# Phase Diagrams, Distinct Conformal Anomalies and Thermodynamics of Spin-1
Bond-Alternating Heisenberg Antiferromagnetic Chain in Magnetic Fields

Xin Yan†, Wei Li†, Yang Zhao, Shi-Ju Ran, and Gang Su  Email<EMAIL_ADDRESS>Theoretical Condensed Matter Physics and Computational Materials Physics
Laboratory, College of Physical Sciences, Graduate University of Chinese
Academy of Sciences, P. O. Box 4588, Beijing 100049, China

###### Abstract

The ground state phase diagrams and thermodynamic properties of a spin-1 bond-
alternating Heisenberg antiferromagnetic chain with a single-ion anisotropy in
longitudinal and transverse magnetic fields are investigated jointly by means
of the infinite time evolving block decimation, the linearized tensor
renormalization group and the density matrix renormalization group methods. It
is found that in the magnetic field-bond alternating ratio plane six phases
such as singlet dimer, Haldane, two Tomonaga-Luttinger liquid, 1/2
magnetization plateau and spin polarized phases are identified in a
longitudinal field, while in a transverse field there are fives phases
including singlet dimer, Haldane, Z2 symmetry breaking, quasi-1/2
magnetization plateau and quasi spin polarized phases. A reentrant behavior of
the staggered magnetization in a transverse field is observed. The quantum
critical behaviors in longitudinal and transverse magnetic fields are
disclosed to fall into different universality classes with corresponding
conformal field central charge $c=1$ and $1/2$, respectively. The experimental
data of the compound NTENP under both longitudinal and transverse magnetic
fields are nicely fitted, and the Luttinger liquid behavior of low temperature
specific heat experimentally observed is also confirmed.

###### pacs:

75.10.Jm, 75.40.Mg, 05.30.-d, 02.70.-c

##  I Introduction

Low dimensional quantum spin systems have been active subjects in quantum
many-body physics for many years. Owing to strong quantum fluctuations and
competitions between various interactions in these systems, a number of exotic
and fascinating quantum emergent phenomena are expected to occur, which thus
arouses persistently considerable interest not only in condense matter physics
but also in other fields such as quantum information and quantum computation
andrew .

Among others, the spin-1 bond alternating Heisenberg antiferromagnetic chain
(BAHAFC) in longitudinal and transverse magnetic fields is of particular
interest, for a series of materials, such as NENP S.Ma , NMOAP Narumi , NDOAP
Narumi2 , and NTENP escuer  can be well described by the spin S=1 BAHAFC, and
the ideal model compound Ni(C9H24N4)(NO2)ClO4 (NTENP) has been extensively
studied both experimentally and theoretically in recent years [Narumi, ;
Narumi3, ; Narumi2, ; Zheludev, ; Hagiwara, ; Suzuki, ; regnault, ; Hagiwara2,
; glazkov, ], where the effects of magnetic field and the bond alternating
ratio on low-lying magnetic excitations, spin correlations and low-temperature
specific heat of the compound NTENP have been investigated. It is known that
in a magnetic field the one-dimensional (1D) Heisenberg quantum spin chains
with periodic ground states would exhibit a topological quantization of
magnetization according to the necessary condition $n(S-m)=integer$ Oshikawa ,
where $n$ is the period of the system, $S$ is the spin and $m$ is the
magnetization per site. Such magnetic plateau states have been addressed in
some polymerized Heisenberg antiferromagnetic or ferrimagnetic spin chains
(e.g. Refs. [su, ]). High-field magnetization measurements on NTENP also
revealed that a $m=1/2$ magnetization plateau appears around 700 kOe Narumi ;
Narumi2 ; the neutron scattering experiments indicated that the spin dynamics
of NTENP quite differs from that of a Haldane spin chain Zheludev ; Hagiwara ;
glazkov ; and the low-temperature specific heat measurement showed a Tomonaga-
Luttinger liquid (TLL) behavior in NTENP in a longitudinal magnetic field
Hagiwara2 . Although the experimental studies on NTENP gain obvious advances,
some ambiguities still remain on further understanding its physical
properties. For instance, the compound NTENP was experimentally studied under
both longitudinal and transverse magnetic fields, but only the data in a
longitudinal field have been fitted by quantum Monte Carlo simulations; no
direct numerical evidence of low-temperature specific heat of this material
was reported to verify the TLL behavior; the sharp peaks of the specific heat
in the NTENP were observed experimentally in both longitudinal and transverse
magnetic fields, where the peak positions move to high temperature side with
the increase of the transverse field, while retain almost intact with
increasing the longitudinal field; and so on. The remaining issues are very
worthy to address.

Apart from that the spin-1 BAHAFC with a single-ion anisotropy [Eq. (1) below]
is believed to be a pertinent model in describing the physical characters of
NTENP, which has been studied previously Zheludev ; Gomez ; Tsvelik ; Kitazawa
; Singh ; Kohno , this model itself has also fascinating properties that
deserve to investigate. When the bond alternating ratio $\alpha=0$, it reduces
to the singlet dimers; when $\alpha=1$, it becomes a spin-1 uniform chain
(Haldane chain). In these two special cases, the system shows an excitation
gap from the singlet dimer or Haldane ground state to the triplet excited
state. The latter uniform chain under both longitudinal ($h_{z}$) and
transverse ($h_{x}$) magnetic fields has been extensively discussed (e.g.
Refs. [Affleck, ; X.G.Wen, ; Xing, ]). However, the overall phase diagrams in
$\alpha-h_{z,x}$ plane for the S=1 BAHAFC with a single-ion anisotropy are
still absent. Therefore, it is really necessary to tackle these above
questions.

In this paper, by means of jointly the infinite time evolving block decimation
(iTEBD) G. Vidal , the linearized tensor renormalization group (LTRG) Li , and
the density matric renormalization group (DMRG) methods White , we shall study
systematically the ground state phase diagrams, magnetic and thermodynamic
properties of the S=1 BAHAFC with a single-ion anisotropy in longitudinal and
transverse magnetic fields. The iTEBD, LTRG and DMRG methods are the numerical
algorithms with very good accuracy and high efficiency that were recently
developed for low-dimensional quantum lattice systems, which allow for
calculating with nice precision the critical properties and the extremely low
temperature behaviors of quantum many-body lattice systems. These methods can
assist us to fit well the experimental data on the NTENP with the accurately
calculated results, thereby capable of reasonably determining the material
parameters and better understanding the fundamental features of the NTENP in
both longitudinal and transverse magnetic fields.

##  II Model and Ground State Phase Diagrams

###  II.1 Model Hamiltonian and Calculational Method

Let us start with the model Hamiltonian given by

$\displaystyle H$ $\displaystyle=$ $\displaystyle
J\sum_{i}^{L/2}(\mathbb{S}_{2i-1}\cdot\mathbb{S}_{2i}+\alpha\mathbb{S}_{2i}\cdot\mathbb{S}_{2i+1})$
(1) $\displaystyle+$
$\displaystyle\sum_{i}^{L}[\Delta(S_{i}^{z})^{2}-g_{\|}\mu_{B}h_{z}S_{i}^{z}-g_{\bot}\mu_{B}h_{x}S_{i}^{x}],$

where $J$ is the coupling constant, $L$ (even) is the number of lattice sites,
$\mathbb{S}_{i}$ is the spin operator at $i$th site, $\Delta$ is the single-
ion anisotropy, $g_{\|}$, $g_{\bot}$ are the Landé g-factors, $\mu_{B}$ is the
Bohr magneton, $h_{z}$ and $h_{x}$ are the longitudinal and transverse
magnetic fields, respectively.

In the following calculations, $g_{\|}\mu_{B}=g_{\bot}\mu_{B}=1$ is assumed
unless the fittings to experimental data are concerned, and $J$ is taken as
the energy scale. The ground state properties were studied by utilizing the
iTEBD G. Vidal  imaginary time projection scheme along with DMRG method White
, and the thermodynamic properties and fittings to the experimental data of
NTENP were performed by invoking the LTRG approach Li . In iTEBD calculations
we take the smallest Trotter step $\tau=10^{-7}$.

###  II.2 Magnetization and Phase Diagram in a Longitudinal Field

Figure 1: (Color online) (a) The magnetic curves of spin-1 bond alternating
Heisenberg antiferromagnetic chain (BAHAFC) with a single-ion anisotropy in
longitudinal magnetic fields $h_{z}$, where three magnetization plateaux
($m=0$, 1/2, and 1) are observed. The inset shows the singularities of the
susceptibility as a function of magnetic field, which signal the occurrence of
quantum phase transitions. (b) The ground state phase diagram in the plane of
the bond alternating ratio ($\alpha$) _vs._ longitudinal magnetic field
($h_{z}$), where the Haldane, singlet-dimer (SD), m=1/2 magnetic plateau,
Tomonaga-Luttinger liquid (TLL), and spin polarized phases are identified. The
single-ion anisotropy $\Delta=0.25$ is taken.

Figure 1(a) presents the field dependence of magnetization per site $m_{z}$
for the system defined by Eq. (1) in the longitudinal magnetic field for
$\alpha=0.45$ and $0.8$, where the bond dimension $D_{c}=120$ of a matrix
product state is presumed. When $\alpha=0.45$, it is clear to see that there
are three magnetization plateaux occurring at $m$=0, 1/2 and 1 in some regions
of $h_{z}$. The $m$=0 plateau suggests that a gap remains in the absence of
magnetic field, implying that the system is in the singlet-dimer phase, as it
should be in the same phase as $\alpha=0$. Based on the valence-bond solid
picture, the $m=1/2$ plateau corresponds to that one of the two bonds in each
coupled spin-1 dimer is broken su . The $m=1$ plateau is in the spin fully
polarized state. Near the critical fields, the magnetization per site $m_{z}$
depends on the magnetic field in a square root behavior, similar to the case
of spin-1 HAFC in a magnetic field. In the ranges between the plateaus, the
spin correlations should be in a power law decay, indicating a quasi long-
range order. The critical magnetic fields for closure of the gap can be
accurately determined by the field dependence of the susceptibility, where the
sharp peaks appear, as shown in the inset of Fig. 1(a). When $\alpha=0.8$, the
three magnetic plateaux still remain, but the width of $1/2$ plateau
decreases, indicating that the corresponding gap becomes narrower. For the
spin-1 BAHAFC, as the ground state period is $n=2$, the topological
quantization condition $n(S-m)=integer$ is satisfied by $m$=0, 1/2 and 1,
i.e., there should exist at most three plateaux in the longitudinal field for
$0\leq\alpha<1$, which is well confirmed in Fig. 1(a). At the special case of
$\alpha=1$, $n$ becomes 1, and there should be two plateaus at $m$=0 and 1.

For $0\leq\alpha\leq 1$, we have calculated the magnetization curves and the
corresponding susceptibilities in the ground state, and collected all critical
values of magnetic fields at which the susceptibility exhibits obvious
singularities, which allows us for readily obtaining a global ground state
phase diagram in $\alpha-h_{z}$ plane for the spin-1 BAHAFC with a single-ion
anisotropy $\Delta=0.25$, as shown in Fig. 1(b). There are six phases,
including the singlet-dimer (SD) phase, Haldane phase, two TLL phases
separated by the $m=1/2$ magnetic plateau phase, and the spin fully polarized
phase. At a critical bond alternating ratio $\alpha_{c}\approx 0.6$, there
exists a quantum phase transition from the gapped SD state into the gapped
Haldane phase. The $m$=1/2 plateau state persists for all values of $\alpha$
until it disappears at $\alpha=1$. At all six phase boundaries, the gap
closes. In these phases, the spin-spin correlation function $\langle
S_{i}^{z}S_{j}^{z}\rangle$ reveals different spatial behaviors, which is
short-range ordered except that in the TLL phase it decays in an algebraic
way.

###  II.3 Magnetization and Phase Diagram in a Transverse Field

Figure 2: (Color online) (a) The magnetic curves of S=1 BAHAFC with a single-
ion anisotropy in transverse magnetic fields $h_{x}$. The up-left inset shows
the staggered magnetization along the $y$ axis, and the down-right inset
presents the corresponding susceptibility as a function of magnetic field. For
$\alpha=0.1$, three plateau-like steps are observed. (b) The ground state
phase diagram in the plane of the bond alternating ratio ($\alpha$) _vs._
transverse magnetic field ($h_{x}$), where the Haldane, singlet-dimer (SD), Z2
symmetry breaking Néel ordered, quasi 1/2 magnetic plateau, and quasi spin
polarized phases are identified. The single-ion anisotropy $\Delta=0.25$ is
taken.

In a transverse magnetic field $h_{x}$, the field dependence of magnetization
per site $m_{x}$ and $m_{y}$ and the corresponding susceptibility of the model
under interest are presented in Fig. 2(a) for $\alpha=0.1$ and 0.45. Three
steps around $m_{x}$=0, 1/2 and 1 in magnetic curves ($m_{x}$ _vs._ $h_{x}$)
are observed for $\alpha=0.1$. The corresponding susceptibility $\chi$ is
illustrated in the lower inset of Fig. 2(a), where one may notice that, due to
the non-conservation of total $S_{x}$, at these three steps $\chi$ does not
exactly vanish although it is negligibly small. Thus, we call these steps as
quasi magnetic plateaus. In the regions between the quasi plateaus, $m_{y}$
has large values as shown in the upper inset of Fig. 1(a). The reason is that
the existence of the single-ion anisotropy makes the spins tend to arrange in
$xy$ plane, and when the magnetic field is applied along the $x$ direction,
the staggered magnetization $m_{y}$ is induced to form a canted Ising order.
In addition, for $\alpha=0.1$ we observed from the upper inset of Fig. 2(a)
that with increasing $h_{x}$ the staggered magnetization $m_{y}$ first is
negligible small, and then exhibits a sharp valley structure, which
illustrates a reentrant behavior of the staggered magnetization Diep ; Hieida
. For $\alpha=0.45$, only two steps appear in the magnetic curve, and the
quasi $m=1/2$ step disappears. In this case, $m_{y}$ has even larger values in
the region between the two steps. For other larger $\alpha$, the magnetic
curves in the transverse field $h_{x}$ have the behaviors similar to that of
$\alpha=0.45$.

Utilizing the method similar to Fig. 1(b) and sweeping various values of
$\alpha$, we can collect all critical magnetic fields by finding the singular
positions of susceptibility, and then obtain the whole ground state phase
diagram in $\alpha-h_{x}$ plane for the system under investigation, as
depicted in Fig. 2(b). It can be seen that there are five phases, namely, the
SD, Haldane, quasi $m=1/2$ magnetic plateau, Z2 symmetry breaking (or spin
canted) and the quasi spin polarized phases. The quantum critical point
$\alpha_{c}\approx 0.6$ is recovered here. In contrast to the case under a
longitudinal field where the $m=1/2$ plateau phase persists into the whole
region of $\alpha<1$ and separates two TLL phases, the quasi $m=1/2$ plateau
phase sets only in a small region ($\alpha\leq 0.24$). Starting from the SD
and Haldane phases, when we increase the field $h_{x}$, the gap is gradually
closed at the phase boundaries. If one continues to increase the magnetic
field $h_{x}$, another gap opens because of the breaking of the discrete
symmetry, giving rise to a Z2 symmetry breaking phase, which displays a long
range behavior.

##  III Entanglement Entropy and Conformal Anomalies

Figure 3: (Color online) The normalized Schmidt coefficient $\lambda_{\nu}$ as
a function of the number of kept states $\nu$ for different phases, where
$D_{c}=80$.

This present system demonstrates quite different behaviors in longitudinal and
transverse magnetic fields. To gain further insight into the underlying
physics behind this character, we have studied the conformal anomalies of this
system at the critical regimes. The conformal field theory (CFT) tells us that
the conformal invariance at the critical point sets useful constraints on the
critical behaviors of two-dimensional classical or 1D quantum systems
conformal , and the universality class can be characterized by the conformal
anomaly or central charge $c$ of the Virasoro algebra. For a system with a
continuous symmetry $G$, if G=SU(N), the possible conformal central charge is
given by $c=(N^{2}-1)k/(N+k)$, $k=1,2,3,...$ Affleck0 . For the spin-S uniform
antiferromagnetic quantum chains, G=SU(2), $c=3S/(1+S)$. For S=1/2, $c=1$;
S=1, $c=3/2$. For the present spin-1 BAHAFC with a single-ion anisotropy, the
SU(2) symmetry is not satisfied, and the conformal central charge should be
calculated via other ways. The von Neumann entropy $S$, defined by

$S=-Tr(\hat{\rho}_{sys}\log_{2}\hat{\rho}_{sys})=-Tr(\hat{\rho}_{env}\log_{2}\hat{\rho}_{env}),$
(2)

offers a possible way to get the central charge of quantum spin chains, where
$\hat{\rho}_{sys(env)}$ is the reduced density matrix (DM) of system
(environment). In critical and noncritical regimes, the entanglement entropy
has different asymptotic behaviors arealaw ; Vidal2 . In critical regimes, the
CFT predicts wilczek

$S\approx\frac{c}{3}\log_{2}(L)+k,$ (3)

where $L$ is the number of spins for a block embedding in an infinite chain,
$c$ is the central charge and $k$ is a non-universal constant. In noncritical
regimes, $S$ vanishes for all L or grows monotonically with L until
saturation. In the following, we shall invoke the iTEBD method to study the
entanglement entropy for the case in a transverse magnetic field. Because of
the nearly unitary evolution, the iTEBD algorithm gives the canonical form of
infinite matrix product states (iMPS). We can make a Schmidt decomposition on
the ground state wave function such that

$|\psi\rangle=\sum_{\nu=1}^{D_{c}}\lambda_{\nu}|\psi^{A}_{\nu}\rangle|\psi^{B}_{\nu}\rangle$
(4)

on each bond, where $|\psi_{\nu}^{A,B}\rangle$ is the orthonormalized basis
states (Schmidt bases) for two parts $A$ and $B$ of the infinite chain, and
$\lambda_{\nu}$ is the Schmidt coefficient (SC). Fig. 3 presents
$\lambda_{\nu}$ as a function of $\nu$ for different phases. It is seen that
in the TLL phase, the SC attenuates much slower than those in

Figure 4: (Color online) The entanglement entropy $S$ as a function of chain
length $L$ calculated by the iTEBD in a transverse field for $\alpha=0.45$ and
$D_{c}=100$, and by the DMRG in a longitudinal magnetic field for $\alpha=0.6$
where the optimal states were kept as 400. In both cases, the single-ion
anisotropy $\Delta=0.25$. The solid line is the fitting curve to Eq. (3),
giving the central charge $c=1/2$ at a critical transverse field
$h_{x}^{c}=0.349$; the inset shows that the entanglement entropy as a function
of $h_{x}$ for a semi-infinite chain length is singular at $h_{x}^{c}=0.349$.
The dashed line is the fitting curve for the case (TLL phase) at a
longitudinal magnetic field $h_{z}=0.3$, giving the central charge $c=1$.

other gapped phases. The double degeneracy of SC only appears in the Haldane
phase, indicating the existence of the topological string order
entang_spectrum . Given a canonical form of iMPS, if the system in Eq. (2) is
chosen as the semi-infinite chain, it is easy to directly calculate the von
Neumann entropy,
$S=-\sum_{\nu=1}^{D_{c}}\lambda_{\nu}^{2}\log_{2}(\lambda_{\nu}^{2})$. In
addition, we can also calculate the entanglement entropy when the system is
successive L spins embedding in an infinite chain, where we may set $L=2$ for
an example. In particular, we can write

$\displaystyle|\psi\rangle=\sum_{\alpha,\beta,\gamma=1}^{D_{c}}\sum_{i,j=1}^{d}\lambda_{\alpha}\Gamma^{i}_{\alpha,\beta}\lambda_{\beta}\Gamma^{j}_{\beta,\gamma}\lambda_{\gamma}|\Phi_{\alpha}\rangle|i\rangle|j\rangle|\Phi_{\gamma}\rangle,$
(5)

where $\Gamma^{i}_{\alpha,\beta}$ is the tensor on the $i$-th site,
$|i\rangle(|j\rangle)$ is the spin basis states on the $i(j)$-th site,
$|\Phi_{\alpha}\rangle$ and $|\Phi_{\gamma}\rangle$ are the left Schmidt basis
of the $i$-th site and the right Schmidt basis of the $j$-th site,
respectively. By contracting two tensors, we will obtain the reduced DM

$\displaystyle\hat{\rho}$ $\displaystyle=$ $\displaystyle
Tr_{ij}|\psi\rangle\langle\psi|$ (6) $\displaystyle=$
$\displaystyle\sum_{\alpha,\gamma=1}^{D_{c}}\sum_{\alpha^{\prime},\gamma^{\prime}=1}^{D_{c}}[\sum_{i,j=1}^{d}\sum_{\beta,\beta^{\prime}=1}^{D_{c}}\lambda_{\alpha}\Gamma^{i}_{\alpha,\beta}\lambda_{\beta}\Gamma^{j}_{\beta,\gamma}\lambda_{\gamma}$
$\displaystyle\lambda_{\alpha^{\prime}}(\Gamma^{i}_{\alpha^{\prime},\beta^{\prime}})^{*}\lambda_{\beta^{\prime}}(\Gamma^{j}_{\beta^{\prime},\gamma^{\prime}})^{*}\lambda_{\gamma^{\prime}}]|\alpha\rangle|\gamma\rangle\langle\alpha^{\prime}|\langle\gamma^{\prime}|.$

One can diagonalize this matrix and calculate the entanglement entropy. For a
larger $L$, we need to contract $L$ tensors and select Schmidt basis to
construct the reduced DM. For all $L$, the dimension of the reduced DM is
$D_{c}^{2}\times{D_{c}^{2}}$. By calculating the von Neumann entropy for a
semi-infinite chain in a transverse magnetic field, we observe that the cusp
position of the entanglement entropy just gives the phase transition point
$h_{x}^{c}=0.349(2)$. By fitting our calculated results to Eq. (3), we find
$c=1/2$ for $h_{x}^{c}=0.349$, as shown in Fig. 4. In the TLL phase of the
case in a longitudinal magnetic field, the iTEBD algorithm gives the ground
state with a staggered magnetization perpendicular to $z$ direction so that it
cannot yield a correct wave function. The reason is that there does not have
an excitation gap and the iTEBD algorithm cannot project the right ground
state wave function. In order to calculate the central charge in the TLL
phase, we utilize the DMRG algorithm with open boundary condition at
$h_{z}=0.3$, $\alpha=0.6$, and $\Delta=0.25$. The result is also included in
Fig. 4, where the fitting result gives the central charge $c=1$ in this TLL
phase note . Consequently, this implies that the critical behaviors of the
spin-1 BAHAFC with a single-ion anisotropy in transverse fields quite differ
from those in longitudinal fields, and the universality falls into two
distinct classes with conformal central charges $c=1/2$ and $c=1$,
respectively.

##  IV Thermodynamic Properties and Comparison to Experiments

###  IV.1 Susceptibility, Magnetization and Comparison to Experiments

Figure 5: (Color online) The temperature dependence of susceptibility of NTENP
measured experimentally is well fitted to the LTRG calculated data for both
longitudinal and transverse magnetic fields. The experimental data are taken
from Ref. [Narumi2, ].

The thermodynamic properties of this system were explored by using recently
proposed LTRG algorithm Li  that allows for accurately calculating the
thermodynamic quantities at very low temperature. In the following LTRG
calculations we keep the Trotter step $\tau=0.1$.

Fig. 5 gives the temperature dependence of susceptibilities of the spin-1
BAHAFC in longitudinal and transverse magnetic fields, where the fittings to
the experimental data of NTENP under both fields are also included. One may
see that the theoretical results are nicely fitted to the experimental data,
generating a set of material parameters of NTENP: $\alpha=0.45$,
$\Delta/J=0.25$, $J=54.2K$, $g_{\|}=2.14$, and $g_{\bot}=2.27$. It should be
noted that in a longitudinal field, these parameters are in agreement with the
previous results from the quantum Monte Carlo calculations Narumi2 , while in
a transverse field the fitting to the experimental data of NTENP is for the
first time done. The magnetization curves of NTENP were measured at T=1.3 K up
to 700 kOe in Ref. [Narumi2, ], which are well fitted to our LTRG calculated
results, giving the same set of fitting parameters as those obtained from the
data of susceptibility except that $g_{\bot}=2.24$ here, as shown in Fig. 6.
Considering that the two sets of experimental data from susceptibility and
magnetic curves are independent, a slight difference on $g_{\bot}$ is
reasonable. Thus, $g_{\bot}$ of NTENP should be around 2.24 $\sim$ 2.27. The
high-field magnetization curve is also well fitted with our LTRG results
(inset of Fig. (6)) using the same parameters.

Figure 6: (Color online) The magnetic curves of NTENP are well fitted to the
LTRG calculated data for both longitudinal and transverse fields, where the
fitting parameters are consistent with those from the susceptibilities. Inset
is the high-field magnetic curve up to 700 kOe fitted with the corresponding
LTRG calculated data. The experimental data are taken from Ref. [Narumi2, ].

###  IV.2 Specific Heat in Magnetic Fields

The temperature dependence of the specific heat $C(T)$ of the S=1 BAHAFC in
longitudinal and transverse magnetic fields is obtained by the LTRG method
down to very low temperatures, as shown in Fig. 7 for $h_{z}=0.5$ and
$h_{x}=0.5$. In a transverse field, there are one round peak and a low
temperature shoulder in $C(T)$, while in a longitudinal field the specific
heat exhibits only one broad peak. It appears that at low temperature the
specific heat displays distinct behaviors. To show this point clearly, we have
carefully calculated the specific heat of this model at extremely low
temperatures by the LTRG algorithm that is very powerful and efficient for
calculating the low-temperature properties than other numerical methods gubo .
Shown in the lower inset of Fig. 7, two different behaviors in both fields at
low temperature are clearly demonstrated, where in a longitudinal field, the
specific heat displays a linear T-dependent character, showing a TLL behavior,
while in a transverse field, $C(T)$ exhibits an exponential decay (the fitting
curve is given in the upper inset of Fig. 7), that can be ascribed to the
$Z_{2}$ symmetry breaking with an open of an Ising gap.

Figure 7: (Color online) The temperature dependence of the specific heat of
S=1 BAHAFC under both longitudinal and transverse magnetic fields. Although
the whole profiles look similar for both cases, the extremely low temperature
behaviors shown in the down-left inset quite differ, where the linear-T
dependence of the specific heat in a longitudinal field is clearly seen at low
temperature, suggesting a Tomonaga-Luttinger liquid behavior, while an
exponential decay for the T-dependence of the specific heat is observed in a
transverse field owing to the appearance of an Ising gap, as seen in the up-
right inset. The single-ion anisotropy $\Delta=0.25$ and the bond alternating
ratio $\alpha=0.45$ are taken.

The specific heat $C(T)$ of NTENP was also measured experimentally Hagiwara2 ,
where the linear T-dependence of $C(T)$ at low temperature in a longitudinal
field above the critical field $h{{}_{z}}^{c}=9.3$ T was observed, which was
identified as a TLL behavior. Our low temperature LTRG calculations at
$h_{z}=0.5$ ($>h{{}_{z}}^{c}$) on the S=1 BAHAFC model strongly supports this
experimental observation. In a transverse field, the experiment on NTENP gives
a distinct nonlinear low-temperature behavior of $C(T)$ from that in a
longitudinal field, which is also backed up by our LTRG results. We should
remark here that the sharp peaks in the temperature dependent specific heat of
NTENP were observed in both fields, which cannot be explained by using this
spin-1 BAHAFC model, as revealed by our present studies, because we do not
find any field-dependent sharp peaks of $C(T)$ in this 1D model. Those sharp
peaks may signal the field-induced long-range orders from the 3D effect of
inter-chain interactions in NTENP. It is this reason that makes us not
directly fit our LTRG results with the experimental data of specific heat of
NTENP. Nevertheless, our present studies on the spin-1 BAHAFC model may give a
possible clue to understand the experimentally observed low-temperature sharp
peaks of $C(T)$ Hagiwara2 . In a longitudinal field, as long as the magnetic
field is higher than the critical field ($h{{}_{z}}^{c}$), the system will
enter the TLL phase that has a quasi LRO, and the true LRO can be established
only by interchain couplings. Thus the peak position is almost determined by
the strength of interchain interactions and hardly moves with the increase of
the magnetic field. In a transverse field, in the range that the experiment
was performed, the staggered magnetization that could enhance the interchain
couplings increases monotonously with increasing the magnetic field, so the
low-temperature sharp peak moves to the high temperature side with the
increase of the transverse field.

##  V Summary and Conclusion

In summary, we have investigated the spin-1 BAHAFC model with a single-ion
anisotropy in longitudinal and transverse magnetic fields by employing the
iTEBD and LTRG methods. The ground state phase diagrams in the plane of field
versus the bond alternating ratio under both fields are obtained, where
various phases are identified for two cases. From the entanglement entropy,
the conformal central charges in both critical longitudinal and transverse
fields are determined to be $c=1$ and $1/2$, respectively, suggesting that the
universality in critical regimes falls into different classes for both fields.
The TLL behavior at low $T$ observed experimentally in NTENP is verified via
our accurate calculations, while an exponential decay of low-temperature
specific heat is uncovered in a transverse field. The experimental data of the
model material NTENP are well fitted with our LTRG results, and the parameters
for characterizing NTENP are determined.

###### Acknowledgements.

We are indebted to Bo Gu, Fei Ye, Shou-Shu Gong, Bin Xi, J. Sirker, and Qing-
Rong Zheng for stimulating discussions. This work is supported in part by the
NSFC (Grants No. 10934008, No. 90922033), the MOST (Grant No. 2012CB932900)
and the Chinese Academy of Sciences.

## References

  * (1) † These authors contribute equally to this work.
  * (2) A. S. Darmawan, and S. D. Bartlett, Phys. Rev. A 82, 012328 (2010).
  * (3) S. Ma, C. Broholm, D. H. Reich, B. J. Sternlieb, and R. W. Erwin, Phys. Rev. Lett. 69, 3571 (1992)
  * (4) Y. Narumi, M. Hagiwara, R. Sato, K. Kindo, H. Nakano, and M. Takahashi, Physica B 509, 246-247 (1998).
  * (5) Y. Narumi, K. Kindo, M. Hagiwara, H. Nakano, A. Kawaguchi, K. Okunishi, M. Kohno, Phys. Rev. B 69, 174405 (2004).
  * (6) A. Escuer, R. Vicente, and X. Solans, J. Chem. Soc. Dalton Trans. (1997), 531.
  * (7) Y. Narumi, M. Hagiwara, M. Kohno, and K. Kindo, Phys. Rev. Lett. 86, 324 (2001).
  * (8) A. Zheludev, T. Masuda, B. Sales, D. Mandrus, T. Papenbrock, T. Barnes, and S. Park, Phys. Rev. B 69, 144417 (2004).
  * (9) M. Hagiwara, L. P. Regnault, A. Zheludev, A. Stunault, N. Metoki, T. Suzuki, S. Suga, K. Kakurai, Y. Koike, P. Vorderwisch, and J. -H. Chung, Phys. Rev. Lett. 94, 177202 (2005).
  * (10) T. Suzuki and S. Suga, Phys. Rev. B 72, 014434 (2005).
  * (11) L. P. Regnault, A. Zheludev, M. Hagiwara, and A. Stunault, Phys. Rev. B 73, 174431 (2006).
  * (12) M. Hagiwara, H. Tsujii, C.R. Rotundu, B. Andraka, Y. Takano, N. Tateiwa, T.C. Kobayashi, T. Suzuki, and S. Suga, Phys. Rev. Lett. 96, 147203 (2006).
  * (13) V. N. Glazkov, A. I. Smirnov, A. Zheludev and B. C Sales, Phys. Rev. B 82, 184406 (2010).
  * (14) M. Oshikawa, M. Yamanaka, and I. Affleck, Phys. Rev. Lett. 78, 1984 (1997).
  * (15) B. Gu, G. Su, and S. Gao, J. Phys.: Condens. Matter 17, 6081 (2005); B. Gu, G. Su, and S. Gao, Phys. Rev. B 73, 134427 (2006); B. Gu and G. Su, Phys. Rev. Lett. 97, 089701 (2006); Phys. Rev. B 75, 174437 (2007); S. S. Gong and G. Su, ibid. 78, 104416 (2008); S. S. Gong, S. Gao, and G. Su, _ibid_. 80, 014413 (2009); S. S. Gong and G. Su, Phys. Rev. A 80, 012323 (2009); Y. Zhao, S. S. Gong, W. Li, and G. Su, Appl. Phys. Lett. 96, 162503 (2010); S. S. Gong, W. Li, Y. Zhao, and G. Su, Phys. Rev. B 81, 214431 (2010).
  * (16) G. Gómez-Santos, Phys. Rev. Lett. 63, 790 (1989).
  * (17) A. M. Tsvelik, Phys. Rev. B 42, 10499 (1990).
  * (18) R. R. P. Singh and M. P. Gelfand, Phys. Rev. Lett. 61, 2133 (1988).
  * (19) M. Kohno and M. Takahashi, Phys. Rev. B 57, 1046 (1998).
  * (20) A. Kitazawa, K. Nomura, and K. Okamoto, Phys. Rev. Lett. 76, 4038 (1996).
  * (21) I. Affleck, Phys. Rev. B 43, 3215 (1991).
  * (22) Z. C. Gu and X. G. Wen, Phys. Rev. B 80, 155131 (2009).
  * (23) H. Z. Xing, G. Su, S. Gao, and J. H. Chu, Phys. Rev. B 66, 054419 (2002).
  * (24) G. Vidal, Phys. Rev. Lett. 98, 070201 (2007); R. Orús and G. Vidal, Phys. Rev. B 78, 155117 (2008).
  * (25) W. Li, S.-J. Ran, S.-S. Gong, Y. Zhao, B. Xi, F. Ye, and G. Su, Phys. Rev. Lett. 106, 127202 (2011).
  * (26) S. R. White, Phys, Rev. Lett. 96, 2863 (1992).
  * (27) P. Azaria, H. T. Diep, and H. Giacomini, Phys. Rev. Lett. 59, 1629 (1987).
  * (28) Y. Hieida, K. Okunishi, and Y. Akutsu, Phys. Rev. B 64, 224422 (2001).
  * (29) A. A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov, J. Stat. Phys. 34, 763 (1984); Nucl. Phys. B 241, 333 (1984); D. Friedan, Z, Qiu, and S. Shenker, Phys. Rev. Lett. 52, 1575 (1984).
  * (30) I. Affleck, Phys. Rev. Lett. 56, 746 (1986).
  * (31) H. W. G. Blöte, J. L. Cardy, and M. P. Nightingale, Phys. Rev. Lett. 56, 742(1986).
  * (32) G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Phys. Rev. Lett. 90, 227902 (2003).
  * (33) J. Eisert, M. Cramer, and M. B.Plenil, Rev. Mod. Phys. 82, 277 (2010).
  * (34) C. Holzhey, F. Larsen, and F. Wilczek, Nucl. Phys. B 424, 443 (1994).
  * (35) F. Pollmann, A. M. Turner, E. Berg, M. Oshikawa, Phys. Rev. B 81, 064439 (2010).
  * (36) In the infinite chain DMRG algorithm with open boundary condition, the entanglement entropy in critical regimes should take the form of $S_{L}\approx(c/6)\log_{2}(L)+k^{\prime}$; see also P. Calabrese, and J. Cardy, J. Stat. Mech. P06002 (2004).
  * (37) B. Gu, Ph. D. Thesis, Graduate University of Chinese Academy of Sciences, Beijing, China, (2006).