# Random Ising chain in transverse and longitudinal fields:
Strong disorder RG study

Tamás Pető<EMAIL_ADDRESS>Institute of Theoretical Physics, Szeged
University, H-6720 Szeged, Hungary  Ferenc Iglói<EMAIL_ADDRESS>Wigner Research Centre for Physics, Institute for Solid State Physics and
Optics, H-1525 Budapest, P.O. Box 49, Hungary  Institute of Theoretical
Physics, Szeged University, H-6720 Szeged, Hungary  István A. Kovács
<EMAIL_ADDRESS>Department of Physics and Astronomy,
Northwestern University, Evanston, IL 60208, USA  Northwestern Institute on
Complex Systems, Northwestern University, Evanston, IL 60208, USA

###### Abstract

Motivated by the compound ${\rm LiHo}_{x}{\rm Y}_{1-x}{\rm F}_{4}$ we consider
the Ising chain with random couplings and in the presence of simultaneous
random transverse and longitudinal fields, and study its low-energy properties
at zero temperature by the strong disorder renormalization group approach. In
the absence of longitudinal fields, the system exhibits a quantum-ordered and
a quantum-disordered phase separated by a critical point of infinite disorder.
When the longitudinal random field is switched on, the ordered phase vanishes
and the trajectories of the renormalization group are attracted to two
disordered fixed points: one is characteristic of the classical random field
Ising chain, the other describes the quantum disordered phase. The two
disordered phases are separated by a separatrix that starts at the infinite
disorder fixed point and near which there are strong quantum fluctuations.

##  I Introduction

Quantum fluctuations play an important role in characterising the physical
properties of systems in the vicinity of a quantum phase-transition point,
which takes place at zero temperature by varying a control parameter, such as
the strength of a transverse field[1]. In the presence of quenched disorder,
which is connected to the energy density, such as having random
(ferromagnetic) couplings and/or transverse fields, the combined action of
quantum and disorder fluctuations generally results in a new type of a phase
transition. This is often controlled by a so called infinite disorder fixed
point (IDFP)[2]. The prototype of such systems is the random transverse-field
Ising (RTFI) chain with nearest neighbour interactions, the low-energy
properties of which has been calculated asymptotically exactly[3, 4] by a so
called strong disorder renormalization group (SDRG) method[5, 6], for reviews
see[7, 8]. It has been shown that at the IDFP the renormalized couplings and
transverse fields have an extremely broad distribution, so that the ratio of
renormalized parameters at neighbouring sites is either zero or infinity, thus
the critical properties are fully dominated by disorder fluctuations. Using an
efficient numerical implementation of the SDRG technique[9] the RTFI model has
been studied in higher dimensions[10, 11, 12, 13, 14, 15, 16, 17] and IDFP
behaviour is observed in these cases, too. On the contrary, the RTFI model
with long-range interactions has conventional random critical behaviour[18,
19].

Experimental realization of the RTFI model is the compound ${\rm LiHo}_{x}{\rm
Y}_{1-x}{\rm F}_{4}$, which is a dipole coupled random Ising ferromagnet[20,
21, 22, 23, 24]. If this system is placed in a magnetic field, which is
transverse to the Ising axis, it acts as an effective transverse field, which,
however, also induces a random longitudinal field[25, 26, 27]. This
experimental example motivates our study in this paper about the properties of
the Ising model with random couplings and in the presence of simultaneous
random transverse and longitudinal fields. Here we consider the one-
dimensional problem with nearest neighbour interactions and use an SDRG
approach, in which the well known decimation rules for the random couplings
and the random transverse fields[7, 8] are extended with those for the random
longitudinal fields[28, 29]. We apply an efficient numerical algorithm and
study the RG phase-diagram from the properties of the RG flow. In the vicinity
of the IDFP we also investigate the scaling behaviour of the excitation energy
and the magnetization moment for large finite samples.

The rest of the paper is organised in the following way. The model is
introduced in Sec.II and then in Sec.III the SDRG method is explained.
Numerical results for finite random longitudinal fields are presented in
Sec.IV and discussed in Sec.V.

##  II The model

We consider the Ising chain with random couplings and in the presence of
simultaneous random transverse and longitudinal fields, defined by the
Hamiltonian:

$\displaystyle\begin{split}\hat{H}&=-\sum_{i=1}^{L}J_{i}\sigma_{i}^{z}\sigma_{i+1}^{z}\\\
&-\sum_{i=1}^{L}\Gamma_{i}\sigma^{x}_{j}-\sum_{i=1}^{L}h_{i}\sigma^{z}_{i}\;,\end{split}$
(1)

Here the $\sigma_{i}^{x,z}$ are Pauli matrices at site $i$ and we use periodic
boundary conditions: $\sigma_{L+1}^{z}\equiv\sigma_{1}^{z}$. The nearest
neighbour couplings are ferromagnetic, $J_{i}>0$ and random, the transverse
fields, $\Gamma_{i}>0$ are random, the longitudinal fields $h_{i}$ are random
and are taken from a symmetric distribution: $p(h)=p(-h)$. In the numerical
work we generally used the following box-like distributions:

$\displaystyle\begin{split}\pi(J)&=\begin{cases}1&\hskip 18.49411pt\text{for
}0<J\leq 1\,,\\\ 0&\hskip 18.49411pt\text{otherwise.}\end{cases}\\\
\pi_{2}(\Gamma)&=\begin{cases}1/\Gamma_{0}&\text{for
}0<\Gamma\leq\Gamma_{0}\,,\\\ 0&\text{otherwise.}\end{cases}\\\
p(h)&=\begin{cases}1/h_{0}&\text{for }-h_{0}/2\leq h\leq h_{0}/2\,,\\\
0&\text{otherwise.}\end{cases}\\\ \end{split}$ (2)

Note that the non-random model, with $J_{i}=J$ and $\Gamma_{i}=\Gamma$ in the
presence of a staggered longitudinal field $h_{i}=h(-1)^{i}$ is equivalent to
the antiferromagnetic Ising model in transverse and longitudinal fields. This
model has been studied theoretically in Refs.[30, 31, 32, 33] and
experimentally in Ref.[34]. For random couplings and random transverse fields,
but with non-random staggered longitudinal fields it is studied in Refs.[35,
28].

##  III SDRG calculations

In the SDRG procedure[7, 8], we consider the separated local degrees of
freedom say at position $i$. These are couplings or sites, having the value of
the characteristic parameters: $J_{i}$ and

$\gamma_{i}=\sqrt{\Gamma_{i}^{2}+h_{i}^{2}}\;,$ (3)

respectively. These parameters and the related gaps are sorted in descending
order and the largest one, denoted by $\Omega$, which sets the energy-scale in
the problem, is eliminated and between remaining degrees of freedom new terms
in the Hamiltonian are generated through perturbation calculation. This
procedure is successively iterated, during which $\Omega$ is monotonously
decreasing. At the fixed point, with $\Omega^{*}=0$ one makes an analysis of
the distribution of the different parameters and calculates the scaling
properties. In the following, we describe the elementary decimation steps.

####  III.0.1 Strong-coupling decimation

In this case, the largest local term in the Hamiltonian is a coupling, say
$\Omega=2J_{i}$, connecting sites $i$ and $i+1$ and the two-site Hamiltonian
is given by:

$\hat{H}_{cp}=-J_{i}\sigma^{z}_{i}\sigma^{z}_{i+1}-\Gamma_{i}\sigma^{x}_{i}-\Gamma_{i+1}\sigma^{x}_{i+1}-(h_{i}\sigma^{z}_{i}+h_{i+1}\sigma^{z}_{i+1})\;.$
(4)

The spectrum of $\hat{H}_{cp}$ contains four levels. We omit the two higher
levels, which corresponds to a merger of the two strongly coupled sites into a
spin cluster in the presence of a (renormalized) transverse field
$\tilde{\Gamma}$ and a longitudinal field $\tilde{h}$ . The magnetic moment of
the cluster is given by $\tilde{\mu}=\mu_{i}+\mu_{i+1}$, where the magnetic
moments at the start are $\mu_{i}=\mu_{i+1}=1$. The remaining two lowest
energy levels are given by second-order degenerate perturbation method as the
eigenvalues of the matrix:

$\begin{bmatrix}-J_{i}-h_{i}-h_{i+1}&-\Gamma_{i}\Gamma_{i+1}/J_{i}\\\
-\Gamma_{i}\Gamma_{i+1}/J_{i}&-J_{i}+h_{i}+h_{i+1}\end{bmatrix}$ (5)

having a gap

$\Delta
E_{cp}=2\sqrt{\left(\Gamma_{i}\Gamma_{i+1}/J_{i}\right)^{2}+\left(h_{i}+h_{i+1}\right)^{2}}\;.$
(6)

Comparing it with the gap of the spin cluster, $2\tilde{\gamma}$ in Eq.(3) we
obtain for the renormalised values of the parameters:

$\tilde{\Gamma}=\frac{\Gamma_{i}\Gamma_{i+1}}{J_{i}},\quad\tilde{h}=h_{i}+h_{i+1}\;.$
(7)

####  III.0.2 Strong field decimation

In this case, the largest local term is associated to a site, say $i$ and the
corresponding parameter of the one-site Hamiltonian is $\gamma_{i}$. Due to
the large gap this site does not contribute to the longitudinal magnetisation
and therefore it is eliminated. The longitudinal magnetic field, however
should be transformed at the remaining neighbouring sites. To calculate the
new renormalized coupling between the remaining sites $i-1$ and $i+1$ we
calculate energy levels with fixed spins at these sites. Denoting by $s_{i\pm
1}=+$ ($-$) a $\uparrow$ ($\downarrow$) boundary state, the eigenvalue problem
with different boundary conditions has the lowest energy as:

$E_{s_{i-1},s_{i+1}}=-\sqrt{\Gamma_{i}^{2}+(s_{i-1}J_{i-1}+s_{i+1}J_{i}+h_{i})^{2}}\;.$
(8)

The renormalised coupling between the remaining sites is given by:

$\tilde{J}=-(E_{\uparrow\uparrow}+E_{\downarrow\downarrow}-E_{\uparrow\downarrow}-E_{\downarrow\uparrow})/4\approx\frac{J_{i-1}J_{i}\Gamma_{i}^{2}}{(\Gamma_{i}^{2}+h_{i}^{2})^{3/2}}\;,$
(9)

where the last relation is calculated perturbatively.

We obtain similarly for the excess longitudinal fields:

$\Delta
h_{i-1}=-(E_{\uparrow\uparrow}-E_{\downarrow\downarrow}+E_{\uparrow\downarrow}-E_{\downarrow\uparrow})/4\approx\frac{J_{i-1}h_{i}}{\sqrt{\Gamma_{i}^{2}+h_{i}^{2}}}\;,$
(10)

and

$\Delta
h_{i+1}=-(E_{\uparrow\uparrow}-E_{\downarrow\downarrow}-E_{\uparrow\downarrow}+E_{\downarrow\uparrow})/4\approx\frac{J_{i}h_{i}}{\sqrt{\Gamma_{i}^{2}+h_{i}^{2}}}\;,$
(11)

so that

$\tilde{h}_{i\pm 1}={h}_{i\pm 1}+\Delta{h}_{i\pm 1}\;.$ (12)

We note that setting the longitudinal field $h_{i}=0$ the decimation equations
in Eqs.(7) and (9) are in the standard form, see in Refs.[7, 8].

###  III.1 Infinite disorder fixed point at $h_{0}=0$

The properties of the system in the absence of random longitudinal fields have
been studied previously and several asymptotically exact results are known
through the solution of the SDRG equations[3, 4].

The control parameter is defined as:

$\delta=\frac{\overline{\ln\Gamma}-\overline{\ln J}}{\text{var}(\ln
h)+\text{var}(\ln J)}\,,$ (13)

where $\overline{x}$ and $\text{var}(x)$ stands for the average and the
variance of the random variable $x$, respectively. The critical point is at
$\delta=0$, which corresponds to $\Gamma_{0}=1$ in the distribution in Eq.(2).
The energy scale, $\epsilon$, which is the smallest gap, scales with the
length $L$ at the critical point as:

$\ln\epsilon\sim L^{\psi}$ (14)

with a critical exponent

$\psi=1/2\;.$ (15)

The magnetization moment, $\mu$ has also a power-law $L$-dependence at the
critical point:

$\mu\sim L^{d_{f}}\;,$ (16)

where the fractal dimension of the magnetization is given by:

$d_{f}=(1+\sqrt{5})/4\;,$ (17)

which is just half of the golden ratio.

In the disordered phase $\delta>0$, the average correlations decay
exponentially with the true correlation length $\xi\sim 1/\delta^{2}$,
implying the correlation length exponent

$\nu=2\;$ (18)

for the random chain. On the other hand the decay of the typical correlations
involves a different exponent:

$\nu_{\rm typ}=1\;.$ (19)

Next to the critical point in the disordered phase, in the so called Griffiths
phase, the energy-scale goes to zero as:

$\epsilon\sim L^{-z}\;,$ (20)

which results in singularities in the dynamical quantities, such as in the
magnetic susceptibility or in the autocorrelation function. In Eq.(20) $z$ is
the dynamical exponent, which is given by the positive root of the
equation[36, 37, 38]:

$\overline{\left(\frac{J}{\Gamma}\right)^{1/z}}=1\,,$ (21)

which is an exact expression in the entire Griffiths region. In the vicinity
of the critical point, the dynamical exponent to the leading order is given by

$z\approx\frac{1}{2|\delta|},\qquad|\delta|\ll 1\,.$ (22)

##  IV Numerical results for $h_{0}>0$

###  IV.1 Properties of the RG flow

Figure 1: Renormalized value of the log-energy excitations as a function of
remaining sites, $n$, for the non-perturbed model with $h_{0}=0$ (green
symbols) and for the model with a random longitudinal field ($\ln h_{0}=-6$,
blue symbols). The value of the log random field at a site decimation is shown
by orange symbols. The results are obtained on a chain with $L=1024$ and the
log-variables are averaged over $10000$ random samples. Upper panel: starting
from the ordered nonperturbed phase $\Gamma_{0}=0.5$; middle panel: starting
from the nonperturbed critical point $\Gamma_{0}=1.$; lower panel: starting
from the disordered nonperturbed phase $\Gamma_{0}=1.5$. The horizontal line
at $\ln h_{0}=-6$ shows the parameter of the original distribution of the
random longitudinal fields.

In this section, we switch on the random longitudinal fields and study their
effect on the renormalization flow. First, we study the behavior of the
logarithmic excitation energy as a function of the remaining sites, $n$, and
compare its behavior in the unperturbed system with $h_{0}=0$ to that with
finite perturbation $\ln h_{0}=-6$. The results for a chain of length $L=1024$
and averaged over $10000$ samples are shown in Fig.1 at three different
points: $\Gamma_{0}=0.5$ (starting from the ordered nonperturbed phase),
$\Gamma_{0}=1.$ (critical point) and $\Gamma_{0}=1.5$ (disordered nonperturbed
phase). In these figures, we also monitor the renormalized value of the
logarithmic longitudinal fields, which are involved in a site decimation.

We can identify an initial period of the renormalization, in which the
excitation energies look similar for the two models, which is due to the fact
that the renormalized longitudinal fields are comparatively small and do not
influence the decimation rules. This behaviour is changed when the value of
the renormalized longitudinal fields becomes comparable with the excitation
energy and the system in the concluding RG steps renormalizes to the final
state. If we start from the ordered nonperturbed phase, then in the initial
period dominantly couplings are decimated and the transverse fields will be
renormalized to an exponentially small value. In the second part of the
renormalization, in the concluding RG steps (which means small values of $n$)
the random longitudinal fields play the dominant role and the system finally
renormalizes to a classical random-field Ising chain. On the contrary, if we
start from the disordered phase, then in the initial period dominantly fields
are decimated and the couplings will be renormalized to an exponentially small
value. In the concluding RG steps the renormalization is influenced, but not
dominated by the random longitudinal fields and the system approaches a
quantum disordered phase. Finally, if we start from the critical point of the
unperturbed system, then in the initial period couplings and fields are
decimated in a symmetric way. In the concluding RG steps the grown-up
longitudinal fields will stop the further rapid decrease of the log excitation
energy and the final state will be the result of all three parameters in the
Hamiltonian.

The evolution of the average cluster moment during renormalization is shown in
Fig.2 for three values of the starting transverse field distributions. As seen
in this figure, in the initial period there is a slow increase of
$\overline{\mu}$, which will turn to a fast increase when the random
longitudinal fields show also an increase. In the final state, where couplings
are very rarely decimated $\overline{\mu}$ shows a plateau, the hight of which
is larger for larger values of $\Gamma_{0}$.

Figure 2: Evolution of the average cluster moment as a function of remaining
sites, $n$, for the model with a random longitudinal field ($\ln h_{0}=-6$)
and for three values of the starting transverse field distributions:
$\Gamma_{0}=0.5,1.0$ and $1.5$. The calculation details correspond to that in
Fig.1.

###  IV.2 RG phase-diagram

Figure 3: Schematic RG phase-diagram in the $L\to\infty$ limit. In the absence
of random longitudinal fields at $h/J=0$ the infinite disorder fixed-point
(denoted by red circle) separates the quantum ordered phase ($\Gamma/J<0$)
from the quantum disordered phase ($\Gamma/J>0$). At finite random
longitudinal fields there is a separatrix, indicated by a dashed red line,
which starts from the infinite disorder fixed point and bends downwards. Below
the separatrix the RG flows (indicated by green lines) are attracted by the
fixed point of the classical random field Ising model, which is shown by a
green circle. Above the separatrix the RG flows (indicated by blue lines)
scale towards the quantum disordered phase.

Based on the previously observed properties of the RG flow, we suggest an RG
phase-diagram which is shown in Fig.3. In the absence of a random longitudinal
field, at $h/J=0$ the phase-diagram has been described in Sec.III.1. It
consists of an ordered phase, with a fixed point at $\Gamma/J=0$ and indicated
by a black circle in Fig. 3 and a quantum disordered phase, with a fixed point
at $\Gamma/J\to\infty$, illustrated by a blue circle. The infinite disorder
fixed point at $\Gamma/J=1$, which is shown by a red circle separates the two
phases. Switching on the random longitudinal field the ordered phase will
disappear. In the classical limit, at the line $\Gamma/J=0$ the properties of
the system are controlled by the fixed point of the classical random field
Ising model, which is located at $h/J\to\infty$ and illustrated by a green
circle. For general values of the parameters the system has two disordered
phases, which are separated by a separatrix indicated by a dashed red line,
which starts from the infinite disorder fixed point and bends downwards, due
to the fact that the gap increases with increasing $h$, see in Eq.(3). Below
the separatrix the RG flows are attracted by the fixed point of the classical
random field Ising model, above the separatrix the RG flows scale towards the
quantum disordered phase.

In the following, we are going to study the properties of the system along the
separatrix in Fig.3 in more detail.

###  IV.3 Scaling behaviour in the vicinity of the IDFP

In the following, we perform a detailed numerical analysis of the
renormalization results on finite periodic chains with lengths $L=2^{n}$,
$n=7,8\dots,14$ and for each length $100000$ random samples are considered. To
treat such large samples, we used an efficient numerical algorithm which works
in time linearly with $L$ with some logarithmic correction.

Figure 4: Average magnetization moment (upper panel) and average log-gap
(lower panel) as a function of $\ln h_{0}$ for different lengths of the chain.

Having the observation in the previous subsection in mind we consider the
critical point of the system at $h_{0}=0$, which corresponds to
$\Gamma_{0}=1$, and monitored the behaviour of the system at small values of
$h_{0}$. It turned out that the appropriate variable is $\ln h_{0}$ and we
made the calculations at $-\ln h_{0}=0.,3.,6.,\dots,30$. We have measured the
average value of the magnetization moment, $\overline{\mu}$, as well as the
average value of the log-gap, $\overline{\ln\epsilon}$, where $\epsilon$ is
given by the last decimated site value:
$\epsilon=\sqrt{\tilde{\Gamma}^{2}+\tilde{h}^{2}}$.

According to the calculated results (see in Fig.4) the final state is
observable for relatively large values of the length of the chain, which means
that large enough number of RG steps are performed. Equivalently, for a fixed
chain length, $L$, the random longitudinal field should be sufficiently large,
$h_{0}>\tilde{h}_{0}(L)$. For relatively small values of the longitudinal
fields, $h_{0}<\tilde{h}_{0}(L)$ the influence of the original fixed point
with $h_{0}=0$ is still strong and the system stays in the initial phase. Deep
in the disordered phase the average quantities have an approximately linear
dependence on $\ln h_{0}$, thus between two close points, $h_{0}^{(2)}$ and
$h_{0}^{(1)}$ we have the relations:

$\displaystyle\overline{\mu}_{L}(h_{0}^{(2)})-\overline{\mu}_{L}(h_{0}^{(1)})$
$\displaystyle\approx-\kappa\ln(h_{0}^{(2)}/h_{0}^{1})$
$\displaystyle\overline{\ln\epsilon}_{L}(h_{0}^{(2)})-\overline{\ln\epsilon}_{L}(h_{0}^{(1)})$
$\displaystyle\approx\alpha\ln(h_{0}^{(2)}/h_{0}^{(1)})\;.$ (23)

The prefactors, $-\kappa$ and $\alpha$ have a weak $\ln h_{0}$ dependence, as
shown in the estimated values in Fig.5.

Figure 5: Upper panel: Estimated values of the local exponents in Eqs.(23) and
(26) for different parameters of the distribution of the random longitudinal
field, $h_{0}$. Lower panel: Ratio of the local exponents $\kappa/\delta$ and
$\alpha/z$ for different values of $\ln h_{0}$.

This asymptotic regime ends at a limiting point, $\tilde{h}_{0}(L)$, which is
approximately signalled by an inflection point, so that:

$\left.\frac{{\partial}^{2}\overline{\mu}_{L}}{{\partial}\ln
h_{0}^{2}}\right|_{\tilde{h}_{0}(L)}\approx\left.\frac{{\partial}^{2}\overline{\ln\epsilon}_{L}}{{\partial}\ln
h_{0}^{2}}\right|_{\tilde{h}_{0}(L)}\approx 0\;.$ (24)

The position of these inflection points are shown in Fig.6.

Figure 6: The boundary of the disordered phase as signalled by an inflexion
point in the $\overline{\mu}$ and $\overline{\ln\epsilon}$ vs $\ln h_{0}$
curves, see in Eq.(24). The full line illustrates an $\sim L^{1/2}$
dependence. Inset: Size dependence of the average magnetization moment and the
average log-gap at a fixed value of the longitudinal field $\ln h_{0}=-6$. The
straight lines have the slopes $\delta=1.2$ and $z=0.817$, see in Eq.(26).

We notice an approximate relation:

$|\ln\tilde{h}_{0}(L)|<L^{1/2}\;,$ (25)

which can be explained as follows. At the infinite disorder fixed point (with
$h_{0}=0$) the value of the typical excitation energy $\epsilon_{0}$ scales as
$|\ln\epsilon_{0}|\sim L^{1/2}$, see in Eq.(15). If
$|\ln\tilde{h}_{0}(L)|>|\ln\epsilon_{0}|$, then the perturbation caused by the
random longitudinal field is much smaller than the smallest excitation energy,
consequently it cannot influence the critical properties of the system. If we
try to fit the measured values as $|\ln\tilde{h}_{0}(L)|\sim L^{\omega}$, we
have $\omega<0.5$. The effective exponent is observed to decrease with $L$,
c.f. at $L=2048$ we have $\omega_{\rm eff}\approx 0.3$. This is due to the
fact that during renormalization the typical value of
$\sqrt{\Gamma^{2}+h^{2}}$ will be larger than the typical value of $J$, thus
the system moves to the disordered phase. Here, almost always strong field
decimations take place (see in Sec.III.0.2) and as a consequence $\ln\epsilon$
will have a weaker (presumably logarithmic) $L$ dependence.

If we consider the curves at a fixed value of $h_{0}$ for different values of
$L$ these are merely shifted. Thus for two lengths, $L_{2}$ and $L_{1}$ we
have the relations:

$\displaystyle\overline{\mu}_{L_{2}}(h_{0})-\overline{\mu}_{L_{1}}(h_{0})$
$\displaystyle\approx\delta\ln(L_{2}/L_{1})$
$\displaystyle\overline{\ln\epsilon}_{L_{2}}(h_{0})-\overline{\ln\epsilon}_{L_{1}}(h_{0})$
$\displaystyle\approx-z\ln(L_{2}/L_{1})\;.$ (26)

This behaviour is illustrated in the inset of Fig.6 and the estimated
prefactors are shown in Fig.5, which have also a weak $\ln h_{0}$ dependence.

Combining the relations in Eqs.(23) and (26) we obtain for the magnetic
moment:

$\overline{\mu}_{L_{\mu}}(h_{0})\approx\overline{\mu}_{L_{1}}(h_{0}^{(1)})-\kappa\ln(h_{0}/h_{0}^{(1)})+\delta\ln(L_{\mu}/L_{1})\;,$
(27)

and similarly for the average log-gap:

$\overline{\ln\epsilon}_{L_{\epsilon}}(h_{0})\approx\overline{\ln\epsilon}_{L_{1}}(h_{0}^{(1)})-\kappa\ln(h_{0}/h_{0}^{(1)})+\delta\ln(L_{\epsilon}/L_{1})\;,$
(28)

Let us fix $L_{1}$ and $h_{0}^{(1)}$ at some reference value and keep the same
value for the average magnetic moment,
$\overline{\mu}_{L_{\mu}}(h_{0})\approx\overline{\mu}_{L_{1}}(h_{0}^{(1)})$.
This is satisfied, if we have a relation between the length associated to
magnetic moments, $L_{\mu}$ and the distance from the fixed point, $h_{0}$ as:

$L_{\mu}\sim h_{0}^{-\nu_{\mu}},\quad\nu_{\mu}=\kappa/\delta\;.$ (29)

Similar analysis of the expression for the log-gap in Eq.(28) leads to the
relation:

$L_{\epsilon}\sim h_{0}^{-\nu_{\epsilon}},\quad\nu_{\epsilon}=\alpha/z\;,$
(30)

where $L_{\epsilon}$ is the length associated to the energy gap.

Estimates for the correlation length exponents $\nu_{\mu}=\kappa/\delta$ and
$\nu_{\epsilon}=\alpha/z$ are shown in the lower panel of Fig.5. For small
values of $h_{0}$, within the error of the approximation, these are in
agreement with the values $\nu_{\mu}\approx 2$ and $\nu_{\epsilon}\approx 1$.
These relations can be interpreted in the following way. Let us first consider
the magnetization. At a given value of $h_{0}>0$ the sites after $L\to\infty$
RG steps are typically decimated out so that the average magnetization
vanishes: $m=\lim_{L\to\infty}\overline{\mu}_{L}(h_{0})/L=0$. After a finite
number ($L<\infty$) of RG steps, however, effective spin clusters are formed
having an average moment: $\overline{\mu}_{L}(h_{0})$. The typical distance
between two such clusters defines the length-scale $L_{\mu}(h_{0})$ and
according to the previous measurement
$L_{\mu}^{1/\nu_{\mu}}h_{0}\sim\tilde{h}={\cal O}(1)$. Consequently
$\nu_{\mu}$ is the critical exponent of the (average magnetization)
correlation length. We can estimate $\nu_{\mu}$ assuming that the increase of
$\tilde{h}$ during renormalization is dominantly due to the process in Eq.(7),
which would mean the sum of $\sim L_{\mu}$ random numbers with the
distribution in Eq.(2). As the consequence $\nu_{\mu}=2$ according to the
central limit theorem. This result agrees with the correlation length exponent
at $h_{0}=0$, see in Eq.(18). The finite-size magnetization is related to
average fluctuations of the parameters, while the smallest energy gap is due
to extreme fluctuations of the parameters in a rare region. In such a rare
region, the renormalized longitudinal field grows in an extreme way, which
means a linear relation with its size, thus the $h_{i}$ parameters here have
the same sign. With this assumption we arrive at the relation: $\tilde{h}\sim
h_{0}L_{\epsilon}={\cal O}(1)$, thus $\nu_{\epsilon}=1$, which is just the
critical exponent of the typical correlations in Eq.(19).

##  V Discussion

In this paper, we considered the Ising model with random couplings and in the
presence of simultaneous random transverse and longitudinal fields. This
investigation is motivated by experimentally realized compounds, such as the
${\rm LiHo}_{x}{\rm Y}_{1-x}{\rm F}_{4}$ which is placed into a magnetic
field, transverse to the Ising axis. Here we considered the one-dimensional
system with nearest neighbour interaction and studied its low-energy
properties by the strong disorder renormalization group approach. A similar
model, however, with a staggered non-random longitudinal field has been
studied recently[35, 28].

In the absence of longitudinal field, the model exhibits an ordered and a
quantum disordered phase, which are separated by an infinite disorder critical
point. Switching on the random longitudinal field, the ordered phase
disappears and the RG trajectories are attracted by two disordered phases, as
shown in Fig.3. There is a separatrix, which is connected to the infinite
disorder fixed point and below which the RG trajectories are attracted by the
classical random field Ising chain fixed point. Above the separatrix the
trajectories approach a disordered phase, dominated by transverse-field
effects.

We have studied the scaling properties of the system in the vicinity of the
IDFP in more detail, since the SDRG is expected to be more accurate in this
region. Approaching the IDFP on the separatrix the correlation length,
$\xi_{\mu}$, measured as the distance between two non-decimated magnetic
moments diverges as $\xi_{\mu}\sim h_{0}^{\nu_{\mu}}$, with $\nu_{\mu}\approx
2$. On the other hand, the distance between two non-decimated rare events of
low-energy excitations, $\xi_{\epsilon}$ diverges with a different exponent:
$\nu_{\epsilon}\approx 1$. Since the value of $\nu_{\mu}$ and $\nu_{\epsilon}$
correspond to the exponents along the $h_{0}=0$ line for the average and
typical correlation lengths, respectively, we expect that scaling at the IDFP
is isotropic. We have also measured the value of the dynamical exponent, which
is found to be $h_{0}$ dependent and which approaches a value $z\approx 0.9$,
as $h_{0}\to 0$. Since $z(h_{0}=0)$ is formally infinity, this means that the
dynamical exponent has a discontinuity at $h_{0}=0$.

Figure 7: Expected schematic RG phase-diagram for the three-dimensional model
with short-range interactions. In the absence of random longitudinal fields,
at $h/J=0$, the infinite disorder fixed-point (denoted by red circle)
separates the quantum ordered phase ($\Gamma/J<0$) from the quantum disordered
phase ($\Gamma/J>0$). Without transverse fields, at $\Gamma/J=0$ the ordered
phase survives until $(h/J)_{c}$, indicated by a yellow circle. The red line
connecting the red and yellow circles indicates the border between the ordered
and the disordered phases. The RG flows, blue and black lines, are
illustrations, their concrete form depends on the stability of the fixed
points.

One might ask the question of how the RG phase-diagram will be modified in
higher dimensions. In $d=2$ the phase-diagram is expected to be qualitatively
the same, as in Fig.3, since the random-field Ising model has no ordered
phase[39] in this case, too. In $d=3$, however, where the ordered phase
survives for small values of $h_{0}/J<(h_{0}/J)_{c}$[40, 41] the phase-diagram
should contain an ordered and a disordered phase. The expected phase-diagram
is shown see Fig.7. Another problem, to study is if the random interaction is
long-ranged, as c.f. in the compound ${\rm LiHo}_{x}{\rm Y}_{1-x}{\rm F}_{4}$,
having dipole couplings. This problem in the absence of random longitudinal
fields has a conventional random fixed point[18, 19] and further detailed
studies are needed to obtain the phase-diagram for the complete model.

###### Acknowledgements.

This work was supported by the National Research Fund under Grants No. K128989
and No. KKP-126749, and by the National Research, Development and Innovation
Office of Hungary (NKFIH) within the Quantum Information National Laboratory
of Hungary.

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