# Quantum critical fans from critical lines at zero temperature

Hui Yu and Sudip Chakravarty  Mani L.Bhaumik Institute for Theoretical Physics
Department of Physics and Astronomy, University of California Los Angeles, Los
Angeles, California 90095, USA

###### Abstract

Quantum critical phenomena influences the finite temperature behavior of
condensed matter systems through quantum critical fans whose extents are
determined by the exponents of the zero temperature criticality. Here we
emphasize the aspects of quantum critical lines, as discussed previously, and
study an exactly solved model involving a transverse field Ising model with
added three-spin interaction. This model has three critical lines. We compute
the spin-spin correlation function and extract the correlation length, and
identify the crossovers: quantum critical to quantum disordered, or
renormalized classical regimes. We construct the quantum critical fans along
one of the critical lines. In addition, we also construct finite temperature
dynamic structure factors. We hope this model will become experimentally
realizable in the future, and our results could stimulate studies in many
similar models

##  I INTRODUCTION

Quantum critical point (QCP) [1, 2, 3, 4, 5] is a point in the parameter space
where a continuous phase transition takes place at zero temperature. A
significant part of research in condensed matter physics is focused on
describing various quantum phases and transitions between them. Thus, QCP has
become a widely studied subject. Exactly at the critical point, we have
quantum fluctuations taking place at all length scales. It is interesting to
probe these fluctuations and the implied quantum behavior of the ground
states. However, in reality, all experiments are carried out at finite
temperatures and it is necessary to learn how such ground state properties can
be deduced from finite temperature measurements. This is accomplished by
measuring the finite temperature correlation lengths, dynamic structure
factors, and other physical properties. The QCP leaves fingerprints at nonzero
temperatures of these properties. Its influence can be felt in a broad regime,
called the quantum critical fan, whose extent depends on the quantum critical
exponents. This idea was successfully exploited in [6, 7] in the context of
two-dimensional quantum antiferromagnets.

As an example, suppose we have a cusp-like quantum critical fan with
$T_{x}\sim|g-g_{c}|^{z\nu}$, where $g$ is a tuning parameter, $z$ is a
dynamical critical point, and $\nu$ is a critical exponent, and $T_{x}$ is the
crossover temperature between two different quantum phases. If $z\nu$ is very
large, this will make the quantum critical fan very narrow and thus limit the
ability to probe quantum fluctuations at finite temperatures. On the other
hand, If we have quantum critical points with smaller values of $z\nu$,
experimental evidence of quantum criticality could be more easily observed at
finite temperatures.

The one-dimensional transverse field Ising model (TFIM) is a classic example
of QCP. Theoretically, the integrability of the model gives us the power to
study its properties in detail. A complete discussion on that topic can be
found in [4]. Experimentally, this model is well-captured by
$\mathrm{CoNb_{2}O_{6}}$ [8], which illustrates the nature of quantum
criticality.

Here, as in our recent paper, we consider an exactly solved model which has
three interesting critical lines that goes beyond the notion of a critical
point. It’s a three-spin extension of the more familiar Ising model in a
transverse field, TFIM. The model is solved by Jordan-Wigner and Bogoliubov
transformations. The corresponding phase diagram was introduced by Kopp and
Chakravarty [9]. Later on, the critical lines in this model were studied and
their topological aspects were discussed by Niu et al [10]. Different phases
were identified by the number of Majorana modes on each end of an open chain.
Subsequently, we calculated the momentum, $k$, and frequency, $\omega$,
dependent dynamical structure factor $S(k,\omega)$ in pure and disordered
versions of the model at zero temperature in a recent paper [11]. However, the
influence of temperature on the phase diagram remained unexplored, especially
from the perspective of the quantum critical fans. This is what we aim in this
paper. It will be important in understanding experimental observations, if and
when such a model is realized and studied in practice.

The paper is organized as follows. In Sec. II, we introduce our model and the
phase diagram, and discuss a few of the properties in the context of quantum
critical lines. In Sec. III, we discuss three regimes that appear in finite
temperatures and review how correlation length behaves in each such regime. In
Sec. IV, we discuss the method for calculating the correlation function using
Pfaffian method. In Sec. V we discuss our results. The final Section, Sec. VI,
is a summary.

##  II The Model and its phase diagram

The model we consider here is a 3-spin extension of the TFIM, studied
previously by [9, 10, 11]. We first discuss its phase diagram and several
characteristics of its critical lines. We will focus on topics that were not
studied previously. The Hamiltonian, $H$, is

$H=-\sum_{i}(h_{i}\sigma_{i}^{x}+\lambda_{2}^{\prime}\sigma_{i}^{x}\sigma_{i-1}^{z}\sigma_{i+1}^{z}+\lambda_{1}^{\prime}\sigma_{i}^{z}\sigma_{i-1}^{z})$
(1)

$\sigma^{x}$ and $\sigma^{z}$ are the standard Pauli matrices. Presently, we
shall set $h_{i}=h=cst$. The Hamiltonian after Jordan-Wigner transformation
[12, 13] is

$\displaystyle\sigma_{i}^{x}=1-2c_{i}^{\dagger}c_{i}$ (2)
$\displaystyle\sigma_{i}^{z}=-\prod_{j<i}(1-2c_{j}^{\dagger}c_{j})(c_{i}+c_{i}^{\dagger})$
(3)

is

$\begin{split}H&=-\sum_{i=1}^{N}h(1-2c_{i}^{\dagger}c_{i})-\lambda_{1}^{\prime}\sum_{i=1}^{N-1}(c_{i}^{\dagger}c_{i+1}+c_{i}^{\dagger}c_{i+1}^{\dagger}+h.c.)\\\
&-\lambda_{2}^{\prime}\sum_{i=2}^{N-1}(c_{i-1}^{\dagger}c_{i+1}+c_{i+1}c_{i-1}+h.c.).\end{split}$
(4)

In contrast to the spin model, the spinless fermion Hamiltonian is actually a
one-dimensional mean-field model of a $p$-wave superconductor [14], when there
are both nearest- and next-nearest neighbor hopping, as well as
condensates—note the pair creation and destruction operators. The solution of
the corresponding spin Hamiltonian through Jordan-Wigner transformation is,
however, exact and includes all possible fluctuation effects and is not a
mean-field solution of any kind.

Imposing periodic boundary condition, the Hamiltonian can be diagnolized by a
Bogoliubov transformation

$H=\sum_{k}2E_{k}\left(\eta_{k}^{\dagger}\eta_{k}-\frac{1}{2}\right).$ (5)

As usual, the anticommuting fermion operators $\eta_{k}$’s are suitable linear
combinations in the momentum space of the original Jordan-Wigner fermion
operators. The spectra of excitations are (lattice spacing will be set to
unity throughout the paper unless stated otherwise)

$E_{k}=h\sqrt{1+\lambda_{1}^{2}+\lambda_{2}^{2}+2\lambda_{1}(1-\lambda_{2})\cos
k-2\lambda_{2}\cos 2k}$ (6)

Here $\lambda_{1}=\lambda_{1}^{\prime}/h$ and
$\lambda_{2}=\lambda_{2}^{\prime}/h$ are the scaled coupling constants.
Quantum phase transitions of this model are given by the nonanalyticities of
the ground state energy:

$E_{0}=-\sum_{k}E_{k}.$ (7)

The derivative of the ground state energy vanishes at $k=0,\pm\pi$ and $\cos
k=\lambda_{1}(1-\lambda_{2})/4\lambda_{2}$. The nonanalyticites are defined by
the critical lines where the energy gaps collapse. The phase diagram can also
be understood from the Majorana zero modes, which we explain below. For the
time being refer to Fig. 1.

Figure 1: The phase diagram: $n=$ 0,1,2, corresponds to regions with $n$
Majorana modes at each end of an open chain. Three quantum critical lines
$\lambda_{2}=\lambda_{1}+1$, $\lambda_{2}=1-\lambda_{1}$ and
$\lambda_{2}=-1$($0<\lambda_{1}<2$). These lines denote the collapse of the
energy gaps of the energy gaps. Points $a$ and $b$ are multicritical points
with dynamical critical exponent $z=1$ and $z=2$ respectively.

  1. 1.

For TIFM without a three-spin interaction, the gaps collapse at the Brillouin
zone boundaries, $k=\pm\pi$ at the self-dual point $\lambda_{1}=1$ and
$\lambda_{2}=0$.

  2. 2.

As we move along the critical line $\lambda_{2}=1-\lambda_{1}$, there are no
additional critical points until we reach a multicritical point
$\lambda_{2}=1$, where the gaps collapse at $k=0$. At exactly $\lambda_{1}=0$
and $\lambda_{2}=1$, we have the dynamical critical exponent $z=1$ due to the
linearly vanishing spectrum at $k=0$. Then $\lambda_{2}=1+\lambda_{1}$
constitutes a critical line with criticality at $k=0$.

  3. 3.

Moreover, the gaps also collapse at incommensurate points
$k=\cos^{-1}(\lambda_{1}/2)$ for $\lambda_{2}=-1$ and $0<\lambda_{1}<2$. This
constitutes an unusual incommensurate critical line. Right at $\lambda_{1}=2$
and $\lambda_{2}=-1$, we have a non-Lorentz invariant multicritical point with
dynamical critical exponent $z=2$. The spectra vanish quadratically at
$\pm\pi$ due to confluence of two Dirac points.

In the spin representation, our model exhibits two phases - ordered and
disordered. These phases are distinguished by the presence of long-range
order. As shown in Fig, 3 the long-range order is reflected in the equal-time
correlation function $C(r,0)$ from Eq. 19. Both $\lambda_{2}=1+\lambda_{1}$
and $\lambda_{2}=1-\lambda_{1}$ separate these phases. However, the line
$\lambda_{2}=-1$ ($0<\lambda_{1}<2$) can not be understood from symmetry
breaking quantum phase transition since it separates two quantum disordered
phases.

In the fermion language, the phase transitions are best described by the
number of Majorana zero modes, $n$, at each end of an open chain, which can be
determined numerically. This was discussed in great detail in a previous paper
[10]. So $\lambda_{2}=-1$ ($0<\lambda_{1}<2$) is a line of topological
transition seperating $n=0$ and $n=2$ Majorana zero modes at each end of the
chain. The number of Majorana modes are also winding numbers [15, 16]
explained in terms of Anderson pseudospin Hamilitonian [17], when the time-
reversal symmetry is preserved. More recently, the topological nature of the
model was explained from the notion of a curvature renormalization group [18,
19], which may be useful in higher-dimensional systems.

Finally, this model has a dual representation in which it is equivalent to an
anistropic $XY$-model with a magnetic field in the $z$-direction. It is
possible that the $XY$-version is better realized in experimental systems.
This dual representation is defined by the dual spin operators $\mu_{x}$,
$\mu_{y}$, and $\mu_{z}$ such that

$\displaystyle\mu_{x}(n)$ $\displaystyle=$
$\displaystyle\sigma_{z}(n+1)\sigma_{z}(n),$ (8) $\displaystyle\mu_{z}(n)$
$\displaystyle=$ $\displaystyle\prod_{m\leq n}\sigma_{x}(m),$ (9)

which implies that

$\displaystyle[\mu_{z}(n),\mu_{x}(n)]$ $\displaystyle=$
$\displaystyle-2i\mu_{y}(n),$ (10) $\displaystyle\mu_{y}(n)$ $\displaystyle=$
$\displaystyle-i\bigg{(}\prod_{m\leq
n}\sigma_{x}(m)\bigg{)}\sigma_{z}(n+1)\sigma_{z}(n).$ (11)

The Hamiltonian under duality transforms to

$\begin{split}H_{D}&=-\frac{2}{1+r}\sum_{n}\bigg{[}\frac{1+r}{2}\mu_{x}(n)\mu_{x}(n+1)\\\
&+\frac{1-r}{2}\mu_{y}(n)\mu_{y}(n+1)+h_{z}\mu_{z}(n))\bigg{]},\end{split}$
(12)

where we have carried out the rotations : $\mu_{x}(n)\to\mu_{z}(n)$,
$\mu_{z}(n)\to\mu_{x}(n)$, $\mu_{y}(n)\to-\mu_{y}(n)$. The parameters are
related by

$\lambda_{1}=\frac{2h_{z}}{1+r},\;\lambda_{2}=\frac{r-1}{1+r}.$ (13)

The critical line in the $XY$-model, separating the disordered phase from the
ordered phase, is at $h_{z}=1$, which corresponds to
$\lambda_{1}+\lambda_{2}=1$, separating the ordered phase from the disordered
phase. Since the ordered and the disordered phases are exchanged under
duality, the disordered phase of the three-spin model is
$\lambda_{1}+\lambda_{2}<1$.

##  III Finite Temperature Crossovers

In this section, we discuss quantum critical fans and their crossovers at
finite temperatures. Generically, as we raise the temperature, the effect of
quantum criticality from a QCP can be felt in an extended region (quantum
critical fan) of the parameter space. The width and the shape of the quantum
critical fan depend on the critical exponents $\nu$ and $z$. In the two-
dimensional parameter space, one can approach a critical point in any
direction. This leads to a cone-like quantum critical fan for a critical
point. In our model, with added three-spin interaction, we have a quantum
critical line made out of a line of critical points. Referring to Fig. 2, the
quantum critical fan looks like a valley along the critical line in this case.
The blue planes denote the fans. These fans are crossover lines that separate
different regimes. These regimes can be distinguished by the temperature
dependence of the correlation length $\xi$ and the relative magnitudes of the
energy scales.

Figure 2:  A sketch of a quantum critical fan of a quantum critical line
$\lambda_{2}=1-\lambda_{1}$. The black solid lines are quantum critical fans.
The blue dashed lines are the usual quantum critical fan from a critical point
which is a point of intersection between the critical line and the blue dashed
line. QC: quantum critical. RC: renormalized classical. QD: quantum
disordered.

The following delineates the regimes pertaining to quantum criticality;

  1. 1.

Quantum Critical ($\Delta_{gap}\ll T$): In this regime, the physical
properties of the model at finite temperatures are completely determined by
the quantum critical point at zero temperature. Tthe correlation length
behaves as a power law in $T$.

$\xi\sim\frac{1}{T^{1/z}}$ (14)

where $z$ is the dynamical critical exponent of the QCP.

  2. 2.

Renormalized Classical ($\Delta_{gap}\ll T$) : In this regime, $\xi$ goes to
$\infty$ exponentially fast as $T$ goes to zero due to the presence of long-
ranged order at zero temperature. In general, we expect the correlation length
to have the following form.

$\xi\sim C_{1}(T)e^{C_{2}/T}$ (15)

where $C_{2}$ is a positive constant and $C_{1}(T)$ is a function of $T$. The
exact form of $C_{1}(T)$ is not important to us since we are only interested
in the general form of $\xi$.

  3. 3.

Quantum Disordered ($\Delta_{gap}\gg T$): Since there is no long-ranged order
at $T=0$, we expect the correlation length to become temperature independent
as $T$ goes to aero, saturating to a value of order unity.

$\xi\sim{\rm Const.}$ (16)

There is one quantum disordered regime called the oscillatory disordered
regime that we will come back to later.

##  IV Finite temperature Correlation function

The signature of the quantum criticality may be discovered by obtaining the
correlation length in the neutron scattering experiments [20, 21]. For this
purpose, we compute spin-spin correlation function. Here we discuss the method
for calculating the correlation function and the correlation length. One can
consult our previous paper [11] if one is interested in a full-detailed
derivation.

Figure 3: Representative equal-time correlation function $C(r,0)$ at $T=0$ in
various regions which are labelled by the number of Majorana zero modes $n$ in
Fig.1. The distance $r$ is expressed in units of lattice spacing. The
calculation of the correlation function follows from the Pfaffian
representation discussed in detail below. Blue: ($\lambda_{1}=1.5$,
$\lambda_{2}=0$), Red: (${\lambda}_{1}=0.5$, ${\lambda}_{2}=2$). Black:
($\lambda_{1}=0.5$, $\lambda_{2}=-0.5$). Green: ($\lambda_{1}=2$,
$\lambda_{2}=-1.5$.)

Quite generally, the spin-spin correlation function $C_{ij}(t)$ is defined as

$C(r,t)=\left<\sigma_{i}^{z}(t)\sigma_{j}^{z}(0)\right>$ (17)

where $i$, $j$ are lattice sites and $r$ is the separation between them. And
the angular brackets represent a thermodynamic average
$\langle(...)\rangle={\rm Tr}(e^{-\beta H}(...))/{\rm Tr}(e^{-\beta H})$. The
equal-time correlation function is

$C(r,0)=\left<\sigma_{i}^{z}\sigma_{j}^{z}\right>$ (18)

In a finite system of length $L$, we choose $i$, $j$ in the middle of the
chain to reduce the boundary effects. Using the Jordan-Wigner transformation,
Eq. 3, we get

$C(r,0)=\left<\Biggl{(}\prod_{m=1}^{i-1}(c_{m}^{\dagger}+c_{m})(c_{m}^{\dagger}-c_{m})\Biggl{)}(c_{i}^{\dagger}+c_{i})\Biggl{(}\prod_{l=1}^{j-1}(c_{l}^{\dagger}+c_{l})(c_{l}^{\dagger}-c_{l})\Biggl{)}(c_{j}^{\dagger}+c_{j})\right>$
(19)

Because of the free fermion nature of the Jordan-Wigner transformed
Hamiltonian, we can apply Wick’s theorem [22] to $C(r,0)$. After collecting
all terms in Wick expansion, we get a Pfaffian:

$C(r,0)=Pf(S)$ (20)

Here $S$ is a $2(i+j-1)$ dimensional skew-symmetric matrix. If we identify
$A_{m}=c_{m}^{\dagger}+c_{m}$ and $B_{n}=c_{n}^{\dagger}-c_{n}$, The matrix
$S$ is

$S=\begin{pmatrix}0&<A_{1}B_{1}>&<A_{1}A_{2}>&<A_{1}B_{2}>&...&<A_{1}A_{j}>\\\
-<A_{1}B_{1}>&0&<B_{1}A_{2}>&<B_{1}B_{2}>&...&<B_{1}A_{j}>\\\
-<A_{1}A_{2}>&-<B_{1}A_{2}>&0&<A_{2}B_{2}>&...&<A_{2}A_{j}>\\\
\vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\\
-<A_{1}A_{j}>&-<B_{1}A_{j}>&-<A_{2}A_{j}>&-<B_{2}A_{j}>&...&0\end{pmatrix}$
(21)

All we need to compute is the two-point correlation function such as

$\left<[c_{m}^{\dagger}\pm c_{m}](c_{l}^{\dagger}\pm c_{l})\right>.$ (22)

This can be done by utilizing the free fermion operators $\eta_{\mu}$ and
$\eta_{\mu}^{\dagger}$ from Eq.5. The results are the following

$<A_{i}A_{j}>=\sum_{p=1}^{L}\phi_{pi}\phi_{pj}$ (23)
$<A_{i}B_{j}>=\sum_{p=1}^{L}\phi_{pi}\psi_{pj}tanh(\beta E_{p})$ (24)
$<B_{i}A_{j}>=-\sum_{p=1}^{L}\psi_{pi}\phi_{pj}\tanh(\beta E_{p})$ (25)
$<B_{i}B_{j}>=-\sum_{p=1}^{L}\psi_{pi}\psi_{pj}$ (26)

Figure 4: Finite temperature $S(k,\omega)$. Thermal brodening is evident. Top
Left: ($\lambda_{1}=0.5$, $\lambda_{2}=-1$, $T=0.01$). Bottom Left:
($\lambda_{1}=0.5$, $\lambda_{2}=-1$, $T=0.1$). Top Right: ($\lambda_{1}=2$,
$\lambda_{2}=-1$, $T=0.01$). Bottom Top: ($\lambda_{1}=2$, $\lambda_{2}=-1$,
$T=0.1$).

where $\beta$ is $1/T$. Three $L\times L$ matrices $\phi$,$\psi$ and $E$ come
from singular value decomposition (SVD) [23] and $M$ obtained by rewriting $H$
from Eq. 4 into a matrix with $c^{\dagger}+c$ and $c^{\dagger}-c$ as its
basis. Here
$c^{\dagger}+c\equiv(c_{1}^{\dagger}+c_{1},c_{2}^{\dagger}+c_{2},...,c_{L}^{\dagger}+c_{L})$.

$M=\phi E\psi^{T}$ (27)
$H=\begin{pmatrix}c^{\dagger}+c&c-c^{\dagger}\end{pmatrix}\begin{pmatrix}0&M^{T}\\\
M&0\end{pmatrix}\begin{pmatrix}c^{\dagger}+c\\\ c^{\dagger}-c\end{pmatrix}$
(28)

The exact form of $M$ is

$M=\begin{bmatrix}h&-\lambda_{1}&-\lambda_{2}&&\\\
&h&-\lambda_{1}&-\lambda_{2}&&\\\ &&h&-\lambda_{1}&\ddots&\\\
&&&\ddots&\ddots&-\lambda_{2}\\\ &&&&\ddots&-\lambda_{1}\\\
&&&&&h\end{bmatrix}$ (29)

the exact diagonalization of $H$ is not numerically stable (suffers from large
errors) if the eigenvalues of $H$ are close to $0$. Thus, instead of directly
diagonalizing our $2L\times 2L$ Hamiltonian, we chose to use SVD to
diagonalize a $L\times L$ matrix.

We still need to deal with one last step. The computation of a Pfaffian
consumes a lot of time by standard methods for a large-size system. One of the
authors in collaboration invented an efficient method for dealing with such
Pfaffian in [24]. Let $X$ be a $2N\times 2N$ skew-symmetric matrix which has
the following form

$X=\left[{\begin{array}[]{cc}A&B\\\ -B^{T}&C\\\ \end{array}}\right]$ (30)

where $A$ is a $2\times 2$ matrix, and $B$ and $C$ are matrices of appropriate
dimensions.Then we have the identity

$\left[{\begin{array}[]{cc}I_{2}&0\\\ B^{T}A^{-1}&I_{2N-2}\\\
\end{array}}\right]X\left[{\begin{array}[]{cc}I_{2}&-A^{-1}B\\\ 0&I_{2N-2}\\\
\end{array}}\right]=\left[{\begin{array}[]{cc}A&0\\\ 0&C+B^{T}A^{-1}B\\\
\end{array}}\right]$ (31)

where $I_{n}$ is a $n\times n$ identity matrix, and

$\mathrm{det}(X)=\mathrm{det}(A)\mathrm{det}(C+B^{T}A^{-1}B)$ (32)

This gives us an iterative method. We will get a $2\times 2$ matrix $A$ at
each iteration step; then we treat $C+B^{T}A^{-1}B$ to be our next $X$ and
keep doing this. Our $\det(X)$ eventually becomes a product chain of $2\times
2$ matrices.

Generically, the equal-time correlation function $C(r,0)$ decays exponentially
at finite temperatures. This allows us to determine the correlation length
$\xi$ by fitting $C(r,0)$ to an exponential function.

$C(r,0)\sim e^{-r/\xi}$ (33)

where the prefactor could be a constant or an oscillatory function of $r$, as
shown below in Sec. V.

##  V Computational Results

###  V.1 T=0, correlation function

First, we provide some results for $C(r,0)$ at $T=0$. The most remarkable
result is the oscillatory quantum disordered phase. A complex calculation [25]
of the instantaneous spin-spin correlation function showed that within the
ferromagnetic phase in the dual representation, Eq. 12, there is an
oscillatory phase in which the connected correlation function has oscillatory
decay. The oscillatory phase in the $XY$-model is bounded by $r^{2}+h^{2}\leq
1$, which corresponds to $\lambda_{2}\leq-\lambda_{1}^{2}/4$ in the three-spin
model. Some of the representative $C(r,0)$ are shown in Fig. 3.

Figure 5: The correlation length $\xi$ vs temperature $T$ at two quantum
critical points $\lambda_{1}=1$, $\lambda_{2}=0$ ($z=1$) and $\lambda_{1}=2$,
$\lambda_{2}=-1$ ($z=2$). $C_{top}\sim 2.38$ and $C_{bottom}\sim 2.25$.

###  V.2 Finite temperature $S(k,\omega)$

We now turn to discussion of the finite temperature results. This requires
unequal time correlation functions, $C(r,t)$.[11] The calculations in the
following sections were performed on a chain that has 300 lattice sites with
free boundary conditions at a number of temperatures; we just show only the
results at two different temperatures. The temperature $T$ is measured in
units of $h$. We show our finite temperature dynamical structure factor at two
critical points $\lambda_{1}=0.5$, $\lambda_{2}=-1$ and $\lambda_{1}=2$,
$\lambda_{2}=-1$ in Fig. 4.

Figure 6: Quantum Critical to Renormalized Classical. The correlation length
$\xi_{CL}$ vs temperature $T$ at $\lambda_{1}=2.075$, $\lambda_{2}=-0.925$
($\Delta_{gap}=0.3$). $C\sim 3.81$,$C_{1}\sim 22$ and $C_{2}\sim 0.25$. Figure
7: Quantum critical fan diagrams. Black: Quantum Critical. Yellow:
Renormalized Classical. Top: $d$ is the distance to the multicritical point
$\lambda_{1}=1$,$\lambda_{2}=0$ ($z=1$). Bottom: $d$ is the distance to the
multicritical point $\lambda_{1}=2$, $\lambda_{2}=-1$ ($z=2$). Two white lines
are the energy gaps $\Delta_{gap}$ at different $d$. Figure 8: Quantum
Critical to Quantum Disordered. The correlation length $\xi$ vs temperature
$T$ at $\lambda_{1}=1.8$,$\lambda_{2}=-1.2$ ($\Delta_{gap}\sim 0.22$). $C\sim
1.69$ Figure 9: The saturated correlation length $\xi^{sat}$ vs distance $d$
in quantum disordered regime. Top: $d$ is the distance to the multicritical
point $\lambda_{1}=1$,$\lambda_{2}=0$ ($z=1$). Bottom: $d$ is the distance to
the multicritical point $\lambda_{1}=2$,$\lambda_{2}=-1$ ($z=2$). $C_{top}\sim
0.25$. $C_{bottom}\sim 2.34$ Figure 10: Quantum critical fan diagrams. Black:
Quantum Critical. Yellow: Quantum Disordered. Top: $d$ is the distance to the
multicritical point $\lambda_{1}=1$,$\lambda_{2}=0$ ($z=1$). Bottom: $d$ is
the distance to the multicritical point $\lambda_{1}=2$,$\lambda_{2}=-1$
($z=2$).

###  V.3 Quantum Critical to Renormalized Classical

Full construction of quantum critical fans along the three critical lines is
not necessary since the quantum critical fan diagrams are quantitatively the
same. Here we choose to construct the quantum fan diagrams at two points
$\lambda_{1}=2$, $\lambda_{2}=-1$ along the line $\lambda_{2}=\lambda_{1}-3$
and $\lambda_{1}=1$, $\lambda_{2}=0$ along the line
$\lambda_{2}=\lambda_{1}-1$. Referring to Fig. 5, at two critical points, we
see the correlation length $\xi$ scale as $1/T^{1/z}$ with $z$ the theoretical
values $1$($2$). Moving away from the critical point ($\lambda_{1}=2$,
$\lambda_{2}=-1$) into the ordered phase along the line
$\lambda_{2}=\lambda_{1}-3$, we see a crossover happens when we plot $\xi$ vs
$T$ in Fig. 6. At high $T$ ($T>\Delta_{gap}$), the temperature dependence of
the correlation length is characterized by a power law. At low $T$
($T<\Delta_{gap}$), the correlation length grows exponentially to infinity.
Between these regimes, we have a crossover. From that, we construct quantum
critical fan diagrams along the lines $\lambda_{2}=\lambda_{1}-1$ and
$\lambda_{2}=\lambda_{1}-3$. These are shown in Fig. 7. The colors in these
plots represent the relative deviation from the power law scaling. The white
lines are obtained by computing the energy gap at different $d$. Since we are
considering crossovers, not phase transitions, we do accept some misalignments
between the places where the color change happens and the white lines.

###  V.4 Quantum Critical to Quantum Disordered

Moving away from the critical point ($\lambda_{1}=2$,$\lambda_{1}=-1$) into
the disordered phase, we also see a crossover happens when we plot $\xi$ vs
$T$ in Fig. 8. At high $T$ ($T>\Delta_{gap}$), the temperature dependence of
the correlation length is again a power law. Interestingly, we see a bump in
the intermediate region ($T\sim\Delta_{gap}$). We can not differentiate
whether the occurrence of the bump is due to the model itself or uncertainty
from the fitting. At low $T$ ($T<\Delta_{gap}$), we see a completely different
behavior. The correlation length saturates to a finite value. We denote this
value as $\xi^{sat}$, which depends on the distance $d$ to the critical point.

$\xi^{sat}\sim d^{-\nu}$ (34)

where $\nu$ is the critical exponent. It is clear from Fig. 9 that $\nu$ are
close to 1 ($z=1$) and 1/2 ($z=2$). Similarly, we construct the other parts of
the quantum critical fan diagrams along the lines $\lambda_{2}=\lambda_{1}-1$
and $\lambda_{2}=\lambda_{1}-3$. These are shown in Fig. 10. The colors in
these contour plots again represent the relative deviations from the power law
scaling.

##  VI Summary and Discussion

In this paper, we have discussed several properties of an exactly solved model
that exhibits three interesting quantum critical lines and two multi-critical
points. Two multicritical points have different dynamical critical exponents
$z$. The three critical lines have their own unique characteristics. On one
line, the criticality is located at $k=\pm\pi$. The other line has its
criticality located at $k=0$. The criticality on the third line is located at
incommensurate $k$ points. At finite temperatures, quantum critical fans are
built upon these critical lines so the phase diagram splits into three regimes
(quantum critical, quantum disordered and renormalized classical) The
correlation length $\xi$ obtained from the calculation in each regime has its
special behavior on temperature. In quantum critical regime, $\xi$ scales as
$1/T^{1/z}$ with $z$ depends on the critical point. In quantum disordered
regime, $\xi$ becomes temperature independent. But the saturated value of
$\xi^{sat}$ scales as $d^{-\nu}$. Both $z$ and $\nu$ determine the size of the
critical fan. In renormalized classical regime, $\xi$ grows as an exponential
function in terms of $T$ as we approach zero temperature. Finally, we
construct the quantum critical fan along two different lines.

In the future, one could add further neighbor interaction while still
maintaining the integrability of the model. This will make fine-tuning a lot
easier. Then one could explore the quantum critical fan from a critical
surface. However, such a model may be difficult to be realized in experiments.

## ACKNOWLEDGMENTS

H.Y. was supported by M.L Bhaumik Institute for Theoretical Physics at UCLA.
S.C. was supported by funds from the David S. Saxon Presidential Term Chair.

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