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The dataset generation failed
Error code: DatasetGenerationError
Exception: CastError
Message: Couldn't cast
text: string
meta: struct<redpajama_set_name: string>
child 0, redpajama_set_name: string
__index_level_0__: int64
to
{'text': Value('string'), 'meta': {'redpajama_set_name': Value('string')}}
because column names don't match
Traceback: Traceback (most recent call last):
File "/usr/local/lib/python3.12/site-packages/datasets/builder.py", line 1872, in _prepare_split_single
for key, table in generator:
^^^^^^^^^
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 609, in wrapped
for item in generator(*args, **kwargs):
^^^^^^^^^^^^^^^^^^^^^^^^^^
File "/usr/local/lib/python3.12/site-packages/datasets/packaged_modules/json/json.py", line 265, in _generate_tables
self._cast_table(pa_table, json_field_paths=json_field_paths),
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
File "/usr/local/lib/python3.12/site-packages/datasets/packaged_modules/json/json.py", line 120, in _cast_table
pa_table = table_cast(pa_table, self.info.features.arrow_schema)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
File "/usr/local/lib/python3.12/site-packages/datasets/table.py", line 2272, in table_cast
return cast_table_to_schema(table, schema)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
File "/usr/local/lib/python3.12/site-packages/datasets/table.py", line 2218, in cast_table_to_schema
raise CastError(
datasets.table.CastError: Couldn't cast
text: string
meta: struct<redpajama_set_name: string>
child 0, redpajama_set_name: string
__index_level_0__: int64
to
{'text': Value('string'), 'meta': {'redpajama_set_name': Value('string')}}
because column names don't match
The above exception was the direct cause of the following exception:
Traceback (most recent call last):
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1342, in compute_config_parquet_and_info_response
parquet_operations, partial, estimated_dataset_info = stream_convert_to_parquet(
^^^^^^^^^^^^^^^^^^^^^^^^^^
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 907, in stream_convert_to_parquet
builder._prepare_split(split_generator=splits_generators[split], file_format="parquet")
File "/usr/local/lib/python3.12/site-packages/datasets/builder.py", line 1739, in _prepare_split
for job_id, done, content in self._prepare_split_single(
^^^^^^^^^^^^^^^^^^^^^^^^^^^
File "/usr/local/lib/python3.12/site-packages/datasets/builder.py", line 1922, in _prepare_split_single
raise DatasetGenerationError("An error occurred while generating the dataset") from e
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text string | meta dict |
|---|---|
\section{Introduction}
Spallation reactions, i.e. proton-induced reactions on heavy targets at a few hundred MeV, have been the subject of many studies since 1950. They are known to be a valuable tool for the study of the de-excitation of hot nuclei because, contrarily to reactions between heavy ions, they lead to the formation of hot prefragments with only a limited excitation of the collective degrees of freedom such as rotation or compression. Their study has also been motivated by astrophysics, as cosmic rays undergo spallation reactions with the hydrogen and helium nuclei of the interstellar medium.
Recently, progresses in high-power accelerator technologies have made possible the realisation of intense neutron sources based on spallation reactions. Such sources are needed for Accelerator-Driven Systems~\cite{Bowmann,Rubbia}, and also find applications in nuclear physics~\cite{NToF} and for material physics and biology~\cite{ESS}. Furthermore, spallation reactions can also be used to produce exotic nuclei and, hence, secondary beams~\cite{ISOLDE,EURISOL}. Those new applications have motivated a large number of experimental works and created a strong demand for high-precision calculation codes.
In order to bring new data and therefore new constraints for the codes, measurements of reaction residues have been undertaken at GSI by an international collaboration. These experiments are based on the inverse-kinematics method. Fragments are identified in-flight using the FRS spectrometer~\cite{FRS}, making possible the first measurements of complete nuclide distributions. In the frame of those studies, production cross sections have already been published for several systems: Au+p at 800A~MeV~\cite{Au_Fanny,Au_Pepe}, Pb+p at 1A~GeV~\cite{Wlazlo,Pb_p}, Pb+d at 1A~GeV~\cite{Pb_d}, U+p at 1A~GeV~\cite{Taieb,Monique} and U+d at 1A~GeV\cite{U_Enrique,U_Jorge}.
These results have helped to partially discriminate between the respective influence of the two main steps of the spallation process, the intranuclear cascade and the fission/evaporation process. The behavior of several codes (the ISABEL~\cite{ISABEL}, INCL4~\cite{INCL4} or BRIC~\cite{BRIC} intranuclear cascades, the ABLA fission/evaporation code~\cite{ABLA}) has proved to be now overall satisfactory for proton energies around 1 GeV. On the other hand, important failures in the description of the emission of charged particles in the Dresner evaporation code~\cite{Dresner} have been put in evidence~\cite{Au_Fanny}. In the 1 GeV energy region, the only serious, remaining deficiency is the underestimation of the lightest evaporation products, which are related to the most violent collisions. Despite large differences in the description of the spallation process, all the codes mentioned above present this weakness. This indicates that some phenomena have not been taken into account. In recent experiments conducted at GSI on lighter nuclei ($^{56}$Fe, $^{136}$Xe), our collaboration explored a range of nuclear temperatures higher than in the systems mentioned in the previous paragraph. Indications were found that fast break-up decay may play an important role in high-energy spallation reactions~\cite{Paolo}.
The question of understanding of the evolution of the reaction mechanisms with decreasing energy also remains open. This is an important point in the perspective of technical applications, because nuclear reactions in thick targets happen in a broad energy range: beam particles are subject to electronic slowing down, and also fast particles emitted in the first stage of the reactions are likely to produce additional nuclear reactions, giving rise to an {\it internuclear} cascade. To address the question of the dependance of the reaction on the energy of the incident particle, an experiment has been performed at GSI aiming at measuring production cross sections of residues in the spallation of lead by protons at 500A~MeV.
The present paper deals with the experimental results on the fragmentation-evaporation residues obtained in this experiment. It completes the results already published obtained during the same experiment for the fission products~\cite{NPA_Bea}. Detailed confrontations between the results from codes dedicated to the description of the spallation process and these data as well as other data on evaporation residues obtained by our group and other related measurements on spallation reactions like light particle production are postponed to a forthcoming paper.
The energy chosen for this experiment, which is low in comparison to the FRS standards for experiments involving nuclei as heavy as lead~\cite{achromat}, was a source of difficulties for the identification of the fragments in the spectrometer. The modified setting and the analysis methods developed for this experiment have been presented in a dedicated paper~\cite{NIM_loa}. We will briefly recall their main features in section~\ref{chap:setup}. We will then discuss in section~\ref{chap:secondary} the influence of multiple reactions taking place in the liquid hydrogen target and modifying the observed fragment production, and the method which has been employed to remove their contribution. In section~\ref{chap:kinematics} we will present the results on the reaction kinematics. Finally, in section~\ref{chap:results} we will present the production cross sections.
\section{Experimental setup and analysis process} \label{chap:setup}
The GSI synchrotron (SIS) was used to produce a 500A~MeV $^{208}$Pb pulsed beam with a pulse duration of 4 seconds and a repetition time of 8 seconds. The beam was sent onto a 87.3~mg/cm$^2$~liquid hydrogen target~\cite{target} located at the entrance of the FRagment Separator (FRS). The target window consisted of two 9~mg/cm$^2$~Ti foils on each side. The beam intensity was monitored during all the experiment by a beam-intensity monitor~\cite{SEETRAM}. In order to maximise the proportion of fully stripped fragments in the spectrometer, a 60~mg/cm$^2$~Nb foil was placed after the target.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.9\textwidth]{frs_2.eps}
\caption{Schematic view of the FRagment Separator. Each magnetic section between focal planes (the target location, $S_2$ and $S_4$) consists of two dipoles plus several quadrupoles and sextupoles (the latter are not represented here as they were not used during this experiment).}
\label{fig:FRS}
\end{center}
\end{figure}
Fragments were identified in-flight using the FRS spectrometer (see figure~\ref{fig:FRS}). The rigidity of the fragments in each of the two magnetic sections is given by:
\begin{equation}
B \rho = \frac{m_0 c}{e} \frac{A}{q} ( \beta \gamma )
\label{eqn:brho}
\end{equation}
where $B$ is the magnetic field, $\rho$ the curvature radius of the fragment trajectory, $A$ the mass number, $q$ the ionic charge of the fragment, and $\beta$ and $\gamma$ are the Lorentz relativistic coefficients. The presence of $q$ in equation~\ref{eqn:brho} is a critical point, as a large part of the fragments produced at the energy of 500A~MeV chosen for this experiment were not fully stripped.
In order to measure the nuclear charge ($Z$) of the fragment, 4 MUlti-Sampling Ionisation Chambers (MUSIC; see~\cite{MUSIC} for complete description) were placed at the exit of the spectrometer, each one filled with 2 bar of P10 gas mixture (90\% Ar, 10\% CH$_4$). The total gas thickness was 800~mg/cm$^2$, with the measurement of the energy loss effectively performed only in some 2/3 of the length of each chamber. Using such a large gas thickness was necessary in order to maximise the charge exchanges (electron pick-up and stripping) of the fragment in the gas, thus washing out the influence of the incoming ionic charge state of the fragment on the energy loss, and ensuring that the latter represents the nuclear charge of the fragment with sufficient resolution~\cite{NIM_loa}. The obtained resolution, $\Delta$Z{\it (FWHM)}/Z, ranged between 0.6\% for the lightest fragments and 0.9\% for the heaviest.
The horizontal position of the fragments at the intermediate and final focal planes (respectively $S_2$~and $S_4$; see figure~\ref{fig:FRS}) was measured using 3~mm thick plastic scintillators. The signals of these detectors were also used to measure the time of flight of the fragments in the second part of the spectrometer. The $A/q$~ratio of the fragment was deduced from the combination of these measurements, according to equation~\ref{eqn:brho}.
The thick aluminium degrader (1700~mg/cm$^2$) located at the intermediate focal plane ($S_2$) was used as a passive energy-loss measurement device. As the energy loss can be related to the variation of the magnetic rigidity and the ionic charge states in the two magnetic sections, the latter may thus be deduced from the energy loss as obtained from the nuclear-charge identification and the velocity measurement~\cite{NIM_loa}.
The resolution obtained in this measurement was not sufficient to discriminate the ionic charge states on an event-by-event basis. The only information obtained was the charge-state changing between the first and the second part of the spectrometer, an integer value that we will note $\Delta q$. Besides, the evaluation of the variation of magnetic rigidity was also used to reject fragments that underwent a nuclear reaction at $S_2$.
The mass of each fragment was determined assuming that its number of electrons in the FRS was the minimum required by its $\Delta q$ value~\cite{NIM_loa}. The production rate for each nuclide was then calculated by constructing its full velocity distribution in the first part of the FRS, using formula~\ref{eqn:brho}. For many nuclides, the momentum width was larger than the momentum acceptance of the FRS ($\pm1.5\%$); in this case several settings of the magnets were used in order to cover the full momentum distribution of the fragment. Due to the hypothesis made on the ionic charge state, some fragments were misidentified; the corresponding correction factor for the production rates was deduced from ionic charge-state probabilities calculated using the code GLOBAL~\cite{GLOBAL}.
The above procedure was performed separately for each group of fragments characterised by a given $\Delta q$ value. The probability of each $\Delta q$ value was then deduced from the scaling factor necessary for all isotopic distributions obtained for a given element to match with each other. The obtained values were found to be in good agreement with the GLOBAL calculations (discrepancies lower than 10\% for the most abundant ionic charge-state combinations, and less than 20\% for other combinations)~\cite{NIM_loa}.
The production rates were corrected for the losses in the different layers of matter located in the path of the fragments after the target area (degrader, plastic scintillators, MUSIC chambers). The total reaction cross sections were calculated using the Karol optical-model-based code~\cite{Karol}. Losses were found to be of the order of 30\%, including reactions in the MUSIC chambers (the latter being characterised by signals of first and last chambers being improperly correlated). The reaction rates in the different layers of matter are presented in table~\ref{table:reac_prob}. The dead time of the acquisition and the detector efficiencies were also taken into account. Fragment losses due to limited angular acceptance of the FRS were found to be negligible. All the measurements and the analysis procedure above were repeated with an empty target, and the resulting production rates were subtracted from the total production rates. Finally, the production cross sections were obtained by normalising the production rates to the number of atoms in the liquid hydrogen target, which had been measured in a previous experiment, and to the beam intensity.
\renewcommand{\arraystretch}{0.4}
\begin{table}[ht]
\begin{center}
{\footnotesize
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
& & & & & \\
Layer & Target windows & Stripper & Scintillator & Degrader & MUSICs \\
& & & & & \\
\hline
& \multicolumn{2}{c|}{} & \multicolumn{2}{c|}{} & \\
Focal plane & \multicolumn{2}{c|}{$S_0$}&\multicolumn{2}{c|}{$S_2$}& $S_4$ \\
& \multicolumn{2}{c|}{} & \multicolumn{2}{c|}{} & \\
\hline
& & & & & \\
Reaction & 2.1\% & 2.1\% & 8.4\% & 16.6\% & 4.9\% \\
probability & & & & & \\
& & & & & \\
\hline
& \multicolumn{2}{c|}{} & \multicolumn{2}{c|}{} & \\
Method of & \multicolumn{2}{c|}{{\it none}} & \multicolumn{2}{c|}{$B\rho$ change} & $\Delta E$ change\\
rejection & \multicolumn{2}{c|}{} & \multicolumn{2}{c|}{} & \\
& \multicolumn{2}{c|}{} & \multicolumn{2}{c|}{} & \\
\hline
& \multicolumn{2}{c|}{} & \multicolumn{3}{c|}{} \\
Method of & \multicolumn{2}{c|}{Dedicated measurement} & \multicolumn{3}{c|}{Calculation} \\
correction & \multicolumn{2}{c|}{(empty target)} & \multicolumn{3}{c|}{(Karol model)} \\
& \multicolumn{2}{c|}{} & \multicolumn{3}{c|}{} \\
\hline
\end{tabular}
}
\caption{Reaction probabilities of the beam ($^{208}Pb$ at $500A~MeV$ at the entrance of the FRS) in the various layers of matter of the FRS beam line. Rejection method of the formed nuclei and correction method of the production rates are also mentioned. See text for details.}
\label{table:reac_prob}
\end{center}
\end{table}
\renewcommand{\arraystretch}{1}
\section{Secondary reactions in the target} \label{chap:secondary}
Any fragment formed in a collision with a proton of the target may undergo additional nuclear collisions, in the target as well as in surrounding material (target window, stripper foil). These secondary reactions are expected to play an important role in the production of nuclides far from the projectile: at relativistic energies, proton-induced reactions mainly produce nuclei lighter than the heavy partner of the reaction, therefore in most cases multiple reactions will remove more nucleons than a single reaction.
There is no way to identify such multiple reactions during the analysis process. Therefore, one has to unfold their contribution by using calculated reaction cross sections or by performing a self-consistent calculation. This section is dedicated to the presentation of a new method developed for this experiment aiming at estimating the contribution of the multiple reactions with a high precision while minimising the input from codes.
\subsection{Unfolding method}
The production cross section of a nuclide $f$ from projectile (indexed as $0$ in the following) is written as:
\begin{equation}
\sigma_{0\rightarrow f} = \frac{e^{\frac{\sigma_0+\sigma_f}{2}x}}{x} \;
\left(T_f(x) - \frac{x^2}{2} \sum_{A_0 < A_i < A_f}^{} \;
\sigma_{0\rightarrow i} \; \sigma_{i\rightarrow f} \;
e^{-\frac{\sigma_0+\sigma_i+\sigma_f}{3}x} \right)
\label{eqn:secondary}
\end{equation}
where $\sigma_i$ is the total reaction cross sections of a nuclide $i$ on a nuclide of the target, $\sigma(i,j)$ is the production cross section of a nuclide $(A_j,Z_j)$ from a nuclide $(A_i,Z_i)$, $T$ is the observed production rate, and $x$ is the thickness of the target. A derivation of this equation is presented in appendix~\ref{chap:sec_calc}. This corresponds to a first-order approximation (i.e. only double reactions in the hydrogen are taken into account), but it is easily extended to higher orders. In our calculations, we actually accounted for the second order reactions: triple reactions in hydrogen, and reactions involving one reaction in a target window and one in the hydrogen (or the reverse). We found that those second order terms actually accounted for less than 20\% of the multiple reactions.
Solving this system of equations (one equation for each observed nuclide) requires the calculation of all the $\sigma$ terms. For the total cross sections, several reliable codes exist; we used the optical-model-based code of Karol~\cite{Karol}, the same one we used for the probability of nuclear reactions at the intermediate focal plane of the FRS. For the partial cross sections involving heavy target nuclei (target windows and stripper foil), the EPAX parametrisation~\cite{EPAX} offers reliable results with minimal calculation time. In the case of proton-induced reactions, the Monte-Carlo cascade-evaporation codes would seem an obvious choice, but they could hardly be used here for two reasons. First, the calculation required to evaluate the hundreds of possible reactions would have been very time consuming. Second, as one of the goals of this experiment was to produce data to constrain these codes in this poorly-known energy region, the use of those codes might have introduced an artificial consistency between the data and the codes.
In order to calculate the isotopic cross sections, we can decompose each cross section of proton-induced reactions in a product of 3 factors:
\begin{equation}
\sigma(x,y) = \sigma_x \; P_A((A_x,Z_x) \rightarrow A_y )
\; P_Z((A_x,Z_x,A_y)\rightarrow Z_y)
\end{equation}
Here, the first term is the total reaction cross section of a nuclide $x$ (we have already stated that it could be calculated using the Karol formula~\cite{Karol}), the second term is the probability to form a nuclide of mass $A_y$ from a nuclide of mass $A_x$, and the third term is the probability that the nuclide formed with a mass $A_y$ has an atomic number $Z_y$.
In order to estimate the second term, we used a property of proton-induced spallation reactions: nearly all nuclides $b$ formed from a nuclide $a$ have a mass strictly smaller than the one from $a$. For example, in the 500A~MeV experiment on $^{208}$Pb we observed no formation of any nuclide of mass 209, and nuclides of mass 208 ($^{208}$Bi) are formed in less than 0.1\% of the reactions. Furthermore, we assumed that, as far as only the probability of mass loss is concerned, the influence of the isospin of the target nuclide is weak enough to be neglected. Using these assumptions, one can solve the system of equations~\ref{eqn:secondary} isobar by isobar, in the decreasing masses order, because the term $P_A(A_x \rightarrow A_y )$ required by each equation is immediately obtained from the previously corrected data as $P_A(A_0-(A_x-A_y))$ (where $A_0$ is the projectile mass).
For the third term, this kind of simple scaling law cannot be applied because, although $P_Z$ depends only on $A_y$ for large values of $A_x-A_y$. Indeed, this well-known property defines the so-called residue corridor~\cite{Dufour}: the statistical nature of the evaporation process favors its ending close to nuclei for which the probability to evaporate a neutron and a proton (in other words, the neutron and proton separation energies), are the closest, a property which is completely independent of the entrance channel. But, in the case of short evaporation chains, the influence of the entrance channel is not suppressed; in other words, for low values of $A_x-A_y$, $P_Z$ depends not only on $A_y$ but also on $Z_x$ . This memory effect is fully taken into account in the EPAX parametrisation~\cite{EPAX}. We will describe hereafter how the EPAX formula can be used even though it is out of its energy domain applicability. Please note that, as EPAX does not take into account the fission process, this method would not be appropriate for reactions involving highly fissile nuclei such as uranium.
\subsection{Calculation of isobaric distributions using EPAX}
Some characteristics of the EPAX parametrisation~\cite{EPAX} correspond to the requirements for a multiple reactions calculation: it needs very little computing time, and it proved to be reliable, not only for reactions involving nuclides close to the stability valley, but also for proton-rich nuclides~\cite{112Sn}.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.9\textwidth]{epax_a.eps}
\caption{Comparison of production rates obtained in the Pb+p at 500A~MeV experiment (dots) with calculations performed with both the standard (discontinuous lines) and a version we modified (continuous lines) of the EPAX parametrisation~\cite{EPAX}. For each isobaric spectrum, calculations have been renormalized to the data.}
\label{fig:EPAX}
\end{center}
\end{figure}
EPAX has been written in order to reproduce residues from reactions in the limiting-fragmentation regime~\cite{limit_frag}, which is reached in spallation only for projectile energies of several GeV. The mass distribution of residues formed in 1A~GeV proton-induced spallation reactions exhibits a very different shape from the one formed in the limiting-fragmentation regime~\cite{Pb_p}. Therefore, the residues from the same reaction with half the incident energy may certainly not be reproduced by the mass-loss formula of EPAX.
On the other hand, the shape of the isobaric spectra is mainly a consequence of the sequential evaporation mechanism. Therefore, there is no reason why its validity should be limited to high incident energies. To check this assumption we extracted the isobaric component of the EPAX formula and compared it to our data after proper renormalization for each isobaric spectrum. Only minor adjustments were necessary to obtain a very satisfactory reproduction of the measured production rates, as it can be seen in figure~\ref{fig:EPAX}. Such a comparison makes sense as the secondary reactions are not expected to play an important role in the production of nuclides with a mass loss of less than, roughly, 30 mass units with respect to the projectile. Unexpectedly, we observed that the charge-pickup reactions were also reasonably well reproduced by the parametrisation, despite the fact that the EPAX authors didn't take this phenomena into account during the development process.
\subsection{Contribution of the multiple reactions in the target}
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.45\textwidth]{secondary72.eps}
\includegraphics[width=0.45\textwidth]{secondary78.eps}
\caption{Production cross sections in hydrogen before and after the subtraction of the multiple reactions (empty and full dots, respectively), and contribution of the multiple reactions in the target (continuous lines), for Hf (left) and Pt (right) isotopes.}
\label{fig:secondary}
\end{center}
\end{figure}
The results of multiple-reaction calculations are presented in figure~\ref{fig:secondary} for 2 isotopic distributions, each one corresponding to an extreme situation regarding the contribution of the multiple reactions. For $Z$ around 78, the multiple reactions are an important contributor for very proton-rich nuclides only. Their contribution increases and spreads towards neutron-rich nuclides with decreasing $Z$. The very proton-rich part of the isotopic distributions of the light elements such as Hf is reproduced by our calculation with differences being less than 20\% in most cases. This demonstrates the validity of our approach. As the uncertainty on this calculation could not be estimated in a systematic way, we quoted the value of 20\% mentioned above.
We chose to consider as results of the experiment only the cross sections deduced from production rates for which multiple reactions contributed for less than 50\%. This discards nearly all nuclides with $Z<70$, which represent only a very small fraction of the fragmentation residues.
\section{Kinematics of the reaction} \label{chap:kinematics}
Once nuclei are identified, their velocity in the first part of the FRS can be calculated using the equation~\ref{eqn:brho}:
\begin{equation}
( \beta \gamma )_1 = \frac{(B\rho)_1}{A/q_1}
\end{equation}
Here the index 1 stands for the first part of the FRS (before $S_2$). Using this technique, the resolution is expected to be of the same order as the one obtained for the magnetic rigidity, roughly $5.10^{-4}$. At the energy used in this experiment, this is by a factor of 3 better than what can be achieved by a time-of-flight measurement in the second part of the FRS. This high resolution makes the FRS a remarkable tool to study the kinematics of nuclear reactions.
We have already pointed out that, because of the limited momentum acceptance of the FRS, the reconstruction of the full velocity spectra of each nuclide is a necessary step in order to evaluate the production rates (section~\ref{chap:setup}). The measured velocity spectra are Lorentz transformed into the reference frame of the beam, and corrected for the contribution of the beam width and for the velocity scattering due to the passage through the target and the surrounding materials. The resulting spectra give direct access to the longitudinal momentum transfer and to the longitudinal momentum spread caused by the nuclear reactions.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.48\textwidth]{p_parrallel_mean.eps}
\includegraphics[width=0.48\textwidth]{sigma_mean.eps}
\caption{Momentum transfer (left) and momentum width (right) measured in the reactions Pb+p at 500A~MeV (triangles), Pb+p at 1A~GeV (squares) and Au+p at 800A~MeV (circles). Data are compared to Morrissey systematics~\cite{Morrissey} (continuous lines) and Goldhaber formula~\cite{Goldhaber}, the later being computed with a Fermi momentum of 118~MeV.c$^{-1}$ (dashed line) and 95~MeV.c$^{-1}$ (dashed-dotted line). All data have been normalised following the Morrissey prescription.}
\label{fig:kin}
\end{center}
\end{figure}
In figure~\ref{fig:kin} these quantities are compared to results of previous spallation experiments as well as to the well-known Morrissey systematics~\cite{Morrissey} and to the Goldhaber formula~\cite{Goldhaber}. The data were averaged over each isobaric distribution using the production cross sections as weighting factor.
A very similar tendency is obtained for all the experimental data regarding the momentum transfer. The simple linear dependence proposed by the Morrissey systematics is not fulfilled by the experiment. The longitudinal momentum transfer is underestimated for fragments corresponding to a mass loss of 10 to 45 units with respect to the projectile. This underestimation vanishes with increasing mass losses.
The momentum width exhibit a linear dependance to the square root of the mass loss. The Morrissey systematics offers a fair reproduction of the data. Using the result of a direct measurement of the Fermi momentum (118~MeV/c)~\cite{fermi_mes}, the Goldhaber formula overestimates the momentum width. This is not unexpected as this formula only takes into account the nucleons removed during the cascade stage, which lead to larger momentum fluctuations with respect to the nucleons emitted in the evaporation phase~\cite{Hanelt}. Nevertheless, a better agreement with data can be obtained by using an arbitrary Fermi momentum value of 95~MeV/c, as often done in heavy-ion calculations. The dispersion between the different data sets is probably related to the delicate corrections applied to the data, namely the beam width in momentum and position, which are difficult to estimate.
\section{Production cross sections} \label{chap:results}
\subsection{Isotopic cross sections}
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.9\textwidth]{isotopic.eps}
\caption{Isotopic production cross-section distributions of residues in the reaction Pb+p at 500A~MeV.}
\label{fig:cs}
\end{center}
\end{figure}
Figure~\ref{fig:cs} shows the measured distributions of isotopic production cross-sections for elements between erbium and bismuth (see appendix~\ref{chap:annexe_xs} for the full list of cross sections). Some 250 spallation-evaporation cross sections have been measured. One observes that the cross sections of isotopic chains vary smoothly. As no even-odd fluctuations are expected~\cite{Valentina}, and considering the statistical nature of the evaporation process, this corroborates that the measured production rates do not hide any other source of fluctuations. The upper limit of 50\% production of multiple reactions in the target, that we have decided to set, removed from the distributions a growing part of the lightest isotopes when $Z$ is decreasing. The most neutron-rich Pt and Ir isotopes have not been measured because of missing settings of the magnetic fields during the experiment.
\subsection{Total cross sections}
\begin{table}[ht]
\begin{center}
\begin{tabular}{l c c c}
\hline
Reaction & $^{208}$Pb+p & $^{208}$Pb+p & $^{208}$Pb+d \\
& (500A MeV) & (1A GeV) & (1A GeV) \\
\hline
\hline
Spallation-evaporation (measured) & 1.44 (0.21) & 1.68 (0.22) & 1.91 (0.24)\\
Total (measured) & 1.67 (0.23) & 1.84 (0.23) & 2.08 (0.24) \\
Total (calculated) & 1.70 & 1.80 & 2.32 \\
\hline
\end{tabular}
\caption{Spallation cross sections (in barns) for reactions $^{208}$Pb+p at 500A~MeV and 1A~GeV, and $^{208}$Pb+d at 1A~GeV. The measured spallation-evaporation and total cross sections (adding fission) are compared to a Glauber calculation performed using updated density distributions. Values in parenthesis are the total uncertainty of the measurements.}
\label{tab:total_cs}
\end{center}
\end{table}
We have estimated the total production cross-section of evaporation residues by summing all the measured residue cross sections, obtaining a value of (1.44$\pm$0.21)~b. Our measurement does not strictly cover all the range of the possible residues. However, the nuclides for which we have no measurement are mostly the lightest fragmentation products. Considering the steepness of the mass curve in this region (see figure~\ref{fig:cs_mass}), we can assume that the contribution of these nuclides is small, and probably much smaller than the error bars.
Adding the fission cross-section for this reaction~\cite{NPA_Bea} we estimated the total cross section to (1.67 $\pm$ 0.23) b.
This value is very close to the 1.70~b found by a Glauber-type calculation performed with updated density distributions. On table~\ref{tab:total_cs} we compare those values with the ones obtained in the reactions Pb+p and Pb+d at 1A~GeV. Although the slight decrease of the total cross section with respect to 1A~GeV measurements is in agreement with the expected trend, the 500A~MeV fission cross section is higher than previous measurements conducted at this energy, and also higher than the systematics of Prokofiev~\cite{Prokofiev}. This question has been discussed in detail in the corresponding paper~\cite{NPA_Bea}. Let us only underline that the agreement of a well-established model with our experimental total cross section comes in support of our measurement.
\subsection{Comparison to radiochemical measurements}
In recent years, a large number of measurements of spallation residues have been performed by the team of R. Michel. Of special relevance to our work is the measurement of residues of spallation of natural lead by protons at 550~MeV published by Gloris \etal~\cite{Gloris}. Cross sections with independent yields ({\it i.e.} nuclides that are not produced by $\beta$ decay) can be compared directly, while cross sections corresponding to accumulation of $\beta$-decaying nuclei require a summation of our data along the decay chain.
\begin{figure}[ht]
\begin{center}
\includegraphics*[width=0.9\textwidth]{gloris.eps}
\caption{Ratio between production cross-sections of residues measured in the reaction $^{208}$Pb+p at 500A~MeV (this work) and in the reaction p+$^{nat}$Pb at 550~MeV as a function of the mass of the residue for the isotopes measured in~\cite{Gloris}. Filled and empty circles represent nuclides with independent yields and cumulated yields, respectively. Calculations of the ratio between production cross sections at 550 and 500 MeV have been performed in two systems: with the same $^{208}$Pb target nuclei (continuous line) and with a different nucleus, $^{207}$Pb (dashed line), in order to study the effect of the use of natural lead in the Gloris experiment (see text).}
\label{fig:chemistry}
\end{center}
\end{figure}
The ratio between our data and those of Gloris \etal\ are presented in figure~\ref{fig:chemistry}. In the case of the cumulated yields, cross sections measured at GSI have been summed along the decay chain in order to be comparable with the radiochemical measurements. The agreement between the two data sets is overall fair for heavy residues, although a systematic shift of roughly 10\% may be guessed. Considering the error bars, all the measurements seem to be compatible, with the exception of $^{203}$Pb and $^{202}$Tl. As we have already underlined, our data points are very consistent with respect to one another. This makes such a large error in our measurement for these two nuclides rather unlikely, since it should have been clearly visible on our isotopic distributions.
For lighter nuclei the ratio decreases rapidly with increasing mass loss with respect to the projectile. This effect is the direct consequence of the differences between the measured systems: the 10\% higher energy strongly favors the production of lighter residues, up to a factor of 3 for mass losses around 35 nucleons.
This statement can be checked by using Monte-Carlo calculations. For this purpose we used the ISABEL~\cite{ISABEL} intranuclear cascade and the ABLA~\cite{ABLA} evaporation code. Although one of the purposes of these measurements is precisely to check the validity of those codes in the few hundreds of MeV region, we can assume that they are reliable enough if one only wants to calculate variations in a very limited energy and mass range, as it is the case here.
Results of the calculation of the ratio between isobaric cross sections for the two systems are also represented in figure~\ref{fig:chemistry}. A calculation that only takes into account the different incident energies offers a satisfactory reproduction of the decrease of the ratio for the light fragments. Replacing $^{208}$Pb by $^{207}$Pb (in order to mock the isotopic mixing of natural lead of which the targets of Gloris experiment were made) leads to a slight reduction of the calculated ratio for light nuclides, which improves the agreement with the data in this mass range. For heavy nuclides the calculations indicate that results at 500A~MeV should be larger than at 550A~MeV, which is not what we observed for most of the points. However, only 3 points are not compatible with the calculations when one considers the error bars. This leads us to conclude that, taking into account the differences between the systems measured in the Gloris experiment and our experiment, the agreement between these data sets is satisfactory.
\subsection{Mass spectra and comparison to previous GSI experiments}
\begin{figure}[ht]
\begin{center}
\includegraphics*[width=0.95\textwidth]{mass_loss.eps}
\caption{Production cross-sections of the residues of the reaction Pb+p at 500A~MeV as a function of the mass loss with respect to the projectile (full circles). Data obtained in previously mentioned experiments are also represented: Au+p at 800A~MeV (triangles), Pb+p (squares) and Pb+d (diamonds) at 1A~GeV. The isolated points at $\Delta A=0$ correspond to a single nuclide, $^{208}$Bi.}
\label{fig:cs_mass}
\end{center}
\end{figure}
Figure~\ref{fig:cs_mass} presents the production cross sections, summed for all isobars, as a function of the mass loss with respect to the projectile. The data obtained from several experiments performed at the FRS are presented here: Pb+p at 500A~MeV, Pb+p at 1A~GeV~\cite{Pb_p}, Pb+d at 1A~GeV~\cite{Pb_d}, and Au+p at 800A~MeV~\cite{Au_Fanny}.
For small mass losses, each spectrum has a nearly constant value. In this mass range, the lower the incident energy, the higher the cross sections. With increasing mass losses, the cross sections start to decrease. Here, the lower the energy, the earlier and the steeper the fall. This is easily understood as the direct consequence of the exploration by the prefragment of all the possible range of excitation energy available in each system. In this respect, the measurement with deuterons gives insights about what would be obtained in a measurement conducted with protons at twice the energy. The shape of the mass-loss curve obtained from the measurement on gold is fully compatible with the tendencies observed for lead.
In the 500A~MeV experiment, a clear separation exists between the group of the evaporation products (which does not extend beyond mass losses of 40 mass units) and the group of the fission product (which starts around mass losses of 70~\cite{NPA_Bea}). This absence of mixing could also be demonstrated by studying the velocity spectra of the light fragments, which all exhibit a quasi-perfect Gaussian shape, while the presence of fission products would have introduced a characteristic double-bumped shape due to the forward-backward selection of the fission fragments by the FRS~\cite{Pb_p}.
\subsection{Charge pick-up}
\begin{figure}[ht]
\begin{center}
\includegraphics*[width=0.45\textwidth]{charge_pickup.eps}
\includegraphics*[width=0.45\textwidth]{iso_pickup.eps}
\caption{Left figure: charge-pickup cross sections measured on lead (full symbols) and gold (empty symbols) as a function of the incident energy. The sum of the partial cross sections measured at the FRS (this work, full dots; Keli\'c \etal~\cite{Kelic}, full squares; Rejmund \etal~\cite{Au_Fanny}, upward triangles) is compared to elemental cross sections from Waddington \etal~\cite{Waddington} (downward triangles) and Binns \etal~\cite{Binns} (diamonds), which were both extracted from CH$_2$ and C measurements. Right figure: isotopic charge-pickup cross sections at 3 energies as a function of the mass loss with respect to the heavy partner of the reaction.}
\label{fig:pickup}
\end{center}
\end{figure}
An especially interesting result of this experiment is the measurement of the production cross section of 15 isotopes of Bi (see figure~\ref{fig:pickup}). Those nuclides are formed by charge-pickup reactions. In the energy range considered here, the capture of the incident proton is not initially possible, as the incident proton energy is well above the Fermi energy of the target nuclide. Therefore the formation of $^{209}$Bi is improbable, and the formation of $^{208}$Bi is only possible via, either the formation of a resonant state ($\Delta$ and pions), or a quasi-elastic collision between the incident proton and a neutron from the target nuclide, in which the neutrons leaves with an energy very close to the initial energy of the proton. The cross section for the charge-pickup is one of the few data that bring direct constraints for the intranuclear-cascade codes.
In the left part of figure~\ref{fig:pickup} we compare our measurement of the total charge-pickup cross section to previous measurements performed by our collaboration~\cite{Au_Fanny,Kelic}, Waddington \etal~\cite{Waddington} and Binns \etal~\cite{Binns}. For a qualitative discussion we do not need to discriminate between gold and lead as those nuclides are close to one another, both in atomic number and mass. Our measurement confirms the trend of a strong increase of the total cross section of the charge-pickup with decreasing energy.
This increase of the Bi production in Pb+p experiments concerns all Bi isotopes, as it can be seen in the right part of figure~\ref{fig:pickup}. The shapes of the 500A~MeV and 1A~GeV distributions are overall similar, but the overproduction at 500A~MeV increases slowly from a factor of 2 for the heaviest isotopes to a factor of 4 for the lightest. Problems in the separation of the ionic charge states~\cite{NIM_loa} prevented us to use the kinematic spectra to distinguish between the respective contribution of the $\Delta$ resonance and the quasi-elastic reactions in the formation of the heaviest Bi isotopes, as it was done by Keli\'c \etal~\cite{Kelic}.
The shape of the Hg spectrum (obtained in the Au+p at 800A~MeV measurement) is slightly different from the lead spectra. On one hand, for mass losses up to 7 mass units, the shape of the isotopic distribution is nearly identical to the Pb spectra, with values in-between the two Pb experiments, which is consistent with a smooth evolution as a function of the projectile energy. On the other hand the production of the lightest isotopes decreases faster than in the Pb+p experiments. This difference in shape can be explained by the lower Coulomb barrier and the shorter distance from the residue corridor~\cite{Dufour} for Hg nuclides with respect to Bi nuclides, which favor the emission of protons by the excited prefragments~\cite{Summerer_pickup}.
\subsection{Isobaric cross sections}
\begin{figure}[t]
\begin{center}
\includegraphics*[width=0.95\textwidth]{isobaric3.eps}
\caption{Isobaric spectra of production cross-sections (in mb) in the reactions Pb+p and Pb+Ti at 500A~MeV (full circles and crosses, respectively). The data obtained in the experiments Au+p at 800A~MeV \cite{Au_Fanny}, Pb+p and Pb+d at 1A~GeV \cite{Pb_p,Pb_d} are also plotted (triangles, squares and diamonds, respectively). }
\label{fig:cs_isobaric}
\end{center}
\end{figure}
In figure~\ref{fig:cs_isobaric} the data from the same experiments as in previous sections are plotted as isobaric spectra.
For heavy fragments, all distributions issued from reactions of Pb with p or d are very similar, both in shape and in magnitude. Low-energy reactions slightly dominate the cross sections for masses down to 185. With decreasing masses, the isobaric spectra behave in accordance with the mass distributions: the spectra for the highest-energy reaction scale down very slowly, whereas this scaling is steeper and steeper when one considers reactions at decreasing incident energy. However, for each isobar, the shapes of the different spectra remains extremely similar, as does its centroid (this can be seen in the left part of figure~\ref{fig:cs_means}).
Data from the reactions of Pb on the dummy target (which consists mainly of Ti in the target itself and Nb for the stripper foil placed after the target) at 500A~MeV have been added to the figure~\ref{fig:cs_isobaric} in order to illustrate the so-called limiting-fragmentation regime~\cite{limit_frag}. We observe no difference of shape or centroid between the spectra issued from the reaction on heavy ions and from the reaction on hydrogen isotopes.
The gold data offer an interesting point of comparison with the lead data. The gold fragments with mass close to 197 are associated with rather cold reactions and have therefore kept a $A/Z$ ratio very close to the initial system, while Pb fragments close to the same mass have lost roughly 10 nucleons, mostly neutrons because of the hindrance of charged-particle emission due to the Coulomb barrier. Therefore the gold and lead residue spectra are strongly shifted with respect to one another. This shift slowly vanishes with the increasing mass loss, which is easily understood as the slow move of the gold fragment distributions towards the residue corridor~\cite{Dufour}. This corridor is clearly visible on the left part of figure~\ref{fig:cs_means}: the barycenter of the isobaric distributions of the residues of all reactions converge on the same line.
\begin{figure}[t]
\begin{center}
\includegraphics*[width=0.45\textwidth]{mean_z.eps}
\includegraphics*[width=0.45\textwidth]{mean_a.eps}
\caption{Average atomic number as a function of the mass of the residue (left figure) and average mass of the residue as a function of the atomic number (right figure) in the reactions Pb+p at 500A~MeV (full circles, this work), Au+p at 800A~MeV (triangles, \cite{Au_Fanny}) and Pb+p at 1A~GeV (squares, \cite{Pb_p}). In the calculation of the average values, points have been weighted according to their cross section.}
\label{fig:cs_means}
\end{center}
\end{figure}
This universal behavior, well known for reactions between heavy ions, is here demonstrated to be valid in a very broad energy range, even for fragments which are at the very end of the mass distribution. In other words, the isobaric distributions are independent of the incident energy in the system studied if properly renormalized. This is an experimental proof that the factorisation hypothesis is valid at energies as low as a few hundreds of MeV. This further strengthens the discussion regarding the agreement between EPAX and the data obtained from proton-induced reactions in systems in which fission does not play a major role (section~\ref{chap:secondary}).
Conversely, various projectile energies lead to variations of the center of the residues isotopic distributions, as it can be seen on the right part of the figure~\ref{fig:cs_means} as a deviation of the average mass value of the residues produced at 500A MeV. If this effect is washed out by the slow variations of the mass curve for higher energy reactions, it becomes noticeable at lower energies when the fall of the mass distribution becomes so steep that the production of the most neutron-deficient isotopes is strongly inhibited. Therefore, the reproduction of the isotopic and elemental cross-sections using a scaling factor between different systems is not appropriate at energies below the fragmentation limit.
\section{Conclusion}
The production cross sections and the momentum distributions have been measured for about 250 nuclei formed in the reaction of $^{208}$Pb on protons at 500A~MeV, covering most of the nuclides created down to a mass loss of 40 units with respect to the projectile, and with cross sections as low as 5~$\mu$b. The reaction products were identified in-flight in atomic number and mass-over-ionic-charge using the FRS spectrometer. The large proportion of non-fully stripped ions in the spectrometer was accounted for in detail, thus allowing to calculate the production cross section for each nuclide. The contribution of multiple reactions in the target to the residue production was carefully subtracted.
The production cross sections are in good agreement with previous radiochemical measurements. The isobaric distributions of the production cross sections are found to be very close to the ones measured at higher energies, thus extending the validity range of the factorisation hypothesis to energies of a few hundreds of MeV. The large variations observed on the isotopic cross sections can be nearly fully ascribed to the variations of the residue distributions with mass-loss at decreasing energy. Kinematical data are consistent with previous measurements.
The data obtained in this experiment, combined with previous measurements performed with the same technique (especially in the same system at 1A~GeV), constitute a set of information that is highly relevant for the development of reliable nuclear-reaction codes and, thus, the design of ADS.
| {
"redpajama_set_name": "RedPajamaArXiv"
} |
\section{Introduction}
The uniqueness problem of meromorphic mappings under a condition
on the inverse images of divisors was first studied by Nevanlinna \cite{Ne}. He
showed that for two nonconstant meromorphic functions $f$ and $g$ on the
complex plane $\mathbb{C}$, if they have the same inverse images for five
distinct values, then $f\equiv g.$ In 1975, Fujimoto \cite{F1} generalized
Nevanlinna's result to the case of meromorphic mappings of $\mathbb{C}^{m}$
into $\mathbb{C}P^{n}$. He showed that for two linearly nondegenerate meromorphic
mappings $f$ and $g$ of $\mathbb{C}^{m}$ into $\mathbb{C}P^{n}$, if they
have the same inverse images counted with multiplicities for $(3n+2)$
hyperplanes in general position in $\mathbb{C}P^{n},$ then $f\equiv g$.
In 1983, Smiley \cite{Sm} showed that
\begin{theorem}\label{thSmiley}
Let $f, g$ be linearly nondegenerate meromorphic mappings of $\mathbb{C}^m$ into $\mathbb{C} P^n.$ Let $\{H_j\}_{j=1}^q$ ($q\geq 3n+2)$ be hyperplanes in $\mathbb{C} P^n$ in general position. Assume that
$\text{a)}\quad$ $f^{-1}(H_j)= g^{-1}(H_j)\ ,\quad \text{for all}\ \ 1\leq j\leq q$ (as sets),
$\text{b)}\quad$ $\dim \big(f^{-1}(H_i)\cap f^{-1}(H_j)\big)\le m-2\ \ \text{for all}\ 1\le i<j\le q\,$,
$\text{c)}\quad$ $f=g$ on \ \
$\bigcup_{j=1}^{q}f^{-1}(H_j)\,$.
\noindent Then $f\equiv g.$
\end{theorem}
\hbox to 5cm {\hrulefill }
\noindent {\small Mathematics Subject
Classification 2000: Primary 32H30; Secondary 32H04, 30D35.}
\noindent {\small Key words : Meromorphic mappings, uniqueness theorems.}
\noindent {\small Acknowledgement: This work was done during a stay of the third named author at the Institut des Hautes \'Etudes Scientifiques, France. He
wishes to express his gratitude to this institute.}
\newpage
\noindent In 2006 Thai-Quang \cite{TQ} generalized this result of Smiley to the case where $q\geq 3n+1$ and $n\geq 2.$ In 2009, Dethloff-Tan \cite{DT} showed that for every nonnegative integer $c$ there exists a positive integer $N(c)$ depending only on $c$ such that Theorem \ref{thSmiley} remains valid if $q\geq (3n+2 -c)$ and $n\geq N(c)$. They also showed that the coefficient of $n$ in the formula of $q$ can be replaced by a number which is smaller than 3 for all $n>>0.$ Furthermore, they established a uniqueness theorem for the case of $2n+3$ hyperplanes and multiplicities are truncated by $n.$ At the same time, they strongly generalized many uniqueness theorems of previous authors such as Fujimoto \cite{F2}, Ji \cite{Ji} and Stoll \cite{St}. Recently, by using again the technique of Dethloff-Tan \cite{DT}, Chen-Yan \cite{CY} showed that the assumption ``multiplicities are truncated by $n$" in the result of Dethloff-Tan can be replaced by ``multiplicities are
truncated by $1$". In \cite{Q}, Quang examined the uniqueness problem for the case of $2n+2$ hyperplanes.
We would like to note that so far, all results on the uniqueness problem have still been restricted to the case where meromorphic mappings are sharing a common family of hyperplanes. The purpose of this paper is to introduce a uniqueness theorem for the case where the family of hyperplanes depends on the meromorphic mapping. We also will allow that the meromorphic mappings may be degenerate. For this purpose we introduce some new techniques which can also be used to obtain simpler proofs for many other uniqueness theorems.\\
We shall prove the following uniqueness theorem:
\begin{theorem} \label{TQT}
Let $f, g$ be nonconstant meromorphic mappings of $\mathbb{C}^m$ into $\mathbb{C} P^n.$ Let $\{H_j\}_{j=1}^q$ and $\{L_j\}_{j=1}^q$ $(q>2n+2)$ be families of hyperplanes in $\mathbb{C} P^n$ in general position. Assume that
$\text{a)}\quad$ $f^{-1}(H_j)= g^{-1}(L_j)\ \quad \text{for all}\ \ 1\leq j\leq q\,,$
$\text{b)}\quad$ $\dim \big(f^{-1}(H_i)\cap f^{-1}(H_j)\big)\le m-2\ \ \text{for all}\ 1\le i<j\le q$\,,
$\text{c)}\quad$ $\frac{(f,H_i)}{(g,L_i)}=\frac{(f,H_j)}{(g,L_j)}$ on
$\bigcup_{k=1\atop }^qf^{-1}(H_k)\setminus\big(f^{-1}(H_i)\cup f^{-1}(H_j)\big)$ for all $1\leq i<j\leq q\,.$
Then the following assertions hold :
$i)\quad\dim\langle \text{Im}f\rangle=\dim\langle \text{Im}g\rangle\stackrel{\text{Def.}}{=:}p,$
\noindent where for a subset $X\subset \Bbb C P^n,$ we denote by $\langle X\rangle$ the smallest projective subspace of $\Bbb C P^n$ containing $X.$
$ii) \quad$ If
$$(*)\quad q>\frac{2n+3-p+\sqrt{(2n+3-p)^2+8(p-1)(2n-p+1)}}{2}\;(\geq 2n+2),\,$$
then $$\frac{(f,H_1)}{(g,L_1)}\equiv\cdots\equiv\frac{(f,H_q)}{(g,L_q)}\,.$$ Furthermore, there exists a linear projective transformation $\mathcal L$ of $\Bbb C P^n$ into itself such that $\mathcal L(f)\equiv g$ and $\mathcal L(H_j\cap \langle \text{Im}f\rangle)=L_j\cap \mathcal L(\langle \text{Im}f\rangle)$ for all $j\in\{1,\dots,q\}.$
\end{theorem}
\noindent {\bf Remark.}
1.) In Theorem \ref{TQT} condition $c)$ is well defined since, by condition $a)$,
$\frac{(f,H_i)}{(g,L_i)}$ is a (nonvanishing) holomorphic function outside $f^{-1}(H_i)$.
2.) The condition $(*)$ is satisfied in the following cases:
$+) \quad q\geq 2n+3$ and $p\in\{1,2,n-1,n\}, n\in\Bbb Z^+.$
$+) \quad q\geq2n+p+1$ and $ p\in\{2, 3, \dots,n\}, n\in \Bbb Z^+.$
3.) If there exists a subset $\{j_0,\dots,j_n\}\subset\{1,\dots,q\}$ such that $H_{j_i}\equiv L_{j_i}$ for all $i\in\{0,\dots,n\},$ then the proof of Theorem \ref{TQT} implies that $f\equiv g.$
4.) For the special case where $f, g$ are linearly nondegenerate
(i.e. $p=n$) and $H_j\equiv L_j\,$, from Theorem \ref{TQT} we get again the results of Dethloff-Tan \cite{DT} and Chen-Yan \cite{CY}.
\section{Preliminaries}
We set $\Vert z \Vert := \big(|z_1|^2 + \cdots + |z_m|^2\big)^{1/2}$
for $z = (z_1, \dots, z_m) \in \Bbb C^m$ and define
$$B(r) := \big\{z \in \Bbb C^m : \Vert z \Vert < r\big\},\qquad
S(r) := \big\{z \in \Bbb C^m : \Vert z \Vert = r\big\}
$$
for all $0 < r < \infty$.
Define
\begin{align*}
d^c &:= \frac{\sqrt{-1}}{4\pi}(\overline \partial - \partial),\quad
\upsilon := \big(dd^c\Vert z \Vert^2 \big)^{m-1}\quad \\
\sigma &:= d^c \text{log}\Vert z\Vert^2 \land
\big(dd^c \text{log}\Vert z\Vert^2\big)^{m-1}.
\end{align*}
Let $F$ be a nonzero holomorphic function on $\Bbb C^m$.
For each $a \in \Bbb C^m$, expanding $F$ as $F = \sum P_i(z-a)$
with homogeneous polynomials $P_i$ of degree $i$ around $a$,
we define
$$ \nu_F(a) := \min \big\{ i : P_i \not\equiv 0 \big\}.$$
Let $\varphi$ be a nonzero meromorphic function on $\Bbb C^m$.
We define the zero divisor $\nu_\varphi$ as follows: For each $z \in
\Bbb C^m$, we choose nonzero holomorphic functions $F$ and $G$ on a
neighborhood $U$ of $z$ such that $\varphi = {F}/{G}$ on
$U$ and $\text{dim}\,\big(F^{-1}(0) \cap G^{-1}(0)\big) \leqslant m-2$.
Then we put $\nu_\varphi(z) := \nu_F(z)$.
\noindent Let $\nu$ be a divisor in $\mathbb{C}^m$ and $k$ be positive integer or $+\infty $. Set \ $
|\nu|:=\overline{\big\{z:\ \nu(z)\neq 0
\big\}}$ and $\nu^{[k]}(z) := \min \{ \nu (z), k\}.$
The truncated counting function of $\nu$ is defined by
$$ N^{[k]}(r, \nu) := \int\limits_1^r
\frac{n^{[k]}(t)}{t^{2m-1}} dt \quad (1 < r < + \infty), $$
where
\begin{align*}
n^{[k]}(t) = \begin{cases}
\displaystyle{\int\limits_{|\nu| \cap B(t)}}
\nu^{[k]}\cdot \upsilon \ &\text{for}\ m \geqslant 2,\\
\sum\limits_{|z| \leqslant t} \nu^{[k]}(z)
&\text{for}\ m = 1.\end{cases}
\end{align*}
We simply write \ $N(r, \nu)$ for $N^{[+\infty]}(r,\nu)$.
\noindent For a nonzero meromorphic function $\varphi$ on $\mathbb{C}^{m},$ we set \quad $N_{\varphi}^{[k]}(r):=N^{[k]}(r, \nu_\varphi)$ and $ N_{\varphi}(r):=N^{[+\infty]}(r, \nu_\varphi).$
We have the following Jensen's formula:
$$ N_\varphi (r) - N_{\frac{1}{\varphi}}(r) =
\int\limits_{S(r)} \text{log}|\varphi| \sigma
- \int\limits_{S(1)} \text{log}|\varphi| \sigma .$$
Let $f : \Bbb C^m \longrightarrow \Bbb C P^n$ be a meromorphic mapping.
For an arbitrary fixed homogeneous coordinate system
$(w_0 : \cdots : w_n)$ in
$\Bbb C P^n$, we take a reduced representation $f = (f_0 : \cdots : f_n)$,
which means that each $f_i$ is a holomorphic function on $\Bbb C^m$
and $f(z) = (f_0(z) : \cdots : f_n(z))$ outside the analytic set
$\{ f_0 = \cdots = f_n = 0\}$ of codimension $\geqslant 2$.
Set $\Vert f \Vert = \big(|f_0|^2 + \cdots + |f_n|^2\big)^{1/2}$.
The characteristic function $T_f(r)$ of $f$ is defined by
$$ T_f(r) := \int\limits_{S(r)} \text{log} \Vert f \Vert \sigma -
\int\limits_{S(1)} \text{log} \Vert f \Vert \sigma , \quad
1 < r < + \infty . $$
For a meromorphic function $\varphi$ on $\Bbb C^m$, the characteristic
function $T_\varphi(r)$ of $\varphi$ is defined by considering
$\varphi$ as a meromorphic mapping of $\Bbb C^m$ into $\Bbb C P^1$.
We state the First Main Theorem and the Second Main Theorem in Value Distribution
Theory:
For a hyperplane $H : a_0 w_0 + \cdots + a_n w_n = 0$ in $\Bbb C P^n$
with Im$f \not\subseteq H$, we put $(f,H) = a_0 f_0 + \cdots
+ a_n f_n$, where $(f_0 : \cdots : f_n)$ is a reduced
representation of $f$.\\
\medskip
\noindent
{\bf First Main Theorem.}
{\it Let $f$ be a meromorphic mapping of
$\Bbb C^m$ into $\Bbb C P^n$, and $H$ be a hyperplane in $\Bbb C P^n$ such that $(f,H) \not\equiv 0$. Then}
$$ N_{(f,H)}(r) \leqslant T_f(r) + O(1) \quad \text{\it for all}\ r >
1.$$ Let $n,N,q$ be positive integers with $q\geq 2N-n+1$ and $N\geq n.$ We say that hyperplanes $H_1,\dots,H_q$ in $\Bbb C P^n$ are in $N$-subgeneral position if $\cap_{i=0}^N H_{j_i}=\varnothing$ for every subset $\{j_0,\dots,j_N\}\subset\{1,\dots,q\}.$
\medskip
\noindent
{\bf Cartan-Nochka Second Main Theorem (\cite{No}, Theorem 3.1).} {\it Let $f$ be a linearly nondegenerate
meromorphic mapping of $\Bbb C^m$ into $\Bbb C P^n$ and
$H_1, \dots, H_q$ hyperplanes in $\Bbb C P^n$
in $N$-subgeneral position $(q\geq 2N-n+1).$ Then
$$ (q-2N+n-1) T_f(r) \leqslant \sum_{j=1}^q N_{(f,H_j)}^{[n]}(r) +
o\big(T_f(r)\big) $$
for all $r$ except for a subset $E$ of $(1, +\infty)$ of finite
Lebesgue measure.}
\section{Proof of Theorem 3 }
We first remark that $f^{-1}(H_j)=g^{-1}(L_j)\ne\Bbb C P^n$ for all $j\in\{1,\dots,q\},$ and that therefore $\{H_j\cap \langle\text{Im}f\rangle\}_{j=1}^q$ (respectively $\{L_j\cap \langle\text{Im}g\rangle\}_{j=1}^q)$ are hyperplanes in $\langle\text{Im}f\rangle$ (respectively $\langle\text{Im}g\rangle$) in $n-$subgeneral position: Indeed, otherwise there exists $t\in\{1,\dots,q\}$ such that $f^{-1}(H_t)=\Bbb C P^n.$ Then by the assumption $b)$ we have dim$f^{-1}(H_j)\leq m-2$ for all $j\in\{1,\dots,q\}\setminus\{t\}.$ Therefore, $f^{-1}(H_j)=\varnothing$ for all $j\in\{1,\dots,q\}\setminus\{t\}.$ Then $ \langle\text{Im}f\rangle\not\subset H_j$ for all $j\in\{1,\dots,q\}\setminus\{t\}.$ Thus, $ \{H_j\cap \langle\text{Im}f\rangle\}_{j=1\atop j\ne t}^q$ are hyperplanes in $\langle\text{Im}f\rangle$ in $n$-subgeneral position.
\noindent By the Cartan-Nochka Second Main Theorem, we have
\begin{align*}
(q-2n+\dim \langle\text{Im}f\rangle-2)T_f(r)\leq \sum_{j=1\atop j\ne t}^q N_{(f, H_j)}^{[\dim \langle\text{Im}f\rangle]}(r)+o(T_f(r)) =o(T_f(r)).
\end{align*}
This is a contradiction to the fact that $q > 2n+2$.
Since $\{H_j\}_{j=1}^{n+1}$ and $\{L_j\}_{j=1}^{n+1}$ are families of hyperplanes in general position, $\tilde{f}:=\big((f,H_1):\cdots :(f,H_{n+1})\big)$ and $\tilde{g}:=\big((g,L_1):\cdots:(g,L_{n+1})\big)$ are reduced representations of meromorphic mappings $\tilde{f}$ and $\tilde{g}$ respectively of $\Bbb C^m$ into $\Bbb C P^n.$ Furthermore, $\dim\langle\text{Im}f\rangle=\dim\langle\text{Im}\tilde{f}\rangle,$ $\dim\langle\text{Im}g\rangle=\dim\langle\text{Im}\tilde{g}\rangle,$ $T_{\tilde{f}}(r)=T_f(r)+O(1)$ and $T_{\tilde{g}}(r)=T_g(r)+O(1).$
By assumptions $a)$ and $c)$ we that
\begin{align}\label{p1}
\tilde{f}=\tilde{g}\quad \text{on}\;\cup_{j=1}^qf^{-1}(H_j).
\end{align}
We now prove that
\begin{align}\label{p2}
\dim \langle\text{Im}f\rangle=\dim\langle\text{Im}g\rangle\stackrel{\text{Def.}}{=}p.
\end{align}
This is equivalent to prove that $\dim \langle\text{Im}\tilde{f}\rangle=\dim\langle\text{Im}\tilde{g}\rangle .$
Therefore, it suffices to show that for any hyperplane $H$ in $\Bbb C P^n$ then
\begin{align*}
(H,\tilde{f})\equiv 0\quad\text{if and only if}\quad (H,\tilde{g})\equiv 0.
\end{align*}
Suppose that the above assertion does not hold. Without loss of the generality, we may assume that there exists a hyperplane $H$ such that $(H, \tilde{f})\not\equiv 0$ and $(H,\tilde{g})\equiv 0.$ Then by (\ref{p1}) we have
\begin{align}\label{p3}
(\tilde{f},H)=0\quad\text{on}\;\cup_{j=1}^qf^{-1}(H_j).
\end{align}
By (\ref{p3}) and by the First Main Theorem and the Cartan-Nochka Second Main Theorem we have
\begin{align*}
(q-2n+\dim \langle\text{Im}f\rangle-1)T_{f}(r)+O(1)
&\leq\sum_{j=1}^qN_{(f,H_j)}^{[\dim \langle\text{Im}f\rangle]}(r)+o(T_f(r))\\
&\leq \dim \langle\text{Im}f\rangle\sum_{j=1}^qN_{(f,H_j)}^{[1]}(r)+o(T_{f}(r))\\
&\stackrel{(\ref{p3})}{\leq}\dim \langle\text{Im}f\rangle N_{(\tilde{f},H)}(r)+o(T_{f}(r))\\
&\leq \dim \langle\text{Im}f\rangle T_{\tilde{f}}(r)+o(T_f(r))\\
&=\dim \langle\text{Im}f\rangle T_{f}(r)+o(T_f(r)).
\end{align*}
This is a contradiction to the fact that $q > 2n+2$.
We complete the proof of (\ref{p2}).
Now we prove that
\begin{align}\label{a0}
\frac{(f,H_1)}{(g,L_1)}\equiv\cdots\equiv\frac{(f,H_q)}{(g,L_q)}.
\end{align}
We distinguish the following two cases:
{\bf Case 1:} There exists a subset $J:=\{j_0,\dots,j_{n}\}\subset\{1,\dots,q\}$ such that $$\frac{(f,H_{j_0})}{(g,L_{j_{0}})}\equiv\cdots\equiv\frac{(f,H_{j_{n}})}{(g,L_{j_{n}})}\stackrel{\text{Def.}}{\equiv}u\,.$$
We have Pole$(u) \cup $Zero$(u)\subset f^{-1}(H_{j_0})\cap f^{-1}(H_{j_1})$, which is an analytic set of codimension at least 2 by assumption $b)$. Hence, $\text{Pole}(u) \cup \text{Zero} (u)=\varnothing.$
Since $H_{j_0},...,H_{j_n}$ are hyperplanes in general position,
$F:=\big((f,H_{j_0}):\cdots:(f,H_{j_{n}})\big)$
is the reduced representation of a meromorphic mapping $F$ of $\Bbb C^m$ into $\Bbb C P^n$. Still by the same reason $T_F(r)=T_f(r)+O(1).$
Suppose that (\ref{a0}) does not hold. Then, there exists $i_0\in\{1,\dots,q\}\setminus\{j_0,\dots,j_n\}$ such that
\begin{align}\label{h1}
\frac{(f,H_{i_0})}{(g,L_{i_0})}\not\equiv u.
\end{align}
Since the families $\{H_j\}_{j=1}^q$ and $\{L_j\}_{j=1}^q$ are in general position, there exist hyperplanes $H^{i_0}: a_0\omega_0+\cdots+a_n\omega_n=0, \;L^{i_0}:b_0\omega_0+\cdots+b_n\omega_n=0$ in $\Bbb C P^n$ such that
$(f,H_{i_0})\equiv (F,H^{i_0}),$ and $(g,L_{i_0})\equiv b_0(g,L_{j_0})+\cdots +b_n(g,L_{j_n})\equiv\frac{(F,L^{i_0})}{u}.$
Therefore, by (\ref{h1}) we have
\begin{align*}
\frac{(F,H^{i_0})}{(F,L^{i_0})}\equiv \frac{(f,H_{i_0})}{u(g,L_{i_0})}\not\equiv 1.
\end{align*}
By assumption $c)$ and since $\text{Pole}(u) \cup \text{Zero} (u)=\varnothing$, we have
$u=\frac{(f,H_{j_0})}{(g,L_{j_0})}=\frac{(f,H_{i_0})}{(g,L_{i_0})}=u\frac{(F,H^{i_0})}{(F,L^{i_0})}$ on $\big(\bigcup_{k=1}^qf^{-1}(H_k)\big)\setminus\big(f^{-1}(H_{i_0})\cup f^{-1}(H_{j_0})\big)$ and $u=\frac{(f,H_{j_1})}{(g,L_{j_1})}=\frac{(f,H_{i_0})}{(g,L_{i_0})}=u\frac{(F,H^{i_0})}{(F,L^{i_0})}$ on $\big(\bigcup_{k=1}^qf^{-1}(H_k)\big)\setminus\big(f^{-1}(H_{i_0})\cup f^{-1}(H_{j_1})\big).$ Then $\frac{(F,H^{i_0})}{(F,L^{i_0})}=1$ on $\big(\bigcup_{k=1}^qf^{-1}(H_k)\big)\setminus f^{-1}(H_{i_0})$.
\noindent Therefore,
\begin{align*}
\sum_{k=1,k\ne i_0}^qN_{(f,H_k)}^{[1]}(r)&\leq N_{\frac{(F,H^{i_0})}{(F,L^{i_0})}-1}(r)\\
&\leq T_{\frac{(F,H^{i_0})}{(F,L^{i_0})}}(r)+O(1)\leq T_F(r)+O(1)=T_f(r)+O(1).
\end{align*}
Therefore, by the Cartan-Nochka Second Main Theorem we have
\begin{align*}
T_f(r)+O(1)\geq \sum_{k=1,k\ne i_0}^qN_{(f,H_k)}^{[1]}(r)\geq \sum_{k=1,k\ne i_0}^q\frac{1}{p}N_{(f,H_k)}^{[p]}(r)\\
\geq \frac{q-2n+p-2}{p}T_f(r)-o(T_f(r)).
\end{align*}
This implies that $q\leq 2n+2.$ This is a contradiction. Hence, we get (\ref{a0}) in this case.
{\bf Case 2:} For any subset $J\subset \{1,\dots,q\}$ with $\#J= n+1,$ there exists a pair $i, j\in J$ such that $$\frac{(f,H_i)}{(g,L_i)}\not\equiv\frac{(f,H_j)}{(g,L_j)}.$$
\noindent We introduce an equivalence relation on $ L:=\{1,\cdots, q\}$ as follows: $i\sim j$ if and only if
\begin{equation*}
\text{det}
\begin{pmatrix}
(f,H_{i}) & & (f,H_{j})\cr& \cr (g,L_{i}) & &
(g,L_{j})
\end{pmatrix}
\equiv 0.
\end{equation*}
Set $\{L_1,\cdots, L_s\}=L/\sim $. It is clear that
$\sharp L_k\leq n$ for all $k\in\{1,\cdots, s\}.$ Without loss of generality, we may assume that $L_k:=\{i_{k-1}+1,\cdots, i_k\}$
($k\in\{1,\cdots, s\})$ where $0=i_0<\cdots <i_s=q.$
\noindent We define the map $\sigma: \{1,\cdots, q\}\to \{1,\cdots, q\}$ by
\begin{equation*}
\sigma (i)=
\begin{cases}
i+n& \text{ if $i+n\leq q$},\\
i+n-q& \text{ if $i+n>q$}.
\end{cases}
\end{equation*}
It is easy to see that $\sigma$ is bijective and $\mid \sigma (i)-i\mid\geq n $ (note that $q> 2n+2).$ This implies that $i$ and $\sigma (i)$ belong to distinct sets of $\{L_1,\cdots, L_s\}.$ This implies that for all $i\in\{1,\dots,q\},$
\begin{equation*}
P_i:=\text{det}
\begin{pmatrix}
(f,H_{i}) & & (f,H_{\sigma(i)})\cr& \cr (g,L_{i}) & &
(g,L_{\sigma(i)})
\end{pmatrix}
\not\equiv 0.
\end{equation*}
By the assumption and by the definition of function $P_i,$ we have
\begin{align}\label{h2}
\nu_{P_i}\geq \min\{\nu_{(f,H_i)},\nu_{(g,L_i)}\}+\min\{\nu_{(f,H_{\sigma(i)})},\nu_{(g,L_{\sigma(i)})}\}+\sum_{j=1\atop j\ne i,\sigma(i)}^q\nu^{[1]}_{(f,H_j)}
\end{align}
outside an analytic set of codimension $\geq 2.$
\noindent On the other hand, since $f^{-1}(H_k)=g^{-1}(L_k)$ we have
\begin{align*}
\min\{\nu_{(f,H_k)},\nu_{(g,L_k)}\}&\geq \min\{\nu_{(f,H_k)},p\} +\min\{\nu_{(g,L_k)},p\}-p\min\{\nu_{(f,H_k)},1\}\\
&=\nu^{[p]}_{(f,H_k)}+\nu^{[p]}_{(g,L_k)}-p\nu^{[1]}_{(f,H_k)}
\end{align*}
for $k\in\{i,\sigma(i)\}.$
\noindent Therefore, by (\ref{h2}) we have
\begin{align*}
\nu_{P_i}\geq \nu_{(f,H_i)}^{[p]}& +\nu_{(g,L_i)}^{[p]}+\nu_{(f,H_{\sigma(i)})}^{[p]}+\nu_{(g,L_{\sigma(i)})}^{[p]}\notag\\
&-p\nu_{(f,H_i)}^{[1]}-p\nu_{(f,H_{\sigma(i)})}^{[1]}+\sum_{j=1\atop j\ne i,\sigma(i)}^q\nu^{[1]}_{(f,H_j)}
\end{align*}
outside an analytic set of codimension $\geq 2.$
\noindent Then for all $i\in\{1,\dots,q\}$ we have
\begin{align}\label{h3}
N_{P_i}(r)\geq N_{(f,H_i)}^{[p]}(r)& +N_{(g,L_i)}^{[p]}(r)+N_{(f,H_{\sigma(i)})}^{[p]}(r)+N_{(g,L_{\sigma(i)})}^{[p]}(r)\notag\\
&-pN_{(f,H_i)}^{[1]}(r)-p N_{(f,H_{\sigma(i)})}^{[1]}(r)+\sum_{j=1\atop j\ne i,\sigma(i)}^q N^{[1]}_{(f,H_j)}(r).
\end{align}
On the other hand, by Jensen's formula
\begin{align*}
N_{P_i}(r)&=\int_{S(r)}\log |P_i| \sigma+O(1)\\
&\leq \int_{S(r)}\log(|(f,H_i)|^2+|(f,H_{\sigma(i)})|^2)^{\frac{1}{2}}\sigma\\
&\quad\quad\quad\quad\quad\quad\quad\quad\quad+\int_{S(r)}\log(|(g,L_i)|^2+|(g,L_{\sigma(i)})|^2)^{\frac{1}{2}}\sigma+O(1)\\
&\leq T_f(r)+T_g(r)+O(1).
\end{align*}
Therefore, by (\ref{h3}) for all $i\in\{1,\dots,q\}$ we have
\begin{align}\label{h3'}
N_{(f,H_i)}^{[p]}(r)& +N_{(g,L_i)}^{[p]}(r)+N_{(f,H_{\sigma(i)})}^{[p]}(r)+N_{(g,L_{\sigma(i)})}^{[p]}(r)\notag\\
&-pN_{(f,H_i)}^{[1]}(r)-p N_{(f,H_{\sigma(i)})}^{[1]}(r)+\sum_{j=1\atop j\ne i,\sigma(i)}^q N^{[1]}_{(f,H_j)}(r)\notag\\
&\leq T_f(r)+T_g(r)+O(1).
\end{align}
By summing-up of both sides of the above inequality for all $ i\in\{1,\dots,q\},$ we have
\begin{align}\label{h4}
2\sum_{j=1}^q\big(N_{(f,H_j)}^{[p]}(r)+N_{(g,L_j)}^{[p]}(r)\big)&+(q-2p-2)\sum_{j=1}^qN^{[1]}_{(f,H_j)}(r)\notag\\
&\leq q\big( T_f(r)+T_g(r)\big)+O(1).
\end{align}
Therefore, since $f^{-1}(H_j)=g^{-1}(L_j)$ we have
\begin{align}\label{h5}
2\sum_{j=1}^q\big(N_{(f,H_j)}^{[p]}(r)+N_{(g,L_j)}^{[p]}(r)\big)&+\frac{q-2p-2}{2}\sum_{j=1}^q\big(N^{[1]}_{(f,H_j)}(r)+N^{[1]}_{(g,L_j)}(r)\big)\notag\\
&\leq q\big( T_f(r)+T_g(r)\big)+O(1).
\end{align}
Then
\begin{align}\label{h6}
\big (2+\frac{q-2p-2}{2p}\big)\sum_{j=1}^q\big(N_{(f,H_j)}^{[p]}(r)+N_{(g,L_j)}^{[p]}(r)\big)
\leq q\big( T_f(r)+T_g(r)\big)+O(1).
\end{align}
By (\ref{h6}) and by the Cartan-Nochka Second Main Theorem we have
\begin{align*}
\frac{(q+2p-2)(q-2n+p-1)}{2p}\big(T_f(r)+T_g(r)\big)
&\leq q\big( T_f(r)+T_g(r)\big)+o\big( T_f(r)+T_g(r)\big).
\end{align*}
It follows that $(q+2p-2)(q-2n+p-1)\leq 2pq.$ Then $q^2-(2n+3-p)q-2(p-1)(2n+1-p)\leq 0.$ This is a contradiction to condition $(*)$ of Theorem \ref{TQT}.
Thus we have completed the proof of (\ref{a0}).\\
Assume that $H_j: a_{j0}\omega_0+\cdots+a_{jn}\omega_n=0,$ $L_j: b_{j0}\omega_0+\cdots+b_{jn}\omega_n=0\quad (j=1,\dots,q).$
Set
$$A:=\begin{pmatrix}
a_{10}& \dots & a_{1n}\\
a_{20}& \dots & a_{2n}\\
\vdots & \ddots & \vdots\\
a_{(n+1)0}& \dots & a_{(n+1)n} \end{pmatrix}, B:= \begin{pmatrix}
b_{10}& \dots & b_{1n}\\
b_{20}& \dots & b_{2n}\\
\vdots & \ddots & \vdots\\
b_{(n+1)0}& \dots & b_{(n+1)n} \end{pmatrix},\;\text{and}\;\mathcal L=B^{-1}\cdot A.$$
\noindent By (\ref{a0}), we have $A(f)\equiv B(g)$, so we get $\mathcal L(f)\equiv g.$
Set $H^*_j=(a_{j0},\dots,a_{jn})\in \Bbb C^{n+1},$ $L^*_j=(b_{j0},\dots,b_{jn})\in \Bbb C^{n+1}.$ We write $H^*_j=\alpha_{j1}H^*_1+\cdots+\alpha_{j(n+1)}H^*_{n+1}$ and $L^*_j=\beta_{j1}L^*_1+\cdots+\beta_{j(n+1)}L^*_{n+1}.$
\noindent By (\ref{a0}) we have
\begin{align*}
\frac{\alpha_{j1}(f,H_1)+\dots+\alpha_{j(n+1)}(f,H_{n+1})}{\beta_{j1}(g,L_1)+\dots+\beta_{j(n+1)}(g,L_{n+1})}\equiv\frac{(f,H_1)}{(g,L_1)}\equiv\cdots\equiv\frac{(f,H_{n+1})}{(g,L_{n+1})}
\end{align*}
for all $j\in\{1,\dots,q\}.$
\noindent This implies that
\begin{align}\label{tp}
(\alpha_{j1}-\beta_{j1})(f,H_1)+\dots+(\alpha_{j(n+1)}-\beta_{j(n+1)})(f,H_{n+1})\equiv 0
\end{align}
for all $j\in\{1,\dots,q\}.$
\noindent On the other hand $f :\Bbb C^m\longrightarrow \langle\text{Im}f\rangle$ is linearly nondegenerate and $\{H_j\}_{j=1}^{n+1}$ are in general position in $\Bbb C P^n.$ Thus, by (\ref{tp}) we have
\begin{align}\label{tp1}
(\alpha_{j1}-\beta_{j1})(\omega, H_1)+\dots+(\alpha_{j(n+1)}-\beta_{j(n+1)})(\omega, H_{n+1})=0
\end{align}
for all $\omega\in \langle\text{Im}f\rangle$ for all $j\in\{1,\dots,q\}.$
Let hyperplanes $\alpha_j: \alpha_{j1}\omega_0+\cdots+\alpha_{j(n+1)}\omega_n=0$ and $\beta_j: \beta_{j1}\omega_0+\cdots+\beta_{j(n+1)}\omega_n=0\quad (j=1,\dots,q).$
\noindent By (\ref{tp1}) we have
\begin{align}\label{tp2}
(A(\omega),\alpha_j)=( A(\omega),\beta_j)
\end{align}
for all $\omega\in \langle\text{Im}f\rangle$ and $ j\in\{1,\dots,q\}.$
For any $j\in\{1,\dots,q\}$ and for any $\omega\in\langle\text{Im}f\rangle $ we have
\begin{align*}
(\omega,H_j)&=\alpha_{j1}(\omega,H_1)+\dots+\alpha_{j(n+1)}(\omega,H_{n+1})\\
&=( A(\omega),\alpha_j)\\
&\stackrel{(\ref{tp2})}{=}( A(\omega), \beta_j)\\
&=( B\cdot\mathcal L(\omega),\beta_j)\\
&=\beta_{j1}(\mathcal L(\omega),L_1)+\dots+\beta_{j(n+1)}(\mathcal L(\omega),L_{n+1})\\
&=(\mathcal L(\omega),L_j).
\end{align*}
This implies that $\mathcal L(\langle\text{Im}f\rangle\cap H_j)=L_j\cap\mathcal L(\langle\text{Im}f\rangle)$ for all $j\in\{1,\dots,q\},$
which completes the proof of Theorem \ref{TQT}.
\hfill$\Box$
| {
"redpajama_set_name": "RedPajamaArXiv"
} |
\section{Introduction}
The entanglement entropy of a subregion in a relativistic quantum field theory is UV divergent because of short range correlations across the entangling surface. This is evident from the continuum limit of the earliest calculations on a lattice \cite{sorkin,srednicki}, or in calculations using the replica trick \cite{Callan:1994py, Holzhey:1994we,Calabrese:2009qy}. Another way to understand this divergence is to realize that the entanglement entropy of e.g. half space can be understood in terms of the standard thermodynamic entropy of quantum fields in Rindler space. The latter is divergent since the local temperature at the horizon diverges and the entropy of relativistic fields increases as a power of the temperature at high temperatures.
It is expected that if one can properly define entanglement entropy in string theory, the result should be finite \cite{Susskind:1994sm}. In string theory, it is not clear how one could go about defining the entanglement of a region in a precise manner. However, if one has access to a dual description in terms of a non-gravitational theory one could try to identify a quantity in the dual theory which provides a notion of entanglement in the gravitational theory in an appropriate approximation.
This issue was addressed in \cite{Das:1995vj} for two-dimensional bosonic non-critical string theory, whose dual formulation is double-scaled gauged quantum mechanics of a single $N \times N$ Hermitian matrix $M$ with a Hamiltonian corresponding to an inverted harmonic oscillator \cite{c1review}.
As is well known, the singlet sector of the model becomes a theory of $N$ non-relativistic fermions in $1+1$ dimensions moving in this potential, whose coordinates are eigenvalues of the matrix. This can be in turn reexpressed as a second-quantized fermionic many-body theory, and in this case the notion of entanglement of a spatial region can be defined unambiguously. It was found in \cite{Das:1995vj} that when the external potential is absent, the leading term in the entanglement entropy for large enough interval $\Delta x$ is $\log (k_F \Delta x)/3$, where $k_F$ is the Fermi momentum. This is equal to the entanglement entropy of a {\em relativistic} massless scalar $1+1$ dimensions where the UV cutoff is replaced by the inverse Fermi momentum $k_F$. It was speculated in \cite{Das:1995vj} that in two-dimensional non-critical string theory, the UV cutoff will be the local Fermi momentum $k_F(x)$. A concrete calculation in the inverted harmonic potential was carried out in \cite{Hartnoll:2015fca} where it was indeed found that the cutoff is the position-dependent Fermi momentum.
From the point of view of the fermion theory, this is easy to understand. The behavior of the entanglement entropy $\sim \log [k_F (x) \Delta x]$ reflects the fact that the low energy excitations around the Fermi level have a linear dispersion relation, exactly like a massless relativistic boson with the speed of light replaced by the Fermi momentum. However, when the momenta becomes of the order of $k_F$ the quadratic term in the dispersion relation becomes important. Furthermore, the presence of a finite Fermi sea means that excitations have an effective UV cutoff given by the Fermi level. Likewise, at high temperatures the entropy of non-relativistic fermions increases logarithmically rather than a power law. In the Rindler calculation this leads to a finite contribution from the region near the horizon \cite{Das:1995vj}.
However, in the dual string theory $1/k_F (x)$ is proportional to the (position dependent) string coupling $g_{st}$. This means that the UV scale which makes the entanglement entropy finite is not simply the string length, but involves the string coupling. For the same reason, as emphasized in \cite{dastalk}, the finiteness of the entanglement entropy should be invisible in any finite order in perturbation theory \footnote{In \cite{dabholkar} the replica trick has been used to define an entanglement entropy in critical string theory. The string theory then lives on a cone, and the perturbative worldsheet partition function is finite with the string length providing the UV cutoff. While the relationship between this calculation and that of \cite{Das:1995vj,Hartnoll:2015fca} is not very clear, it appears that this calculation is quantifying a different kind of entanglement.}.
Generally collective field theory offers a reformulation of matrix and vector, providing a systematic $1/N$ expansion field theory. The two-dimensional string perturbation expansion likewise is generated through the collective field formulation. The dynamical degree of freedom in this case is a massless scalar field (``massless tachyon'') and one has an interacting quantum field theory. Although the higher string modes are non-propagating: they can, however, lead to non-trivial backgrounds. The collective field representation for the dynamics of the massless scalar is simply obtained by rewriting the matrix quantum mechanics
\begin{equation}
H = {\rm Tr} \left[ - \frac{1}{2} \left(\frac{\partial}{\partial M}\right)^2 +V(M) \right]
\label{0-4}
\end{equation}
in terms of the density of eigenvalues - the collective field $\phi(x)$
\begin{equation}
\phi (x) = {\rm Tr} \,\delta (x\cdot I - M) = \psi^\dagger (x) \psi (x)
\label{0-1}
\end{equation}
and its canonically conjugate momentum \cite{Jevicki:1979mb}.
This represents a non-relativistic bosonization and has been
studied thoroughly.
Fluctuations of the collective field around its classical value behave as a massless scalar in $1+1$ dimensions with a position-dependent couplings proportional to inverse of the double-scaled Fermi level \cite{Das:1990kaa}. The space dimension descends from the space of eigenvalues. Thus to the lowest order in perturbation theory in this coupling the result for the entanglement entropy is UV divergent. The form of the answer from the fermionic description indicates that that this would continue to be divergent if one truncates perturbation theory to any finite order, as will be clear in the following. It is therefore natural to ask how does the finiteness of the entanglement entropy shows up in collective field theory. This is the central issue which we address in this paper \footnote{It turns out that the fluctuation of the collective field is related to the massless tachyon of string theory as defined in world-sheet string theory by a spatial transform (the leg pole transform) whose kernel is non-local at the string scale \cite{c1review}. This fact is not relevant to the discussion of whether the result is finite.}.
It should be noted that the finiteness is due to the finiteness of the Fermi momentum $k_F$, which is proportional to the number density $N/L$ where $L$ is the size of a large box in which the fermions live. This means that the entanglement entropy remains finite in the limit $N \rightarrow \infty$, $L \rightarrow \infty$ with $N/L$ fixed. Likewise in the $c=1$ matrix model the quantity is finite in the double-scaling limit. What is important is that the coupling should be finite.
In this paper we calculate the entanglement entropy in the ground state {\em as defined in the fermionic many-body theory} using the collective field theory. For $d$-dimensional mutually non-interacting fermions, this entanglement entropy $S_A$ of a region $A$ has a well known expansion in terms of cumulants of the particle number \cite{klich},
\begin{equation}
S_A = {\rm lim}_{M \rightarrow \infty} \sum_{m=1}^{M} \alpha_{2m} (M) C_{2m},\quad C_m = (-i\partial_\lambda) \log \langle [ \exp(i\lambda N_A)] \rangle
\vert_{\lambda = 0} ,
\label{2-1a}
\end{equation}
where $N_A$ is the particle number operator in the region $A$,
\begin{equation}
N_A = \int_A d^d\vec{x}~ \psi^\dagger (\vec{x}) \psi (\vec{x}) .
\label{2-3}
\end{equation}
The coefficients $\alpha_{2m}$ are pure numbers given in \cite{klich}. In many situations $C_2$ is the leading contribution \footnote{There is no general proof of this: in fact there is no parametric suppression of the higher cumulants. However in many systems, including the systems considered in this paper the higher cumulants are nevertheless suppressed.}, and this is what we evaluate. Since the collective field is \eqref{0-1}, we have (using $\alpha_2 = \pi^2 / 3)$, for a single interval
$a \leq x \leq b$ in one space dimension,
\begin{equation}
S^{(2)}_A = \frac{\pi^2}{3}\int_a^b dx \int_a^b dx^\prime \left[ \langle F| \phi(x) \phi(x^\prime) | F \rangle - \langle F| \phi(x) | F \rangle \langle F |\phi(x^\prime) | F \rangle \right] ,
\label{2-4}
\end{equation}
where $| F \rangle$ is the ground state. This is simply the integral of the connected Green's function. In this paper we calculate this quantity using the collective field theory Hamiltonian \footnote{For Slater determinant states all the terms in the cumulant expansion can be expressed in terms of the expectation value of the fermion phase space density \cite{dhl1,satya1}. This can be easily evaluated in a Thomas-Fermi approximation \cite{satya1,satya4,dhl2}. It remains to be seen if a theory of the phase space density regarded as an operator along the lines of \cite{dhar,son} is useful to proceed further.}. The quantity $S^{(2)}_A$ is finite only if the short distance behavior of the collective field correlator is soft.
As discussed above, in the lowest order in perturbation theory this correlator is exactly the same as that of a free massless relativistic scalar and therefore divergent.
Since the short distance behavior of the correlator is independent of the potential, we examine in detail the theory with no external potential. The interactions of the fluctuations of the collective field are then characterized by a coupling which is proportional to $1/k_F$.
In this case the exact eigenstates and eigenvalues of the collective Hamiltonian have been obtained in \cite{Jevicki:1991yi,nomura}. Using this exact solution we calculate the momentum space correlator and demonstrate agreement with the known answer obtained using fermions. We then calculate this quantity perturbatively and show that the perturbation expansion can be resummed. The resummed answer is in exact agreement with the result in the fermionic many-body theory which then leads to an agreement of the entanglement entropy. The expression (\ref{2-4}) involves an integral over the equal time correlator. We find that in momentum space the exact result is $|k|/\pi$ for $|k| < 2k_F$ which is also the leading perturbative result. For $|k| > 2k_F$ the result is a constant $2k_F / \pi$. The perturbation expansion in the collective field theory is a low momentum expansion in powers of $k/k_F$. The exact result shows that perturbatively there is no correction to the lowest order result which is independent of $k_F$. This means that the entanglement entropy is divergent perturbatively, and the finiteness of the result is a non-perturbative feature.
While our explicit calculation is for the matrix model without a potential, we expect that the same conclusion will hold in the presence of a potential, in particular the double-scaled $c=1$ matrix model. The collective field theory in these cases provides a field theory of strings with $1/N$ expansion being systematically generated \cite{Demeterfi:1991tz, Demeterfi:1991nw, Balthazar:2017mxh}. It needs to be treated with care and singular counter-terms present in the collective Hamiltonian will probably play a role.
As emphasized above, we are calculating the entanglement entropy as defined in the fermionic many-body theory, which we perform using collective field theory.
On the other hand one could define a notion of entanglement in the collective field theory itself. We need to determine if these two notions of entanglement agree with each other since bosonization involves a non-local transformation. This question has been investigated for lattice theories leading to relativistic fermions and conformal field theories in the literature \cite{ee-boson-fermion}, and is non-trivial when the subregion of interest consists of disconnected intervals. We will argue, however, that for non-relativistic fermions with a conserved fermion number, the situation is somewhat different. This is because now there is a first-quantized description, where the entanglement in the fermionic many-body theory becomes a target space entanglement \cite{target1}-\cite{target6}. In this first-quantized description, the operators which make sense are many-body operators involving a sum over all the identical particles. The latter can be in turn expressed either in terms of a second-quantized fermion field or in terms of the collective field and its momentum conjugate. For free fermions, and for Slater determinant states, it was shown in \cite{target2} that the reduced density matrix in the first-quantized description is exactly the same as that obtained in the second-quantized description. In the following we will argue that this implies that the entanglement entropy in terms of fermions is in fact the same as that in terms of the collective field.
While the singlet sector of single matrix quantum mechanics becomes a theory of free fermions, non-singlet sectors lead to models of interacting fermions, notably the Spin-Calogero models, particularly in the study of the long string sector \cite{Maldacena:2005hi,Balthazar:2018qdv}. Collective field theory for Calogero models have been developed in \cite{Andric:1982jk, Aniceto:2006rr,sen, Bardek:2010jg}. In these cases, the entanglement entropy can no longer be expressed in terms of fermion number cumulants. However, the collective formulation should be useful.
Our results should have implications for higher dimensional string theories whose holographic duals are matrix models with multiple matrices, e.g. the BFSS matrix model \cite{bfss} or the BMN matrix model \cite{bmn}. The notion of target space entanglement for multiple matrices has been formulated and explored in \cite{target2}-\cite{target5}. In terms of matrices explicitly, entanglement is discussed in \cite{target7,target8}. On the other hand, a collective formalism for the BMN matrix model has been established in \cite{bmncollective}. Here, the collective variables are ingredients of string fields. Since the gauge invariant matrix operators can be directly expressed in terms of the collective variables, a formulation of entanglement in terms of the latter will provide an understanding of the string theoretic meaning of target space entanglement.
In section (\ref{two}) we calculate the connected correlator of the collective field and hence the leading term in the entanglement entropy of a single interval for fermions without any external potential. The correlator is calculated first by using exact eigenstates and eigenvalues and then by resumming the perturbation expansion as well as exactly. In section (\ref{four}) we discuss the relationship of entanglement in the collective field and fermionic description. We also discuss possible applications to the long string sector which involves non-singlet states and multi-matrix models dual to higher dimensional strings. Section (\ref{five}) contains a discussion. The appendix provides some details of the derivation of the expression of the exact eigenstates and eigenvalues of the Hamiltonian.
\section{Entanglement Entropy for a Vanishing Potential}
\label{two}
In this section we consider the singlet sector of matrix quantum mechanics (\ref{0-4}) and the associated entanglement entropy.
In the collective field formalism, the Hamiltonian is given by \footnote{In addition the general collective Hamiltonian contains a singular subleading counterterm. In this case this counterterm does not play much of a role, except to ensure that the $O(1/N^2)$ corrections to the ground state energy vanish.}
\begin{equation}
H = \frac{1}{2} \int dx ~\left[ \partial_x \Pi\,\phi\,\partial_x\Pi +\frac{\pi^2}{3}\phi^3 - 2\mu_F \phi \right]
\label{2-1}
\end{equation}
where $\Pi(x)$ is the canonically conjugate momentum to $\phi(x)$ defined in (\ref{0-1}) and $\mu_F$ is a Lagrange multiplier which imposes the condition
\begin{equation}
\int dx \,\phi(x) = N .
\end{equation}
The classical solution features a uniform distribution
\begin{equation}
\phi_0=\frac{k_F}{\pi}, \quad\mu_F = \frac{1}{2}k_F^2.
\end{equation}
To study quantum fluctuations, we expand the collective field around the classical solution
\begin{equation}
\phi(x)=\phi_0+\eta(x), \quad \partial_x\Pi(x)\rightarrow \partial_x\Pi(x).
\end{equation}
The fluctuation Hamiltonian becomes
\begin{equation}
H=\frac{1}{2}\int dx \left\{\frac{k_F}{\pi}\left[(\partial_x\Pi)^2+(\pi\eta)^2\right]+\left[\partial_x\Pi\,\eta\,\partial_x\Pi+\frac{1}{3}(\pi\eta)^3\right]\right\}.
\label{hamil}
\end{equation}
Writing $\eta = \partial_x \varphi$ and $\partial_x \Pi = \Pi_\varphi$ we see that it is evident that such a perturbation expansion is essentially a low energy expansion. For a process with momentum $k$ we see that (\ref{hamil}) is a theory of a massless scalar field in $1+1$ dimensions with cubic interactions. The cubic terms are small when the momenta $k$ are small compared to $k_F$ so that there is a perturbative expansion in powers of $k/k_F$
The quantity $S_A^{(2)}$ can be now expressed entirely in terms of the connected equal time
Green's function $G(x,x^\prime) \equiv \braket{\eta(x)\eta(x')}$ leading to
\begin{equation}
S_A^{(2)} = \frac{\pi^2}{3}\int_a^b dx \int_a^b dx' \,G(x,x') .
\label{2-7}
\end{equation}
In lowest order in the perturbation expansion, the Green's function is that of a massless field, so that the coincident integrated Green's functions which appear in (\ref{2-7}) are logarithmically divergent, leading to a logarithmically divergent result for $S_A^{(2)}$ - exactly as expected.
The detailed form of $S_A^{(2)}$ depends on the boundary conditions. For example when the theory lives in a large box of size $L$, the integrated Green's function is
\begin{equation}
\int dx \int dx'\, G(x,x')=-\frac{1}{2\pi^2}\log\left|x-x'\right|.
\end{equation}
leading to the entropy
\begin{equation}
S_A^{(2)}=\frac{1}{3}\log\frac{b-a}{\epsilon}.
\end{equation}
On the other hand, the answer in the fermionic many-body theory is not divergent. The underlying reason is the fact that the fermions are non-relativistic.
Fluctuations of the collective field are particle-hole pair excitations around the Fermi sea. In the exact theory the energy of such an excitation is
\begin{equation}
\omega = k_F \left( k + \frac{1}{2 k_F} k^2 \right) .
\label{2-8}
\end{equation}
The perturbative spectrum of the collective field is linear. As expected, this captures only the low energy spectrum, valid for $k \ll k_F$. On the other hand, the divergence of the entanglement entropy comes from the UV. Since $1/k_F$ is the coupling constant in the collective theory, this would mean that the correct answer with a finite $k_F$ has to be non-perturbative in the collective theory.
In the next subsection we will demonstrate the exact spectrum with eigenstates in the collective formulation \cite{Jevicki:1991yi} and \cite{nomura}. The result is complete agreement with the known result in the fermionic many-body theory, featuring the poles corresponding to (\ref{2-8}). This means that at non-perturbative level the finite entropy is obtained in an exact calculation. We then consider the theory perturbatively. We show that the perturbation expansion can be resummed, again yielding the exact result.
\subsection{Direct evaluation using exact eigenstates}
\label{sec:2.1}
In this section we will obtain the Green's function of the collective field using exact eigenstates of the full Hamiltonian, using \cite{Jevicki:1991yi}. If we express the Hamiltonian as $H = H_2 + H_3$, where $H_2, H_3$ are the quadratic and cubic parts, it follows from the commutation relations of $\alpha_{L,R}$ that
\begin{equation}
[ H_2 , H_3 ] = 0
\end{equation}
so that they can be simultaneously diagonalized. The eigenstates of $H_2$ are characterized by the total momentum $k$ in the emergent space direction $q$ which can be distributed among any number of particles in multiple ways. Thus these eigenstates are degenerate. It is useful to consider the coordinate $q$ to be in a periodic box of length $L$ so that the momenta
\begin{equation}
k = \frac{2\pi n}{L},\quad n = 0, \pm 1, \pm 2, \cdots .
\label{3-10}
\end{equation}
Then the degeneracy of $H_2$ can be characterized by partitions of an integer. The exact eigenstates of $H$ are then obtained by transforming to a basis which also diagonalizes $H_3$.
The construction of exact eigenstates and eigenvalues follows from the connection of the matrix model Hamiltonian and the Laplacian on $U(N)$. Consider the unitary matrix $U$
\begin{equation}
U = {\rm exp} \left( \frac{2\pi i}{L} M \right) .
\label{3-1}
\end{equation}
Then the Hamiltonian is given by
\begin{equation}
H = -\frac{1}{2} {\rm Tr} \left( \frac{\partial}{\partial M}\frac{\partial}{\partial M} \right) = \left(\frac{2\pi}{L} \right)^2 \sum_\alpha C_\alpha C_\alpha ,
\label{3-2}
\end{equation}
where
\begin{equation}
C_\alpha = {\rm Tr} \left( t^\alpha \frac{\partial}{\partial U} \right) .
\label{3-3}
\end{equation}
Here $t^\alpha, \alpha = 1 \cdots N^2$ are the generators of $U(N)$.
The Hamiltonian is therefore the Laplacian on $U(N)$.
Let us introduce the collective variables
\begin{equation}
\phi_n={\rm Tr}\,U^n.
\end{equation}
These are Fourier transforms of the collective field $\phi (x)$ of the previous section. Using the standard procedure in \cite{Jevicki:1979mb} the collective Hamiltonian is
\begin{equation}
\label{collh}
\begin{split}
H_2=&\frac{2\pi(N-1)}{L}\sum_n |n|\phi_n\frac{\partial}{\partial \phi_n},\\
H_3=&\frac{1}{2} \left(\frac{2\pi}{L}\right)^2
\sum_{n,m} nm\phi_{n-m}\frac{\partial}{\partial \phi_n}\frac{\partial}{\partial \phi_m}+\sum_{n,m}|n|\phi_m\phi_{n-m}\frac{\partial}{\partial \phi_n}.
\end{split}
\end{equation}
The eigenstates can be now expressed in terms of characters of representations of $U(N)$. Consider a representation described by a Young tableau with $n$ boxes with $\lambda_j$ boxes in the $j$-th row
\begin{equation}
\lambda \equiv \{\lambda_1,\lambda_2,\cdots \},\quad\lambda_1 > \lambda_2 \geq \lambda_3 \geq \cdots, \quad \sum_j \lambda_j = n.
\label{3-11}
\end{equation}
The eigenstates of $H$ are then given by the Schur polynomials of $(\phi_1 \cdots \phi_n)$. Denote a conjugacy class of the permutation group $S_n$ by
\begin{equation}
\nu=\{1^{\nu_1}, 2^{\nu_2},\cdots\}.
\end{equation}
This corresponds to a partition of $n$ where $j$ appears $\nu_j$ times.
Then the Schur polynomials may be written as
\begin{equation}
\label{egfn}
s_\lambda(\{\phi\})=\sum_{\nu}\chi^\lambda_\nu \prod_{m}\frac{\phi_m^{\nu_m}}{\nu_m!m^{\nu_m}},
\end{equation}
where $\chi^\lambda_\nu$ denotes the character of the irreducible representation $\lambda$ for $\nu$ of $S_n$. This Fock space representation of this state may be obtained by the representation
\begin{equation}
\phi_n \rightarrow \sqrt{n}a^\dagger_n,\quad \frac{\partial}{\partial\phi_n}\rightarrow \frac{1}{\sqrt{n}}a_n,\quad[ a_m , a^\dagger_n ] = \delta_{mn}.
\end{equation}
and the fluctuation of the collective field is
\begin{equation}
\delta \phi_n = \int dx \,e^{-\frac{2\pi i n}{L}}\,\eta(x) = \sqrt{n} (a_n + a_n^\dagger) .
\label{3-15}
\end{equation}
In terms of these annihilation and creation operators the Hamiltonian reads
\begin{equation}
\begin{split}
H_2=&\frac{2\pi}{L}k_F\sum_{n\neq 0}|n|a_n^\dagger a_n ,\\
H_3=&\frac{2\pi^2}{L^2}\sum_{n,m>0;n,m<0}\sqrt{nm|n+m|}(a^\dagger_na^\dagger_ma_{n+m}+a^\dagger_{n+m}a_na_m) .
\label{3-15a}
\end{split}
\end{equation}
The eigenstate in question is then expressed in terms of the Fock vacuum $|0 \rangle$
\begin{equation}
\ket{\lambda}=s_\lambda(\sqrt{j}a^\dagger_j)\ket{0}.
\end{equation}
The eigenvalue of the Hamiltonian $H$ can be then computed to yield
\begin{eqnarray}
E_{\lambda} & =& E_2 + E_3 ,
\nonumber \\
E_2 & = & \frac{1}{2} \left(\frac{2\pi}{L} \right)^2 Nn,\quad E_3 =\frac{1}{2} \left(\frac{2\pi}{L} \right)^2 \sum_{j} \lambda_j (\lambda_j-2j+1).
\label{3-16}
\end{eqnarray}
In this equation $E_2$ is the eigenvalue of $H_2$ and $E_3$ is the eigenvalue of $H_3$.
A particular class of these states play a special role in the following. These are single particle states. For a given $n$ these states are labelled by an integer $m$, leading to a $\lambda$ given by
\begin{equation}
\lambda (n,m) =\{m+n-M,\underbrace{1,1,\cdots,1}_{M-m+1}\}.
\label{3-18}
\end{equation}
Using (\ref{3-16}) the energy of this state above the ground state is given by
\begin{equation}
E_\lambda(n,m) =\frac{1}{2}\left(\frac{2\pi}{L} \right)^2 (n^2+2nm) .
\label{3-17}
\end{equation}
In terms of continuous momenta $k = 2\pi n / L$, $p= 2\pi m / L$ for a large box, we have
\begin{equation}
E_{\lambda (p,k)} = \frac{1}{2} (k^2 + 2 pk) = \frac{1}{2} [ (p+k)^2 - p^2 ] .
\label{3-20}
\end{equation}
Similarly for negative $k$ we have the particle-hole branch which has the dispersion relation (\ref{3-20}) with $k \rightarrow -k$.
The Weyl formula expresses Schur polynomials as ratios of Slater determinants - this means that these exact eigenstates are precisely states of an $N=2M+1$ fermionic many-body theory \cite{Jevicki:1991yi}. The ground state is the filled Fermi sea where the states labelled by $-M,-M+1,\cdots M$ are filled. The Fermi momentum $k_F$ is given by
\begin{equation}
k_F = \frac{\pi(N-1)}{L} = \frac{2\pi M}{L} .
\end{equation}
The state represented by (\ref{3-16}) is a state where the a fermion is removed from the $m$-th level and moved to the $(n+m)$-th level. Note that the collective Hamiltonian (\ref{3-15a}) is the Hamiltonian of fluctuations so that the energies are the excitation energies of the fermionic many-body theory. This correspondence immediately implies that the states $|\lambda(p,k) \rangle$ are the only states which have non-vanishing matrix elements
\begin{equation}
\langle 0 | \delta \phi (k) | \lambda \rangle .
\end{equation}
Without any reference to the fermions, this result can be proven as follows.
According to the Frobenius characteristic formula, in order to give single particle states, the cycle type must be
\begin{equation}
\nu=\{n^1\}.
\end{equation}
Therefore, any Schur polynomial $s_\lambda$ with non-vanishing character $\chi^\lambda_\nu$ of the particular cyclic type $\nu$ contributes to the Dirac bracket. We can compute $\chi^\lambda_\nu$ using the Murnaghan–Nakayama rule
\begin{equation}
\chi^\lambda_\nu=\sum_{Y\in \operatorname{BST}(\lambda,\nu)}(-)^{ht(Y)},
\end{equation}
where $\operatorname{BST}(\lambda,\nu)$ denotes all border-strip tableaux of the shape $\lambda$ and the type $\nu$, and $\operatorname{ht}(Y)$ denotes the sum of the heights of the border strips in $Y$. The height of a border strip is one less than the number of rows it touches. For a given Young tableau of the shape $\lambda$, we start to fill the boxes with $n$ integers $`1'$. Those Young tableaux not of the hook form must contain at least one $2\times 2$ square of $`1'$, thus they fail to form border-strip tableaux, which means the combination of $\lambda$ and $\nu$ gives
\begin{equation}
\chi^\lambda_\nu=0.
\end{equation}
Hence only the Young tableaux of the hook form survive from the integral. In this case, the leading term of the Schur polynomial is equal to
\begin{equation}
s_\lambda=(-)^{k_F-p}\frac{\phi_k}{k}+\cdots\rightarrow(-)^{k_F-p}\frac{1}{\sqrt{k}}a_k^\dagger+\cdots.
\end{equation}
We have identified $\phi_k$ with creation operator $\sqrt{k}a_k^\dagger$.
Consider now the two-point function of collective field fluctuations
\begin{equation}
\tilde{G}(\omega,k)=\int dt \,e^{i\omega \tau}\braket{\rho(\tau,k)\rho(0,-k)}=\int_{-k_F}^{k_F}\frac{dp}{2\pi}\frac{|\bra{0}\delta \phi(k)\ket{\lambda(p,k)}|^2}{i\omega-E_\lambda(p,k)}.
\end{equation}
Using (\ref{3-20}), performing the integral, and adding the contributions for positive and negative $k$ we get
\begin{equation}
\tilde{G}(\omega,k)=\frac{1}{2\pi k}\left(\log\frac{i\omega-k_F k+k^2/2}{i\omega-k_F k-k^2/2}-\log\frac{i\omega+k_F k+k^2/2}{i\omega+k_F k-k^2/2}\right).
\label{8-1}
\end{equation}
After analytic continuation back to real time, this expression clearly displays the dispersion relation (\ref{2-8}) and is in exact agreement with a direct calculation in the fermionic many-body theory (see e.g. \cite{Pereira:2007}).
\subsection{Perturbative calculation and resummation}
It is convenient to define left and right moving chiral bosons
\begin{equation}
\alpha_L=\frac{1}{\sqrt{2\pi}}(\partial_x\Pi+\pi\eta), \quad \alpha_R=\frac{1}{\sqrt{2\pi}}(\partial_x\Pi-\pi\eta),
\end{equation}
with commutation relations
\begin{eqnarray}
\left[\alpha_L(x), \alpha_L(x')\right] &=& -i\partial_x\delta(x-x'),\\
\left[\alpha_R(x), \alpha_R(x')\right] &=& +i\partial_x\delta(x-x'),\\
\left[\alpha_L(x), \alpha_R(x')\right] &=& 0.
\end{eqnarray}
We can rewrite the Hamiltonian in terms of the new fields
\begin{eqnarray}
H&=& H_L+H_R,\\
H_L&=&\frac{k_F}{2}\int dx \left(\alpha_L^2 + \frac{\sqrt{2\pi}g}{3k_F}\alpha_L^3\right),\\
H_R&=&\frac{k_F}{2}\int dx \left(\alpha_R^2 - \frac{\sqrt{2\pi}g}{3k_F}\alpha_R^3\right).
\end{eqnarray}
Here we have introduced a small parameter $g$ to keep track of the terms in an perturbation expansion, which we will set to $1$ at the end of the calculation. As mentioned above the true expansion parameter is $k/k_F$ where $k$ is the momentum in the Green's function. The following calculation is similar to that in \cite{Pereira:2007}.
Using mode expansions
\begin{equation}
\alpha_{L,R}(\tau,x)=i\int_0^\infty dk\sqrt{\frac{k}{2\pi}}\left[a_{L,R}(k)e^{-k(k_F\tau\pm ix)}-a_{L,R}^\dagger(k)e^{k(k_F\tau\pm ix)}\right],
\end{equation}
we can compute the propagators of chiral bosons. In Euclidean signature,
\begin{eqnarray}
D_L(\tau,x)\equiv \braket{\alpha_L(\tau,x)\alpha_L(0,0)}&=&\frac{1}{2\pi}\frac{1}{(k_F \tau+ix)^2},\\
D_R(\tau,x)\equiv \braket{\alpha_R(\tau,x)\alpha_R(0,0)}&=&\frac{1}{2\pi}\frac{1}{(k_F \tau-ix)^2}.
\end{eqnarray}
In momentum space, by doing contour integral we obtain
\begin{equation}
D_{L,R}(\omega,k)=-\int_{-\infty}^\infty d\tau \int_{-\infty}^\infty dx \,e^{i(\omega \tau-kx)}D_{L,R}(\tau,x)=\frac{\mp k}{i\omega\pm k_F k}.
\end{equation}
Therefore we can read off the Feynman rules. Apart from propagators, the left and right vertices are given by $\pm\sqrt{2\pi}g$ respectively.
The main ingredient of calculating entanglement entropy is the Green's function of $\eta$, which we define in the following way
\begin{equation}
\braket{\eta(\tau,x)\eta(0,0)}\equiv G(\tau,x)=-\int_{-\infty}^\infty\frac{d\omega}{2\pi}\int_0^\infty\frac{dk}{2\pi}\,e^{-i(\omega \tau-kx)}\tilde{G}(\omega,k),
\end{equation}
with
\begin{equation}
\tilde{G}(\omega,k)=\frac{1}{2\pi}\left[\tilde{G}_L(\omega,k)+\tilde{G}_R(\omega,k)\right].
\end{equation}
The leading order $\{$\ref{fig:f1}$\}$ of $\tilde{G}_R(i\omega,k)$ is simply
\begin{equation}
\tilde{G}_R^{(0)}(\omega,k)=D_R(\omega,k)=\frac{k}{i\omega-k_F k},
\end{equation}
\begin{figure}[h]
\centering
\includegraphics[width=8cm]{f1.png}
\caption{Leading order}
\label{fig:f1}
\end{figure}
while the subleading order $\{$\ref{fig:f2}$\}$ in $g$ gives
\begin{equation}
\tilde{G}_R^{(1)}(\omega,k)=D_R(\omega,k)\Gamma_R(\omega,k)D_R(\omega,k).
\end{equation}
\begin{figure}[h]
\centering
\includegraphics[width=8cm]{f2.png}
\caption{Subleading order}
\label{fig:f2}
\end{figure}
The self-energy can be computed again using contour integral
\begin{equation}
\begin{split}
\Gamma_R(\omega,k)=&\frac{1}{2}\left(\sqrt{2\pi}g\right)^2\int_{-\infty}^\infty\frac{d\tilde{\omega}}{2\pi}\int_0^k \frac{d\tilde{k}}{2\pi}\,D_R(i\tilde{\omega},\tilde{k})D_R(i\omega-i\tilde{\omega},k-\tilde{k})\\
=&\frac{g^2}{12}\frac{k^3}{i\omega-k_F k}.
\end{split}
\end{equation}
Plugging it back, we get
\begin{equation}
\tilde{G}_R^{(1)}(\omega,k)=\frac{g^2}{24}\frac{k^5}{(i\omega-k_F k)^3}.
\end{equation}
The sub-subleading order $\{$\ref{fig:f3}, \ref{fig:f4}, \ref{fig:f5}$\}$ contains three Feynman diagrams, which give
\begin{equation}
\frac{g^4}{144}\frac{k^9}{(i\omega-k_F k)^5}, \quad\frac{g^4}{504}\frac{k^9}{(i\omega-k_F k)^5}, \quad\frac{g^4}{280}\frac{k^9}{(i\omega-k_F k)^5}
\end{equation}
respectively.
\begin{figure}[h]
\centering
\includegraphics[width=8cm]{f3.png}
\caption{Sub-subleading order:A}
\label{fig:f3}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=8cm]{f4.png}
\caption{Sub-subleading order:B}
\label{fig:f4}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=8cm]{f5.png}
\caption{Sub-subleading order:C}
\label{fig:f5}
\end{figure}
Collecting all of the contributions, we obtain
\begin{equation}
\tilde{G}_R^{(2)}(\omega,k)=\frac{g^4}{80}\frac{k^9}{(i\omega-k_F k)^5}.
\end{equation}
This series can be resummed, leading to
\begin{equation}
\begin{split}
\tilde{G}_R(\omega,k)=&\tilde{G}_R^{(0)}(\omega,k)+\tilde{G}_R^{(1)}(\omega,k)+\tilde{G}_R^{(2)}(\omega,k)+\cdots\\
=&\frac{1}{gk}\left[\frac{gk^2}{i\omega-kk_F}+\frac{1}{3}\left(\frac{gk^2}{i\omega-kk_F}\right)^3+\frac{1}{5}\left(\frac{gk^2}{i\omega-kk_F}\right)^5+\cdots\right]\\
=&\frac{1}{gk}\log\frac{i\omega-k_Fk+g k^2/2}{i\omega-k_Fk-gk^2/2}.
\end{split}
\label{3-21}
\end{equation}
Sending $g$ to 1, we have
\begin{equation}
\tilde{G}_R(\omega,k)=\frac{1}{k}\log\frac{i\omega-k_Fk+k^2/2}{i\omega-k_Fk-k^2/2}.
\end{equation}
Similarly, the Green's function of left chiral bosons is equal to
\begin{equation}
\tilde{G}_L(\omega,k)=-\frac{1}{k}\log\frac{i\omega+k_Fk+k^2/2}{i\omega+k_Fk-k^2/2},
\end{equation}
thus
\begin{equation}
\tilde{G}(\omega,k)=\frac{1}{2\pi k}\left(\log\frac{i\omega-k_Fk+k^2/2}{i\omega-k_Fk-k^2/2}-\log\frac{i\omega+k_Fk+k^2/2}{i\omega+k_Fk-k^2/2}\right).
\label{2-11}
\end{equation}
This is in agreement with the exact answer (\ref{8-1}).
\subsection{Entanglement entropy of a single interval}
We are now ready to obtain an expression for the entanglement entropy of a single interval $(a,b)$, equation (\ref{2-7}).
Notice that the only dependence on $x$ and $x^\prime$ is in the Fourier transformation, so we can integrate out them first,
\begin{equation}
\int_a^b dx dx'\, e^{ik(x-x')}=\frac{4}{k^2}\sin^2\frac{k(b-a)}{2}.
\end{equation}
We now need to integrate (\ref{2-11}) to extract the equal time correlator. Performing a partial integration and using residue theorem, one gets
\begin{equation}
\int_{-\infty}^\infty d\omega\log(\omega+ic)=2\pi|c| +I_1 +I_2 .
\end{equation}
Here
\begin{eqnarray}
I_1 & = &{\rm \lim}_{\Lambda \rightarrow \infty} [\omega \log(\omega + ic) - \omega]^\Lambda_{-\Lambda}, \nonumber \\
I_2 & = & ic\int_{\text{semicircle}} \frac{d\omega}{\omega + ic},
\end{eqnarray}
where the integral in $I_2$ is along an infinite radius semicircle in the lower- or upper-half plane depending on the sign of $c$.
In the integral over the four terms contained in $\tilde{G}(\omega,k)$ it may be checked that these divergent contributions cancel.
After simplifying the expression, we obtain
\begin{equation}
\begin{split}
S_A^{(2)}=&\frac{2}{3}\int_0^\infty \frac{dk}{k^3}\sin^2\frac{k(b-a)}{2}\left(\left|k_Fk+k^2/2\right|-\left|k_Fk-k^2/2\right|\right)\\
=&\frac{2}{3}\int_0^{2k_F}\frac{dk}{k}\sin^2\frac{k(b-a)}{2}+\frac{4k_F}{3}\int^\infty_{2k_F}\frac{dk}{k^2}\sin^2\frac{k(b-a)}{2}.
\end{split}
\end{equation}
After performing the integral, the entanglement entropy can be recast into the form
\begin{equation}
\begin{split}
S_A^{(2)}=&\frac{1}{3}\left\{-\operatorname{Ci}[2k_F(b-a)]-2k_F(b-a)\operatorname{Si}[2k_F(b-a)]+\log [k_F(b-a)]\right.\\
&\left.+\pi k_F(b-a)+2\sin^2 [k_F(b-a)]+\gamma+\log 2\right\},
\end{split}
\label{4-1}
\end{equation}
where $\gamma$ is the Euler–Mascheroni constant. This is our final result. This, of course, is in exact agreement with the result obtained directly in the fermionic many-body theory.
It is now clear that this answer requires a resummation of the perturbative expansion. In fact, rather remarkably, the perturbative corrections to the leading order result exactly vanish. This may be seen explicitly by expanding the Fourier transform of the equal time correlator,
\begin{equation}
G_0(k) = \int_{-\infty}^\infty\frac{d\omega}{2\pi}\,\tilde{G}(\omega,k)=G^{(0)}_0(k)+G_0^{(1)}(k) + \cdots
\end{equation}
and using the expansion of $\tilde{G}(\omega,k)$. Performing the integrals explicitly one finds, for $k > 0$
\begin{eqnarray}
G_0^{(0)} (k) & = & \frac{k}{\pi} , \nonumber \\
G_0^{(m)} (k) & = & 0 , \quad m=1,2,3 \cdots .
\label{3-22}
\end{eqnarray}
This means that there is no perturbative correction to the divergent lowest order result for the entanglement entropy. The answer is inherently non-perturbative. In fact the exact $G_0(k)$ obtained by integrating (\ref{2-11}) over $\omega$ is
\begin{equation}
G_0 (k) =
\left\{ \begin{array}{rcl}
|k|/\pi & \mbox{for}
& |k| < 2 k_F\\
2k_F/\pi & \mbox{for} & |k| > 2k_F
\end{array}\right. .
\label{3-23}
\end{equation}
In position space, the exact equal time correlator is given by
\begin{equation}
G(x-y) = \left(\frac{k_F}{\pi}\right) \delta (x-y) \left( \frac{\sin [k_F(x-y)]}{k_F(x-y)}\right)-\left(\frac{k_F}{\pi}\right)^2 \left( \frac{\sin [k_F(x-y)]}{k_F(x-y)}\right)^2 .
\end{equation}
Integration of this quantity over the interval A yields the result (\ref{4-1}).
The perturbation expansion is in powers of $k/k_F$. Thus for all $k < 2k_F$ the result is indeed given exactly by the lowest order result (\ref{3-22}), consistent with what we found. The result for $k > 2k_F$, which is responsible for the finiteness of the entanglement entropy, is inaccessible in perturbation theory.
In the large interval limit $k_F(b-a)\gg1$, the entanglement entropy is given by
\begin{equation}
S_A^{(2)}=\frac{1}{3}\{\log [k_F(b-a)]+1+\gamma+\log 2+\cdots\}.
\end{equation}
Notice that this result agrees with the lowest order calculation, except that the UV cutoff $\epsilon$ has been replaced by a finite number $1/k_F$. This can be understood as follows. In the large interval limit, the small momentum $G_0(k)$ contributes which can be calculated perturbatively. However the exact result (\ref{3-22}) shows that the low momentum behavior changes at $k \sim k_F$ - thus $k_F$ acts as a cutoff.
In the small interval limit $k_F(b-a)\ll1$, the entanglement entropy is given by
\begin{equation}
S_A^{(2)}=\frac{1}{3}\{\pi k_F(b-a)+k_F^2(b-a)^2+\cdots\}.
\end{equation}
Unlike the large interval limit, this is extensive in the interval size. In this limit $G_0(k)$ is a constant so that the position space equal time correlator is a Dirac delta function, and the expression for the leading entanglement entropy (\ref{2-4}) leads to this extensive behavior.
\section{Entanglement in the Collective and Eigenvalue Descriptions}
\label{four}
As emphasized in the introduction, the preceding calculations are those of entanglement entropies as defined in the fermionic description, but calculated using the collective theory. In this section we discuss the connection of this quantity with the entanglement directly defined in terms of the bosonic collective field. This is the question of the relationship of the notion of entanglement of a region and bosonization.
In bosonization a fermion field is related to the boson field by a transformation which is non-local in space. Therefore, {\em a priori} the notion of locality in terms of bosons could be generally quite different for the notion of locality in terms of fermions and may lead to different entanglement entropies. For relativistic fermions and spin models this issue has been discussed in the literature \cite{ee-boson-fermion}.
For a non-relativistic fermionic many-body theory which is considered in this paper, the situation is rather different. This is because there is conserved number of fermions and a first-quantized description where the fermion coordinates are the dynamical variables. In fact, this is the basic description which comes from matrix quantum mechanics. Second-quantized fermionic many-body theory and collective field theory are two different formulations of this many-body system. The operators in the first-quantized formalism are of the type
\begin{equation}
{\cal O}_{mn} = \sum_{i=1}^N \hat{\lambda}_i^m \hat{p}_i^n ,
\label{6-1}
\end{equation}
where $\hat{p}_i$ are the momenta conjugate to the position operators $\hat{\lambda}_i$, and various orderings of these operators.
In terms of the second-quantized fermion field $\psi (x)$ this is
\begin{equation}
{\cal O}_{mn} = \int dx~ \psi^\dagger x^m (-i\partial_x)^n \psi,
\label{6-2}
\end{equation}
while the expression in terms of the collective field should be obtained by making a change of variables from $\{ \lambda_i \}$ to $\phi(x)$,
\begin{equation}
\phi(x) = \frac{1}{N} \sum_{i=1}^N \delta (x - \hat{\lambda}_i)
\label{6-3}
\end{equation}
and using the chain rule \cite{Jevicki:1979mb}.
The notion of entanglement of a subregion $A$ is best understood in terms of a subalgebra of operators. In the fermionic many-body theory the set of operators are simply those which are made out of the fermion fields $\psi (x), \psi^\dagger (x)$, with $x \in A$ . In the first-quantized language, specifying the subalgebra requires a constraint on the target space. This is best done by defining a projection operator for each $i$ \cite{target3}
\begin{equation}
P_i = \int_A dy~\delta(y - \hat{\lambda}_i) .
\label{6-4}
\end{equation}
The subalgebra of operators are then obtained by replacing
\begin{equation}
(\hat{\lambda}_i, \hat{p}_i) \rightarrow (P_i \hat{\lambda}_i P_i, P_i \hat{p}_i P_i) .
\end{equation}
This projection breaks up the Hilbert space into a direct sum of super-selection sectors characterized by the number of particles $k$ which are in the subregion $A$. This is most easily seen by computing the expectation value of many-body operators of the form $\mathcal{O}_{mn}$ in some state described by a properly anti-symmetrized wavefunction $\Psi (\lambda_1,\lambda_2,\cdots \lambda_N)$. Consider operators of the form
\begin{equation}
{\cal O}_{m} = \sum_{i=1}^N \hat{\lambda}_i^m ,
\label{6-5}
\end{equation}
whose projected version is
\begin{equation}
{\cal O}^P_{m} = \sum_{i=1}^N P_i\hat{\lambda}_i^m .
\label{6-5a}
\end{equation}
It is straightforward to see that the expectation value of the projected operator is
\begin{equation}
\langle \Psi |{\cal O}^P_m | \Psi \rangle
= \sum_{k=1}^N {N \choose k} \sum_{i=1}^k \int_A \prod_{a=1}^k d\lambda_a \int_{\bar{A}} \prod_{\alpha=k+1}^N d\lambda_\alpha~\Psi^\star (\lambda_1 \cdots \lambda_N) \lambda_i^m (\lambda_1 \cdots \lambda_N)
\label{6-7}
\end{equation}
where $\bar{A}$ is the complement of the region $A$. This contains a sum over the sectors mentioned above. In each sector labelled by $k$ the result can be obtained from an (unnormalized) reduced density matrix $\tilde{\rho}_k$ which is an operator in the $k$-particle Hilbert space of particles living in the region $A$. The trace ${\rm tr} \,\tilde{\rho}_k$ is the probability of finding $k$ particles in $A$.
The reduced density matrix of the entire system is block-diagonal
\begin{equation}
\rho = {\rm diag}(\tilde{\rho}_1, \tilde{\rho}_2, \cdots \tilde{\rho}_N)
\label{6-8}
\end{equation}
and normalized (since the sum of probabilities is unity) and the target space entanglement entropy is given by the von Neumann entropy of $\rho$.
In the second-quantized fermionic many-body theory the Hilbert space is a direct product as usual and the reduced density matrix is obtained simply by integrating out the fermion fields in $\bar{A}$. This density matrix which evaluates fermion bilinears in this region is identical to $\rho$ defined in (\ref{6-8}) - as was explicitly shown for Slater determinant states in \cite{target2}.
Let us now come to collective field theory. The subalgebra of operators pertaining to a region $A$ is the subalgebra of operators formed by taking products of $\{ \phi(x), \Pi (x) \}$, with $x \in A$. The restriction to $A$ can be implemented again by a projector, i.e.
\begin{equation}
\phi^P(x) = \int_A dy~ \delta (y-x) \phi (y) ,
\label{6-9}
\end{equation}
and similarly for $\pi^P(x)$. Focusing on many-body operators of the form (\ref{6-5}) the subalgebra now consists of operators of the form
\begin{equation}
{\tilde{\cal O}}^P_m = \int_R dx\, \phi^P(x) x^m = \sum_{i=1}^N \int_A dy~y^m\delta(y-\hat{\lambda}_i) .
\label{6-10}
\end{equation}
Clearly the subalgebra of operators ${\tilde{\cal O}}^P_m$ is identical to the subalgebra of operators ${\cal O}^P_{m}$, as may be checked explicitly by computing expectation values in arbitrary states. Therefore the reduced density matrices which evaluates these are identical as well.
The same projector can be used to obtain the projected versions of many-body operators which involve momenta $p_i$. In terms of the collective field these involve integrals over the collective field and powers of the conjugate momentum, and the discussion above can be generalized.
We now discuss further applications of the collective field approach to the problem of target space entanglement in several problems of direct interest to string theory.
\subsection{Field theory of long strings}
A single matrix quantum mechanics with inverted oscillator potential $V(M)=-M^2/2$ is defined by the Hamiltonian
\begin{equation}
H={\rm Tr} \left[-\frac{1}{2}\left(\frac{\partial}{\partial M}\right)^2-\frac{1}{2}M^2\right].
\end{equation}
Here $M$ is a Hermitian $N\times N$ matrix, which can be polar-decomposed in the form
\begin{equation}
M=\Omega^\dagger\Lambda\Omega
\end{equation}
for some matrix $\Omega\in SU(N)/\mathbb{H}$ with $\mathbb{H}$ being the stablizer, where $\Lambda=\operatorname{diag}(\lambda_1,\cdots,\lambda_N)$ is diagonal. The invariance of the theory under $SU(N)$ transformation implies that we we can rewrite the Hamiltonian as
\begin{equation}
H=-\frac{1}{2}\left(\frac{1}{\Delta}\sum_{i}\frac{\partial^2}{\partial\lambda_i^2}\Delta+\sum_{i}\lambda_i^2\right)+\Xsum-_{i,j}\frac{Q_{ij}^\mathcal{R}Q_{ji}^\mathcal{R}}{(\lambda_i-\lambda_j)^2},
\end{equation}
associated with wavefunction $\Psi(\Lambda,\Omega)$ invariant under $S_N\ltimes U(1)^N$ gauge redundancy, where
\begin{equation}
\Delta=\prod_{i<j}(\lambda_i-\lambda_j)
\end{equation}
is the Vandermonde determinant, and $Q_{ij}^\mathcal{R}$ is the $ij$ generator of $SU(N)$ under the representation $\mathcal{R}$.
While the singlet sector of matrix quantum mechanics reduces to a theory of {\em non-interacting} fermions, non-singlet sectors lead to {\em interacting} fermions whose coordinates are again given by the eigenvalues. In particular the long string sector is described by the adjoint sector and becomes related to the spin-Calogero model \cite{Aniceto:2006rr}. The problem of target space entanglement in the many-body quantum mechanics of these particles can be formulated exactly as above. In fact there is a well known collective field theory formulation of the Calogero model using its bosonized current-algebra representation, so that this can be reformulated in terms of collective fields \cite{Andric:1982jk,sen,Bardek:2010jg,Aniceto:2006rr}
\begin{equation}
\begin{split}
H=&\int dx\left[\frac{1}{2}\partial_x\Pi(x)\phi(x)\partial_x\Pi(x)+\frac{\pi^2}{6}\phi^3(x)-\frac{1}{2}x^2\phi(x)-\partial_x\Pi(x)\bar{\psi}(x)\partial_x\psi(x)\right]\\
&-\Xint- dx dy \,\bar{\psi}(y)\frac{\phi(x)}{(x-y)^2}\psi(x)-\Xint- dx\,\bar{\psi}(x)\left[\partial_x\Xint- dy\,\frac{\phi(y)}{x-y}\right]\psi(x)\\
&+\frac{1}{2}\Xint- dx dy\,\bar{\psi}(x)\bar{\psi}(y)\left[\frac{\psi(x)-\psi(y)}{x-y}\right]^2 ,
\end{split}
\end{equation}
where boson $\phi$ and fermion $\psi$ represent closed string and long string respectively.
Since this is a model of interacting fermions, there is no direct connection between counting statistics \cite{Smith:2020gfl} and entanglement entropy. Nevertheless the short distance behavior of collective field correlators determines the behavior of the entanglement entropy. Preliminary results suggest that this can be obtained in a manner similar to the case detailed in this paper.
\subsection{Multi-matrix models and higher dimensional strings}
In \cite{target1,target2} the notion of target space entanglement has been generalized to multi-matrix models, e.g. the BFSS or the BMN matrix models.
A Kaluza-Klein expansion of the $\mathcal{N}=4$ super-Yang-Mills theory on $\mathbb{R}\times \mathbb{S}^3$ in terms of spherical harmonics on $\mathbb{S}^3$ leads to matrix model reduction. Keeping only the zero mode degrees of freedom, for the Higgs sector the Lagrangian reads
\begin{equation}
L=\operatorname{Tr}\left\{\frac{1}{2}\sum_i\left(\dot{\Phi}_i^2-\frac{1}{R^2}\Phi_i^2\right)+\frac{1}{4}\Xsum-_{i,j}[\Phi_i,\Phi_j]^2\right\},\quad i,j=1,\cdots,6.
\end{equation}
The holomorphic notation introduces $SU(3)$ triplet $Z_i=\Phi_i+i\Phi_{i+3}$ and their complex conjugates $\bar{Z}_i$. Restriction to $1/2$-BPS configurations corresponds to single trace operators involving only the chiral primary operators of the general form $\operatorname{Tr}Z^n$. This model is essentially a one-matrix model described in this work. More generally addressing two-matrix problem, we may consider the simplest case of the complex matrix model with
\begin{equation}
Z=A+B^\dagger, \quad \bar{Z}=A^\dagger+B.
\end{equation}
A gauge invariant notion of target space entanglement can be formulated in the following way. In a theory of several Hermitian matrices $M^I, I = 1 \cdots K$ consider a function $f(M)$ which is itself a Hermitian matrix. Then define a projector
\begin{equation}
P^f_{ij} = \int_A dy~\left[ \delta (y \cdot I - f(M)) \right]_{ij} .
\label{7-1}
\end{equation}
A set of gauge invariant operators are of the form
\begin{equation}
{\cal {O}}^{I_1I_2\cdots} = {\rm Tr} \left[ M^{I_1} M^{I_2} \cdots \right] .
\label{7-2}
\end{equation}
The projector (\ref{7-1}) can be then used to define a subalgebra of operators
\begin{equation}
{\cal {O}}_f^{I_1I_2\cdots} = {\rm Tr} \left[ P^fM^{I_1} P^fM^{I_2}P^f \cdots \right] .
\label{7-3}
\end{equation}
There is a reduced density matrix which evaluates expectation values of operators belonging to this subalgebra, and an associated entanglement entropy. This is a completely gauge invariant specification of the subalgebra.
In a gauge in which $f(M)$ is diagonal, the operator $P^f$ projects out the eigenvalues of $f(M)$ which lie in some specified interval $A$. A simple example involves $f(M) = M^1$. Then the eigenvalues of $M^1$ which lie in the interval $A$ are retained. In a sector where $n$ of the eigenvalues lie in $A$, $P^f$ projects onto an $n \times n$ block of the other matrices $M^I, I\neq 1$ the operator projects out. In the BFSS or BMN model, we have a $K$ dimensional target space $x^1 \cdots x^K$ and the eigenvalues of the matrices represent the locations of D0 branes in this target space. The reduced density matrix then evaluates measurements made on D0 branes whose $x^1$ lies in an interval A and the projection amounts to integrating out the open strings joining branes which do not lie in this region. Target space entanglement provides a concrete notion of entanglement in the string field theory dual to these matrix models.
The BMN matrix model has a collective field formulation \cite{bmncollective}. One has the general collective loops of $W, X$ and $Y$
\begin{equation}
{\rm Tr}\,(W^{n_1}X^{m_1}Y^{k_1}W^{n_2}X^{m_2}\cdots) .
\end{equation}
These invariant loops variables denoted collectively by $\phi_C$ constitute all the observables in the full string field theory, where $C$ stands for word index. The collective Hamiltonian can be expressed in terms of $\phi_C$ and its conjugate $\pi_C$. The emergent spacetime is again seen through collective density \cite{Donos:2005vm}. In a way analogous to our treatment of the one-matrix problem one should then be able to consider entanglement in terms of this collective field. Potential evaluation of entanglement entropy can be done through numerical methods introducing in \cite{Koch:2021yeb}.
\section{Discussion}
\label{five}
In this paper we explored the question of finiteness of entanglement entropy in theories whose spatial dimensions emerge out of matrix degrees of freedom. More specifically, we addressed the question concretely in collective field theory of matrix quantum mechanics which becomes equivalent to two-dimensional non-relativistic fermions in an external potential. When the external potential is a regulated inverted harmonic oscillator this collective field theory is a string field theory of non-critical strings and the perturbation expansion is a string loop expansion. In the fermionic description the entanglement entropy is manifestly finite for a finite particle number density. However the collective field theory fluctuations are described by a self-interacting relativistic massless scalar field whose coupling is proportional to $k/k_F$. Thus to the lowest order in a perturbation expansion the result has the usual logarithmic divergence. The question is to understand how the interactions render the answer finite. This question is independent of the nature of the external potential.
We have answered this question unequivocally for the case where the potential is vanishing. In this case, the collective theory can be solved exactly \cite{Jevicki:1991yi} and the exact eigenvalues of the Hamiltonian are known to reproduce the fermion dispersion relation. Here we verified that the connected two-point function exactly reproduces the fermion four-point function. Since the leading term in the entanglement entropy involves an integral of this correlator this also leads to the correct exact answer. If we treat the interaction perturbatively we show that the leading order divergence is not cured in any finite order of the perturbation expansion. However the series can be resummed (as noted in \cite{Pereira:2007}) yielding the exact answer. The finiteness of the entanglement entropy is therefore essentially non-perturbative \footnote{It should be noted that we have performed a canonical perturbation expansion using the Hamiltonian. The conjugate momenta is non-polynomial in the time derivative of the field, so that there are an infinite number of vertices. A perturbation expansion using the Lagrangian will be rather complicated. However, a careful calculation should display cancellations and establish agreement with the Hamiltonian perturbative expansion.}.
When the external potential is an inverted harmonic oscillator, this system is a description of string field theory of bosonic non-critical string and the perturbation expansion is the string loop expansion. In this case, the collective field theory is more subtle. In particular in Hamiltonian has additional singular counter-terms which are subleading in $1/N$, and these are essential for a detailed correspondence to string theory. Nevertheless, we expect that the same mechanism will work in this case, i.e. the divergence is present in all finite orders of perturbation theory and its cure is non-perturbative. In the string theory this means that the scale which renders the entanglement entropy finite involves the string coupling $g_{st}$.
The main reason why the entanglement entropy is finite in the theories we consider is that the dynamics of the target space of these matrix models is non-relativistic in nature. This drastically alters the short distance behavior of correlations.
We have not performed any explicit calculation for theories with multiple matrices which are relevant to higher dimensional strings. However in known examples, e.g. the BFSS or BMN models, the target space is again non-relativistic. As conjectured in \cite{target2,target3} one would expect a similar mechanism for these models.
In the examples we investigated in this paper, the finiteness of the entanglement entropy persists in a double-scaling limit where $N \rightarrow \infty$ and some other parameter (e.g. the size of the box for fermions in a box with no potential, or the Fermi level measured from the top of an inverted harmonic oscillator potential) also tuned keeping the coupling fixed. The treatment in section \ref{sec:2.1} is valid for any finite $N$. However only $N$ of the $\phi_m$ are independent variables because of trace relations. Naively this fact, also called the ``stringy exclusion principle", did not play a role in the subsequent analysis where we took both $N \rightarrow \infty$ and $L \rightarrow \infty$ keeping $k_F$ fixed. This point demands further investigation. It will be interesting to see if there are similar limits in these higher dimensional models.
Finally it will be interesting to investigate the connection of the origin of finiteness discussed in this paper to other recent discussions based on the types of von Neumann algebras
\cite{liu}.
\section{Acknowledgements} S.R.D. would like to thank Cesar Gomez, Gautam Mandal and Sandip Trivedi for discussions and Instituto de Fisica Teorica, Madrid for hospitality during the completion of this manuscript. The work of S.R.D. is supported by National Science Foundation grants NSF/PHY-1818878 and NSF/PHY-2111673 and by the Jack and Linda Gill Chair Professorship. The work of A.J. and J.Z. is supported by the U.S. Department of Energy under contract DE-SC0010010.
\section{Appendix}
In this appendix we will provide the proof of the exact dispersion relation (\ref{3-16}). This result follows from two fusion rules. The first is the Littlewood-Richardson rule, which states that the fusion of Schur polynomials is determined by the equation
\begin{equation}
s_\lambda s_\mu=\sum_\nu f^\nu_{\lambda,\mu}s_\nu,
\end{equation}
with coefficients $f^\nu_{\lambda,\mu}$ equal to the number of the Littlewood–Richardson tableaux of skew shape $\nu/\lambda$ and of weight $\mu$.
The second is the fusion rule of characters of permutation group $S_n$
\begin{equation}
\frac{C_\rho}{d_\lambda}\chi^\lambda_\rho \frac{C_\mu}{d_\lambda}\chi^\lambda_\mu=\sum_\nu g^\nu_{\rho,\mu}\frac{C_\nu}{d_\lambda}\chi^\lambda_\nu,
\end{equation}
where the depth $d_\lambda$ of a Young tableau $\lambda$ is the number of boxes that do not belong to the first row. And the number of different permutations in the conjugacy class $\nu$ is given by
\begin{equation}
C_\nu=\frac{n!}{\prod_j \nu_j!j^\nu},
\end{equation}
where $n!$ is the total number of elements in the permutation group $S_n$. The idea is to choose
\begin{equation}
\rho=\left(1^{n-2},2^1\right).
\end{equation}
With this choice, we have
\begin{equation}
C_\rho=n(n-1)
\end{equation}
and
\begin{equation}
\frac{C_\rho}{d_\lambda}\chi^\lambda_\rho=\sum_j\lambda_j(\lambda_j-2j+1)
\end{equation}
which is exactly the eigenvalue $E_3(\lambda)$ of $H_3$. We then have the eigenequation representing a special case of the multiplication formula
\begin{equation}
H_3 C_\mu\chi^\lambda_\mu=\sum_\nu g^\nu_{\rho,\mu}C_\nu\chi^\lambda_\nu.
\end{equation}
Working out the special structure constant $g^\nu_{\rho,\mu}$ one gets
\begin{equation}
\sum_k k\sum_{l=1}^{k-1}C_{\nu,s}\chi^\lambda_{\nu,s}+\sum_{k<l}kl C_{\nu,j}\chi^\lambda_{\nu,j},
\end{equation}
where 's' denotes splitting of the conjugacy class $C_\nu$
\begin{equation}
\phi_k\rightarrow\phi_l, \phi_{k-l},
\end{equation}
while 'j' denotes joining of the conjugacy class $C_\nu$
\begin{equation}
\phi_k, \phi_l\rightarrow \phi_{k+l}.
\end{equation}
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\large \bf
Kun Hao${}^{a}$,~Junpeng Cao${}^{b,c}$,~Guang-Liang Li${}^{d}$,~Wen-Li Yang${}^{a,e}\footnote{Corresponding author:
wlyang@nwu.edu.cn}$,
~ Kangjie Shi${}^a$ and~Yupeng Wang${}^{b,c}\footnote{Corresponding author: yupeng@iphy.ac.cn}$
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\newcommand{\Address}{\begin{center}
${}^a$Institute of Modern Physics, Northwest University,
Xian 710069, China\\
${}^b$Beijing National Laboratory for Condensed Matter
Physics, Institute of Physics, Chinese Academy of Sciences, Beijing
100190, China\\
${}^c$Collaborative Innovation Center of Quantum Matter, Beijing,
China\\
${}^d$Department of Applied Physics, Xian Jiaotong University, Xian 710049, China\\
${}^e$Beijing Center for Mathematics and Information Interdisciplinary Sciences, Beijing, 100048, China
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\preprint \thispagestyle{empty}
\bigskip\bigskip\bigskip
\Title{Exact solution of an
$su(n)$ spin torus} \Author
\Address \vspace{1cm}
\begin{abstract}
The trigonometric $su(n)$ spin chain with anti-periodic boundary
condition ($su(n)$ spin torus) is demonstrated to be Yang-Baxter
integrable. Based on some intrinsic properties of the $R$-matrix,
certain operator product identities of the transfer matrix are
derived. These identities and the asymptotic behavior of the
transfer matrix together allow us to obtain the exact eigenvalues in
terms of an inhomogeneous $T-Q$ relation via the off-diagonal Bethe
Ansatz.
\vspace{1truecm} \noindent {\it PACS:} 75.10.Pq, 03.65.Vf, 71.10.Pm
\noindent {\it Keywords}: Spin chain; Bethe
Ansatz; $T-Q$ relation
\end{abstract}
\newpage
\section{Introduction}
\label{intro} \setcounter{equation}{0}
Study on quantum integrable models \cite{bax1,Kor93} has played an essential role in many areas of physics, such as condensed matter physics,
quantum field theory, the AdS/CFT \cite{Mal98, Bei12} correspondence in string theory, nuclear physics, atomic and molecular physics and ultracold atoms.
Strikingly, the algebraic Bethe Ansatz (BA) method \cite{Tak79} to solve quantum integrable models with obvious reference states has inspired and led to remarkable developments in different branches of mathematical physics in the past decades.
While for quantum integrable models without $U(1)$-symmetry, obvious reference states are usually absent,
making the conventional Bethe Ansatz methods \cite{bax1,Tak79,Bet31,Skl78,Alc87,Skl88} almost inapplicable.
Recently, a new approach, i.e., the off-diagonal Bethe Ansatz (ODBA)\cite{Cao1} (for comprehensive introduction we refer the reader to \cite{Wan15}) was proposed to obtain exact solutions of generic integrable models either with or without $U(1)$ symmetry.
Several long-standing models were then solved \cite{Cao1,Cao13NB875,Cao14NB886,Cao13NB877,Cao14JHEP143,Li14,Zha13,Hao14} via this method. It should be remarked that some other interesting methods such as
the q-Onsager algebra method \cite{Bas1,Bas2,Bas3}, the modified algebraic Bethe ansatz method \cite{Bel13,Bel15,Bel15-1,Ava15} and the Sklyanin's separation of variables (SoV) method \cite{Skl92} were also applied to some integrable models related to the $su(2)$ algebra \cite{Fra08,Nic12,Fad14,Kit14}.
Quantum spin models provide a typical setting of quantum fluctuations leading to various exotic spin liquid states \cite{fazekas74,misguich03}.
The Bethe Ansatz solution \cite{Cao1} of the spin-$1/2$ chain with anti-periodic boundary conditions ($su(2)$ topological spin torus or quantum M\"obius stripe) is a well-known example to reveal topological nature of elementary excitations in such kind of systems. An interesting issue is to study the high-rank systems with topological boundaries.
We note that the models of $su(n)$ quantum spin systems are far from merely theoretical
exercises: It could be realized either in cold-atom systems in
optical lattices \cite{honerkamp04,hofstetter05,buchler05},
in quantum dot arrays \cite{onufriev}, or in spin systems with orbital degrees of freedom \cite{mila2003}. The aim of the present work is to study the integrability and exact spectrum of the trigonometric $su(n)$ chain with anti-periodic boundary condition with the nested ODBA \cite{Cao14JHEP143}.
The paper is organized as follows. Section 2 serves as an introduction of our notations and some basic ingredients. The commuting transfer matrix associated with the
$su(n)$ spin torus is constructed to show the integrability of the model.
In section 3, taking the $su(3)$ spin torus as a concrete example, we derive some operator identities based on intrinsic
properties of the $R$-matrix, which allow us to give the eigenvalues of the transfer matrix
in terms of a nested inhomogeneous $T-Q$ relation. The corresponding Bethe Ansatz equations (BAEs) are also given. The generalization
to $su(n)$ case is given in section 4. We summarize our results and give some discussions in Section 5.
The generic integrable twisted boundary conditions are shown in Appendix A. Some details about the $su(4)$ case, which could be crucial to understand the procedure for $n\geq4$, are given in Appendix B.
\section{$su(n)$ spin torus }
\setcounter{equation}{0}
Let ${\rm\bf V}$ be an $n$-dimensional linear space with an
orthonormal basis $\{|i\rangle|i=1,\cdots,n\}$, we introduce the
Hamiltonian \bea H=\sum_{j=1}^Nh_{j\,j+1},\label{Ham} \eea where $N$
is the number of sites and $h_{j\,j+1}$ is the local Hamiltonian
given by \bea h_{j\,j+1}=\frac{\partial}{\partial
u}\lt.\lt\{P_{j\,j+1}\,R_{j\,j+1}(u)\rt\}\rt|_{u=0}. \eea Here $P$
is the permutation operator on the tensor space ${\rm\bf V}\otimes
{\rm\bf V}$; the $R$-matrix $R(u)\in {\rm End}({\rm\bf V}\otimes
{\rm\bf V})$ is the trigonometric $R$-matrix associated with the
quantum group \cite{Cha94} $U_q(\widehat{su(n)})$, which was given in
\cite{Che80,Chu80,Sch81,Bab81,Perk81} and further studied in
\cite{Perk83,Schu83,Perk06,Baz85,Jim86}\footnote{The $R$-matrix
given by (\ref{R-matrix-1}) corresponds to the so-called principal
gradation, which is related to the $R$-matrix in homogeneous
gradation by some gauge transformation \cite{Nep02}.}
\bea R(u) &=&
\sinh({u} + \eta) \sum_{k=1}^{n} E^{k\,, k}\otimes E^{k\,, k}+
\sinh{u}
\sum_{k \ne l}^n E^{k\,, k}\otimes E^{l\,, l} \no \\
&&+ \sinh\eta \left(\sum_{k<l}^ne^{\frac{n-2(l-k)}{n}u} +
\sum_{k>l}^n e^{-\frac{n-2(k-l)}{n}u} \right) E^{k\,, l}\otimes
E^{l\,, k} ,\label{R-matrix-1} \eea where the $n^2$ fundamental
matrices $\{E^{k,l}|k,l=1,\cdots,n\}$ are all $n\times n$ matrices
with matrix entries
$(E^{k,l})^{\a}_{\b}=\delta^k_{\a}\,\delta^l_{\b}$ and $\eta$ is the
crossing parameter. The $R$-matrix satisfies the quantum Yang-Baxter
equation (QYBE)
\begin{eqnarray}
R_{12}(u_1-u_2)R_{13}(u_1-u_3)R_{23}(u_2-u_3)=
R_{23}(u_2-u_3)R_{13}(u_1-u_3)R_{12}(u_1-u_2), \label{QYB}
\end{eqnarray}
and possesses the properties:
\begin{eqnarray}
&&\hspace{-1.45cm}\mbox{
Initial condition}:\hspace{42.5mm}R_{12}(0)= \sinh\eta\, P_{1\,2},\label{Initial}\\[4pt]
&&\hspace{-1.5cm}\mbox{
Unitarity}:\hspace{11.5mm}R_{12}(u)R_{21}(-u)= \rho_1(u)\times {\rm id},\quad \rho_1(u)=-\sinh(u+\eta)\sinh(u-\eta),\label{Unitarity}\\[4pt]
&&\hspace{-1.5cm}\mbox{
Crossing-unitarity}:\,
R^{t_1}_{12}(u)R_{21}^{t_1}(-u-n\eta)
=\rho_2(u)\times \mbox{id},\; \rho_2(u)=-\sinh u\sinh(u+n\eta),
\label{crosing-unitarity}\\[4pt]
&&\hspace{-1.4cm}\mbox{Fusion conditions}:\hspace{22.5mm}\, R_{12}(-\eta)\propto P^{(-)}_{1\,2},\label{Fusion}\\[4pt]
&&\hspace{-1.4cm}\mbox{Periodicity}:\hspace{22.5mm}\, R_{12}(u+i\pi)=-h_1\,R_{12}(u)\,h_1^{-1}=-h_2^{-1}\,R_{12}(u)\,h_2
.\label{Periodicity-R}
\end{eqnarray}
Here $R_{21}(u)=P_{1\,2}R_{12}(u)P_{1\,2}$;
$P^{(-)}_{1\,2}$ is the q-deformed anti-symmetric
project operator in the tensor product space ${\rm\bf
V} \otimes {\rm\bf V} $ (such as below (\ref{P-3}) and (\ref{Fusion-3})); $t_i$ denotes the transposition in the
$i$-th space; $h$ is an $n\times n$ diagonal matrix given by
\bea
h=\lt(\begin{array}{cccc}1&&&\\
&\omega_n&&\\
&&\ddots&\\
&&&\omega_n^{n-1}\\
\end{array}\right),\quad \omega_n=e^{\frac{2i\pi}{n}},\quad {\rm and}\quad h^n=1. \label{h-matrix}
\eea
Here and below we adopt the standard notation: for any
matrix $A\in {\rm End}({\rm\bf V})$, $A_j$ is an embedding operator
in the tensor space ${\rm\bf V}\otimes {\rm\bf V}\otimes\cdots$,
which acts as $A$ on the $j$-th space and as an identity on the
other factor spaces; $R_{ij}(u)$ is an embedding operator of
$R$-matrix in the tensor space, which acts as an identity on the
factor spaces except for the $i$-th and $j$-th ones.
In order to construct the quantum spin chain with integrable twisted boundary condition \cite{deV84}, let us introduce an $n\times n$ twist matrix $g$
\bea
g=\lt(\begin{array}{cccc}&&&1\\
1&&&\\
&\ddots&&\\
&&1&\\
\end{array}\right),\quad g^n=1.\label{g-matrix}
\eea
It can be easily checked that the $R$-matrix (\ref{R-matrix-1}) is invariant with $g$, namely,
\bea
&&R_{12}(u)\,g_1\,g_2=g_1\,g_2\,R_{12}(u),\label{Invariant-R}\\
&&h\,g=\omega_n\,g\,h,\label{hg-relation}
\eea
where $h$ is given by (\ref{h-matrix}). (Generic twist matrix satisfing the above equation
is given in Appendix A.) This property enables us to construct the integrable $su(n)$ spin torus model.
Similar to the $su(2)$ spin torus (or the XXZ spin chain with anti-periodic boundary condition) \cite{Bat95},
the $su(n)$ spin torus is described by the Hamiltonian $H$ given by (\ref{Ham}) with anti-periodic
boundary conditions
\bea
E^{k,l}_{N+1}=g_1\,E^{k,l}_1\,g_1^{-1},\quad k,l=1,\cdots,n.\label{anti-boundary}
\eea
Let us introduce the ``row-to-row" monodromy matrix
$T(u)$, an $n\times n$ matrix with operator-valued elements
acting on ${\rm\bf V}^{\otimes N}$, \bea T_0(u)
=R_{0N}(u-\theta_N)R_{0\,N-1}(u-\theta_{N-1})\cdots
R_{01}(u-\theta_1).\label{Monodromy-1} \eea Here
$\{\theta_j|j=1,\cdots,N\}$ are generic free complex parameters
which are usually called as inhomogeneity parameters.
The transfer matrix $t(u)$ of the associated spin chain describing the Hamiltonian (\ref{Ham}) with the antiperiodic
boundary condition (\ref{anti-boundary}) can be constructed similarly as \cite{Skl88,deV84,Bat95}
\begin{eqnarray}
t(u)&=&tr_0\lt\{g_0\,T_0(u)\rt\}.\label{transfer}
\end{eqnarray}
The QYBE (\ref{QYB}) and the definition (\ref{Monodromy-1}) of the monodromy matrix $T(u)$ imply that the matrix elements of $T(u)$
satisfy the Yang-Baxter algebra:
\bea
R_{12}(u-v)\,T_1(u)\,T_2(v)=T_2(v)\,T_1(u)\,R_{12}(u-v).\label{Yang-Baxter-algebra}
\eea
The above relation and the invariant relation (\ref{Invariant-R})
lead to the fact that the transfer matrices $t(u)$ given by (\ref{transfer}) with
different spectral parameters are mutually commuting:
$[t(u),t(v)]=0$. The Hamiltonian (\ref{Ham}) with the
anti-periodic boundary condition (\ref{anti-boundary})
can be obtained from the transfer matrix as
\begin{eqnarray}
&&H=\sinh\eta\, \frac{\partial \ln t(u)}{\partial u}|_{u=0,\{\theta_j\}=0}.
\end{eqnarray}
This ensures the integrability of the $su(n)$ spin torus.
The aim of this paper is to obtain eigenvalues of the transfer matrix $t(u)$ specified by the twist matrix (\ref{g-matrix}) via ODBA.
\section{ODBA solution of the $su(3)$ spin torus}
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Following the method developed in \cite{Cao14JHEP143}, we apply the fusion techniques \cite{Kar79,Kulish81,Kulish82,Kirillov86} to study the $su(n)$ spin torus. For this purpose, besides the fundamental transfer
matrix $t(u)$ some other fused transfer matrices $\{t_j(u)|j=1,\cdots,n\}$, which commute with each other and include
the original one as $t_1(u)=t(u)$, should be constructed through an anti-symmetric fusion procedure.
In this section, we present the results for the $su(3)$ spin torus.
\subsection{Operator identities of the transfer matrices}
For the $su(3)$ case, the $R$-matrix $R(u)$ given by (\ref{R-matrix-1}) reads
\bea\label{R-matrix-su3q}
R(u)=
\left(
\begin{array}{ccc|ccc|ccc}
\bar a(u) & & & & & & & & \\
&\bar b(u)& &\bar c(u)& & & & & \\
& &\bar b(u)& & & &\bar d(u)& & \\
\hline
&\bar d(u)& &\bar b(u)& & & & & \\
& & & &\bar a(u)& & & & \\
& & & & &\bar b(u)& &\bar c(u)& \\
\hline
& &\bar c(u)& & & &\bar b(u)& & \\
& & & & &\bar d(u)& &\bar b(u)& \\
& & & & & & & &\bar a(u)\\
\end{array}
\right)
\eea
with
\bea
&\displaystyle \bar a(u)=\sinh(u+\eta),\quad &\bar b(u)=\sinh(u),\no\\[4pt]
&\displaystyle \bar c(u)=e^{{u\over3}}\sinh(\eta),\quad &\bar d(u)=e^{-{u\over3}}\sinh(\eta),\no
\eea
and the twist matrix $g$ given by (\ref{g-matrix}) becomes
\bea
g=\lt(\begin{array}{ccc}0&0&1\\
1&0&0\\
0&1&0
\end{array}\right),\quad {\rm and}\,\,g^3=1.\label{g-matrix-3}
\eea
Following \cite{Hao14}, let us introduce the following vectors in the tensor space ${\rm\bf V}\otimes{\rm\bf V}$ associated
with the $R$-matrix (\ref{R-matrix-su3q})
\bea
|\Phi^{(1)}_{12}\rangle&=&\frac{1}{\sqrt{2e^{-\eta\over3}\cosh({\eta\over3})}}(|1,2\rangle-e^{-{\eta\over3}}|2,1\rangle),\no\\[6pt]
|\Phi^{(2)}_{12}\rangle&=&\frac{1}{\sqrt{2e^{\eta\over3}\cosh({\eta\over3})}}(|1,3\rangle-e^{{\eta\over3}}|3,1\rangle),\no\\[6pt]
|\Phi^{(3)}_{12}\rangle&=&\frac{1}{\sqrt{2e^{-\eta\over3}\cosh({\eta\over3})}}(|2,3\rangle-e^{-{\eta\over3}}|3,2\rangle),
\eea
and a vector in the tensor space ${\rm\bf V}\otimes{\rm\bf V}\otimes{\rm\bf V}$,
\bea
|\Phi_{123}\rangle&=&\frac{1}{\sqrt{6e^{-{\eta\over3}}\cosh({\eta\over3})}} (|1,2,3\rangle-e^{-{\eta\over3}}|1,3,2\rangle-e^{-{\eta\over3}}|2,1,3\rangle\no\\[6pt]
&&+|2,3,1\rangle+|3,1,2\rangle-e^{-{\eta\over3}}|3,2,1\rangle).
\eea
From these vectors we can construct the associated projectors\footnote{These operators are $q$-deformed anti-symmetric projectors and in contrast to the rational ones, $P^{(-)}_{21}=P_{12}P^{(-)}_{12}P_{12}\neq P^{(-)}_{12}$.}
\bea
P^{(-)}_{12}&=&|\Phi^{(1)}_{12}\rangle\langle\Phi^{(1)}_{12}|
+|\Phi^{(2)}_{12}\rangle\langle\Phi^{(2)}_{12}|+|\Phi^{(3)}_{12}\rangle\langle\Phi^{(3)}_{12}|,\label{P-3}\\[6pt]
P^{(-)}_{123}&=&|\Phi_{123}\rangle\langle\Phi_{123}|.
\eea
Direct calculation shows that the $R$-matrix given by (\ref{R-matrix-su3q}) at some degenerate points is proportional to the projectors,
\bea
R_{12}(-\eta)=P^{(-)}_{12}\times S^{(-)}_{12},\qquad
R_{12}(-\eta)R_{13}(-2\eta)R_{23}(-\eta)=P^{(-)}_{123}\times S^{(-)}_{123},\label{Fusion-3}
\eea
where $S^{(-)}_{12}\in {\rm End}({\rm\bf V}\otimes{\rm\bf V})$ and $S^{(-)}_{123}\in {\rm End}({\rm\bf V}\otimes{\rm\bf V}\otimes{\rm\bf V})$ are some non-degenerate diagonal matrices.
Now we are in position to derive some operator product identities of the transfer matrices which are crucial to obtain eigenvalues of the transfer matrix. Let us evaluate the products of the monodromy matrices at some special points, which lead to the useful relation (for details we refer the readers to the book \cite{Wan15}, chapter 7),
\bea
T_1(\theta_j)T_2(\theta_j-\eta)&=&P_{21}^{(-)}T_1(\theta_j)T_2(\theta_j-\eta),\label{Fusion-Products-1}\\[6pt]
T_1(\theta_j)P^{(-)}_{32}T_2(\theta_j-\eta)T_3(\theta_j-2\eta)P^{(-)}_{32}&=&
P^{(-)}_{321}T_1(\theta_j)T_2(\theta_j-\eta)T_3(\theta_j-2\eta)P^{(-)}_{32}.\label{Fusion-Products-2}
\eea
The invariance (\ref{Invariant-R}) of the $R$-matrix $R(u)$ and the relations (\ref{Fusion-3}) imply the relations
\bea
[g_1\,g_2,\,P^{(-)}_{21}]=0=[g_1\,g_2\,g_3,\,P^{(-)}_{321}].\label{comm-g-pm}
\eea
With the help of the above relations (\ref{Fusion-Products-1})-(\ref{comm-g-pm}), we can calculate the products of the
fundamental transfer matrices at some special points
\bea
t(\theta_j)t(\theta_j-\eta)&=&tr_{12}\{g_1T_1(\theta_j)g_2T_2(\theta_j-\eta)\}\no\\[4pt]
&=&tr_{12}\{g_1g_2T_1(\theta_j)T_2(\theta_j-\eta)\}\no\\[4pt]
&=&tr_{12}\{g_1g_2P_{21}^{(-)}T_1(\theta_j)T_2(\theta_j-\eta)\}\no\\[4pt]
&\overset{(\ref{comm-g-pm})}{=}&tr_{12}\{P_{21}^{(-)}g_1g_2P_{21}^{(-)}P_{21}^{(-)}T_1(\theta_j)T_2(\theta_j-\eta)P_{21}^{(-)}\}\no\\[4pt]
&=&tr_{12}\{g_{<12>}T_{<12>}(\theta_j)\}\no\\[4pt]
&=&t_2(\theta_j),\label{trans-trans}
\eea
where
\bea
g_{<12>}\equiv P_{21}^{(-)}g_1g_2P_{21}^{(-)},\quad\quad T_{<12>}(u)\equiv P_{21}^{(-)}T_1(u)T_2(u-\eta)P_{21}^{(-)},\no\eea
and the fused transfer matrix $t_2(u)$ is given by
\bea
t_2(u)=tr_{12}\{g_{<12>}T_{<12>}(u)\}.\label{transfer-2}
\eea
Moreover, we can derive
\bea
t(\theta_j)t_2(\theta_j-\eta)&=&tr_{123}\{g_1T_1(\theta_j)P^{(-)}_{32}g_2g_3P^{(-)}_{32}P^{(-)}_{32}
T_2(\theta_j-\eta)T_3(\theta_j-2\eta)\}\no\\[4pt]
&=&tr_{123}\{g_1g_{<23>}T_1(\theta_j)P^{(-)}_{32}T_2(\theta_j-\eta)T_3(\theta_j-2\eta)P^{(-)}_{32}\}\no\\[4pt]
&=&tr_{123}\{g_1g_2g_3T_1(\theta_j)T_{<23>}(\theta_j-\eta)\}\no\\
&\overset{(\ref{Fusion-Products-2})}{=}&tr_{123}\{g_1g_2g_3 P^{(-)}_{321}T_1(\theta_j)T_2(\theta_j-\eta)
T_3(\theta_j-2\eta)P^{(-)}_{32}\}\no\\[4pt]
&=&tr_{123}\{g_{<123>}T_{<123>}(\theta_j)\}\no\\[4pt]
&=&t_3(\theta_j),\label{trans-trans2}
\eea
where
\bea
g_{<123>}&=&P^{(-)}_{321}g_1g_2g_3P^{(-)}_{321},\no\\[6pt]
T_{<123>}(u)&=&P^{(-)}_{321}T_1(u)T_2(u-\eta)T_3(u-2\eta)P^{(-)}_{321}\no\\[6pt]
&=&\prod^N_{l=1}\sinh(u-\theta_l+\eta)\sinh(u-\theta_l-\eta)
\sinh(u-\theta_l-2\eta)P^{(-)}_{321}.\no
\eea
Direct calculation shows that
\bea
t_3(u)=tr_{123}\{g_{<123>}T_{<123>}(u)\}=(-1)^{(3-1)}{\rm Det}_{q}T(u)\times{\rm id},
\eea
where the quantum determinant function ${\rm Det}_{q}T(u)$ reads
\bea
{\rm Det}_{q}T(u)=\prod^N_{l=1}\sinh(u-\theta_l+\eta)\sinh(u-\theta_l-\eta)
\sinh(u-\theta_l-2\eta).\no
\eea
Then the relation (\ref{trans-trans2}) becomes
\bea
t(\theta_j)t_2(\theta_j-\eta)={\rm Det}_{q}T(\theta_j)\times{\rm id}.\label{trans-trans-1}
\eea
Using the initial condition (\ref{Initial}) and the unitarity relation (\ref{Unitarity}),
we can derive that the fused transfer matrix $t_2(u)$ vanishes at the points: $\theta_j+\eta$, i.e.,
\bea
t_2(\theta_j+\eta)=0,\quad j=1,\cdots,N.
\label{fusion-orth}\eea
Moreover, it follows from the fusion procedure and the QYBE (\ref{QYB}) that the fused transfer matrices constitute commutative families, namely,
\bea
[t_i(u),t_j(v)]=0,\qquad i,j=1,2,3.\label{commu-trans-su3}
\eea
Now let us consider the periodicities and the asymptotic behaviors of the transfer matrices $t(u)$ and $t_2(u)$. The periodicity (\ref{Periodicity-R}) of the
the $R$-matrix $R(u)$ and the definition (\ref{Monodromy-1}) of the monodromy matrix give rise to the relation
\bea
T(u+i\pi)=(-1)^N\,h\,T(u)\,h^{-1},\quad h=\left(
\begin{array}{ccc}
1 & & \\
& \omega_3 & \\
& & \omega_3^2 \\
\end{array}
\right), \label{Periodicity-T}
\eea
with $\omega_3=e^{\frac{2i\pi}{3}}$. Keeping the fact that $gh=\omega_3 gh$ and using the relation (\ref{Periodicity-T}), we can derive that
the transfer matrices $t(u)$ and $t_2(u)$ satisfy the periodicities:
\bea
t(u+i\pi)=(-1)^N\,e^{-\frac{2i\pi}{3}}t(u), \quad t_2(u+i\pi)=e^{-\frac{4i\pi}{3}}t_2(u).\label{Periodicity-T-1}
\eea
The explicit expression (\ref{R-matrix-su3q}) of the $R$-matrix and the definitions (\ref{Monodromy-1}), (\ref{transfer}) and (\ref{transfer-2}) allow us to derive that $e^{-\frac{u}{3}}t(u)$ and $e^{\frac{u}{3}}t_2(u)$, as functions of $u$, are polynomials of $e^{\pm u}$ with the asymptotic behaviors:
\bea
e^{-\frac{u}{3}}t(u)&\propto& e^{\pm(N-1)u}+\cdots,\quad u\rightarrow\pm\infty,\label{transfer-asym}\\[6pt]
e^{\frac{u}{3}}t_2(u)&\propto& e^{\pm(2N-1)u}+\cdots,\quad u\rightarrow\pm\infty.\label{transfer-asym-1}
\eea
\subsection{Inhomogeneous $T-Q$ relation and the associated BAEs}
The commutativity (\ref{commu-trans-su3}) of the transfer matrices $t(u)$ and $t_2(u)$ with different spectral parameters implies that they have common eigenstates.
Let $|\Psi\rangle$ be a common eigenstate of $\{t_m(u)\}$, which dose not depend upon $u$, with the eigenvalues $\Lambda_m(u)$,
\bea
t_m(u)|\Psi\rangle=\Lambda_m(u)|\Psi\rangle,\qquad m=1,2, 3.\no
\eea
The fusion relations (\ref{trans-trans}), (\ref{trans-trans-1}) and (\ref{fusion-orth}) imply that the eigenvalues $\L_i(u)$ satisfy the relations
\bea
&&\Lambda(\theta_j)\Lambda_m(\theta_j-{\eta})=\Lambda_{m+1}(\theta_j),~~~~~ m=1,2,\quad j=1,\cdots,N,\label{Eigenvalue-function-1}\\[4pt]
&&\Lambda_3(u)=\prod^N_{l=1}\sinh(u-\theta_l+{\eta})\prod^{2}_{k=1}\sinh(u-\theta_l-k{\eta}),
\label{Eigenvalue-function-2}\\[4pt]
&&\Lambda_2(\theta_j+{\eta})=0,~~~~~~j=1,\cdots,N.\label{Eigenvalue-function-3}
\eea
The periodicity properties (\ref{Periodicity-T-1}) of the transfer matrices enable us to derive that the eigenvalues $\L_i(u)$ satisfy the associated
periodicity relations
\bea
\Lambda(u+i\pi)=e^{-\frac{2i\pi}{3}}(-1)^N\Lambda(u),\quad \Lambda_2(u+i\pi)=e^{-\frac{4i\pi}{3}}\Lambda_2(u).\label{Eigenvalue-function-4}
\eea
In addition, the asymptotic behaviors (\ref{transfer-asym})-(\ref{transfer-asym-1}) of the transfer matrices and their definitions (\ref{transfer})
and (\ref{transfer-2}) lead to the fact that the eigenvalues $\L_i(u)$, as a function of $u$, can be expressed as
\bea
\Lambda(u)&=&e^{\frac{u}{3}}\lt\{I^{(1)}_1e^{(N-1)u}+I^{(1)}_2e^{(N-3)u}+\cdots+I^{(1)}_Ne^{-(N-1)u}\rt\},\label{asym-su3-1}\\[6pt]
\Lambda_2(u)&=&e^{-\frac{u}{3}}\lt\{I^{(2)}_1e^{(2N-1)u}+I^{(2)}_2e^{(2N-3)u}+\cdots+I^{(2)}_{2N}e^{-(2N-1)u}\rt\},\label{asym-su3-2}
\eea
where $\{I^{(1)}_j|j=1,\cdots,N\}$ and $\{I^{(2)}_j|j=1,\cdots,2N\}$ are $3N$ constants which are eigenstate dependent. Then these constants can be
completely determined by the $3N$ equations (\ref{Eigenvalue-function-1})-(\ref{Eigenvalue-function-3}). The above relations
(\ref{Eigenvalue-function-1})-(\ref{asym-su3-2}) allow us to express the eigenvalues
$\L_i(u)$ in terms of some inhomogeneous $T-Q$ relations \cite{Wan15}.
For this purpose, let us introduce some functions:
\bea
&&a(u)=\prod_{l=1}^N\sinh(u-\theta_l+\eta),\quad d(u)= \prod_{l=1}^N\sinh(u-\theta_l)=a(u-\eta),\label{ad-functions}\\[4pt]
&&Q^{(i)}(u)=\prod_{l=1}^{N_i}\sinh(u-\l^{(i)}_l),\quad i=1,2,3,4,\no\\[4pt]
&&f_1(u)=f_1^{(+)}e^u+f_1^{(-)}e^{-u},\quad f_2(u)=f_2^{(-)}e^{-u},\no
\eea
where the $(N_1+N_2+N_3+N_4)$ parameters $\{\l^{(i)}_l|l=1,\cdots, N_i;\,i=1,2,3,4\}$ and the parameters $f_1^{(\pm)}$ and $f_2^{(-)}$ will be specified later by the associated
BAEs (\ref{BAE-1})-(\ref{BAE-8}). For convenience, we introduce further the following notations:
\bea
&&Z_1(u)=e^{\phi_1}e^{\frac{4u}{3}}a(u)\frac{Q^{(1)}(u-\eta)}{Q^{(2)}(u)},\no\\[4pt]
&&Z_2(u)=e^{-\phi_1}\omega_3 e^{-{2(u+\eta)\over3}} d(u)\frac{Q^{(2)}(u+\eta)Q^{(3)}(u-\eta)}{Q^{(1)}(u)Q^{(4)}(u)},\no\\[4pt]
&&Z_3(u)=\omega_3^2 e^{-{2(u+2\eta)\over3}} d(u)\frac{Q^{(4)}(u+\eta)}{Q^{(3)}(u)},\no\\[4pt]
&&X_1(u)=e^{\frac{u}{3}}a(u)d(u)\frac{Q^{(3)}(u-\eta)f_1(u)}{Q^{(1)}(u)Q^{(2)}(u)},\no\\[4pt]
&&X_2(u)=e^{\frac{u}{3}}a(u)d(u)\frac{Q^{(2)}(u+\eta)f_2(u)}{Q^{(3)}(u)Q^{(4)}(u)},
\eea
where $\phi_1$ is a parameter to be determined later. The eigenvalues $\{\Lambda_i(u)\}$ satisfing
(\ref{Eigenvalue-function-1})-(\ref{Eigenvalue-function-3}) and (\ref{asym-su3-1})-(\ref{asym-su3-2})
can be expressed in terms of the inhomogeneous $T-Q$ relations as follows
\footnote{It is well-known that for the $su(n)$ spin chain with the periodic boundary condition the eigenvalues are given by
the usual homogeneous $T-Q$ relations which only have $n-1$ types of $Q$-functions. However, it seems
that for the anti-periodic boundary condition case the associated inhomogeneous $T-Q$ relations (\ref{T-Q-1})-(\ref{T-Q-2})
(or (\ref{T-Q-n-1}) for the generic $n$) have to involve more types of $Q$-functions (c.f. $2(n-1)$ types of $Q$-functions see (\ref{T-Q-n-1}) below)
than that of the periodic case, except the $n=2$ case \cite{Cao1}.
}
\bea
&&\hspace{-1.2truecm}\L(u)=Z_1(u)+Z_2(u)+Z_3(u)+X_1(u)+X_2(u)\no\\[6pt]
&&\hspace{-1.2truecm}~~~~~~=e^{\frac{u}{3}}\lt\{e^{\phi_1}e^u a(u)\frac{Q^{(1)}(u-\eta)}{Q^{(2)}(u)}
+e^{-\phi_1}\omega_3 e^{-u-{2\eta\over3}} d(u)\frac{Q^{(2)}(u+\eta)Q^{(3)}(u-\eta)}{Q^{(1)}(u)Q^{(4)}(u)}\rt.\no\\[4pt]
&&\hspace{-1.2truecm}\quad\quad\quad+\omega_3^2 e^{-u-{4\eta\over3}} d(u)\frac{Q^{(4)}(u+\eta)}{Q^{(3)}(u)}
+a(u)d(u)\frac{Q^{(3)}(u-\eta)f_1(u)}{Q^{(1)}(u)Q^{(2)}(u)}\no\\[4pt]
&&\hspace{-1.2truecm}\quad\quad\quad
+a(u)d(u)\lt.\frac{Q^{(2)}(u+\eta)f_2(u)}{Q^{(3)}(u)Q^{(4)}(u)} \rt\},\label{T-Q-1}\\[4pt]
&&\hspace{-1.2truecm}\L_2(u)=Z_1(u)Z_2^{(1)}(u)+Z_1(u)Z_3^{(1)}(u)+Z_2(u)Z_3^{(1)}(u)+X_1(u)Z_3^{(1)}(u)+Z_1(u)X_2^{(1)}(u)\no\\[6pt]
&&\hspace{-1.2truecm}~~~~~~~=e^{-\frac{u}{3}}d(u-\eta)\lt\{\omega_3 e^{u} a(u)\frac{Q^{(3)}(u-2\eta)}{Q^{(4)}(u-\eta)}
+e^{-\phi_1}e^{-4\eta\over3} e^{-u} d(u)\frac{Q^{(2)}(u+\eta)}{Q^{(1)}(u)}\rt.\no\\[4pt]
&&\hspace{-1.2truecm}\quad\quad\quad+\omega_3^2 e^{\phi_1}e^{-2\eta\over3} e^{u} a(u)\frac{Q^{(1)}(u-\eta)Q^{(4)}(u)}{Q^{(2)}(u)Q^{(3)}(u-\eta)}
+\omega_3^2e^{-2\eta\over3}\hspace{-0.1truecm}a(u)d(u)\frac{Q^{(4)}(u)f_1(u)}{Q^{(1)}(u)Q^{(2)}(u)}\no\\[4pt]
&&\hspace{-1.2truecm}\quad\quad\quad+e^{\phi_1}\hspace{-0.1truecm}e^{2u-{\eta\over3}}a(u)d(u)
\lt.\frac{Q^{(1)}(u-\eta)f_2(u-\eta)}{Q^{(3)}(u-\eta)Q^{(4)}(u-\eta)} \rt\}\hspace{-0.1truecm},\label{T-Q-2}\\[4pt]
&&\hspace{-1.2truecm}\L_3(u)=Z_1(u)\,Z_2^{(1)}(u)\,Z_3^{(2)}(u)\no\\[6pt]
&&\hspace{-1.2truecm}~~~~~~~~=\omega_3^3 a(u)d(u-\eta)d(u-2\eta)=a(u)d(u-\eta)d(u-2\eta).
\eea
Here and below we adopt the conventions
\bea
Z_i^{(l)}(u)=Z_i(u-l\eta),\quad X_i^{(l)}(u)=X_i(u-l\eta).\label{Notation}
\eea
To make the inhomogeneous $T-Q$ relations (\ref{T-Q-1}) and (\ref{T-Q-2}) to fulfill the asymptotic behavior
(\ref{asym-su3-1})-(\ref{asym-su3-2}),
the non-negative integers $N_i$ should satisfy the relations
\bea
N_1=N_2=N_3=N_4=N,\label{N-su(3)}
\eea
and the $4N+4$ parameters $\{\l^{(i)}_l|l=1,\cdots, N;\,i=1,2,3,4\}$, $f_1^{(\pm)}$, $f_2^{(-)}$ and $e^{\phi_1}$ satisfy the associated
BAEs\footnote{It is still an interesting open problem to investigate the structure of the Bethe roots of the BAEs (\ref{BAE-1})-(\ref{BAE-8})
for a large $N$. One promising strategy might be to study the corresponding elliptical model for the large sites with the crossing parameter
taken some special values (which will become dense in the whole complex plan when $N\rightarrow\infty$) for which the inhomogeneous $T-Q$ relation reduce to the usual one \cite{Cao14NB886}. This allows one to study the pattern of the corresponding Bethe roots for the large $N$ and then taking the trigonometric limit we can obtain the pattern of the Bethe root for the trigonometric models for a large $N$. This strategy has proven to be very successful for the studying the thermodynamics of the spin-$\frac{1}{2}$ open XXZ chain \cite{Li14}.}:
\bea
&&\omega_3 e^{-\phi_1}e^{-\lambda^{(1)}_j-{2\eta\over3}} \frac{Q^{(2)}(\lambda^{(1)}_j+\eta)}{Q^{(4)}(\lambda^{(1)}_j)}
+a(\lambda^{(1)}_j)\frac{f_1(\lambda^{(1)}_j)}{Q^{(2)}(\lambda^{(1)}_j)}=0,\quad j=1,\cdots,N,\label{BAE-1}\\[4pt]
&&e^{\phi_1}e^{\lambda^{(2)}_j}Q^{(1)}(\lambda^{(2)}_j-\eta)
+d(\lambda^{(2)}_j)
\frac{Q^{(3)}(\lambda^{(2)}_j-\eta)f_1(\lambda^{(2)}_j)}{Q^{(1)}(\lambda^{(2)}_j)}=0,
\quad j=1,\cdots,N,\label{BAE-2}\\[4pt]
&&\omega_3^2 e^{-\lambda^{(3)}_j-{4\eta\over3}}Q^{(4)}(\lambda^{(3)}_j+\eta)
+a(\lambda^{(3)}_j)
\frac{Q^{(2)}(\lambda^{(3)}_j+\eta)f_2(\lambda^{(3)}_j)}{Q^{(4)}(\lambda^{(3)}_j)}=0,
\,\, j=1,\cdots,N,\label{BAE-3}\\[4pt]
&&\omega_3
e^{-\phi_1}e^{-\lambda^{(4)}_j-{2\eta\over3}}\frac{Q^{(3)}(\lambda^{(4)}_j-\eta)}{Q^{(1)}(\lambda^{(4)}_j)}
+a(\lambda^{(4)}_j)\frac{f_2(\lambda^{(4)}_j)}{Q^{(3)}(\lambda^{(4)}_j)}=0,
\quad j=1,\cdots,N,\label{BAE-4} \eea \bea
&&\hspace{-1.2truecm}e^{\phi_1}e^{-\Theta-\chi^{(1)}+\chi^{(2)}}+e^{-2\Theta+\chi^{(1)}+\chi^{(2)}-\chi^{(3)}}f^{(+)}_1=0,\label{BAE-5}\\[4pt]
&&\hspace{-1.2truecm}\omega_3 e^{-\phi_1}e^{-{2\eta\over3}+\Theta-\chi^{(1)}+\chi^{(2)}+\chi^{(3)}-\chi^{(4)}}
+\omega_3^2 e^{-{4\eta\over3}+\Theta-\chi^{(3)}+\chi^{(4)}-N\eta}\no\\[4pt]
&&+e^{2\Theta-N\eta}\lt\{e^{-\chi^{(1)}-\chi^{(2)}+\chi^{(3)}+N\eta}f^{(-)}_1
+e^{+\chi^{(2)}-\chi^{(3)}-\chi^{(4)}-N\eta}f^{(-)}_2\rt\}=0,\label{BAE-6}\\[4pt]
&&\hspace{-1.2truecm}\omega_3 e^{-\Theta-\chi^{(3)}+\chi^{(4)}}+\omega_3^2 e^{\phi_1}e^{-{2\eta\over3}
-\Theta-\chi^{(1)}+\chi^{(2)}+\chi^{(3)}-\chi^{(4)}+N\eta}\no\\[4pt]
&&+e^{-2\Theta+N\eta}\lt\{\omega_3^2 e^{-{2\eta\over3}+\chi^{(1)}+\chi^{(2)}-\chi^{(4)}}f^{(+)}_1
+e^{\phi_1}e^{{2\eta\over3}-\chi^{(1)}+\chi^{(3)}+\chi^{(4)}+N\eta}f^{(-)}_2\rt\}=0,\label{BAE-7}\\[4pt]
&&\hspace{-1.2truecm}e^{-\phi_1}e^{-{4\eta\over3}+\Theta-\chi^{(1)}+\chi^{(2)}-N\eta}+\omega_3^2
e^{-{2\eta\over3}+2\Theta-\chi^{(1)}-\chi^{(2)}+\chi^{(4)}-N\eta}f^{(-)}_1=0,\label{BAE-8}
\eea
where
\bea
\Theta=\sum_{l=1}^N\theta_l,\quad \chi^{(i)}=\sum_{l=1}^N\l^{(i)}_l,\quad i=1,2,3,4.\no
\eea
It is easy to check that the inhomogeneous $T-Q$ relations (\ref{T-Q-1}) and (\ref{T-Q-2}) fulfill the relations (\ref{Eigenvalue-function-1})-(\ref{Eigenvalue-function-3})
and the periodicity properties (\ref{Eigenvalue-function-4}). The BEAs (\ref{BAE-5})-(\ref{BAE-8})
ensure that the inhomogeneous $T-Q$ relations (\ref{T-Q-1}) and (\ref{T-Q-2}) indeed
satisfy the asymptotic behavior (\ref{asym-su3-1})-(\ref{asym-su3-2}), while the BAEs (\ref{BAE-1})-(\ref{BAE-4}) assure that the inhomogeneous $T-Q$ relations have no
singularity at points $\l^{(i)}_l$.
\section{Results for the $su(n)$ spin torus}
\setcounter{equation}{0}
For the $su(n)$ case, by using the similar method introduced in the previous section, we can derive that the fused transfer matrices $\{t_j(u)|j=1,\cdots, n\}$
satisfy the analogous operator product identities such as (\ref{trans-trans}), (\ref{trans-trans-1}), (\ref{fusion-orth}),
(\ref{Periodicity-T-1}) and (\ref{transfer-asym})-(\ref{transfer-asym-1}). These identities lead to that the corresponding eigenvalues
$\{\L_j(u)|j=1,\cdots,n\}$ satisfy the functional relations:
\bea
&&\Lambda(\theta_j)\Lambda_m(\theta_j-\eta)=
\Lambda_{m+1}(\theta_j),\qquad m=1,\cdots,n-1,\quad j=1,\cdots N,\label{Eigenvalue-n-1}\\[6pt]
&&\Lambda_m(\theta_j+k\eta)=0,\quad k=1,\cdots,m-1,\quad m=1,\cdots,n-1,\quad j=1,\cdots N,\label{Eigenvalue-n-2}\\[6pt]
&&\Lambda_n(u)=(-1)^{n-1}\prod^N_{l=1}\sinh(u-\theta_l+{\eta})\prod^{n-1}_{k=1}\sinh(u-\theta_l-k{\eta})\times {\rm id},\label{Eigenvalue-n-3}\\[6pt]
&&\Lambda_m(u+i\pi)=e^{-m({2\over n})i\pi}((-1)^N)^m\Lambda_m(u),\quad m=1,\cdots,n-1,\label{Eigenvalue-n-4}\\[6pt]
&&e^{-u+2({m\over n})u}\Lambda_m(u) \propto e^{\pm(mN-1)u}+\cdots,\quad u\rightarrow\pm\infty,\quad m=1,\cdots,n-1.\label{Eigenvalue-n-5}
\eea
Similar to the $su(3)$ case, the above relations completely determine the eigenvalues $\L_i(u)$
and thus enable us to express them in
terms of certain inhomogeneous $T-Q$ relations as those given by (\ref{T-Q-1})-(\ref{T-Q-2}). For the $su(n)$ case, let us
introduce the functions:
\bea
&&Q^{(i)}(u)=\prod_{l=1}^{N_i}\sinh(u-\l^{(i)}_l),\quad i=1,\cdots,2n-2,\label{Q-functions-n-1}\\[6pt]
&&Z_1(u)=e^{\phi_1}e^{(2-\frac{2}{n})u}a(u)\frac{Q^{(1)}(u-\eta)}{Q^{(2)}(u)},\no\\[6pt]
&&Z_2(u)=e^{\phi_2}\omega_n e^{-{2(u+\eta)\over{n}}} d(u)\frac{Q^{(2)}(u+\eta)Q^{(3)}(u-\eta)}{Q^{(1)}(u)Q^{(4)}(u)},\no\\[6pt]
&&\qquad\qquad\vdots\no\\[6pt]
&&Z_i(u)=e^{\phi_i}\omega_n^{i-1} e^{-{2(u+(i-1)\eta)\over{n}}} d(u)\frac{Q^{(2i-2)}(u+\eta)Q^{(2i-1)}(u-\eta)}{Q^{(2i-3)}(u)Q^{(2i)}(u)},
\no\\[6pt]
&&\qquad\qquad\vdots\no\\[6pt]
&&Z_{n-1}(u)=e^{-\sum_{j=1}^{n-2}\phi_j}\omega_n^{n-2} e^{-{2({u+(n-2)\eta})\over{n}}} d(u)\frac{Q^{(2n-4)}(u+\eta)Q^{(2n-3)}(u-\eta)}{Q^{(2n-5)}(u)Q^{(2n-2)}(u)},\no\\[6pt]
&&Z_n(u)=\omega_n^{n-1} e^{-{2({u+(n-1)\eta})\over{n}}} d(u)\frac{Q^{(2n-2)}(u+\eta)}{Q^{(2n-3)}(u)},\no
\eea
and
\bea
&&X_1(u)=e^{(1-\frac{2}{n})u}a(u)d(u)\frac{Q^{(3)}(u-\eta)f_1(u)}{Q^{(1)}(u)Q^{(2)}(u)},\no\\[6pt]
&&X_2(u)=e^{(1-\frac{2}{n})u}a(u)d(u)\frac{Q^{(2)}(u+\eta)Q^{(5)}(u-\eta)f_2(u)}{Q^{(3)}(u)Q^{(4)}(u)},\no\\[6pt]
&&\qquad\qquad\vdots\no\\[6pt]
&&X_i(u)=e^{(1-\frac{2}{n})u}a(u)d(u)\frac{Q^{(2i-2)}(u+\eta)Q^{(2i+1)}(u-\eta)f_i(u)}{Q^{(2i-1)}(u)Q^{(2i)}(u)},\no\\[6pt]
&&\qquad\qquad\vdots\no\\[6pt]
&&X_{n-1}(u)=e^{(1-\frac{2}{n})u}a(u)d(u)\frac{Q^{(2n-4)}(u+\eta)f_{n-1}(u)}{Q^{(2n-3)}(u)Q^{(2n-2)}(u)}.\label{X-function-n-1}
\eea
Here $\omega_n=e^{\frac{2i\pi}{n}}$ such that $\omega_n^n=1$, the functions $\{f_i(u)|i=1,\cdots,n-1\}$ are
\bea
&&f_1(u)=f_1^{(+)}e^u+f_1^{(-)}e^{-u},\label{f-function-1}\\[6pt]
&&f_i(u)=f_i^{(-)}e^{-u},\quad i=2,\cdots,n-1.\label{f-function-2}
\eea
The $2(n-1)$ constants $f_1^{(\pm)}$, $\{f_i^{(-)}|i=2,\cdots,n-1\}$ and $\{\phi_i,i=1,\cdots,n-2\}$ are to be determined later. We define further functions $\{Y_l(u)|l=1,\cdots,2n-1\}$,
\bea
\lt\{\begin{array}{ll}Y_{2j-1}(u)=Z_j(u),& j=1,\cdots,n,\\[6pt]
Y_{2j}(u)=X_j(u),& j=1,\cdots,n-1,
\end{array}\rt.\label{Def-Y}
\eea
and take the notation
\bea
Y_j^{(l)}(u)=Y_j(u-l\eta),\quad l=1,\cdots,n,\quad j=1,\cdots,2n-1.\label{Notation-1}
\eea
The eigenvalue $\{\Lambda_m(u)|m=1,\cdots n-1\}$ satisfying the relations (\ref{Eigenvalue-n-1})-(\ref{Eigenvalue-n-5}) can
be given in terms of the inhomogeneous $T-Q$ relations as
\footnote{For the $n=2$ case, the corresponding inhomogeneous $T-Q$ relation reduces to the alternative inhomogeneous $T-Q$ given in \cite{Wan15} (i.e., the equation (4.4.1) of subchapter 4.4).}
\bea
\Lambda_m(u)={\sum}'_{1\leq i_1<i_2<\cdots<i_m\leq 2n-1}Y_{i_1}(u)Y^{(1)}_{i_2}(u)\cdots Y^{(m-1)}_{i_m}(u),\quad
m=1,\cdots,n-1.\label{T-Q-n-1}
\eea
The sum $\sum'$ is over the constrained increasing sequences $1\leq i_1<i_2<\cdots<i_m\leq 2n-1$
such that when any $i_k=2j$ (i.e., $Y_{i_k}^{(k-1)}(u)=Y_{2j}^{(k-1)}(u)=X^{(k-1)}_j(u)$), then $i_{k-1}\leq 2j-3$ and $i_{k+1}\geq 2j+3$.
Namely, when $Y_{i_k}(u)=X_j(u)$, the previous element $Y_{i_{k-1}}(u)$ and the next element $Y_{i_{k+1}}(u)$ can not be chosen as its nearest neighbors
(e.g., $X_{j-1}(u)$, $Z_j(u)$, $Z_{j+1}(u)$ and $X_{j+1}(u)$) in the diagram (\ref{struc-inhomo}).
\bea
\xymatrix{
Z_1\ar@{-}[rr]\ar@{-}[dr]& &Z_2\ar@{-}[dr]\ar@{-}[rr]& &Z_3\ar@{-}[dl]\ar@{-}[rr]& &Z_4\ar@{-}[dl]\ar@{.}[rr]& &Z_{n}\ar@{.}[dl]\\
&X_1\ar@{-}[rr] \ar@{-}[ur]& &X_2\ar@{-}[rr]& &X_3\ar@{-}[ul]\ar@{.}[rr]& &X_{n-1}}
\label{struc-inhomo}\eea
To satisfy the asymptotic behaviors (\ref{Eigenvalue-n-5}), the non-negative integers
$\{N_i|i=1,\cdots,2(n-1)\}$ must be chosen as follows:
\begin{itemize}
\item For odd $n$,
\bea
N_{2i-1}=N_{2i}=N_{2(n-i)-1}=N_{2(n-i)}=\frac{i(n-i)}{2}N,\quad
i=1,\cdots,\frac{n-1}{2}.\label{N-1}
\eea
\item For even $n$ and even $N$,
\bea
N_{2i-1}=N_{2i}=N_{2(n-i)-1}=N_{2(n-i)}=\frac{i(n-i)}{2}N,\quad
i=1,\cdots,\frac{n}{2}.\label{N-2}
\eea
\item For even $n$ and odd $N$,
\bea
N_{2i-1}=N_{2i}=N_{2(n-i)-1}=N_{2(n-i)}=\frac{i(n-i)}{2}N+\frac{i}{2},\quad
i=1,\cdots,\frac{n}{2}.\label{N-3}
\eea
In contrast with (\ref{f-function-2}), the function $f_{\frac{n}{2}}(u)$ now should be adjusted to
\bea
f_{n\over2}(u)=\sinh(u)\,f_{n\over2}^{(-)}\,e^{-u}.\label{f-function-3}
\eea
\end{itemize}
We present the details for the $su(4)$ case in Appendix B, which could be crucial to understand the structure for $n\geq4$.
It is easy to check that if $\{N_i|i=1,\cdots,2(n-1)\}$ are chosen as (\ref{N-1})-(\ref{N-3}) the corresponding inhomogeneous
$T-Q$ relations (\ref{T-Q-n-1}) have the asymptotic behavior
\bea
&&e^{-u+2({m\over n})u}\Lambda_m(u)=F^{(\pm)}_m\,e^{\pm(mN+1)u}+{F^{(\pm)}_m}'\,e^{\pm(mN-1)u}+\cdots, \quad u\rightarrow\pm\infty,\no\\[4pt]
&&\quad\quad m=1,\cdots,n-1.
\eea
The $2(n-1)$ coefficients $F^{(\pm)}_m$ are the functions of the parameters $\{\l^{(i)}_j|i=1,\cdots, 2(n-1);j=1,\cdots,N_i\}$,
$f_1^{(\pm)}$, $\{f_i^{(-)}|i=2,\cdots,n-1\}$ and $\{\phi_i,i=1,\cdots,n-2\}$. The explicit expressions can be obtained by direct
calculation. To make (\ref{Eigenvalue-n-5}) satisfied, the coefficients $F^{(\pm)}_m$ must vanish which leads to the $2(n-1)$
BAEs (for an example, (\ref{BAE-5})-(\ref{BAE-8}) for the $su(3)$ case)
\bea
F_m^{(\pm)}=0,\quad m=1,\cdots,n-1.\label{BAE-n-1}
\eea
Moreover, the vanishing condition of the residues of $\L_m(u)$ at the points $\l^{(i)}_j$ gives rise to the other BAEs:
\bea
&&e^{\phi_2}\omega_n e^{-\lambda^{(1)}_j-{2\eta\over{n}}} \frac{Q^{(2)}(\lambda^{(1)}_j+\eta)}{Q^{(4)}(\lambda^{(1)}_j)}
+a(\lambda^{(1)}_j)\frac{f_1(\lambda^{(1)}_j)}{Q^{(2)}(\lambda^{(1)}_j)}=0,\qquad j=1,\cdots,N_1,\\[6pt]
&&e^{\phi_1}e^{\lambda^{(2)}_j}Q^{(1)}(\lambda^{(2)}_j-\eta)
+d(\lambda^{(2)}_j)
\frac{Q^{(3)}(\lambda^{(2)}_j-\eta)f_1(\lambda^{(2)}_j)}{Q^{(1)}(\lambda^{(2)}_j)}=0,
\quad j=1,\cdots,N_2,
\eea
\bea
&&e^{\phi_i}\omega_n^{i-1}e^{-\lambda_j^{(2i)}-\frac{2(i-1)\eta}{n}}
\frac{Q^{(2i-1)}(\lambda_j^{(2i)}-\eta)}{Q^{(2i-3)}(\lambda_j^{(2i)})}
+a(\lambda_j^{(2i)})
\frac{Q^{(2i+1)}(\lambda_j^{(2i)}-\eta)f_i(\lambda_j^{(2i)})}{Q^{(2i-1)}(\lambda_j^{(2i)})}=0,\no\\[6pt]
&&\qquad\qquad i=2,\cdots,n-2,\quad j=1,\cdots,N_{2i},\\[6pt]
&&e^{\phi_{i+1}}\omega_n^{i}e^{-\lambda_j^{(2i-1)}-\frac{2i\eta}{n}}
\frac{Q^{(2i)}(\lambda_j^{(2i-1)}+\eta)}{Q^{(2i+2)}(\lambda_j^{(2i-1)})}
+a(\lambda_j^{(2i-1)})
\frac{Q^{(2i-2)}(\lambda_j^{(2i-1)}+\eta)f_i(\lambda_j^{(2i-1)})}{Q^{(2i)}(\lambda_j^{(2i-1)})}=0,\no\\[6pt]
&&\qquad \qquad i=2,\cdots,n-3,\quad j=1,\cdots,N_{2i-1},
\eea
\bea
&&e^{-\sum_{j=1}^{n-2}\phi_j}\omega_n^{n-2} e^{-\lambda_j^{(2n-5)}-{2(n-2)\eta\over{n}}} \frac{Q^{(2n-4)}(\lambda_j^{(2n-5)}+\eta)}{Q^{(2n-2)}(\lambda_j^{(2n-5)})}\no\\[6pt]
&&\qquad+a(\lambda_j^{(2n-5)})\frac{Q^{(2n-6)}(\lambda_j^{(2n-5)}+\eta)f_{n-2}(\lambda_j^{(2n-5)})}{Q^{(2n-4)}(\lambda_j^{(2n-5)})}=0,
\quad j=1,\cdots,N_{2n-5},
\eea
\bea
&&\omega_n^{n-1} e^{-\lambda_j^{(2n-3)}-{2(n-1)\eta\over{n}}} \hspace{-0.1truecm}Q^{(2n-2)}(\lambda_j^{(2n-3)}\hspace{-0.1truecm}+\hspace{-0.1truecm}\eta)
\hspace{-0.1truecm}+\hspace{-0.1truecm}a(\lambda_j^{(2n-3)})
\frac{Q^{(2n-4)}(\lambda_j^{(2n-3)}+\eta)f_{n-1}(\lambda_j^{(2n-3)})}{Q^{(2n-2)}(\lambda_j^{(2n-3)})}=0,\no\\[6pt]
&&\qquad\quad\quad\qquad j=1,\cdots,N_{2n-3},\quad\\[6pt]
&&e^{-\sum_{j=1}^{n-2}\phi_j}\omega_n^{n-2} e^{-\lambda_j^{(2n-2)}-{2(n-2)\eta\over{n}}} \frac{Q^{(2n-3)}(\lambda_j^{(2n-2)}-\eta)}{Q^{(2n-5)}(\lambda_j^{(2n-2)})}
+a(\lambda_j^{(2n-2)})\frac{f_{n-1}(\lambda_j^{(2n-2)})}{Q^{(2n-3)}(\lambda_j^{(2n-2)})}=0,
\no\\[6pt]
&&\qquad\quad\quad\qquad j=1,\cdots,N_{2n-2}.\label{BAE-n-2}
\eea
Associated with the BAEs (\ref{BAE-n-1})-(\ref{BAE-n-2}), the inhomogeneous $T-Q$ relation (\ref{T-Q-n-1}) give the eigenvalues of the transfer matrix of the $su(n)$ spin torus.
\section{Conclusions}
In this paper, we have studied the $su(n)$ spin torus described by the Hamiltonian (\ref{Ham}) with
the anti-periodic boundary condition (\ref{anti-boundary}). In the framework of ODBA, we have obtained the
eigenvalues of the corresponding transfer matrix in terms of the inhomogeneous $T-Q$ relation (\ref{T-Q-n-1})
and the associated BAEs (\ref{BAE-n-1})-(\ref{BAE-n-2}). The exact spectrum obtained in this paper allows us further to construct the corresponding eigenstates.
The results will be presented elsewhere \cite{Hao16}.
\section*{Acknowledgments}
The financial supports from the National Natural Science Foundation
of China (Grant Nos. 11375141, 11374334, 11434013, 11425522 and 11547045), BCMIIS and the Strategic Priority Research Program
of the Chinese Academy of Sciences are gratefully acknowledged.
\section*{Appendix A: Generic twist matrices}
\setcounter{equation}{0}
\renewcommand{\theequation}{A.\arabic{equation}}
A generic twist matrix ${\mathcal{G}}$ associated with an integrable boundary satisfies the relation
\bea
R_{12}(u-v)\,{\mathcal{G}}_1\,{\mathcal{G}}_2={\mathcal{G}}_1\,{\mathcal{G}}_2\,R_{12}(u-v).\label{Invariant-R-1}
\eea
Normally, ${\mathcal{G}}$ is a c-number matrix. The solutions of (\ref{Invariant-R-1}) with the $R$-matrix given by (\ref{R-matrix-1}) can be specified as
$n$ classes labeled by $l$
\bea
{\mathcal{D}}\,g^l,\quad l=0,\cdots,n-1,\label{Twist-matrices}
\eea
where ${\mathcal{D}}$ is an arbitrary non-degenerate diagonal $n\times n$ matrix and $g$ is given by (\ref{g-matrix}). All solutions to (\ref{Invariant-R-1}) are
some products of elements of these classes. ${\mathcal{G}}={\mathcal{D}}$
corresponds to the diagonal twisted boundary condition (including the periodic boundary condition ${\mathcal{D}}={\rm id}$ as a special case).
Without losing the generality, in this paper we consider the twist matrix ${\mathcal{G}}=g$ which corresponds to the antiperiodic boundary condition (\ref{anti-boundary}). The generalization to the other cases is
straightforward.
\section*{Appendix B: $T-Q$ relation for the $su(4)$ spin torus}
\setcounter{equation}{0}
\renewcommand{\theequation}{B.\arabic{equation}}
For $n=4$, the functions (\ref{Q-functions-n-1})-(\ref{X-function-n-1}) and the functions $X_j(u)$ read
\bea
&&Q^{(i)}(u)=\prod_{l=1}^{N_i}\sinh(u-\l^{(i)}_l),\quad i=1,\cdots,6,\\[6pt]
&&Z_1(u)=e^{\phi_1}e^{\frac{3u}{2}}a(u)\frac{Q^{(1)}(u-\eta)}{Q^{(2)}(u)},\no\\[6pt]
&&Z_2(u)=e^{\phi_2}\omega_4 e^{-{u+\eta\over2}} d(u)\frac{Q^{(2)}(u+\eta)Q^{(3)}(u-\eta)}{Q^{(1)}(u)Q^{(4)}(u)},\no\\[6pt]
&&Z_3(u)=e^{-\phi_1-\phi_2}\omega_4^2 e^{-{u\over2}-\eta} d(u)\frac{Q^{(4)}(u+\eta)Q^{(5)}(u-\eta)}{Q^{(3)}(u)Q^{(6)}(u)},\no\\[6pt]
&&Z_4(u)=\omega_4^3 e^{-{{u+3\eta}\over2}} d(u)\frac{Q^{(6)}(u+\eta)}{Q^{(5)}(u)},\no\\[6pt]
&&X_1(u)=e^{\frac{u}{2}}a(u)d(u)\frac{Q^{(3)}(u-\eta)f_1(u)}{Q^{(1)}(u)Q^{(2)}(u)},\no\\[6pt]
&&X_2(u)=e^{\frac{u}{2}}a(u)d(u)\frac{Q^{(2)}(u+\eta)Q^{(5)}(u-\eta)f_2(u)}{Q^{(3)}(u)Q^{(4)}(u)},\no\\[6pt]
&&X_3(u)=e^{\frac{u}{2}}a(u)d(u)\frac{Q^{(4)}(u+\eta)f_3(u)}{Q^{(5)}(u)Q^{(6)}(u)}.
\eea
Here $\omega_4=e^{\frac{2i\pi}{4}}$. The corresponding $T-Q$ relations (\ref{T-Q-n-1}) become
\bea
&&\hspace{-1.2truecm}\L(u)=Z_1(u)+Z_2(u)+Z_3(u)+Z_4(u)+X_1(u)+X_2(u)+X_3(u),\\[6pt]
&&\hspace{-1.2truecm}\L_2(u)=Z_1(u)Z_2(u-\eta)+Z_1(u)X_2(u-\eta)+Z_1(u)Z_3(u-\eta)+Z_1(u)X_3(u-\eta)\no\\[6pt]
&&+Z_1(u)Z_4(u-\eta)+X_1(u)Z_3(u-\eta)+X_1(u)X_3(u-\eta)+X_1(u)Z_4(u-\eta)
\no\\[6pt]
&&+Z_2(u)Z_3(u-\eta)+Z_2(u)X_3(u-\eta)+Z_2(u)Z_4(u-\eta)+X_2(u)Z_4(u-\eta)\no\\[6pt]
&&+Z_3(u)Z_4(u-\eta),\\[6pt]
&&\hspace{-1.2truecm}\L_3(u)=Z_1(u)Z_2(u-\eta)Z_3(u-2\eta)+Z_1(u)Z_2(u-\eta)X_3(u-2\eta)\no\\[6pt]
&&+Z_1(u)Z_2(u-\eta)Z_4(u-2\eta)+Z_1(u)X_2(u-\eta)Z_4(u-2\eta)\no\\[6pt]
&&+Z_1(u)Z_3(u-\eta)Z_4(u-2\eta)+X_1(u)Z_3(u-\eta)Z_4(u-2\eta)\no\\[6pt]
&&+Z_2(u)Z_3(u-\eta)Z_4(u-2\eta),\\[6pt]
&&\hspace{-1.2truecm}\Lambda_4(u)=Z_1(u)Z_2(u-\eta)Z_3(u-2\eta)Z_4(u-3\eta).
\eea
\begin{itemize}
\item For even $N$ ,
\bea
N_1=N_2=N_5=N_6=\frac32 N, \quad N_3=N_4=2N,
\eea
and the functions $f_i(u)$ are :
\bea
f_1(u)=f_1^{(+)}e^u+f_1^{(-)}e^{-u},\, f_2(u)=f_2^{(-)}e^{-u},\,
f_3(u)=f_3^{(-)}e^{-u}.
\eea
\item For odd $N$,
\bea N_1=N_2=N_5=N_6=\frac{3N+1}2, \quad
N_3=N_4=2N+1,
\eea
and the functions $f_i(u)$ are :
\bea
f_1(u)=f_1^{(+)}e^u+f_1^{(-)}e^{-u},\, f_2(u)=\sinh(u)\,f_2^{(-)}\,e^{-u},\,
f_3(u)=f_3^{(-)}e^{-u}.
\eea
\end{itemize}
| {
"redpajama_set_name": "RedPajamaArXiv"
} |
"\\section{Examples of Hard Negatives}\n\nFig.~\\ref{fig:sample_outputs} compares the outputs of \\e(...TRUNCATED) | {
"redpajama_set_name": "RedPajamaArXiv"
} |
"\\section{Introduction}\n\\label{intro}\nUnknown intent detection is a realistic and challenging ta(...TRUNCATED) | {
"redpajama_set_name": "RedPajamaArXiv"
} |
"\\section{Introduction}\n\n\nRecently, the advancement of electric battery technology pushes forwar(...TRUNCATED) | {
"redpajama_set_name": "RedPajamaArXiv"
} |
"\\section{Introduction}\n\nWhenever we measure anything using a particular number system, the\ncorr(...TRUNCATED) | {
"redpajama_set_name": "RedPajamaArXiv"
} |
"\\section{Introduction}\nWithin the next few years, powerful lasers may be able to realise\nintensi(...TRUNCATED) | {
"redpajama_set_name": "RedPajamaArXiv"
} |
"\\section{Introduction} \n\nThe field of quantum computation has seen fervent activity in the deve(...TRUNCATED) | {
"redpajama_set_name": "RedPajamaArXiv"
} |
End of preview.
SlimPajama-6B & SlimPajama-30B
Pre-training data sampled from cerebras/SlimPajama-627B, at two scales: ~6B tokens and ~30B tokens. The data is formatted in JSONL and is compatible with LLaMA-Factory training pipelines.
Repository Structure
├── train/
│ ├── 6B/ # ~6B tokens, 7 JSONL files by source
│ │ ├── RedPajamaCommonCrawl-6B.jsonl
│ │ ├── RedPajamaC4-6B.jsonl
│ │ ├── RedPajamaGithub-6B.jsonl
│ │ ├── RedPajamaBook-6B.jsonl
│ │ ├── RedPajamaArXiv-6B.jsonl
│ │ ├── RedPajamaWikipedia-6B.jsonl
│ │ └── RedPajamaStackExchange-6B.jsonl
│ └── 30B/ # ~30B tokens, 7 JSONL files by source
│ ├── RedPajamaCommonCrawl-30B.jsonl
│ ├── RedPajamaC4-30B.jsonl
│ ├── RedPajamaGithub-30B.jsonl
│ ├── RedPajamaBook-30B.jsonl
│ ├── RedPajamaArXiv-30B.jsonl
│ ├── RedPajamaWikipedia-30B.jsonl
│ └── RedPajamaStackExchange-30B.jsonl
└── val/
└── validation.jsonl # validation set
Each JSONL record has the following schema:
{"text": "...", "meta": {"redpajama_set_name": "RedPajamaC4"}, "__index_level_0__": 12345}
Data Source
Both subsets originate from cerebras/SlimPajama-627B, a 627B-token cleaned and deduplicated version of RedPajama. SlimPajama-627B contains data from 7 sources: CommonCrawl, C4, GitHub, Books, ArXiv, Wikipedia, and StackExchange.
SlimPajama-6B
Source & Sampling
Obtained from DKYoon/SlimPajama-6B.
Since the original data was shuffled before chunking, only train/chunk1 (of 10 total) was downloaded and further sampled at 10%, resulting in roughly 6B tokens.
- Training set: 5,489,000 records
- Validation set: 9,347 records
SlimPajama-30B
Source & Sampling
Sampled directly from cerebras/SlimPajama-627B train/chunk1. The sampling targets ~30B tokens while preserving the source proportions of SlimPajama-6B.
Sampling procedure:
- Load all records from
train/chunk1and tokenize with the Qwen-2.5 tokenizer. - For each source, compute a target token count based on the SlimPajama-6B domain weights.
- Randomly shuffle records within each source (seed=42) and greedily select records until the target token count is reached.
Token Statistics (Qwen-2.5 Tokenizer)
| Data Source | Records | Tokens | Proportion % |
|---|---|---|---|
| RedPajamaCommonCrawl | 8,779,572 | 14,973,812,443 | 54.10% |
| RedPajamaC4 | 16,214,931 | 7,943,593,272 | 28.70% |
| RedPajamaGithub | 952,170 | 1,162,483,837 | 4.20% |
| RedPajamaBook | 8,145 | 1,024,205,301 | 3.70% |
| RedPajamaArXiv | 50,277 | 941,068,936 | 3.40% |
| RedPajamaWikipedia | 1,026,174 | 858,018,818 | 3.10% |
| RedPajamaStackExchange | 1,252,229 | 774,984,699 | 2.80% |
| Total | 28,283,498 | 27,678,167,306 | 100% |
Acknowledgement
This dataset is built upon cerebras/SlimPajama-627B and DKYoon/SlimPajama-6B. We thank the authors for their awesome work.
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