
. (2.71)
Then it is better to let O(λ1) ≤ O(α1), since if not, both λ1 and α1 would have an
extra factor λ1/α1 on the exponent which would make the value far smaller than the
case O(λ1) ≤ O(α1). Therefore we finally get the result:
α1 ∼1
∆Γ2e− 1√
∆Γ , λ1 ∼1
∆Γ2e− 1√
∆Γ . (2.72)
They also behave essentially singular at ∆Γ = 0 point so the transition line is also
extremely near (but this time below) the straight line u = −Γ/2. Finally, the asymp-
totic behavior of the paired and unpaired transition is also:
Γ =1
2−∆Γ
u = −14+
1
2∆Γ+O
(
e− 1√
∆Γ
)
,
(2.73)
where ∆Γ ≪ 1.
42
-
2.7.2 Near u = 0 ,Γ = 0 Point
The other interested point is the origin of the phase diagram: u = 0 ,Γ = 0 point.
Also, we will perturb the system a little away from Γ = 0 or equivalently let Γ ≪ 1.
Then the goal is to see what u behaves in terms of Γ asymptotically when Γ → 0.
Thermal Activation and Quantum Tunneling Transition
In this case, we know that Γ2 ≤ − u ≤ Γ/2. First consider two extreme cases:
1. If −u ∼ O(Γ), then πΓ√−u−Γ2 ∼ O(√Γ) , ln
(
Γ−2uΓ+2u
)
� O(1). So we have
πΓ√−u−Γ2 ≪ ln
(
Γ−2uΓ+2u
)
. It is in thermal activation region.
2. If −u ∼ O(Γ2), then πΓ√−u−Γ2 ∼ O(1) , ln(
Γ−2uΓ+2u
)
∼ O(Γ). Then we haveπΓ√
−u−Γ2 ≫ ln(
Γ−2uΓ+2u
)
. It is in quantum tunneling region.
Thus, the transition line must be placed in between where Γ2 ≪ − u ≪ Γ. When
this is true,
πΓ√−u− Γ2
≈ πΓ√−u ,
ln(
Γ− 2uΓ+ 2u
)
≈ −4uΓ
. (2.74)
⇒ πΓ√−u =−4uΓ
,
π
4Γ2 = (−u) 32 ,
−u =(
√
π
4Γ
) 4
3 ∼ Γ 43 . (2.75)
This is a power law behavior when Γ → 0.
Paired and Unpaired States Transition
In this case, we have Γ/2 ≤ −u ≤ 1/8. Also consider the two extreme cases first:43
-
1. If −u ∼ O(1), then xt =√2√
−u− Γ2 − Γ√2u+ s̄2 + Γ2 ≈
√−2u ∼ O(1).
But B =−u− 1
2x2t b
2
Γ
√s̄2−x2t b2
∼ O(−uΓ) → ∞. In addition, arccoshB = ln(B +
√B2 − 1) → ln 2B when B → ∞. So γ = 2
∫ 1
0
(
arccosh−u− 1
2x2t b
2
Γ√
s̄2 − x2t b2)
xt db ∼
− lnΓ → ∞. But Σ � O(1), so it means maxs̄
(Σ− γ) < 0. It is in paired states
region. Thus the transition line tends to reach u = 0 point when Γ → 0.
2. If −u ∼ O(Γ), then we define −u = α2Γ, where α2 ∼ O(1) and α2 ≥ 1/2 since
we have −u ≥ Γ/2. Under this assumption, we have:
xt =√2
√
−u− Γ2 − Γ√2u+ s̄2 + Γ2
=√2
√
α2Γ− Γ2 − Γ√
s̄2 − 2α2Γ+ Γ2
≈√2
√
α2Γ− Γ√
s̄2 − 2α2Γ
=√2 ·
√Γ ·
√
α2 −√
s̄2 − 2α2Γ
� O(√Γ) , (2.76)
whereas,
B =−u− 1
2x2t b
2
Γ√
s̄2 − x2t b2
≈ α2Γ− b2Γ(α2 −
√s̄2 − 2α2Γ)
Γ√
s̄2 − x2t b2
=α2 − b2(α2 −
√s̄2 − 2α2Γ)
√
s̄2 − x2t b2. (2.77)
So for a given finite s̄, we have B ∼ O(1). Thus, γ = 2∫ 1
0
(arccoshB)xt db �
O(√Γ) → 0, but Σ ∼ O(1), so it means max
s̄(Σ − γ) > 0. It is in unpaired
states region.
Thus, the transition line must also be placed in between where Γ ≪ − u ≪ 1.
Then we get xt =√2√
−u− Γ2 − Γ√2u+ s̄2 + Γ2 ∼ O(
√−u) and B = −u−
1
2x2t b
2
Γ
√s̄2−x2t b2
∼44
-
O(−uΓ) ≫ 1.
⇒ γ = 2∫ 1
0
(arccoshB)xt db ∼ O[
ln(−u
Γ
)
·√−u
]
. (2.78)
In order to let γ comparable to Σ which is typically O(1), we should have:
O[
ln(−u
Γ
)
·√−u
]
∼ O(1)
⇒(
ln(−u)− lnΓ)√
−u ∼ 1
⇒ − lnΓ√−u ∼ 1
⇒ − u ∼ 1(lnΓ)2
. (2.79)
This is equivalent to Γ ∼ e−1√−u which is also an essential singularity at u = 0 ,Γ = 0
point. It means that the transition line is extremely near the u-axis but not the
u = −Γ/2 line.
2.8 Numerical Evidence
We now present some numerical results about the thermalization properties and
the distinction between phases F1 and F3 (phase F2 is too narrow to clearly see in
the size systems we can diagonalize). We exactly diagonalize Hamiltonians with up
to N = 15 spins. We first put the Hamiltonian into a block diagonal form using basis
states that are even and odd with respect to the system’s Z2 Ising symmetry, and then
diagonalize each sector numerically to obtain all of the eigenstates within that sector.
For finite N , the coefficient λ in the disorder term H1 in Eq. (2.2) matters. If λ is too
large the system will become a spin glass rather than a ferromagnet. Thus λ needs to
be carefully chosen to be large enough for our finite systems to show thermalization,
45
-
but small enough to avoid the spin glass regime. After some exploration, we chose to
use the parameters λ = 0.7, p = 3/4 and Γ = 1/8 for our exact diagonalizations.
0 0.2 0.4 0.6 0.8 10
0.01
0.02
0.03
0.04
0.05
δ
f
Figure 2.6: The level-spacing statistics using 100 realizations of H at N = 15 in phaseF1 within the even sector. δ is the ratio between the smaller level spacing δ< tothe larger level spacing δ> for three consecutive eigenenergies in the even sector. f isthe relative frequency for each bin in this histogram.
First, we show the level-spacing statistics, which should be GOE if the eigenstates
are thermal. This must be done within one Z2 symmetry sector, since there is no level-
repulsion between states in different sectors. We look at each set of three consecutive
levels in one sector and denote δ< as the smaller level spacing and δ> as the larger
level spacing. Then the histogram of the ratio δ can be compared to GOE level
statistics [2]. The even sector results in the F1 phase for 100 realizations at N = 15
are shown in Fig. 2.6. We see the expected strong level repulsion, consistent with the
46
-
thermalization. All other phases and symmetry sectors were also examined and the
results are also thermal, since phase F1 is the lowest-energy phase and thus the most
difficult to thermalize.
Next, we examine a ‘distance’ between two eigenstates that are adjacent in the
energy spectrum by comparing their probability distributions for the total spin Sz.
We define this distance between eigenstate 1 and eigenstate 2 as
D12 =
N/2∑
Sz=−N/2
∣
∣P1(Sz)− P2(Sz)∣
∣ , (2.80)
where P1(Sz) and P2(Sz) are the probability distributions of Sz in eigenstates 1 and
2, respectively. We tested three different distances: Deo the distance between an even
parity state (eigenstate 1) and the nearest-energy odd parity state (eigenstate 2), and
similarly for Dee and Doo. If a system thermalizes, each eigenstate is equivalent to
a microcanonical ensemble characterized by its energy. For two eigenstates that are
adjacent in energy, the energy difference ∆E ∼ 2−N , therefore we expect the eigen-
state distances Deo, Dee and Doo should decrease exponentially with N . In phase F1,
since the spectrum consists of nearly-degenerate pairs of states, the energy differences
satisfy ∆Eeo ≪ ∆Eee , ∆Eoo. Thus we expect in phase F1, Deo ≪ Dee , Doo. In addi-
tion, if we choose the upper bound of the energy window we average over to be well
within the F1 phase, we would also expect that Eq. (2.84) holds, so the exponential
decay rate of Deo would be greater than those of Dee and Doo. Meanwhile, in phases
other than F1, we expect all three D’s are well coincident. The numerical results are
shown in Fig. 2.7. As we expected, all these eigenstate distances decay exponentially
with N . In addition, Deo in phase F1 is much smaller and decreasing much faster
than the other two distances, and in phase F3 all three D’s are well coincident. This
demonstrates the clear distinction between phases F1 and F3, and further tests the
thermalization in phase F3.
47
-
8 9 10 11 12 13 14 15
�6
�5
�4
�3
�2
�1
N
log(D
)
e�o
e�e
o�o
(a) Eigenstate distance in phase F1
8 9 10 11 12 13 14 15
�3.5
�3
�2.5
�2
�1.5
N
log(D
)
e�o
e�e
o�o
(b) Eigenstate distance in phase F3
Figure 2.7: Averages of log (D) in phases F1 and F3, respectively, where D is the‘eigenstate distance’ defined in the text.. The energy density range we used in F1is from the first excited state in each sector up to uc − 0.02 where uc is the energydensity at the phase boundary between F1 and F2, whereas in F3 we used the phase’sfull energy density range. N is the total number of spins varying from 8 to 15. Theexponential decrease of D with increasing N indicates thermalization. The error barscome from averaging over 100 realizations.
48
-
�0.14 �0.12 �0.1 �0.08 �0.06 �0.04 �0.02 00
0.2
0.4
0.6
0.8
1
1.2
F1
F2
F3
u
αn
ᾱn at N = 11
ᾱn at N = 15
Δαn at N = 11Δαn at N = 15
N → ∞
Figure 2.8: The mean ᾱn and the standard deviation ∆αn of the quantity αn definedin Eq. (2.81). The number of realizations is 1600 for N = 11 (blue dash-dotted lines)and 100 for N = 15 (red dashed lines). The green (solid) line gives the theoreticalquantity α(u,Γ) defined in Eq. (2.32) for the system size N → ∞.
In the ferromagnetic phases, the system spontaneously flips between magnetiza-
tion up and down at a rate that behaves as ∼ exp(−Nα(u,Γ)), where the quantity
α(u,Γ) is defined in Eq. (2.32). The probability of the system having total magneti-
zation zero (or 1/2 for systems with odd N) also behaves as ∼ exp(−Nα(u,Γ)). Thus
from a single many-body eigenstate |n〉 we can obtain an estimate of the quantity α
as
αn =1
Nln
maxSz{Pn(S