# Magnetic phase diagram of a spin $S=1/2$ antiferromagnetic two-leg ladder in
the presence of modulated along legs Dzyaloshinskii-Moriya interaction

N. Avalishvili1,3 B. Beradze2 G.I. Japaridze1,3 1Ilia State University,
Faculty of Natural Sciences and Medicine, 0162, Tbilisi, Georgia 2 Ivane
Javakhishvili Tbilisi State University, Faculty of Exact and Natural Sciences,
Chavchavadze Avenue 3, 0112, Tbilisi, Georgia 3Andronikashvili Institute of
Physics, 0177, Tbilisi, Georgia

(today)

###### Abstract

We study the ground-state magnetic phase diagram of a spin $S=1/2$
antiferromagnetic two-leg ladder in the presence of period two lattice units
modulated, Dzyaloshinskii-Moriya (DM) interaction along the legs. We consider
the case of collinear DM vectors and strong rung exchange and magnetic field.
In this limit we map the initial ladder model onto the effective spin
$\sigma=1/2$ $XXZ$ chain and study the latter using the continuum-limit
bosonization approach. We identified four quantum phase transitions and
corresponding critical magnetic fields, which mark transitions from the spin
gapped regimes into the gapless quantum spin-liquid regimes. In the gapped
phases the magnetization curve of the system shows plateaus at magnetisation
$M=0$ and to its saturation value per rung $M=M_{sat}=1$. We have shown that
the very presence of alternating DM interaction leads to opening of a gap in
the excitation spectrum at magnetization $M=0.5M_{sat}$. The width of the
magnetization plateau at $M=0.5M_{sat}$, is determined by the associated with
the dynamical generation of a gap in the spectrum is calculated and is shown
that its length scales as $(D_{0}D_{1}/J^{2})^{\alpha}$ where $D_{0},D_{1}$
are uniform and staggered components of the DM term, $J$ is the intraleg
exchange and $\alpha\leq 3/4$ and weakly depends on the DM couplings.

###### pacs:

75.10.Jm, 75.10.Pq, 75.70.Tj

###### pacs:

71.10.Pm; 71.27.+a; 71.45.Lr; 75.40.Kb

##  I The model

Low-dimensional quantum magnetism has been the subject of intense research
activity since the pioneering paper by Bethe Bethe_31 . Interest in study of
these systems is determined by their remarkably rich and unconventional low-
energy properties (see for review Refs. Mikeska_Kolezhuk_04 ;
Vasiliev_Volkova_18 . An increased current activity in this field is connected
with the large number of qualitatively new and dominated by the quantum
effects phenomena discovered in these systems Sachdev_NP_08 ; Broholm_et_al_08
; Mila_11 ; Balents_Savary_17  as well as with the opened wide perspectives
for use low-dimensional magnetic materials in modern nanoscale technologies.

A significant fraction of current research in the field of low-dimensional
magnetism is focused on studies of helical structures and chiral order in the
frustrated quantum magnetic systems Perk_76 ; Zviagin ; Oshikawa_Affleck ;
YuLu_2003 ; Aristov_Maleev_00 ; Tsvelick_01 ; Zvyagin_Kolezhuk_04 ;
Zvyagin_Kolezhuk_05 ; Starykh_08 ; Garate_Affeck_10 ; Mila_et-al_11 ; Fazio_14
; Starykh_17 . The key couplings, responsible for stabilization of non-
collinear magnetic configurations in these systems, is the Dzyaloshinskii-
Moriya (DM) interaction DMI

${\cal H}_{DM}=\sum_{n}{\bf D}(n)\cdot[{\bf S}_{n}\times{\bf S}_{n+1}]\,,$ (1)

where ${\bf D}(n)$ is an axial DM vector. Although generally the vectors ${\bf
D}(n)$ may spatially vary both in direction and magnitude, the symmetry
restrictions based on the properties of real solid state materials usually
rule out most of the possibilities and confine the majority of theoretical
discussion to two principal cases – uniform DM interaction, ${\bf D}$ vector
remains unchanged over the system Zviagin ; Aristov_Maleev_00 ; Tsvelick_01 ;
Starykh_08  and the case of staggered DM interaction, with antiparallel
orientation of ${\bf D}$ on adjacent bonds Oshikawa_Affleck ; YuLu_2003 . In
both these cases the DM term can be eliminated by the gauge transformation and
absorbed in a boundary conditions not affecting the structure of the basic
Hamiltonian Perk_76 ; Tsvelick_01 .

The spin S=1/2 two-leg ladders represent the another subclass of low-
dimensional quantum magnets which has been also the subject of perpetual
intensive studies during last three decades Schulz_86 ; Affleck_91 ;
Nersesyan_96 ; Poilblanc_96 ; Dagotto_Rice_96 ; Dagotto_99 ; Mila_98 ;
Totsuka_98 ; Chitra_Giamarchi_97 ; Cabra_Honecker_97-98 ; Usami ;
GiamarchiTsvelik_99 ; Wang_2a ; Wang_2b ; VJM_03 ; Hida_04 ; VJM_04 ; Mila_06
; JP_06 ; JLM_07 ; JaP_Mah_09 ; Zheludev_14 ; Zheludev_15a ; Zheludev_15b ;
Zheludev_16  . Because the gapped character of the excitation spectrum of a
two-leg antiferromagnetic ladder for arbitrary ratio of the intraleg and
interleg couplings Mikeska_Kolezhuk_04 , in the presence of a magnetic field
these systems show dominated by quantum effects complex behavior, which is
manifestly displayed in the magnetic field driven quantum phase transitions in
ladder systems Mila_98 ; Totsuka_98 ; Chitra_Giamarchi_97 ;
Cabra_Honecker_97-98 ; Usami ; GiamarchiTsvelik_99 ; Wang_2a ; Wang_2b ;
VJM_04 ; Mila_06 ; JP_06 ; JLM_07  . During the last few years interest in
$S=1/2$ ladder materials has been enhanced by the discovery of various systems
characterized by the presence of the uniform Dzyaloshinskii-Moriya interaction
along the ladder legs and characterized by unconventional low-temperature
magnetic properties Zheludev_14 ; Zheludev_15a ; Zheludev_15b ; Zheludev_16 .
Among them are as materials characterized by the strong-leg exchange such as
$(C_{7}H_{1}0N)_{2}CuBr_{4}$ Zheludev_15a ; Zheludev_15b  as well as the
strong rung compounds such as $Cu(C_{8}H_{6}N_{2})Cl_{2}$ Zheludev_16 .

Recently it has been demonstrated that the DM interaction can be efficiently
tailored with an substantial efficiency factor by structural modulations
Geometric_tailor_DM  or by external electric field EF_Enhanc_DM_1 ;
EF_Enhanc_DM_2 ; EF_Enhanc_DM_3 . This unveils the possibility to control DM
interaction and magnetic anisotropy via the electric field and opens a wide
area for theoretical consideration of the effects caused by the spatially
modulated DM interaction on the properties of low-dimensional spin systems. In
the recent publication it has been shown that the effect of modulated
spatially DMI on the ground state properties of the spin $S=1/2$ Heisenberg
chain is substantial, leading to the appearance of the additional quantum
phase transition point in the ground state phase diagram, which marks opening
of a gap in the Luttinger-liquid phase of the spin chain and to the formation
of the new gapped phase in the ground state phase diagram characterized by the
coexisting spin dimerization and alternating spin chirality pattern AJR_19 .

In the present work we study the effect of the alternating Dzyaloshinskii-
Moriya interaction on the ground state phase diagram of the spin-1/2 strong-
rung Heisenberg ladder in the presence of magnetic field. The Hamiltonian of
the model under consideration is given by

$\hat{H}=H_{leg}^{1}+H_{leg}^{2}+H_{rung}\,,$ (2)

where the Hamiltonian for leg $\alpha$ is

$\displaystyle H_{leg}^{\alpha}=\sum_{n=1}^{N}\big{\\{}J{{\bf
S}}_{n,\alpha}{{\bf S}}_{n+1,\alpha}+{\bf D}_{\alpha}(n)\cdot[{\bf
S}_{n,\alpha}\times{\bf S}_{n+1,\alpha}]\big{\\}}$ (3)

and the term corresponding to the interleg interaction and coupling with
magnetic field is given by

$\displaystyle H_{rung}$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{N}\big{\\{}J_{\bot}\,{{\bf S}}_{n,1}\cdot{{\bf
S}}_{n,2}-H\,(S^{z}_{n,1}+S^{z}_{n,2})\big{\\}}.$ (4)

Here ${{\bf S}}_{n,\alpha}$ is the spin $S=1/2$ operators at the $n$-th rung,
${\bf D}_{\alpha}(n)$ is a DM vector and the index $\alpha=1,2$ denotes the
ladder legs. Both the intraleg and interleg exchange constant are
antiferromagnetic ($J,J_{\bot}>0$). In what follows we restric our
consideration by the case where the vectors ${\bf D}_{1}(n)$ and ${\bf
D}_{2}(n)$ are collinear and the applied magnetic field is parallel to ${\bf
D}_{\alpha}(n)$. We take ${\bf D}_{\alpha}(n)=\left(0,0,D_{\alpha}(n)\right)$,
where

$\displaystyle D_{\alpha}(n)$ $\displaystyle=$ $\displaystyle
D_{0}^{\alpha}+(-1)^{n}D_{1}^{\alpha}\,.$ (5)

It is obvious, that at $D_{\alpha}(n)\neq 0$ the spin-rotation symmetry of the
Hamiltonian is $U(1)$ and reflects invariance of the system with respect to
the rotation around the vector ${\bf D}$ ($\hat{z}$ axis). In addition, at
$D_{1}^{\alpha}$ the translational symmetry of the Hamiltonian is restored
only after the shift on two lattice units.

The outline of the paper is as follows: In the forthcoming section we derive
transformation which allows to gauge away the DM coupling. Starting from the
section III we restrict our consideration by the limit of strong rung exchange
and magnetic field ($J_{\bot},H\gg\tilde{J}_{\alpha}$) and derive the
effective spin-chain Hamiltonian. In the Section IV we use the continuum-limit
bosonization approach to study the ground state properties of the effective
model. Finally, we conclude and summarize our results in section V.

##  II Gauging away the DM interaction

Because the alternating DM term breaks the translation symmetry, it is
convenient to rewrite the leg Hamiltonian in a form, which explicitly
incorporates doubling of the unit cell of the model. We define the
dimensionless parameters $d_{\pm}^{\alpha}=(d^{\alpha}_{0}\pm
d_{1}^{\alpha})/J$, where $d^{\alpha}_{0}=D^{\alpha}_{0}/J$ and
$d^{\alpha}_{1}=D^{\alpha}_{1}/J$ and rewrite the $\alpha$-leg Hamiltonian in
the following way

$\displaystyle H_{leg}^{\alpha}$ $\displaystyle=$
$\displaystyle\frac{J}{2}\sum_{m=1}^{N/2}\Big{[}\,\left(S^{+}_{2m-1,\alpha}S^{-}_{2m,\alpha}+S^{-}_{2m-1,\alpha}S^{+}_{2m,\alpha}\right)+\left(S^{+}_{2m,\alpha}S^{-}_{2m+1,\alpha}+S^{-}_{2m,\alpha}S^{+}_{2m+1,\alpha}\right)$
(6) $\displaystyle+$ $\displaystyle
i\,d^{\alpha}_{-}\left(S^{+}_{2m-1,\alpha}S^{-}_{2m,\alpha}-S^{-}_{2m-1,\alpha}S^{+}_{2m,\alpha}\right)+i\,d^{\alpha}_{+}\left(S^{+}_{2m,\alpha}S^{-}_{2m+1,\alpha}-S^{-}_{2m,\alpha}S^{+}_{2m+1,\alpha}\right)$
$\displaystyle+$ $\displaystyle
2S^{z}_{2m,\alpha}\left(S^{z}_{2m-1,\alpha}+S^{z}_{2m+1,\alpha}\right)\Big{]}\,,$

where $S^{\pm}_{n,\alpha}=S^{x}_{n,\alpha}\pm S^{y}_{m,\alpha}$. Thus, at
$d_{+}^{\alpha}\neq d_{-}^{\alpha}$, the translation symmetry of a
$\alpha$-leg Hamiltonian is broken by the presence of nonequal amplitudes of
the spin current operator

$j^{\alpha}_{sp}\sim
i\left(S^{+}_{n,\alpha}S^{-}_{n+1,\alpha}-S^{-}_{n,\alpha}S^{+}_{n+1,\alpha}\right)\,.$

on odd and even links of a leg.

The next useful step is to rewrite the Hamiltonian (2) in a physically more
suggestive manner by rotating spins and gauging away the DM interaction term
in each leg. Here we follow the route, developed in the Ref. AJR_19 , in the
case of a single chain with alternating DM interaction. By performing a site-
dependent rotation around the $z$ axis of spins along the leg $\alpha$ with
relative angle $\vartheta^{\alpha}_{-}$ for spins at consecutive odd-even
sites ($2m-1,2m$) and $\vartheta^{\alpha}_{+}$ for spins at consecutive even-
odd sites ($2m,2m+1$), we introduce new spin variables $\bm{\tau}_{2m,\alpha}$
and $\bm{\tau}_{2m+1,\alpha}$ by

$\displaystyle\tau^{+}_{2m-1,\alpha}$ $\displaystyle=$ $\displaystyle
e^{-i(m-1)(\vartheta^{\alpha}_{-}+\vartheta^{\alpha}_{+})}\,\,S^{+}_{2m-1,\alpha}\,,$
(7) $\displaystyle\tau^{+}_{2m,\alpha}$ $\displaystyle=$ $\displaystyle
e^{-im\vartheta^{\alpha}_{-}-i(m-1)\vartheta^{\alpha}_{+}}\,\,S^{+}_{2m,\alpha}\,,$
(8) $\displaystyle\tau^{z}_{2m\pm 1,\alpha}$ $\displaystyle=$ $\displaystyle
S^{z}_{2m\pm 1,\alpha}\,,\quad\tau^{z}_{2m,\alpha}=S^{z}_{2m,\alpha}\ .$ (9)

Using (7)-(9) we map the initial $\alpha$-leg Hamiltonian onto

$\displaystyle H_{leg}^{\alpha}$ $\displaystyle=$
$\displaystyle\frac{J}{2}\sum_{m=1}^{N/2}\Big{[}\left(\cos\vartheta^{\alpha}_{-}+d^{\alpha}_{-}\sin\vartheta^{\alpha}_{-}\right)\left(\tau_{2m-1,\alpha}^{+}\tau_{2m,\alpha}^{-}+\tau_{2m,\alpha}^{-}\tau_{2m-1,\alpha}^{+}\right)$
(10) $\displaystyle\hskip
28.45274pt+\left(\cos\vartheta^{\alpha}_{+}+d^{\alpha}_{+}\sin\vartheta^{\alpha}_{+}\right)\left(\tau_{2m,\alpha}^{+}\tau_{2m+1,\alpha}^{-}\,+\,\tau_{2m,\alpha}^{-}\tau_{2m+1,\alpha}^{+}\right)$
$\displaystyle\hskip
28.45274pt-i\left(\sin\vartheta^{\alpha}_{-}-d^{\alpha}_{-}\cos\vartheta^{\alpha}_{-}\right)\left(\tau_{2m-1,\alpha}^{+}\tau_{2m,\alpha}^{-}\,-\,\tau_{2m,\alpha}^{-}\tau_{2m-1,\alpha}^{+}\right)$
$\displaystyle\hskip
28.45274pt-i\left(\sin\vartheta^{\alpha}_{+}-d^{\alpha}_{+}\cos\vartheta^{\alpha}_{+}\right)\left(\tau_{2m,\alpha}^{+}\tau_{2m+1,\alpha}^{-}\,-\,\tau_{2m,\alpha}^{-}\tau_{2m+1,\alpha}^{+}\right)\,$
$\displaystyle\hskip
28.45274pt+2\tau_{2m}^{z}\,\left(\tau_{2m-1}^{z}+\tau_{2m+1}^{z}\right)\Big{]}\,.$

Choosing $\tan\vartheta^{\alpha}_{\pm}=d_{\alpha}^{\pm}$, gives

$\displaystyle
J(\sin\vartheta^{\alpha}_{\pm}-d_{\alpha}^{\pm}\,\cos\theta_{\alpha}^{\pm})=0\,,$
$\displaystyle
J(\cos\vartheta^{\alpha}_{\pm}+d_{\pm}\,\sin\theta_{\alpha}^{\pm})=J\sqrt{1+(d_{\alpha}^{\pm})^{2}}\,\equiv
J_{\alpha}^{(\pm)}$ (11)

and thus we eliminate the modulated DM interaction and obtain the Hamiltonian
of a each leg characterized by the alternating transverse exchange interaction

$\displaystyle H^{leg}_{\alpha}$ $\displaystyle=$
$\displaystyle\sum_{m=1}^{N/2}\Big{[}\,\frac{J_{\alpha}^{(-)}}{2}\left(\tau^{+}_{2m-1,\alpha}\tau^{-}_{2m,\alpha}+\tau^{-}_{2m-1,\alpha}\tau^{+}_{2m,\alpha}\right)+\,\frac{J_{\alpha}^{(+)}}{2}\left(\tau^{+}_{2m,\alpha}\tau^{-}_{2m+1,\alpha}+\tau^{-}_{2m,\alpha}\tau^{+}_{2m+1,\alpha}\right)$
(12) $\displaystyle\hskip
28.45274pt+\,J\,\tau^{z}_{2m,\alpha}\left(\tau^{z}_{2m-1,\alpha}+\tau^{z}_{2m+1,\alpha}\right)\Big{]}\,.$

It is instructive to rewrite the Hamiltonian (12) in the following, more
common, form

$\displaystyle H^{leg}_{\alpha}$ $\displaystyle=$ $\displaystyle
J_{\alpha}\sum_{n}\Big{[}\,\frac{1}{2}(1+(-1)^{n}\delta_{\alpha})\left(\tau^{+}_{n,\alpha}\tau^{-}_{n+1,\alpha}+\tau^{-}_{n,\alpha}\tau^{+}_{n+1,\alpha}\right)\,+\gamma_{\alpha}\tau^{z}_{n,\alpha}\tau^{z}_{n+1,\alpha}\Big{]}\,,$
(13)

where, at $d^{\alpha}_{i},\ll 1$ ( $i=\pm$),

$\displaystyle J_{\alpha}$ $\displaystyle=$
$\displaystyle\frac{J_{\alpha}^{(+)}+J_{\alpha}^{(-)}}{2}\simeq
J/\gamma_{\alpha}+{\cal O}\left((d^{\alpha}_{i})^{4}\right)\,,$ (14)
$\displaystyle\delta_{\alpha}$ $\displaystyle=$
$\displaystyle\frac{J_{\alpha}^{(+)}-J_{\alpha}^{(-)}}{J_{\alpha}^{(+)}+J_{\alpha}^{(-)}}\simeq
d^{\alpha}_{0}d^{\alpha}_{1}\gamma^{2}_{\alpha}+{\cal
O}\left((d^{\alpha}_{i})^{4}\right)\,$ (15)

and

$\displaystyle\gamma_{\alpha}=J/J_{\alpha}=\frac{1}{\sqrt{1+(d_{0}^{\alpha})^{2}+(d_{1}^{\alpha})^{2}}}\,.$
(16)

Thus at $J_{\alpha}^{(+)}\neq J_{\alpha}^{(-)}$ the Hamiltonian (13) is
recognized as a Hamiltonian of the $XXZ$ chain with easy-plane anisotrophy
($\gamma^{\ast}_{\alpha}<1$) and alternating transverse exchange. Note that
the alternation of the transverse exchange $\delta_{\alpha}\neq 0$ only for
finite $D^{\alpha}_{1}\neq 0$ and $D^{\alpha}_{0}\neq 0$. In the following we
will discard ${\cal O}\left(d_{i}^{4}\right)$ corrections.

Gauging away of the DM interaction does not affect coupling with the magnetic
field and, as it can easily check by inspection, results only via appearance
of the site dependents phase factor in the transverse part of the on-rung
exchange. Inserting (7) -(9) in (4) we obtain

$\displaystyle H_{rung}$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{N}\Big{[}\,\frac{J_{\bot}}{2}\left(\,e^{-i\Phi_{n}}\,\tau^{+}_{n,1}\tau^{-}_{n,2}+h.c\right)+J_{\bot}\tau^{z}_{n,1}\tau^{z}_{n,2}-H\left(\tau^{z}_{n,1}+\tau^{z}_{n,2}\right)\Big{]},$
(17)

where

$\displaystyle\Phi_{2m-1}=(m-1)(\vartheta_{1}^{+}+\vartheta_{1}^{-}-\vartheta_{2}^{+}-\vartheta_{2}^{-}),$
(18)
$\displaystyle\Phi_{2m}=m(\vartheta_{1}^{+}+\vartheta_{1}^{-}-\vartheta_{2}^{+}-\vartheta_{2}^{-})-(\vartheta_{1}^{+}-\vartheta_{2}^{+})\,.$
(19)

In the particular case of a ladder with identical legs and in-phase modulation
of the DM interaction ($D_{1}(n)=D_{2}(n))$, $\Phi_{2m-1}=0$ and
$\Phi_{2m}=\vartheta^{1}_{+}-\vartheta^{2}_{+}$.

Thus, after gauging away the DM interaction the Hamiltonian under
consideration takes the following form

$\displaystyle{\cal H}$ $\displaystyle=$
$\displaystyle\sum_{n,\alpha}\Big{[}\,\frac{J_{\alpha}}{2}(1+(-1)^{n}\delta_{\alpha})\left(\tau^{+}_{n,\alpha}\tau^{-}_{n+1,\alpha}+\gamma^{\ast}_{\alpha}\tau^{-}_{n,\alpha}\tau^{+}_{n+1,\alpha}\right)+\,\tau^{z}_{n,\alpha}\tau^{z}_{n+1,\alpha}-H(\tau^{z}_{n,1}+\tau^{z}_{n,1})\Big{]}$
(20) $\displaystyle+$ $\displaystyle
J_{\bot}\sum_{n=1}^{N}\Big{[}\,\frac{1}{2}\left(\,e^{-i\Phi_{n}}\,\tau^{+}_{n,1}\tau^{-}_{n,2}+h.c\right)+\tau^{z}_{n,1}\tau^{z}_{n,2}\,\Big{]}.$

##  III Effective Hamiltonian in the case of strong rung exchange and magnetic
field

In what follows we restrict ourselves by consideration of the limit of strong
rung exchange and magnetic field ($J_{\bot},H\gg J_{\alpha}$) and follow the
route developed already decades ago to study magnetic phase diagram of a two-
leg ladder in this limit Mila_98 ; Totsuka_98 .

We start from the case $J_{\alpha}=0$, where the system decouples into a set
of noninteracting rungs in a magnetic field

$\displaystyle{\cal H}$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{N}\Big{[}\,\frac{J_{\bot}}{2}\left(\,e^{-i\Phi_{n}}\,\tau^{+}_{n,1}\tau^{-}_{n,2}+h.c\right)+J_{\bot}\tau^{z}_{n,1}\tau^{z}_{n,2}-H(\tau^{z}_{n,1}+\tau^{z}_{n,2})\,\Big{]}$
(21)

and all eigenstates of the Hamiltonian (21) can be written as a product of
rung states. The phase factor $\Phi_{n}$ does not change the eigenvalues of a
pair of coupled spins on a rung – at each rung, two spins form either a
singlet state $|s\rangle$ with energy $E_{s}=-0.75J_{\bot}$ or in one of the
triplet states $|t^{+}\rangle$, $|t^{0}\rangle$ and $|t^{-}\rangle$ with
energies $E_{t^{+}}=0.25J_{\bot}-H$, $E_{t^{0}}=0.25J_{\bot}$ and
$E_{t^{-}}(n)=0.25J_{\bot}+H$, respectively. When $H$ is small, the ground
state consists of a product of rung singlets. As the field $H$ increases, the
energy of the triplet state $|t^{+}\rangle$ decreases and at $H\simeq
J_{\perp}$ forms, together with the singlet state, a doublet of almost
degenerate low energy state, split from the remaining high energy two triplet
states. To project the Hamiltonian (20) on the corresponding low-energy sector
we introduce the effective spin operator $\sigma$ which act on these states as
Mila_98

$\displaystyle\sigma_{n}^{z}|\,s_{0}>_{n}~{}=-\frac{1}{2}|\,s_{0}>_{n}\,,~{}~{}~{}~{}\sigma_{n}^{z}|\,t^{+}>_{n}~{}=\frac{1}{2}|t^{+}>_{n}\,,$
$\displaystyle\sigma_{n}^{+}|\,s_{0}>_{n}~{}=~{}~{}~{}|\,t^{+}>_{n}\,,~{}~{}~{}~{}~{}\sigma_{n}^{+}|t^{+}>_{n}~{}=~{}~{}~{}0\,,$
(22)
$\displaystyle\sigma_{n}^{-}|\,s_{0}>_{n}~{}=~{}~{}~{}~{}0\,,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\sigma_{n}^{-}|\,t^{+}>_{n}=|\,s_{0}>_{n}\,.$

The relation between the initial ladder spin operator
${\mbox{\boldmath$\tau$}}_{n}$ and the pseudo-spin operator
${\mbox{\boldmath$\sigma$}}_{n}$ in this restricted subspace can be easily
derived by inspection,

$\tau^{\pm}_{n,\alpha}=(-1)^{\alpha}\frac{1}{\sqrt{2}}\sigma^{\pm}_{n}\,,\quad\tau^{z}_{n,\alpha}=\frac{1}{2}\left(\frac{1}{2}+\sigma^{z}_{n}\right)\,.$
(23)

Using (23), to the first order and up to a constant, we easily obtain the
following effective Hamiltonian

$\displaystyle\mathcal{H}_{eff}$ $\displaystyle=$ $\displaystyle
J^{\ast}\sum_{n=1}^{N}\big{[}\frac{1}{2}(1+(-1)^{n}\delta^{\ast})(\sigma_{n}^{+}\sigma_{n+1}^{-}+\sigma_{n}^{-}\sigma_{n+1}^{+})+\gamma^{\ast}\,\sigma_{n}^{z}\sigma_{n+1}^{z}-h\,\sigma_{n}^{z}\big{]}$
(24)

which describes a single spin 1/2 chain with a fixed XY anisotropy of
$\gamma^{\ast}$ and alternating transverse exchange in an effective uniform
magnetic field $H_{eff}$. Here

$\displaystyle J^{\ast}$ $\displaystyle=$
$\displaystyle\frac{J_{1}+J_{2}}{2}\,,$ (25) $\displaystyle\delta^{\ast}$
$\displaystyle=$
$\displaystyle\frac{\delta_{1}J_{1}+\delta_{2}J_{2}}{J_{1}+J_{2}}\,,$ (26)
$\displaystyle\gamma^{\ast}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\frac{\gamma_{1}J_{1}+\gamma_{2}J_{2}}{J_{1}+J_{2}}$
(27)

and

$\displaystyle H_{eff}=J^{\ast}\,h$ $\displaystyle=$ $\displaystyle
H-J_{\perp}-\gamma^{\ast}J^{\ast}\,.$ (28)

It is worth to notice that a similar problem has been studied intensively in
past years Chitra_Giamarchi_97 ; Cabra_Honecker_97-98 ; Derzhko_07 ;
Dmitriev_Krivnov_12 .

In absence of the DM interaction, ($\delta_{\alpha}=0$ and
$\gamma_{\alpha}=1$) the Hamiltonian (24) is the Hamiltonian of XXZ chain with
a fixed XY anisotropy of $1/2$ and coincides with the corresponding effective
Hamiltonian for a standard ladder derived in the strong-rung and limit Mila_98
. As it follows from (26) in the strong on-rung coupling limit the effect of
modulated DM interaction, displayed via the alternation of the transverse
exchange, is most pronounced in the case of in-phase modulation of the DM
coupling in both legs, where $\delta_{1}$ and $\delta_{2}$ are of the same
sign and is the weakest in the case of anti-phase modulation where
$\delta_{1}$ and $\delta_{2}$ are of different signs.

##  IV Magnetic properties

###  IV.1 The first critical field $H_{on}$ and the saturation field $H_{sat}$

The performed mapping allows to determine critical fields $H_{on}$
corresponding to the onset of magnetization in the system and the saturation
field $H_{sat}$ Mila_98 . The easiest way to express $H_{on}$ and $H_{sat}$ in
terms of ladder parameters is to perform the Jordan-Wigner transformation
which maps the problem onto a system of interacting spinless fermions JW_1928
:

$\displaystyle\sigma_{n}^{-}=e^{i\pi\sum_{m<n}c_{n}^{{\dagger}}c_{n}}\,c_{n},$
$\displaystyle\sigma_{n}^{+}=c_{n}^{{\dagger}}\,e^{-i\pi\sum_{m<n}c_{n}^{{\dagger}}c_{n}},$
(29) $\displaystyle\sigma_{n}^{z}=c_{n}^{{\dagger}}c_{n}-1/2\equiv\rho_{n},$

the model (24) could be rewritten in the following form:

$\displaystyle H_{sf}$ $\displaystyle=$ $\displaystyle
t\sum_{n}(1+(-1)^{n}\delta^{\ast})(a_{n}^{+}a_{n+1}+h.c.)+V\sum_{n}\rho_{n}\rho_{n+1}-\mu\,\sum_{n}\rho_{n},$
(30)

where $t=V=J^{\ast}/2$ and $\mu=H-J_{\perp}$. The lowest critical field
$H_{on}$ corresponds to that value of the chemical potential $\mu_{c}$ for
which the band of spinless fermions starts to fill up. In this limit we can
neglect the interaction term in Eq. (30) and obtain the model of free massive
particles with spectrum

$E^{\pm}(k)=-\mu\pm J^{\ast}(cos^{2}(k)+\delta^{\ast\,2}\sin^{2}(k))^{1/2}.$

This gives

$\displaystyle H_{on}=J_{\perp}-J^{\ast}\,.$ (31)

Thus at $H<H_{on}$ the effective spin-chain model is fully polarized with
magnetization (per site) $m=-1/2$. This corresponds to the state, where on all
rungs pairs of spins form singlets and respectively, as it follows from (23)
the net magnetization (per site) of the of the initial ladder system

$M=\frac{1}{N}\sum_{n,\alpha}\tau^{z}_{n,\alpha}=0\,.$

A similar argument can be used to determine $H_{sat}$. In the limit of almost
saturated magnetization, it is useful to make a particle-hole transformation
and estimate $H_{sat}$ from the condition where the transformed hole band
starts to fill, what gives

$\displaystyle H_{sat}=J_{\perp}+J^{\ast}.$ (32)

Therefore for $H>H_{sat}$ the effective spin-chain model is in the fully
polarized state with $m=1/2$, all on-rung pairs of spins form triplets and
respectively, as it follows from (23) the net magnetization of the of the
initial ladder system

$M=\frac{1}{N}\sum_{n,\alpha}\tau^{z}_{n,\alpha}=1\,.$

###  IV.2 Magnetization plateau at $M=0.5$

To characterize excitation spectrum and magnetic properties of the system at
intermediate values of the magnetic fields $H_{on}<H<H_{sat}$ we use the
continuum-limit bosonization treatment of the model (24). Following the usual
procedure in the low energy limit, we bosonize the spin degrees of freedom at
fixed magnetization $m$ and the interaction term becomes GNT

$\displaystyle\sigma_{n}^{z}$ $\displaystyle=$ $\displaystyle
m+\sqrt{\frac{K}{\pi}}\partial_{x}\phi(x)+\frac{A_{1}}{\pi}\sin\left(\sqrt{4\pi
K}\phi(x)+(2m+1)\pi n\right)\,,$ (33) $\displaystyle\sigma_{n}^{+}$
$\displaystyle=$ $\displaystyle
e^{-i\sqrt{\pi/K}\theta(x)}+\frac{B_{1}}{\pi}e^{-i\sqrt{\frac{\pi}{K}}\theta(x)}\sin\left(\sqrt{4\pi
K}\phi(x)+(2m+1)\pi n\right),$ (34)

where $A_{1}$ and $B_{1}$ are non-universal real constants of the order of
unity Hikihara01 . and $m$ is the magnetization (per site) of the chain. Here
$\phi(x)$ and $\theta(x)$ are dual bosonic fields,
$\partial_{t}\phi=v_{s}\partial_{x}\theta$, and satisfy the following
commutation relations

$\displaystyle[\phi(x),\theta(y)]=i\Theta(y-x)\,,$
$\displaystyle[\phi(x),\theta(x)]=i/2\,,$ (35)

and $K(\gamma^{\ast},m)$ is the spin-stiffness parameter for a chain with
anisotropy $\gamma^{\ast}$ and magnetization $m$. At zero magnetization LP_75

$K(\gamma^{\ast},0)=\frac{\pi}{2\left(\pi-\arccos\gamma^{\ast}\right)}\,.$
(36)

Using (33) and (34) we get the following bosonized Hamiltonian

$\displaystyle{\cal H}$ $\displaystyle=$ $\displaystyle u\int
dx\,\Big{[}\frac{1}{2}(\partial_{x}\phi)^{2}+\frac{1}{2}(\partial_{x}\theta)^{2}-h\,\sqrt{\frac{K}{\pi}}\partial_{x}\phi+\frac{\delta^{\ast}}{2\pi^{2}\alpha^{2}}\cos\left(\sqrt{4\pi
K}\phi+2\pi mx\right)\,\Big{]}\,,$ (37)

where $u\simeq J^{\ast}/K$. In deriving (37) the strongly irrelevant at
$\gamma^{\ast}<1/2$ (i.e for $K>3/4$) term $\sim\cos\left(\sqrt{16\pi
K}\phi+2\pi mx\right)$ has been omitted.

As we observe, for arbitrary $m\neq 0$ the cosine term in (37) contains
oscillating factor and therefore has to be ignored in the continuum limit.
Therefore in this case the effective spin chain model is critical and its
long-wavelength excitations are described by the standard Gaussian theory and
the magnetic field term can be easily absorbed by a shift
$\partial_{x}\phi\rightarrow\partial_{x}\phi)+\frac{K}{\pi}h$ resulting to
linear in $h$ magnetization of a gapless critical phase shift .

At $m=0$ i.e. $M=0.5$ the oscillating factor in the cosine term is absent and
the continuum-limit bosonized version of the effective spin-chain model is
given by the Hamiltonian

$\displaystyle{\cal H}$ $\displaystyle=$ $\displaystyle u\int
dx\,\Big{[}\frac{1}{2}(\partial_{x}\phi)^{2}+\frac{1}{2}(\partial_{x}\theta)^{2}-h\,\sqrt{\frac{K}{\pi}}\partial_{x}\phi+\frac{\delta^{\ast}}{2\pi^{2}\alpha^{2}}\cos\sqrt{4\pi
K}\phi\,\Big{]}\,.$ (38)

The Hamiltonian (38) is easily recognized as the standard Hamiltonian for the
commensurate-incommensurate phase transition C_IC_transition_1 ;
C_IC_transition_2  which has been intensively studied in the past using
bosonization approach JN_79  and the Bethe ansatz technique JNW_84 . Below we
use the results obtained in JN_79 ; JNW_84  to describe magnetization plateau
and the transitions from a gapped (plateau) to gapless paramagnetic phases.

Let us first consider $h=0$. In this case, the continuum theory of the initial
spin-chain model in the magnetic field $H=J_{\perp}+\gamma^{\ast}J^{\ast}$ is
given by the quantum sine-Gordon (SG) model with a massive term
$\sim\delta^{\ast}\cos(\sqrt{4\pi K}\phi)$ with $3/4\leq K<1$. From the exact
solution of the SG model DHN_74 ; KF_75 ; Zamolodchikov_95 , it is known that
at $3/4\leq K<1$ the excitation spectrum of the model consists of solitons and
antisolitons with masses Zamolodchikov_95

$\Delta=J^{\ast}{\cal C}(K)\left(\delta^{\ast}\right)^{1/(2-K)}\,,$ (39)

where ${\cal C}(K)$ is a constant of the order for all $K$ from the considered
sector $0.75\leq K<1$.

${\cal
C}(K)\\!\\!=\\!\\!\frac{2}{\sqrt{\pi}}\frac{\Gamma(\frac{K}{8-2K})}{\Gamma(\frac{2}{4-K})}\left[\frac{\Gamma(1-K/4)}{2\Gamma(K/4)}\right]^{2/(4-K)}\\!\\!.$

and $\Gamma$ is the Gamma function.

In the gapped phase, the ground state properties of the system are determined
by the dominant potential energy term $\sim\cos(\sqrt{4\pi K}\phi)$ and
therefore in the gapped phases, the field $\phi$ is pinned in one of the vacua

$\langle 0|\sqrt{4\pi K}\phi|0\rangle=(2n+1)\pi\,,$ (40)

to ensure the minimum of the energy. From (33)-(34) it is clearly seen in the
ordered phase local spin degrees of freedom are fully suppressed and only the
on link dimer order

$D_{n}=(-1)^{n}\left(\sigma_{n}^{+}\sigma_{n+1}^{-}+\sigma_{n}^{-}\sigma_{n+1}^{+}\right)\sim\cos(\sqrt{4\pi
K}\phi)$

show a long range order. In terms of initial ladder system it corresponds to
the formation by singlet and triplet located on neighboring rungs of a
plaquette of the coherent entangled pair.

At $h\neq 0$ (i.e. $H\neq J_{\perp}+\gamma^{\ast}J^{\ast}$), the presence of
the gradient term in the Hamiltonian (38) makes it necessary to consider the
ground state of the sine-Gordon model in sectors with nonzero topological
charge. The effective chemical potential

$\sim h_{eff}^{0}\sqrt{\frac{K}{\pi}}\partial_{x}\phi$

tends to change the number of particles in the ground state i.e. to create
finite and uniform density of solitons. It is clear that the gradient term in
(38) can be again eliminated by a gauge transformation

$\phi\rightarrow\phi+h_{eff}^{0}\sqrt{\frac{K}{\pi}}\,x\,,$

however this immediately implies that the vacuum distribution of the field
$\phi$ will be shifted with respect of the minima (40). This competition
between contributions of the smooth and modulated components of the magnetic
field is resolved as a continuous phase transition from a gapped state at
$|h|<\Delta$ to a gapless (paramagnetic) phase at $|h|>\Delta$, where $\Delta$
is the soliton mass C_IC_transition_1 . This condition gives two critical
values of the magnetic field for each plateau

$\displaystyle H_{c}^{\pm}$ $\displaystyle=$ $\displaystyle
J_{\perp}+\gamma^{\ast}J^{\ast}\pm J^{\ast}{\cal
C}(K)\left(\delta^{\ast}\right)^{1/(2-K)}$ (41)

and respectively determines the width of the magnetization plateau by

$H_{c}^{+}-H_{c}^{-}\simeq 2J^{\ast}\left(\delta^{\ast}\right)^{1/(2-K)}\,.$
(42)

As usual in the case of C-IC transition, the magnetic susceptibility of the
system shows a square-root divergence at the transition points:

$\chi(H)\sim\sqrt{(H/H_{c}^{-})-1}\quad{\mbox{f}or}\quad H\geq H_{c}^{-}$

and

$\chi(H)=\sqrt{1-(H/H_{c}^{+})}\quad{\mbox{f}or}\quad H\leq H_{c}^{+}\,.$

Thus we obtain the following magnetic phase diagram for a ladder with
alternating DMI along the legs. For $H\leq H_{on}$, the system is in a rung-
singlet phase with zero magnetization and vanishing magnetic susceptibility.
For $H>H_{on}$ some of singlet rungs melt and the magnetization increase as
$M\sim\sqrt{(H/H_{on})-1}$ . With further increase of the magnetic field the
system gradually crosses to a regime with linearly increasing magnetization.
However, in the vicinity of the magnetization plateau at $M=0.5$, for $H\leq
H_{c}^{-}$ this linear dependence changes again into a square-root behavior
$M=0.5-\sqrt{1-(H/H_{c}^{-})}$. For fields in the interval between
$H_{c}^{-}<H<H_{c}^{+}$ the magnetization is constant $M=0.5$. At
$H>H_{c}^{+}$the magnetization increases as $M=0.5+\sqrt{(H/H_{c}^{+})-1}$,
then passes again through a linear regime until, in the vicinity of the
saturation field $H<H_{sat}$ , it becomes $M=1-\sqrt{1-(H/H_{sat})}$.

##  V Summary

We have studied the ground state magnetic phase diagram of a spin $S=1/2$
antiferromagnetic two-leg ladder in the case where spins along the legs are
affected by the modulated with period of two lattice units Dzyaloshinskii-
Moriya interaction. The model is studied in the limit of strong rung exchange
and magnetic field. It is shown, that the very presence of modulated DM term
opens a gap in the excitation spectrum of the ladder at magnetization equal to
half of its saturated value $M_{sat}$. The value of a gap is determined
simultaneously by the uniform and staggered components of the DMI and is
absent if one of these components is zero. Respectively, at $M=0.5M_{sat}$ the
plateau at magnetization curve appears of a width which determined by the gap
in the excitation spectrum. We have also shown, that in the gapped phase
unconventional magnetic order, where triplet and singlet states localized on
neighboring rungs form an entangled dimer states and this dimerized plaquettes
form a long-range order phase with period two lattice sites wavelength.

To conclude we have to stress that the obtained phase diagram is not a
particularity of the considered case of two lattice unit modulation of the
DMI. The gapped phases, characterized by magnetization plateau and dimerized
plaquette order, appear in the case of arbitrary, but commensurate with the
lattice unit modulation of the DMI interaction
$D_{\alpha}(n)=D_{0}^{\alpha}+\cos(qn)D_{1}^{\alpha}$ at corresponding values
of the magnetization $M=p/qM_{sat}$, where $p$ and $q$ are natural numbers and
$p<q$.

##  VI Acknowledgment

We are grateful to Prof. A.A. Nersesyan for his interest in this work and
helpful comments.

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# Test Coverage for Network Configurations

Paper #377, 12 pages body, 14 pages total

###### Abstract.


# Reconsideration of feasibility of Hall amplifier

Abhimanyu Kumar  Electrical and Instrumentation Engineering Department, Thapar
Institute of Engineering and Technology, Patiala–147004, Punjab, India. Email:
<EMAIL_ADDRESS>Om Prakash Pandey  School of Physics and Material
Sciences, Thapar Institute of Engineering and Technology, Patiala–147004,
Punjab, India. Email<EMAIL_ADDRESS>
###### Abstract

In 1955, it was first suggested that Hall effect can be employed for
amplification purposes by using semiconductor material with very high
mobility. While this idea was limited at that time, yet it was not entirely
discarded expecting eventual progress. We revisit this idea and discuss it in
the light of current literature. This manuscript kindles this 65 year old
amazing idea and views it with modern understanding, which will aid in
realizing Hall amplifiers.

Ross and Thompson [1], in 1955, suggested to design amplifiers employing the
Hall effect principle. Their idea relied on using a material with high
mobility in the Hall device which might be able to amplify input signal at the
expense of applied magnetic field. While this paper appears as the top result
when searched with the keyword “Hall amplifier”, interestingly, it has no more
than 18 citations since its inception. In fact, there seems to be no scholarly
work being done in this direction.

In the same year, Prof. Barlow [2] indicated that their idea was amazing but
the literature then (and infact even today) was (is) limited to accomplish the
task. For a given input in volts, the Hall device produces output in
millivolts or lower. This makes it suitable for sensor applications and
measurement devices [3], but obviously not for amplification. In view of this,
it is easy to understand why the idea of Hall amplifier was never valued.

Since 1950s, the industry has undergone major changes. We have witnessed the
semiconductor revolution which not only caused the extinction of vacuum tubes
and several other primitive devices, but has also led to the miniaturization
of circuits and development of efficient control apparatus. Several advances
have been made in materials employed for Hall devices. Two-dimensional (2D)
materials like graphene [4, 5, 6] have gained popularity for sensing
applications. Table 1 summarizes the Hall sensitivity ($H_{c}$) values for
various materials.

Table 1: Hall sensitivity values for different materials Material |  $H_{c}$ ($\Omega$/T)
---|---
InAs [7] | 370
InSb [8] | 370
Graphene [9] | 800
Graphene [10] | 1200
Graphene [11] | 2093
Graphene [12] | 2745
MoS2/h-BN [13] | 2996

Transition metal chalcogenides and dichalcogenides are also studied [14].
Materials like Al0.20Ga0.80N/GaN [15] and Au/MoSe2/Au structure [16] with
sensitivities $113~{}\Omega/\text{T}$ and $118.5~{}\Omega/\text{T}$ are
currently emerging in this area. As reported in the table above, higher Hall
sensitivity will produce high Hall voltage.

In addition to appropriate choice of material, it is also required to ensure
that Hall material has small thickness. Devices based on semiconductors in
their 2D forms offer nano-scale thickness and consume low power. A promising
solution would be a material with thin conduction layer and high surface
mobility, which are the main requirements for a high magnetic sensitivity
[13]. With the advent of micro-Hall sensors, this requirement of small
dimensions is conveniently satisfied.

Even with huge strides made in the field of material science, we know that the
requirement of high mobility is not enough for development of Hall amplifiers.
One might now think of increasing the magnetic field, but this would require
tens of Tesla to even get the output voltage equal to the input. In reality,
the output gets saturated at such high magnetism because the linearity range
is usually in the order of tens of milli-Tesla. Therefore, in addition to high
Hall sensitivity, a low series resistance is necessary. For most materials,
the series resistance is of the order of kilo-Ohms which limits their
usefulness to sensor applications only [16]. We have identified two approaches
to this problem.

One approach is to mount several Hall sensors together such that their input
ports are in parallel while output ports are in series, as depicted in Figure
1. The parallel connection on the input side reduces the series resistance,
infact it becomes $\frac{R_{s}}{n}$, where $R_{s}$ is the series resistance of
one Hall element and $n$ is the number of Hall elements. The series connection
at output port increases the net Hall voltage given by $V_{H}=nE_{H}$, where
$E_{H}$ is the Hall voltage of one Hall element. In case micro-Hall sensors
are employed then one can mount hundreds of them on just a unit square-
centimeter area. Using a NbFeB permanent magnet (which produces upto 32 mT
field) or an electromagnet (for a controllable amplification) to apply
magnetic field perpendicular to the sensor plane, one can achieve the desired
purpose.

Figure 1: Plate composing of several Hall elements

The second, perhaps more simpler, approach is to use negative resistance
elements such as tunnel diode [17, 18, 19] or Gunn diode [20, 21, 22] such
that the net series resistance can be decreased. From our experience, we
prefer tunnel diodes, majorly due to easiness to use as well as their
longevity (as reported in [23]). The amplification can probably be better
controlled by changing the net series resistance. The decision to use a
particular approach is left to the users and the requirement of their specific
application.

For design engineers or experimental physicists, the implementation now is not
a difficult task. We believe, in future, the Hall amplifiers will turn into a
technology which will be able to compete with the currently available
amplifiers.

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# Addendum to: Guts, Volume and Skein Modules of 3-Manifolds

Brandon Bavier  Department of Mathematics, Michigan State University, East
Lansing, MI, 48824, USA<EMAIL_ADDRESS>
###### Abstract.

In Guts, Volume, and Skein Modules of 3-Manifolds [1], we showed that the
twist number of certain hyperbolic weakly generalized alternating links can be
recovered from a Jones-like polynomial, and offers a lower bound for the
volume of the link complement. Here, we modify the proof to work for a larger
class of links.

##  1\. Introduction

The goal of this paper is to generalize the results of our recent paper [1],
which showed that, under certain hypothesis, the volume of a hyperbolic link
in a compact, irreducible 3-manifold $M$ that admits an alternating projeciton
on a closed surface $F\subset M$, is bounded below in terms of the Kauffman
bracket function defined on link diagrams on $F$. In the paper, they
generalize the proofs and results of Dasbach and Lin [4], as well as Futer,
Kalfagianni, and Purcell [5, 6] to a subset of weakly generalized alternating
knots, as defined by Howie and Purcell [7].

In particular, in [1], we worked with hyperbolic links that admit a prime,
twist-reuced alternating projection diagram $D$ on a surface $F$ in a manifold
$M$, with the additional properties that any essential curve of $F$ intersects
our knot at least three times and that $F\setminus D$ is checkerboard
colorable. In this paper, we will generalize our result to all links admiting
twist-reduced, reduced alternating projections. We do this by looking at
additional coefficients coming from the bracket function. We will recall all
these definitions in Section 2.

Let $M$ be an irreducible, compact 3-manifold with or without boundary, and
let $F\subset M$ be a closed, orientable, embedded surface. Then, if we have a
link $L\subset F\times[-1,1]$, we can get a projection under the map
$\pi:F\times[-1,1]\to F=F\times\\{0\\}$. We can then define, for connected
$F$, a Kauffman bracket function, and construct a family of Jones-like
polynomials

$J_{X}(\pi(L))=a_{m,X}t^{m}+a_{m-1,X}t^{m-1}+\cdots+b_{n+1,X}t^{n+1}+b_{n,X}t^{n},$

where $X$ is a collection of simple disjoint closed curves on $F$. We will
give the precise definitions for these polynomials in the next section. We can
consider the sums of these coefficients,

$a_{k}=\sum_{X}a_{k,X}\qquad b_{k}=\sum_{X}b_{k,X},$

where we sum over all possible collections of simple disjoint closed curves on
$F$. Then we are able to say the following:

######  Theorem 1.

Let $M$ be an irreducible, compact 3-manifold with empty or incompressible
boundary. Let $F\subset M$ be an incompressible, closed, orientable surface
such that $M\setminus N(F)$ is atoroidal and $\bndry-anannular$. Suppose that
a link $L$ admits a weakly generalized alternating projection $\pi(L)\subset
F$ that is reduced, twist-reduced and with all regions of $F\setminus\pi(L)$
disks. Finally, suppose that $r(\pi(L),F)>4$. Then $L$ is hyperbolic and

$\operatorname{vol}(M\setminus L)\geq
v_{8}\mathrm{max}\\{|a_{m-1}|,|b_{n+1}|\\}-\frac{1}{2}\chi(\bndry M),$

where $v_{8}=3.66386\ldots$ is the volume of an ideal octahedron.

Just as in [1], we are able to get the following result involving the Jones-
like polynomial, the twist number, and the guts of our link complement:

######  Theorem 2.

Let $M$ and $F$ be as in Theorem 1 Suppose that a link $L$ admits a weakly
generalized alternating projection $\pi(L)\subset F$ that is twist-reduced and
with all regions of $F\setminus\pi(L)$ disks. Then we have the following.

  1. (1)

$\chi(\mathrm{guts}(M_{A}))=-|a_{m-1}|+\frac{1}{2}\chi(M)$

  2. (2)

$\chi(\mathrm{guts}(M_{B}))=-|b_{n+1}|+\frac{1}{2}\chi(M)$

  3. (3)

$t_{F}(\pi(L))=|a_{m-1}|+|b_{n+1}|+\chi(F)$.

Further, if $\pi(L)$ is a cellularly embedded, twist-reduced, reduced
alternating diagram, we still get (3).

As in [1], Theorem 1 follows directly from 2. Also, as in [1], we get the
following result for links in thickened surfaces.

######  Corollary 3.

Let $L$ be a link in $F\times[-1,1]$ that admits a reduced diagram
$\pi(L)\subset F$ that is twist reduced and has $F\setminus\pi(L)$ disk
regions. Then any two diagrams for $L$ have the same twist number. That is,
$t_{F}(L)$ is an isotopy invariant of $L$.

The remainder of this paper is split into two sections. First, in Section 2,
we recall some key definitions related to generalized link diagrams, including
the weakly generalzied alternating diagrams of Howie and Purcell [7] and
reduced diagrams. Here, we will also introduce the bracket function and Jones-
like polynomial we use in our results. Then, in Section 3, we prove the
Theorem 2.

######  Remark 4.

In addition to extending our recent paper [1], these results should also be
compared to Champanerkar and Kofman [3], who prove similar results for
strongly reduced weakly generalized alternating diagrams using a
specialization of the Krushkal polynomial. Another similar result, Will was
able to give two sided bounds for the twist number of a reduced alternating
diagram in a thickened surface [8]. However, this approach does not lead to a
proof of the invariance of the twist number

##  2\. Definitions

###  2.1. Generalized Link Projections

We start first by reviewing the defintion of a weakly generalized alternating
link diagram, as defined by Howie and Purcell [7]. Let $M$ be a compact,
irreducible 3-manifold, $F\subset M$ a embedded, orientable, connected
surface, and $\pi(L)\subset F$ a projection of a link $L$ onto $F$.

######  Definition 5.

We say the representativity of $\pi(L)$, denoted $r(\pi(L),F)$, is the minimum
number of times a compression disk for $F$ intersects the diagram $\pi(L)$.

A diagram $\pi(L)$ is called weakly prime if, whenever a disk $D\subset F$ has
$\bndry D$ intersecting $\pi(L)$ transversely exactly twice, then either $F$
has positive genus and $D\cap\pi(L)$ is a single embedded arc, or $F=S^{2}$
and either $\pi(L)\cap D$ is a single embedded arc or $\pi(L)\cap(F\setminus
D)$ is.

With these two defintions in mind, we can fully define a weakly generalized
alternating knot:

######  Definition 6.

Let $M$ be a compact, irreducible 3-manifold with an embedded, orientable
surface $F\subset M$. Then $\pi(L)\subset F$ is a weakly generalized
alternating knot if

  1. (1)

$\pi(L)$ is alternating on $F$,

  2. (2)

$\pi(L)$ is weakly prime,

  3. (3)

$\pi(L)$ has at least one crossing,

  4. (4)

$\pi(L)$ is checkerboard colorable, and

  5. (5)

$r(\pi(L),F)\geq 4$.

Note that, in the original paper [1], it is also assumed that every essential
simple closed curve $\gamma$ on $F$ that intersects $\pi(L)$ at exactly two
points, one of the subarcsof $\pi(L)$ with endpoints on $\gamma$ has no
crossings. We will not need that assumption for this paper, and can instead
just work with weakly generalized alternating diagrmas.

First, we recall the definition of a reduced knot diagram, as defined by
Boden, Karimi, and Sikora [2]

######  Definition 7.

A crossing of $\pi(L)\subset F$ is nugatory if there is a a simple loop in $F$
which separates $F$ and interesects $\pi(L)$ only at the crossing. Such a
crossing is removable if the loop can be choosen to bound a disk.

A link diagram $\pi(L)$ on $F$ is reduced if it has no removable nugatory
crossings.

When we start working with the coefficients, we will work with reduced
alternating diagrams that have disk regions. Note, though, that all weakly
generalized alternating diagrams must be reduced alternating as well; if not,
we could find a loop that violates the weakly prime condition.

###  2.2. Jones-like Polynomials

We will use a modification to the Kauffman bracket function, and thus Jones-
like polynomial, used in [1]. The bracket function is defined in the same way,
except when we have a single unknot component in our diagram, we assign it
$(-A^{2}-A^{-2})$. This will allow us to more easily work with states that
have no contractible state circles. We can then, given a diagram $D$ on $F$,
split $\langle D\rangle$ into several bracket polynomials based on the
different Kauffman states. Recall that, given a Kauffman state $s$, $a(s)$ is
the number of $A$ resolutions, $b(s)$ the number of $B$ resolutions, $s_{t}$
the collection of contractible state circles, and $s_{nt}$ the collection of
non contractible circles. We will also denote the number of contractible state
circles by $|s|_{t}$. Then we can define the following polynomials:

(1) $\langle
D\rangle_{0}=\sum_{\\{s|s_{nt}=\emptyset\\}}A^{a(s)-b(s)}(-A^{2}-A^{-2})^{|s|_{t}}$

(2) $\langle
D\rangle_{X}=\sum_{\\{s|s_{nt}=X\\}}A^{a(s)-b(s)}(-A^{2}-A^{-2})^{|s|_{t}}$

where $X\in X_{F}$ is a collection of simple closed disjoint curves on $F$. We
can then define the Jones-like polynomials as follows:

$\displaystyle J_{0}(\pi(L))$
$\displaystyle=\left((-1)^{w(D)}A^{-3w(D)}\langle\pi(L)\rangle_{0}\right)|_{t=A^{4}}$
$\displaystyle J_{X}(\pi(L))$
$\displaystyle=\left((-1)^{w(D)}A^{-3w(D)}\langle\pi(L)\rangle_{X}\right)|_{t=A^{4}}.$

We can write these out with coefficients:

$\displaystyle J_{0}(\pi(L))=a_{m,0}t^{m}+a_{m-1,0}t^{m-1}+$
$\displaystyle\cdots+b_{n+1,0}t^{n+1}+b_{n,0}t^{n}$ $\displaystyle
J_{X}(\pi(L))=a_{m,X}t^{m}+a_{m-1,X}t^{m-1}+$
$\displaystyle\cdots+b_{n+1,X}t^{n+1}+b_{n,0}t^{n}.$

As $J_{0}(\pi(L))$ and $J_{X}(\pi(L))$ might have different highest or lowest
degrees, we take $m$ to be the highest degree over all possible
$J_{X}(\pi(L))$ and $J_{0}(\pi(L))$, and $n$ to be the lowest. Then, if a
polynomial has lower maximum degree, we take all higher coefficients to be 0.

Note the use of $\pi(L)$ in these definitions. In general, these are not link
invariants, but instead diagram invariants. Finally, we define

$a_{k}=a_{k,0}+\sum_{X\in X_{F}}a_{k,X}\qquad b_{k}=b_{k,0}+\sum_{X\in
X_{F}}b_{k,X}.$

###  2.3. Reduced Graphs

Finally, we define certain graphs associated to $\pi(L)$, which we will use to
connect the twist number to the coefficients.

######  Definition 8.

Let $\pi(L)\subset F$ be a diagram for a link on a surface $L$. Let $s$ be
some state for $\pi(L)$. Then the state graph $G_{s}$ is defined as having
vertices corresponding to the state circles of $s$, and edges connecting state
circles that are adjacent across crossings of $\pi(L)$.

The reduced graph of $G_{s}$, $G^{\prime}_{s}$, can be obtained from $G_{s}$
by, whenever two edges of $G_{s}$ bound a disk on $F$, we remove one of them.
Then $e^{\prime}_{a}$ denotes the number of edges of $G^{\prime}_{s}$.

##  3\. Proofs

We begin now by examining the coefficients for reduced alternating diagrams.

######  Lemma 9.

Suppose $\pi(L)$ is twist-reduced, reduced alternating diagram on a closed
orientable surface $F$, with $F\setminus\pi(L)$ disks. Suppose also that
$\pi(L)$ is checkerboard colorable. Then $|a_{m}|=|b_{n}|=1$.

######  Proof 3.1.

As regions of $F\setminus\pi(L)$ are disks, the all $A$ state, $s_{A}$, has
all state circles compressible disks (in particular, we can view $s_{A}$ as
the collection of all regions of $F\setminus\pi(L)$ colored a single color).
Then $s_{A}$ has highest degree $\left(c+2|s_{A}|_{t}\right)$, and the highest
term will be $\pm 1$.

Suppose we have a state $s_{k}$ with exactly $k$ $B$-resolutions, and a state
$s_{k-1}$ which differs from $s_{k}$ by exactly one resolution change from an
$A$-resolution to a $B$-resolution. This revolution change can, at most,
increase the number of contractible state circles by 1. Look, then, at any
state that has exactly one $B$ resolution, $s_{1}$. As $s_{A}$ has only
contractible state circles, there are exactly three ways a resolution change
from $s_{A}$ to a state $s_{1}$ can act.

  1. (1)

We could merge two contractible circles into a single contractible circle.
Then $|s_{1}|_{t}=|s_{A}|_{t}-1$.

  2. (2)

We could split a single contractible circle into two noncontractible circles.
Then $|s_{1}|_{t}=|s_{A}|_{t}-1$.

  3. (3)

We could split a single contractible circle into two contractible circles.
Then $|s_{1}|_{t}=|s_{A}|_{t}+1$.

Note that the highest degree $s_{1}$ can contribute to a polynomial is
$\left(c+2|s_{1}|_{t}-2\right)$. The only way $s_{1}$ could contribute to the
highest degree of $J(\pi(L))$ is if the last case, (3), happens. However, note
that for this to happen, the resolution change must occur at a crossing that
meets a single state circle of $s_{A}$ twice. Then we can construct a loop
$\gamma$ that intersects the diagram at exactly this crossing, and then
follows along the boundary of the state circle. As there are no nugatory
crossings, this must mean that $\gamma$ is a non-separating curve. In
particular, $\gamma$ must be essential. After the resolution change, we will
get two state circles that must have essential boundary, and thus be
noncontractible. But then we must be in the second case, (2), and have a
contradiction. So no $s_{1}$ can contribute to the highest degree.

Finally, look at any state $s_{k}$ with exactly $k$ $B$ resolutions. This must
have highest degree $\left(c+2|s_{k}|_{t}-2k\right)$. In order to contribute
to the highest degree, we must have $|s_{k}|_{t}=|s_{A}|_{t}+k$. However, as
each resolution change can, at most, introduce one contractible state circle,
and we know that the change from $s_{A}$ to $s_{1}$ does not, we must have
$s_{k}$ not contribute to the highest degree. So then $a_{m}=\pm 1$, and we
are done.

Before we begin, we introduce one last definition that we will use in the
later half of the proof for the second coeffient.

######  Definition 10.

Let $\pi(L)\subset F$ be a reduced alternating diagram, and $s$ a Kauffman
state, with reduced graph $G^{\prime}_{s}$. We say two edges, $\ell$ and
$\ell^{\prime}$ in $G^{\prime}_{s}$ are parallel if they are adjacent to the
same vertices (or, equivalently, connect the same state circles) and, when
placed on $F$ with the state circles, there is a disk in $F$ with boundary
containing $\ell$, $\ell^{\prime}$, and a portion of the state circles they
connect.

######  Lemma 11.

Suppose $\pi(L)$ is a twist-reduced, reduced alternating diagram on a closed
orientable surface $F$, with $F\setminus\pi(L)$ disks. Suppose also that
$\pi(L)$ is checkerboard colorable. Then

  1. (1)

$|a_{m-1}|=e^{\prime}_{A}-|s_{A}|_{t}$.

  2. (2)

$|b_{n+1}|=e^{\prime}_{A}-|s_{B}|_{t}$.

######  Proof 3.2.

First, note that $s_{A}$ contributes to the second highest degree. In
particular, it contributes $(-1)^{|s_{A}|_{t}}|s_{A}|_{t}$. Next, we also know
that any state with exactly one $B$ resolution, $s_{1}$, will contribute to
the second highest degree, by work above. Further, if we have a sequence of
states $s_{A},s_{1},\cdots,s_{k}$, each differing from the previous state by a
resolution change from an $A$-resolution to a $B$-resolution, then for $s_{k}$
to contribute, we must have each resolution change after the first introducing
a new contractible state circle. We know that $s_{1}$ either has all
contractible state circles, or has exactly two non-contractible state circles
that are parallel to each other. Focus first on the states $s_{1}$ and
$s_{2}$. There are three possible ways a resolution change can act and still
have $s_{2}$ contribute to the second highest degree:

  1. (1)

We split a single contractible circle into two contractible circles.

  2. (2)

We merge two noncontractible circles into a single contractible circle.

  3. (3)

We split a single noncontractible circle into a noncontractible circle and a
contractible circle.

We focus on each of these cases individually. In the case of (1), where we
split a contractible circle into two contractible circles and don’t change the
pattern of noncontractible state circles, we can use work by Boden, Karimi,
and Sikora [2, Theorem 11], which says that any state with a single $B$
resolution and the same type of nonctractible state circles as $s_{A}$ must
have, at most, the same number of contractible circles as $s_{A}$, to get a
contradiction.

In the case of (2), where we merge two noncontractible circles into a single
contractible circle, denote the edge changed in the resolution change from
$s_{A}$ to $s_{1}$ as $\ell_{1}$, and the edge changed from $s_{1}$ to $s_{2}$
as $\ell_{2}$. We know that the two noncontractible circles must be parallel,
and so, to merge them to get a contractible circle, we must have $\ell_{1}$
parallel to $\ell_{2}$. Look at $\ell_{2}$ in $s_{A}$, and construct a new
state $s^{\prime}_{1}$, which has all $A$-resolutions except at $\ell_{2}$. In
Figure 1, the leftmost state is $s_{1}$, the center $s_{A}$, and the right
$s^{\prime}_{1}$.

$\ell_{2}$
---
$\ell_{1}$
---
Figure 1.  (Left) The state $s_{1}$ gotten from $s_{A}$ by mergining two
noncontractible circles into a single contractible circle; (Center) The all
$A$ state, $s_{A}$; (Right) The modified state, $s^{\prime}_{1}$, with exactly
one $A$ resolution.

Note that $s^{\prime}_{1}$ must, then, introduce a new contractible state
circle. In addition, it will have the same types of non-contractible state
circles as $s_{A}$. Then, once again, By Boden, Karimi, and Sikora [2, Theorem
11], any state of a reduced alternating diagram with one $B$ resolution must
have, at most, the same number of contractible state circles as $s_{A}$, so we
get a contradiction.

Finally, in the case of (3), we split a noncontractible state into a
contractible circle and a noncontractible circle. Label the edges as before:
$\ell_{1}$ is the edge first changed in the resolution change from $s_{A}$ to
$s_{1}$, and $\ell_{2}$ the edge changed from $s_{1}$ to $s_{2}$. There are
two possibilities for $\ell_{2}$: either it is parallel to $\ell_{1}$ in
$s_{A}$, or it is not. If the two edges are not parallel, then $\ell_{2}$ must
have it’s endpoints both to the same side of $\ell_{1}$, and cut off a
contractible circle disjoint from $\ell_{1}$. Then we can construct a new
state, $s^{\prime}_{1}$ that has all $A$ resolutions except for at $\ell_{2}$,
where we have a $B$ resolution. As $\ell_{2}$ cuts off a contractible circle
disjoint from $\ell_{1}$, $s^{\prime}_{1}$ must have more contractible circles
than $s_{A}$, a contradiction. So then $\ell_{1}$ and $\ell_{2}$ must be
parallel in $s_{A}$.

Note that, in order for $s_{2}$ to contribute to the second highest degree,
the second edge must be parallel to the first. If it isn’t, it either does not
introduce a new contractible circle (like in the first case), or we can find a
state with 1 B resolution that has more contractible circles than $s_{A}$. We
can do this same process for any state $s_{k}$ with $k$ B resolutions, and so
all states that contribute to $|a_{m-1}|$ must have $B$ resolutions at only
parallel edges.

Also note, if any two edges are parallel, they must be part of the same twist
region. To see this, as parallel edges bound a disk on $F$, we get a disk
whose boundary intersects the link diagram at exactly two crossings. As the
diagram is twist reduced, this must mean both of these crossings, and thus the
edges, belong to the same twist region.

So then we get a contribution to $|a_{m-1}|$ only when we make resolution
changes to edges that, in $s_{A}$, are all parallel to each other. As we are
asking about $|a_{m-1}|$, we will leave off the essential curve variable $X$
during our calculations. Each such state will contribute
$(-1)^{|s_{A}|-2}(-1)^{j}$, where $j$ is the number of changed edges, and
there are $\binom{k}{j}$ possible states with exactly $j$ resolution changes,
where $k$ is the total number of parallel edges. Using the binomial theorem,
we get that each family of parallel edges must contribute $(-1)^{|s_{A}|-1}$
to the second highest coefficient. As each family of parallel edges
corresponds to a single edge in $G^{\prime}_{A}$, and factoring in the
contribution to the second coefficient from $s_{A}$, we get that

$\displaystyle|a_{m-1}|$
$\displaystyle=|(-1)^{|s_{A}|_{t}-1}e^{\prime}_{A}+(-1)^{|s_{A}|_{t}}(|s_{A}|_{t})$
$\displaystyle=|(-1)^{|s_{A}|_{t}-1}(e^{\prime}_{A}-|s_{A}|_{t})|$
$\displaystyle=e^{\prime}_{A}-|s_{A}|_{t}$

and so we are done with part (1) of the theorem.

To prove part (2), we can apply the same argument to the diagram $D^{*}$.

We can now use Lemmas 9 and 11 to prove one of our main results:

######  Theorem 12.

Suppose that $\pi(L)$ is a twist-reduced, reduced alternating link projeciton
with twist number $t_{F}(\pi(L))$ on a projection surface $F\subset M$ of
genus at least 1. Suppose also that the regions of $F\setminus\pi(L)$ are all
disks. Then

$|a_{m-1}|+|b_{n+1}|=t_{F}(\pi(L))-\chi(F).$

######  Proof 3.3.

First, note that we have

$t_{F}(\pi(L))=c-(c-e^{\prime}_{B})-(c-e^{\prime}_{B})=e^{\prime}_{A}+e^{\prime}_{B}-c.$

Next, as the regions of $F\setminus\pi(L)$ are disks, we also have

$|s_{A}|+|s_{B}|=c+2-2g(F)=c+\chi(F).$

It is important to note that, as $F\setminus\pi(L)$ are all disks, then
$|s_{A}|=|s_{A}|_{t}$ and $|s_{B}|=|s_{B}|_{t}$, as there are only
contractible circles. Then we get

$\displaystyle|a_{m-1}|+|b_{n+1}|$
$\displaystyle=e^{\prime}_{A}-|s_{A}|_{t}+e^{\prime}_{B}-|s_{B}|_{t}$
$\displaystyle=e^{\prime}_{A}+e^{\prime}_{B}-|s_{A}|-|s_{B}|$
$\displaystyle=t_{F}(\pi(L))+c-c-\chi(F)$
$\displaystyle=t_{F}(\pi(L))-\chi(F),$

and we are done.

The proofs for Theorems 1 and 2 follow their original proofs in [1], with no
modifications except for the new numbers. We will sketch their proofs here for
completeness.

######  Proof 3.4 ((Proof of 2)).

We have already proved the twist number result above, in Theorem 12, as well
as the last last statement about reduced alternating diagrams. So we now focus
on the other two statements.

Let $S_{A}$ and $S_{B}$ be the checkerboard surfaces of $\pi(L)$,
$X=M\setminus L$, and $M_{A}=X\cut S_{A}$ and $M_{B}=X\cut S_{B}$. As $\pi(L)$
is a weakly generalized alternating diagram, [7] gives us

(3) $\chi(\guts(M_{A}))=\chi(F)+\frac{1}{2}\chi(\bndry M)-|s^{\prime}_{B}|,$

where $|s^{\prime}_{B}|$ is the number of non-bigon $B$ regions of $S_{A}$.
Then note that

$\chi(F)=|s_{A}|-e^{\prime}_{A}+|s^{\prime}_{B}|.$

In particular, by rearranging, we get

$\chi(F)-|s^{\prime}_{B}|=|s_{A}|-e^{\prime}_{A}$

Finally, by Lemma 11, we get

(4) $\chi(F)-|s^{\prime}_{B}|=-|a_{m-1}|.$

Then by combining Equations 3 and 4, we immediately get our result. The proof
for $\guts(M_{B})$ follows similarly, by swapping $A$ and $B$.

######  Proof 3.5 ((Proof of 1)).

Let $S_{A}$ and $S_{B}$ be the checkerboard surfaces of $\pi(L)$,
$X=M\setminus L$, and $M_{A}=X\cut S_{A}$ and $M_{B}=X\cut S_{B}$. Then, by
[7][Theorem 9.1], we have

$\operatorname{vol}(M)\geq-
v_{8}\chi(\guts(M_{A}))\quad\mathrm{and}\quad\operatorname{vol}(M)\geq-
v_{8}\chi(\guts(M_{B})).$

Then, by 2, we immediately get our result.

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