Bethe’s Wave-Function Revisited

1. Introduction

Bethe’s wave-function was initially devised to diagonalise the Heisenberg[1,2] Hamiltonian for an infinite, one-dimensional ($d=1$) lattice and was later extended[3,4] to the $d=1$ Hubbard Hamiltonian. However, since those analyses resort heavily to technical peculiarities, associated with the concerned Hamiltonians, Bethe’s wave-function could not be applied to any $d>1$ model[5], nor to other $d=1$ Hamiltonians. Therefore this article is aimed at showing that Bethe’s wave-function gives access to the eigenspectrum of every realistic $d=1$ Hamiltonian. Then the groundstate energy will be compared with data[6], obtained previously for the Hubbard Hamiltonian[3,4]. An additional comparison will be carried out with the data, resulting from the correlated Fermi gas model, introduced recently[7] to account for the properties of interacting electrons in normal metals. This method will be further applied to electrons interacting on neighbouring sites[8], which was believed so far to lie out of the scope of Bethe’s wave-function[1,2,3,4].

Here is the outline : the proof of Bethe’s wave-function being a many-electron eigenstate is laid out in section I and the general expression of the corresponding eigenvalue is worked out; the two-body scattering is studied in section II for two different $d=1$ Hamiltonians, whereas section III will be concerned with the prominent role of a boundary condition; the groundstate energy, associated with each of the two mentioned models, is reckoned and compared with data, obtained by other methods, in sections IV and V; the various many-electron states, discussed here, are analysed comparatively in section VI; at last, the main results are summarised in the conclusion.

2. 1-Diagonalisation

Let us consider $n>>1$ of electrons moving on a $d=1$ lattice, comprising an even number $N>>1$ of atomic sites, labelled by the index $i=1,..N$. The lattice parameter is taken equal to unity and each site can accommodate at most two electrons of opposite spin $\sigma$, which implies $n\le 2N$. The Hamiltonian H, governing the electron motion, then reads

$$H=h_e+h_{e-e},$$

wherein $h_e,h_{e-e}$ describe the one-electron hopping between first neighbours and the two-electron interaction, respectively. Hence $h_e$ reads[9]

$$h_e=-t\sum _{i,j,\sigma }\left(c_{i,\sigma }^{+}{c}_{j,\sigma }+c_{j,\sigma }^{+}{c}_{i,\sigma }\right),$$

wherein the sum is carried out over $\sigma =\pm 1$, $i=1,..N-1$ with $j=i+1$ and $c_{i,\sigma }^{+},c_{i,\sigma }$ are one-electron creation and annihilation operators on the Wannier[9] state $\left|i,\sigma \right.〉$

$$\left|i,\sigma \right.〉=c_{i,\sigma }^{+}\left|0\right.〉,\left|0\right.〉=c_{i,\sigma }\left|i,\sigma \right.〉,$$

with $\left|0\right.〉$ referring to the no electron state. Then t designates the hopping integral

$$t=\left|\left.〈0\right|c_{j,\sigma }{V}_{e-n}{c}_{i,\sigma }^{+}\left|0\right.〉\right|,$$

with $V_{e-n}$ standing for the electron-nucleus Coulomb potential. The resulting one-electron energy dispersion $ϵ(k\in \left[-\pi ,\pi \right])$ reads[9]

$$ϵ\left(k\right)=-2tcosk.$$

(1)

$h_{e-e}$ will be referred to as $h_0,h_1$ for two electrons sitting on the same site, which characterises the Hubbard model, or on first neighbours. They read[8]

$$\begin{array}{c}{h}_0=U_0\sum _{i=1}^N{c}_{i,\sigma }^{+}{c}_{i,-\sigma }^{+}{c}_{i,-\sigma }{c}_{i,\sigma }\hfill \\ U_0=\left.〈0\right|c_{i,-\sigma }{c}_{i,\sigma }{V}_{e-e}{c}_{i,\sigma }^{+}{c}_{i,-\sigma }^{+}\left|0\right.〉\hfill \\ H_0=h_e+h_0\hfill \\ h_1=\sum _{i,j,\sigma }\left(\left(U_1-J\right)c_{i,\sigma }^{+}{c}_{j,\sigma }^{+}{c}_{j,\sigma }{c}_{i,\sigma }\right.\hfill \\ \left.+U_1{c}_{i,\sigma }^{+}{c}_{j,-\sigma }^{+}{c}_{j,-\sigma }{c}_{i,\sigma }-J{c}_{i,\sigma }^{+}{c}_{j,-\sigma }^{+}{c}_{j,\sigma }{c}_{i,-\sigma }\right)\hfill \\ U_1=\left.〈0\right|c_{j,\sigma }{c}_{i,\sigma }{V}_{e-e}{c}_{i,\sigma }^{+}{c}_{j,\sigma }^{+}\left|0\right.〉\hfill \\ J=\left.〈0\right|c_{j,\sigma }{c}_{i,-\sigma }{V}_{e-e}{c}_{i,\sigma }^{+}{c}_{j,-\sigma }^{+}\left|0\right.〉\hfill \\ H_1=h_e+h_1\hfill \end{array},$$

(2)

wherein $U_0,U_1$ and J designate[9] Coulomb and exchange integrals, respectively, and $V_{e-e}$ refers to the electron-electron Coulomb potential.

The n-electron states make up a Hilbert space of dimension $d_0=\left(\begin{array}{c}2N\\ n\end{array}\right)$, subtended by the basis $\left\{\left|I=1,..d_0\right.〉\right\}$ with each $\left|I\right.〉$ reading $\left|I\right.〉={\prod }_{j=1}^n{c}_{i_j,{\sigma }_j}^{+}\left|0\right.〉$. The $j^{th}$ electron, having spin ${\sigma }_j$, sits on the site $i_j$. The sites are ordered such that $i_j<i_{j+1}$, except in case of double occupancy, characterised by $i_j=i_{j+1}$ and ${\sigma }_j{\sigma }_{j+1}<0$. Let us introduce now a sequence of n real numbers $k_{j=1,..n}\in \left[-\pi ,\pi \right]$ and ${\varphi }_1$ defined as

$${\varphi }_1=\sum _{I=1}^{d_0}{e}^{ik.I}\left|I\right.〉,e^{ik.I}=e^{i{\sum }_{j=1}^n{k}_j{r}_{i_j}},$$

with $r_{i=1,..N}$ being the position vector of site i.

The group of permutations P of n objects is assumed to act on ${\varphi }_1$ as follows

$${\varphi }_P=P{\varphi }_1=\sum _{I=1}^{d_0}{e}^{iPk.I}\left|I\right.〉,e^{iPk.I}=e^{i{\sum }_{j=1}^{n}P\left(k_j\right)r_{i_j}},$$

wherein there is a one-to-one correspondence $k_j\stackrel{P}{\to }{k}_l=P\left(k_j\right)$ between $k_{j=1,..n}$ and $k_{l=1,..n}$. At last, the Hamiltonian H is projected onto the subspace, subtended by $n!$ of ${\varphi }_P$’s, so that every eigenvector of H is found to read as a linear combination of the ${\varphi }_P$’s

$$\psi =\sum _P{x}_P{\varphi }_P=\sum _I\sum _P\left(x_P{e}^{iPk.I}\right)\left|I\right.〉,$$

(3)

wherein $n!$ of $x_P$’s are complex numbers to be determined below. This last term on the right-hand side of Eq.(3) is recognised to be identical to the expression of Bethe’s wave-function in Eq.(9) of [4] and extends thereby its validity to any$d=1$ Hamiltonian, as illustrated below for $H_1$. However noteworthy is that $n!$ may become bigger than $d_0$ for n close to N, in which case the ${\varphi }_P$’s are no longer linearly independent from one another.

The expression of the eigenvalue ${\epsilon }_e$, associated with $\psi$ in Eq.(3), will now be worked out for $n\le N$ in the case of the Hubbard Hamiltonian $H_0$. To that end, the particular n-electron state $\left|I_0\right.〉$, sketched in Fig.1, is defined as

$$\left|I_0\right.〉=\prod _{j=1}^n{c}_{j,{(-1)}^j}^{+}\left|0\right.〉,$$

(4)

while assuming the boundary conditions

$$\begin{array}{c}{h}_e\left|1,\sigma \right.〉=-t\left(\left|2,\sigma \right.〉+|N,\sigma \right.〉)\hfill \\ h_e\left|N,\sigma \right.〉=-t\left(\left|1,\sigma \right.〉+|N-1,\sigma \right.〉)\hfill \end{array}.$$

(5)

Figure 1. Sketch of $\left|I_0\right.〉$, defined by Eqs.(4,8) for $H_0,H_1$, respectively, and sites $i=1,..6$; dots and $+,-,\pm$ signs refer to an empty site or one accommodating either one electron of spin 1 or $-1$ or two electrons of opposite spin $\pm 1$, respectively.

Figure 1. Sketch of $\left|I_0\right.〉$, defined by Eqs.(4,8) for $H_0,H_1$, respectively, and sites $i=1,..6$; dots and $+,-,\pm$ signs refer to an empty site or one accommodating either one electron of spin 1 or $-1$ or two electrons of opposite spin $\pm 1$, respectively.

Thus $\left|I_0\right.〉$ is seen to have the properties

$$\begin{array}{c}〈I_0\left|H_0\right|I_0〉=0,〈I_0\left|H_0\right|I_j^{\pm }〉\ne 0\Rightarrow \hfill \\ \left|I_{j=1,..n}^{\pm }\right.〉=c_{j\pm 1,{(-1)}^j}^{+}{c}_{j,{(-1)}^j}\left|I_0\right.〉\Rightarrow \hfill \\ 〈I_0\left|H_0\right|I_j^{\pm }〉=-t,e^{ik.I_j^{\pm }}=e^{\pm i{k}_j}{e}^{ik.I_0}\hfill \end{array}.$$

(6)

Eq.(6) then entails

$$\begin{array}{c}〈I_0\left|H_0\right|{\varphi }_1〉=-t{e}^{ik.I_0}\sum _{j=1}^n\left(e^{i{k}_j}+e^{-i{k}_j}\right)\hfill \\ =e^{ik.I_0}\sum _{j=1}^nϵ\left(k_j\right)\hfill \end{array},$$

with $ϵ\left(k_j\right)$ defined in Eq.(1). Extending this equation to every ${\varphi }_P$ and applying it further to the Schrödinger equation $\left(H_0-N{\epsilon }_e\right)\psi =0$, with $\psi$ defined by Eq.(3), yields finally the energy per site ${\epsilon }_e$ as

$$\begin{array}{c}〈I_0\left|H_0-N{\epsilon }_e\right|\psi 〉=0\Rightarrow \\ \sum _{j=1}^n\frac{ϵ\left(k_j\right)}{N}\sum _P{x}_P{e}^{iPk.I_0}={\epsilon }_e\sum _P{x}_P{e}^{iPk.I_0}\Rightarrow \\ {\epsilon }_e=\sum _{j=1}^n\frac{ϵ\left(k_j\right)}{N}\end{array}.$$

(7)

Proceeding for $H_1$ similarly as done for $H_0$, $\left|I_0\right.〉$, sketched in Fig.1, will be defined for even$n\le N$ as

$$\left|I_0\right.〉=\prod _{j=1}^{n/2}{c}_{2j,+}^{+}{c}_{2j,-}^{+}\left|0\right.〉.$$

(8)

Assuming again the conditions in Eq.(5), $\left|I_0\right.〉$ is inferred to have the properties

$$\begin{array}{c}〈I_0\left|H_1\right|I_0〉=0,〈I_0\left|H_1\right|I_{j,\pm \pm }〉\ne 0\hfill \\ \Rightarrow \left\{\begin{array}{c}\left|I_{j=1,..\frac{n}{2},+}^{\pm }\right.〉=c_{2j\pm 1,+}^{+}{c}_{2j,+}\left|I_0\right.〉\hfill \\ \left|I_{j=1,..\frac{n}{2},-}^{\pm }\right.〉=c_{2j\pm 1,-}^{+}{c}_{2j,-}\left|I_0\right.〉\hfill \end{array}\right.\hfill \\ \Rightarrow \left\{\begin{array}{c}〈I_0\left|H_1\right|I_{j=1,..\frac{n}{2},\pm }^{\pm }〉=-t\hfill \\ e^{ik.I_{j=1,..\frac{n}{2},+}^{\pm }}=e^{\pm i{k}_{2j}}{e}^{ik.I_0}\hfill \\ e^{ik.I_{j=1,..\frac{n}{2},-}^{\pm }}=e^{\pm i{k}_{2j-1}}{e}^{ik.I_0}\hfill \end{array}\right.\hfill \end{array},$$

(9)

which entails for every P

$$\begin{array}{c}〈I_0\left|H_1\right|{\varphi }_P〉=-t{e}^{iPk.I_0}\sum _{j=1}^n\left(e^{iP\left(k_j\right)}+e^{-iP\left(k_j\right)}\right)\hfill \\ =e^{iPk.I_0}\sum _{j=1}^nϵ\left(k_j\right)\hfill \end{array}.$$

Taking advantage of $〈I_0\left|H_1-N{\epsilon }_e\right|\psi 〉=0$, as done in Eq.(7), yields for ${\epsilon }_e$ the same expression as already given in Eq.(7). This latter expression of ${\epsilon }_e$ is recognised to be identical to Eq.(11) of [4]. Likewise, since it consists in a sum over one-particle energies, it is typical of a many-electron scattering state, so that the many bound electron states of $H_0,H_1$, addressed elsewhere[8], are left out of the purview of this work. Nevertheless it is worth noticing that ${\epsilon }_e$ could be achieved thanks to a careful, model dependent choice of $\left|I_0\right.〉$ (see Eqs.(4,8)). Though ${\epsilon }_e$ is independent from the two-electron coupling, $h_{e-e}$ will prove below quite instrumental in assigning the $x_P$ values.

But before doing that, it is in order to derive the expression of the energy ${\epsilon }_h$, valid for $n\in \left[N,2N\right]$, by replacing electrons by holes. To that end, we begin with recasting $h_e,h_0,h_1$ as

$$\begin{array}{c}{h}_e=t\sum _{i,j,\sigma }\left(c_{j,\sigma }{c}_{i,\sigma }^{+}+c_{i,\sigma }{c}_{j,\sigma }^{+}\right)\hfill \\ h_0=U_0\sum _{i=1}^N{c}_{i,\sigma }{c}_{i,-\sigma }{c}_{i,-\sigma }^{+}{c}_{i,\sigma }^{+}\hfill \\ h_1=\sum _{i,j,\sigma }\left(\left(U_1-J\right)c_{i,\sigma }{c}_{j,\sigma }{c}_{j,\sigma }^{+}{c}_{i,\sigma }^{+}\right.\hfill \\ \left.+U_1{c}_{i,\sigma }{c}_{j,-\sigma }{c}_{j,-\sigma }^{+}{c}_{i,\sigma }^{+}-J{c}_{i,-\sigma }{c}_{j,\sigma }{c}_{j,-\sigma }^{+}{c}_{i,\sigma }^{+}\right)\hfill \end{array}.$$

Any n-hole state $\left|I=1,..d_0\right.〉$ reads $\left|I\right.〉={\prod }_{j=1}^n{c}_{i_j,{\sigma }_j}\left|2\right.〉$, with $\left|2\right.〉$ characterised by each of N sites accommodating 2 electrons. Then substituting $\left|2\right.〉$ to $\left|0\right.〉$ in Eqs.(4,8) and $c_{j,\sigma }$ to $c_{j,\sigma }^{+}$ in Eqs.(4,6,8,9), and proceeding as done above for electrons yields the following expression for the energy per site of a n-hole state

$${\epsilon }_h={\epsilon }_h\left(2\right)-\sum _{j=1}^n\frac{ϵ\left(k_j\right)}{N},$$

(10)

with ${\epsilon }_h\left(2\right)=U_0$ and ${\epsilon }_h\left(2\right)=2(1-\frac{1}{N})(2U_1-J)$ being the energy per site of $\left|2\right.〉$ for $H_0$ and $H_1$, respectively.

3. 2-Two-body scattering

The $x_P$ coefficients will be assessed by analysing the two-body scattering, embodied by $h_0,h_1$. As a matter of fact, the latter turns out to be model dependent and to differ according to whether both electrons, partaking in the scattering process, have parallel or anti-parallel spin, which is referred to below as triplet or singlet case, respectively, for the sake of comparison with a previous work[7]. The general procedure, used to calculate the $x_P$’s, is to be sketched now. First a n-electron state $\left|J_0\right.〉$ is culled in order to illustrate the two-body scattering. It can be seen in Fig.2 to differ from $\left|I_0\right.〉$ in Fig.1, merely by both electrons, involved in the scattering process and located on sites $r_{j<4}$. Accordingly it is convenient to make use of transpositions $T_{ij}$, defined by $T_{ij}\left(k_i\right)=k_j,T_{ij}\left(k_j\right)=k_i,T_{ij}\left(k_{l\ne i,j}\right)=k_l$, which enables us to recast $A=〈J_0\left|H-N{\epsilon }_e\right|\psi 〉=0$, with $\psi$ given by Eq.(3), as

$$\begin{array}{c}A=\sum _P{B}_P=0\hfill \\ B_P=〈J_0\left|H-N{\epsilon }_e\right|x_P{\varphi }_P+x_{T_{12}P}{\varphi }_{T_{12}P}〉\hfill \\ =x_P{z}_P+x_{T_{12}P}{z}_{T_{12}P}\hfill \\ z_Q=\sum _j{e}^{iQk.J_j}〈J_0\left|H\right|J_j〉-N{\epsilon }_e{e}^{iQk.J_0}\hfill \end{array},$$

(11)

wherein Q stands for $P,T_{12}P$ and the sum over P is to be carried out on $\left(n-2\right)!$ of pairs $\left\{P,T_{12}P\right\}$, whereas the subset of n-electron states $\left|J_j\right.〉$ is characterised by $〈J_0\left|H\right|J_j〉\ne 0$. Then $A=0$ in Eq.(11) will be fulfilled by requiring

$$B_P=0,\forall P\Rightarrow \frac{x_{T_{12}P}}{x_P}=-\frac{z_P}{z_{T_{12}P}}.$$

(12)

Figure 2. Sketch of $\left|J_0\right.〉$ in Eqs.(13,16,18) and $\left|J_0^{+}\right.〉$ in Eq.(20); the various symbols have the same meaning as in Fig.1.

Figure 2. Sketch of $\left|J_0\right.〉$ in Eqs.(13,16,18) and $\left|J_0^{+}\right.〉$ in Eq.(20); the various symbols have the same meaning as in Fig.1.

• $H_0$-triplet scattering Taking $P\left(k_{j=1,..n<N-1}\right)=k_j$ for simplicity, because the final result will prove P independent, yields the following properties for the $\left|J_j\right.〉$ states

  $$\begin{array}{c}\left|J_0\right.〉=c_{2,+}^{+}{c}_{3,+}^{+}\prod _{j=5}^{n+2}{c}_{j,{(-1)}^j}^{+}\left|0\right.〉\\ 〈J_0\left|H_0\right|J_0〉=0,〈J_0\left|H_0\right|J_j^{(\pm )}〉\ne 0\Rightarrow \\ \left\{\begin{array}{c}\left|J_1\right.〉=c_{1,+}^{+}{c}_{2,+}\left|J_0\right.〉\hfill \\ \left|J_4\right.〉=c_{4,+}^{+}{c}_{3,+}\left|J_0\right.〉\hfill \\ \left|J_{j>4}^{\pm }\right.〉=c_{j\pm 1,{\sigma }_j}^{+}{c}_{j,{\sigma }_j}\left|J_0\right.〉\hfill \end{array}\right.\\ 〈J_0\left|H_0\right|J_{1,4}〉=〈J_0\left|H_0\right|J_{j>4}^{\pm }〉=-t\\ e^{ik.J_1}=e^{-i{k}_1}{e}^{ik.J_0}{e}^{ik.J_4}=e^{i{k}_2}{e}^{ik.J_0}\\ e^{ik.J_{j>4}^{\pm }}=e^{\pm i{k}_{j-2}}{e}^{ik.J_0}\end{array}.$$

  (13)
  In addition an important relationship between $e^{ik.J_0},e^{i{T}_{12}k.J_0}$, valid also for $H_1$ triplet and singlet scattering, can be deduced from Eq.(13)

  $$\frac{e^{ik.J_0}}{e^{i{T}_{12}k.J_0}}=\frac{e^{\frac{i}{2}\left(k_1{r}_2+k_2{r}_3\right)}}{e^{\frac{i}{2}\left(k_2{r}_2+k_1{r}_3\right)}}=\frac{e^{\frac{i}{2}\left(k_2-k_1\right)}}{e^{\frac{i}{2}\left(k_1-k_2\right)}}.$$

  (14)
  Replacing $x_P,x_{T_{12}P},z_P,z_{T_{12}P}$ by $x\left(k_1,k_2\right)$, $x\left(k_2,k_1\right)$, $z\left(k_1,k_2\right),z\left(k_2,k_1\right)$ and taking advantage of Eqs.(12,13,14) give

  $$\begin{array}{c}\frac{z\left(k_1,k_2\right)}{z\left(k_2,k_1\right)}=\frac{e^{\frac{i}{2}\left(k_2-k_1\right)}}{e^{\frac{i}{2}\left(k_1-k_2\right)}}\frac{t\left(e^{-i{k}_1}+e^{i{k}_2}\right)+ϵ\left(k_1\right)+ϵ\left(k_2\right)}{t\left(e^{-i{k}_2}+e^{i{k}_1}\right)+ϵ\left(k_1\right)+ϵ\left(k_2\right)}\hfill \\ =\frac{e^{\frac{i}{2}\left(k_2-k_1\right)}}{e^{\frac{i}{2}\left(k_1-k_2\right)}}\frac{e^{i{k}_1}+e^{-i{k}_2}}{e^{i{k}_2}+e^{-i{k}_1}}=1\Rightarrow \frac{x\left(k_2,k_1\right)}{x\left(k_1,k_2\right)}=-1\hfill \end{array}.$$
  Applying this equation with any P, such that $P\left(k_1\right)=k_i,P\left(k_2\right)=k_j$, leads finally to $\frac{x\left(k_i,k_j\right)}{x\left(k_j,k_i\right)}=-1,\forall i,j\ne i$. Noteworthy is that $z\left(k_1,k_2\right),z\left(k_2,k_1\right)$ are found to be complex conjugate of each other, a property which will be seen to hold for all other kinds of two-body scattering investigated in this work. Therefore it is convenient to recast $\frac{x\left(k_i,k_j\right)}{x\left(k_j,k_i\right)}=-1$ as

  $$\frac{x\left(k_i,k_j\right)}{x\left(k_j,k_i\right)}=e^{2i{\theta }_t\left(k_i,k_j\right)}\Rightarrow {\theta }_t\left(k_i,k_{j\ne i}\right)=\frac{\pi }{2}.$$

  (15)
  Remarkably, ${\theta }_t$ is constant and thence independent from $t,U_0$.
• $H_0$-singlet scattering Here are the properties of the $\left|J_j\right.〉$ states

  $$\begin{array}{c}\left|J_0\right.〉=c_{2,+}^{+}{c}_{2,-}^{+}\prod _{j=4}^{n+1}{c}_{j,{(-1)}^j}^{+}\left|0\right.〉\\ 〈J_0\left|H_0\right|J_j^{\pm }〉\ne 0\Rightarrow \\ \left\{\begin{array}{c}\left|J_1^{\pm }\right.〉=c_{1,\pm }^{+}{c}_{2,\pm }\left|J_0\right.〉\hfill \\ \left|J_3^{\pm }\right.〉=c_{3,\pm }^{+}{c}_{2,\pm }\left|J_0\right.〉\hfill \\ \left|J_{j>3}^{\pm }\right.〉=c_{j\pm 1,{\sigma }_j}^{+}{c}_{j,{\sigma }_j}\left|J_0\right.〉\hfill \end{array}\right.\\ 〈J_0\left|H_0\right|J_0〉=U_0,〈J_0\left|H_0\right|J_{1,3,j>3}^{\pm }〉=-t\\ e^{ik.J_1^{\pm }}=e^{-i{k}_1}{e}^{ik.J_0},e^{ik.J_3^{\pm }}=e^{i{k}_2}{e}^{ik.J_0}\\ e^{ik.J_{j>3}^{\pm }}=e^{\pm i{k}_{j-1}}{e}^{ik.J_0}\end{array}.$$

  (16)
  It is inferred from $e^{ik.J_0}=e^{i{T}_{12}k.J_0}$ and Eq.(16)

  $$\begin{array}{c}\left.\begin{array}{c}\frac{z\left(k_1,k_2\right)}{z\left(k_2,k_1\right)}=\hfill \\ \frac{U_0-2t\left(e^{-i{k}_1}+e^{i{k}_2}\right)-ϵ\left(k_1\right)-ϵ\left(k_2\right)}{U_0-2t\left(e^{-i{k}_2}+e^{i{k}_1}\right)-ϵ\left(k_1\right)-ϵ\left(k_2\right)}=\hfill \\ \frac{U_0/t-e^{-i{k}_1}+e^{i{k}_1}-e^{i{k}_2}+e^{-i{k}_2}}{U_0/t-e^{i{k}_1}+e^{-i{k}_1}-e^{-i{k}_2}+e^{i{k}_2}}=\hfill \\ \frac{U_0/\left(2t\right)+i(sin{k}_1-sin{k}_2)}{U_0/\left(2t\right)-i(sin{k}_1-sin{k}_2)}\hfill \end{array}\right\}\Rightarrow \hfill \\ \frac{x\left(k_1,k_2\right)}{x\left(k_2,k_1\right)}=-\frac{z\left(k_2,k_1\right)}{z\left(k_1,k_2\right)}=\frac{sin{k}_2-sin{k}_1+i{U}_0/\left(2t\right)}{sin{k}_2-sin{k}_1-i{U}_0/\left(2t\right)}\hfill \end{array}.$$
  Applying this equation with any P, as done in Eq.(15), leads finally to

  $$\begin{array}{c}\frac{x\left(k_i,k_j\right)}{x\left(k_j,k_i\right)}=e^{2i{\theta }_s\left(k_i,k_j\right)}\Rightarrow \hfill \\ tan{\theta }_s\left(k_i,k_{j\ne i}\right)=\frac{U_0}{2t\left(sin{k}_j-sin{k}_i\right)}\hfill \end{array}.$$

  (17)
  Substituting the short-cut $x\left(k_i,k_j\right)$ to $x_P$ is seen to be vindicated by ${\theta }_{t,s}\left(k_i,k_j\right)$ showing up indeed independent from every $k_{l\ne i,j}$ in Eqs.(15,17). This fortunate property could be established owing to a dedicated choice of $\left|J_0\right.〉$.
• $H_1$-triplet scattering The properties of the $\left|J_j\right.〉$ states are as follows for $n<N-1$

  $$\begin{array}{c}\left|J_0\right.〉=c_{2,+}^{+}{c}_{3,+}^{+}\prod _{j=3}^{\frac{n}{2}+1}{c}_{2j,+}^{+}{c}_{2j,-}^{+}\left|0\right.〉\hfill \\ 〈J_0\left|H_1\right|J_0〉=U_1-J\hfill \\ 〈J_0\left|H_1\right|J_{j>0,(\pm )}^{(\pm )}〉\ne 0\Rightarrow \hfill \\ \left\{\begin{array}{c}\left|J_1\right.〉=c_{1,+}^{+}{c}_{2,+}\left|J_0\right.〉\hfill \\ \left|J_4\right.〉=c_{4,+}^{+}{c}_{3,+}\left|J_0\right.〉\hfill \\ \left|J_{j>2,+}^{\pm }\right.〉=c_{2j\pm 1,+}^{+}{c}_{2j,+}\left|J_0\right.〉\hfill \\ \left|J_{j>2,-}^{\pm }\right.〉=c_{2j\pm 1,-}^{+}{c}_{2j,-}\left|J_0\right.〉\hfill \end{array}\right.\hfill \\ 〈J_0\left|H_1\right|J_{1,4}〉=〈J_0\left|H_1\right|J_{j>2,\pm }^{\pm }〉=-t\hfill \\ e^{ik.J_1}=e^{-i{k}_1}{e}^{ik.J_0},e^{ik.J_4}=e^{i{k}_2}{e}^{ik.J_0}\hfill \\ e^{ik.J_{j>2,+}^{\pm }}=e^{\pm i{k}_{2j-2}}{e}^{ik.J_0}\hfill \\ e^{ik.J_{j>2,-}^{\pm }}=e^{\pm i{k}_{2j-3}}{e}^{ik.J_0}\hfill \end{array}.$$

  (18)
  Since Eq.(14) applies to this case too, it is inferred from Eq.(18)

  $$\begin{array}{c}\frac{z\left(k_1,k_2\right)}{z\left(k_2,k_1\right)}=\frac{e^{\frac{i}{2}\left(k_2-k_1\right)}}{e^{\frac{i}{2}\left(k_1-k_2\right)}}\frac{t\left(e^{-i{k}_1}+e^{i{k}_2}\right)-U_1+J+ϵ\left(k_1\right)+ϵ\left(k_2\right)}{t\left(e^{-i{k}_2}+e^{i{k}_1}\right)-U_1+J+ϵ\left(k_1\right)+ϵ\left(k_2\right)}\hfill \\ =\frac{e^{\frac{i}{2}\left(k_2-k_1\right)}-\frac{2t}{U_1-J}cos\left(\frac{k_1+k_2}{2}\right)}{e^{\frac{i}{2}\left(k_1-k_2\right)}-\frac{2t}{U_1-J}cos\left(\frac{k_1+k_2}{2}\right)}\Rightarrow \frac{x\left(k_j,k_i\right)}{x\left(k_i,k_j\right)}\hfill \\ =\frac{sin\left(\frac{k_i-k_j}{2}\right)+i\left(cos\left(\frac{k_i-k_j}{2}\right)-\frac{2t}{U_1-J}cos\left(\frac{k_i+k_j}{2}\right)\right)}{sin\left(\frac{k_i-k_j}{2}\right)-i\left(cos\left(\frac{k_i-k_j}{2}\right)-\frac{2t}{U_1-J}cos\left(\frac{k_i+k_j}{2}\right)\right)}\hfill \\ =e^{2i{\theta }_t\left(k_i,k_j\right)}\hfill \end{array},$$

  which yields finally for ${\theta }_t\left(k_i,k_j\right)$

  $$\begin{array}{c}tan{\theta }_t\left(k_i,k_{j\ne i}\right)=\frac{cos\frac{k_i-k_j}{2}-\frac{2t}{U_1-J}cos\left(\frac{k_i+k_j}{2}\right)}{sin\left(\frac{k_i-k_j}{2}\right)}\hfill \end{array}.$$

  (19)
• $H_1$-singlet scattering The properties of the $\left|J_j\right.〉$ states are as follows

  $$\begin{array}{c}\left|J_0^{+}\right.〉=c_{2,+}^{+}{c}_{3,-}^{+}\prod _{j=3}^{\frac{n}{2}+1}{c}_{2j,+}^{+}{c}_{2j,-}^{+}\left|0\right.〉\\ \left|J_0^{-}\right.〉=c_{2,-}^{+}{c}_{3,+}^{+}\prod _{j=3}^{\frac{n}{2}+1}{c}_{2j,+}^{+}{c}_{2j,-}^{+}\left|0\right.〉\\ 〈J_0^{+}\left|H_1\right|J_0^{+}〉=U_1〈J_0^{+}\left|H_1\right|J_0^{-}〉=-J\\ 〈J_0^{+}\left|H_1\right|J_{j>0,(\pm )}^{(\pm )}〉\ne 0\Rightarrow \\ \left\{\begin{array}{c}\left|J_1\right.〉=c_{1,+}^{+}{c}_{2,+}\left|J_0^{+}\right.〉\hfill \\ \left|J_2\right.〉=c_{2,-}^{+}{c}_{3,-}\left|J_0^{+}\right.〉\hfill \\ \left|J_3\right.〉=c_{3,+}^{+}{c}_{2,+}\left|J_0^{+}\right.〉\hfill \\ \left|J_4\right.〉=c_{4,-}^{+}{c}_{3,-}\left|J_0^{+}\right.〉\hfill \\ \left|J_{j>2,+}^{\pm }\right.〉=c_{2j\pm 1,+}^{+}{c}_{2j,}^{+}\left|J_0^{+}\right.〉\hfill \\ \left|J_{j>2,-}^{\pm }\right.〉=c_{2j\pm 1,-}^{+}{c}_{2j,}^{-}\left|J_0^{+}\right.〉\hfill \end{array}\right.\\ 〈J_0^{+}\left|H_1\right|J_{1,..4}〉=〈J_0^{+}\left|H_1\right|J_{j>2,\pm }^{\pm }〉=-t\\ e^{ik.J_0^{-}}=e^{ik.J_0^{+}}\\ e^{ik.J_1}=e^{-i{k}_1}{e}^{ik.J_0^{+}},e^{ik.J_3}=e^{i{k}_1}{e}^{ik.J_0^{+}}\\ e^{ik.J_2}=e^{-i{k}_2}{e}^{ik.J_0^{+}},e^{ik.J_4}=e^{i{k}_2}{e}^{ik.J_0^{+}}\\ e^{ik.J_{j>2,+}^{\pm }}=e^{\pm i{k}_{2j-2}}{e}^{ik.J_0}\\ e^{ik.J_{j>2,-}^{\pm }}=e^{\pm i{k}_{2j-3}}{e}^{ik.J_0}\end{array}.$$

  (20)
  Since Eq.(14) is valid in this case too, it is inferred from Eq.(20)

  $$\begin{array}{c}\frac{z\left(k_1,k_2\right)}{z\left(k_2,k_1\right)}=\frac{e^{\frac{i}{2}\left(k_2-k_1\right)}}{e^{\frac{i}{2}\left(k_1-k_2\right)}}\times \hfill \\ \frac{U_1-J-t\left(e^{i{k}_1}+e^{-i{k}_1}+e^{i{k}_2}+e^{-i{k}_2}\right)-ϵ\left(k_1\right)-ϵ\left(k_2\right)}{U_1-J-t\left(e^{i{k}_1}+e^{-i{k}_1}+e^{i{k}_2}+e^{-i{k}_2}\right)-ϵ\left(k_1\right)-ϵ\left(k_2\right)}\hfill \\ =\frac{e^{\frac{i}{2}\left(k_2-k_1\right)}}{e^{\frac{i}{2}\left(k_1-k_2\right)}}\Rightarrow \frac{x\left(k_j,k_i\right)}{x\left(k_i,k_j\right)}=-\frac{e^{\frac{i}{2}\left(k_j-k_i\right)}}{e^{\frac{i}{2}\left(k_i-k_j\right)}}=e^{2i{\theta }_s\left(k_i,k_j\right)}\hfill \end{array},$$

  which yields finally for ${\theta }_s\left(k_i,k_j\right)$

  $${\theta }_s\left(k_i,k_{j\ne i}\right)=\frac{\pi -k_i+k_j}{2}.$$

  (21)
  Unlike ${\theta }_t$ in Eq.(19), ${\theta }_s$ shows up independent from $t,U_1,J$.

4. 3-Boundary condition

The hereabove results will be taken advantage of, in order to show that the $k_{i=1,..n}$’s are related to one another through a boundary condition, ensueing from Eq.(5). To that end, the permutations $P_1,P_N$ and n-electron states $\left|I_1\right.〉,\left|I_N\right.〉$ are needed

$$\begin{array}{c}\left.\begin{array}{c}{P}_1\left(k_{i=1,..n}\right)=k_i\hfill \\ P_N\left(k_{i=1,..n-1}\right)=k_{i+1},P_N\left(k_n\right)=k_1\hfill \end{array}\right\}\Rightarrow \hfill \\ P_N=\prod _{j=2}^n{T}_{1j}{P}_1\Rightarrow x_{P_N}=e^{2i{\sum }_{j=2}^n{\theta }_{\beta }\left(k_1,k_j\right)}{x}_{P_1}\hfill \\ \left.\begin{array}{c}\left|I_1\right.〉=\prod _{j=1}^n{c}_{j,{\sigma }_j}^{+}\left|0\right.〉\hfill \\ \left|I_N\right.〉=c_{N,{\sigma }_1}^{+}{c}_{1,{\sigma }_1}\left|I_1\right.〉\hfill \end{array}\right\}\Rightarrow \frac{e^{i{P}_{N}k.I_N}}{e^{i{P}_{1}k.I_1}}=e^{iN{k}_1}\hfill \end{array},$$

(22)

with ${\sigma }_{j=1,..n-1}{\sigma }_{j+1}>0$ ($<0$) for $\beta =t$ ($\beta =s$). Besides, it is convenient to take the sequence $\left\{k_{i=1,..n}\right\}$, such that $k_{i=1,..n-1}<k_{i+1}$. Then the sought boundary condition reads by substituting $T_{1i}{P}_1,T_{i1}{P}_N$ to $P_1,P_N$ in Eq.(22) and applying it further to any $k_{i>1}$

$$\begin{array}{c}{x}_{P_1}{e}^{i{P}_{1}k.I_1}=x_{P_N}{e}^{i{P}_{N}k.I_N}\Rightarrow \hfill \\ e^{i\left(N{k}_{i=1,..n}+2{\sum }_{j=1}^n{\theta }_{\beta }\left(k_i,k_{j\ne i}\right)\right)}=1\Rightarrow \hfill \\ k_i=2\pi {\alpha }_{i=1,..n}-\frac{2}{N}\sum _{j=1}^n{\theta }_{\beta }\left(k_i,k_{j\ne i}\right)\hfill \end{array},$$

(23)

with ${\alpha }_{i=1,..n}=\frac{m_i}{N}$ and the integers $m_{i=1,..n}\in \left[1,N\right]\Rightarrow {\alpha }_{i=1,..n}\in \left[0,1\right]$. At last taking the limit $N\to \infty ,n\to \infty ,\frac{n}{N}\to c_0\in \left[0,1\right]$ yields

$$\begin{array}{c}k(\alpha \in \left[0,1\right])=2\pi \alpha -2{\int }_{-\pi }^{\pi }{\theta }_{\beta }\left(k,u\right){\rho }_{\beta }\left(u\right)du\hfill \\ {\rho }_{\beta }\left(k\right)dk=d\alpha \Rightarrow \hfill \\ {\rho }_{\beta }\left(k\right)=\frac{d\alpha }{dk}=\frac{1}{2\pi }+{\int }_{-\pi }^{\pi }\frac{\partial {\theta }_{\beta }}{\partial k}\left(k,u\right){\rho }_{\beta }\left(u\right)\frac{du}{\pi }\hfill \\ {\epsilon }_{\beta }={\int }_{-\pi }^{\pi }ϵ\left(k\right){\rho }_{\beta }\left(k\right)dk,c_0={\int }_{-\pi }^{\pi }{\rho }_{\beta }\left(k\right)dk\hfill \end{array},$$

(24)

for which $c_0,{\epsilon }_{\beta =t,s},{\rho }_{\beta }\left(k\right)$ stand for the whole electron concentration, triplet or singlet energy per site and corresponding one-electron density of states, respectively. Though each eigenvalue ${\epsilon }_{\beta =t,s}$ is seen to read as a sum over one-particle energies $ϵ\left(k\right)$, as is the case for independent electrons[9], there is a one to one correspondence between ${\epsilon }_{\beta }$ and its associated, $h_{e-e}$-dependent ${\rho }_{\beta }\left(k\right)$, whereas the density of states is unique for all many independent electron states. Moreover, since each eigenvector $\psi$ is defined by a unique sequence $\left\{k_{i=1,..n}\right\}$, two different eigenvectors do not even belong in the same vector space, because their respective $\left\{k_{i=1,..n}\right\}$-dependent bases $\left\{{\varphi }_P\right\}$ are thence linearly independent from each other.

Solving Eq.(24) for the groundstate energy will be done below as follows

$$\begin{array}{c}{\rho }_{\beta }\left(k\right)=\frac{1}{2\pi }+{\int }_{-k_{\beta }}^{k_{\beta }}\frac{\partial {\theta }_{\beta }}{\partial k}\left(k,u\right){\rho }_{\beta }\left(u\right)\frac{du}{\pi }=\frac{1}{2\pi }+\hfill \\ {\int }_0^{k_{\beta }}\left(\frac{\partial {\theta }_{\beta }}{\partial k}\left(k,u\right){\rho }_{\beta }\left(u\right)+\frac{\partial {\theta }_{\beta }}{\partial k}\left(k,-u\right){\rho }_{\beta }(-u)\right)\frac{du}{\pi }\hfill \end{array},$$

(25)

with $k_{\beta }\in \left[0,\pi \right]$. Besides Eqs.(15,17,19,21) imply

$$\frac{\partial {\theta }_{\beta }}{\partial k}\left(k,u\right)=\frac{\partial {\theta }_{\beta }}{\partial k}\left(-k,-u\right),\frac{\partial {\theta }_{\beta }}{\partial k}\left(k,-u\right)=\frac{\partial {\theta }_{\beta }}{\partial k}\left(-k,u\right),$$

whence it is inferred ${\rho }_{\beta }\left(k\right)={\rho }_{\beta }(-k)$, so that Eq.(25) can be recast as

$${\rho }_{\beta }\left(k\right)=\frac{1}{2\pi }+{\int }_0^{k_{\beta }}{g}_{\beta }\left(k,u\right){\rho }_{\beta }\left(u\right)\frac{du}{\pi },$$

(26)

with $g_{\beta }\left(k,u\right)=\frac{\partial {\theta }_{\beta }}{\partial k}\left(k,u\right)+\frac{\partial {\theta }_{\beta }}{\partial k}\left(k,-u\right)$. By discretising $\left[0,k_{\beta }\right]\Rightarrow k_{j=1,..m}=\frac{j-1}{m-1}{k}_{\beta }$ and calculating the integral in Eq.(26) with help of Simpson’s rule, Eq.(26) will be eventually solved below for ${\rho }_{\beta }\left(k\right)$, as a Cramer system, comprising m equations in terms of the unknowns ${\rho }_{\beta }\left(k_{j=1,..m}\right)$, while $\frac{\partial {\theta }_t}{\partial k}\left(k,u\right),\frac{\partial {\theta }_s}{\partial k}\left(k,u\right)$ are taken from Eq.(15),Eq.(17) (Eq.(19),Eq.(21)), respectively, in case of $H_0$ ($H_1$). Finally the groundstate energy ${\epsilon }_g$ is achieved by reckoning $c_t,{\epsilon }_t,c_s,{\epsilon }_s$ as follows

$$\begin{array}{c}{c}_{\beta }\left(k_{\beta =t,s}\right)=2{\int }_0^{k_{\beta }}{\rho }_{\beta }\left(k\right)dk\hfill \\ {\epsilon }_{\beta }\left(k_{\beta }\right)=2{\int }_0^{k_{\beta }}ϵ\left(k\right){\rho }_{\beta }\left(k\right)dk\hfill \end{array}.$$

(27)

Assigning the values of $k_t\left(c_0\right),k_s\left(c_0\right)$ thanks to the constraint $c_t=c_s=c_0$ gives finally access to ${\epsilon }_g\left(c_0\right)$ as the lower of ${\epsilon }_t\left(k_t\left(c_0\right)\right),{\epsilon }_s\left(k_s\left(c_0\right)\right)$.

5. 4-Groundstate energy for $H_0$

${\theta }_t$ being constant in Eq.(15) entails via Eq.(26) ${\rho }_t\left(k\right)=\frac{1}{2\pi },\forall k$, which further implies thanks to Eq.(27)

$$\left.\begin{array}{c}{c}_t=\frac{k_t}{\pi }\hfill \\ {\epsilon }_t=-\frac{2t}{\pi }sin{k}_t\hfill \end{array}\right\}\Rightarrow {\epsilon }_t\left(c_t\right)=-\frac{2t}{\pi }sin\left(\pi c_t\right),$$

(28)

which shows up independent from $U_0$. The ${\epsilon }_t\left(c_0\right),{\epsilon }_s\left(c_0\right)$ data have been plotted in Fig.3. The inequality ${\epsilon }_s(c_0\in \left[0,1\right])<{\epsilon }_t\left(c_0\right)$ can be seen to hold for any $\frac{U_0}{t}$ value. A comparison with groundstate energies ${\eta }_1\left(c_0\right)$, reckoned with a previous version of Bethe’s wave-function[6], reveals that the inequality ${\epsilon }_s\left(c_0\right)<{\eta }_1\left(c_0\right)$ is found to hold for $\frac{U_0}{t}<4$, whereas the opposite one ${\epsilon }_s\left(c_0\right)>{\eta }_1\left(c_0\right)$ is observed for $\frac{U_0}{t}>4$. Actually this discrepancy results from ${\epsilon }_s\left(c_0\right)$ being calculated for singlet electrons only, whereas ${\eta }_1\left(c_0\right)$ has been worked out for a mixture of singlet and triplet electrons. Therefore it can be explained as follows: the smaller $\frac{U_0}{t}$ is, the bigger ${\rho }_s\left(k\right)$ is with respect to ${\rho }_t\left(k\right)=\frac{1}{2\pi }$ for $k<\frac{\pi }{2}$, so that refraining from mixing singlet and triplet electrons causes ${\epsilon }_s\left(c_0\right)<{\eta }_1\left(c_0\right)$. Contrarily, ${\epsilon }_s\left(c_0\right)$ is seen to merge into ${\epsilon }_t\left(c_0\right)$ for $\frac{U_0}{t}\to \infty$, because of ${\theta }_s(k_i,k_j)\to {\theta }_t=\frac{\pi }{2}$ as seen in Eqs.(15,17). Consequently, $\frac{U_0}{t}\to \infty$ entails that ${\rho }_s(k<\frac{\pi }{2})\to {\rho }_t,{\rho }_s(k>\frac{\pi }{2})<{\rho }_t$, so that the singlet-triplet mixture favours eventually the opposite conclusion ${\eta }_1\left(c_0\right)<{\epsilon }_s\left(c_0\right)$ for $\frac{U_0}{t}>4$.

Figure 3. Plots of ${\epsilon }_t\left(c_0\right)$, as given in Eq.(28) (white square), ${\epsilon }_s\left(c_0\right)$ (solid line, dashed line, dotted line, dashed-dotted line) and ${\eta }_1\left(c_0\right)$ (white triangle, white circle, white diamond, ×), calculated for $H_0$ with $\gamma =\frac{U_0}{t}=1,2,4,8$, as said in the text.

Figure 3. Plots of ${\epsilon }_t\left(c_0\right)$, as given in Eq.(28) (white square), ${\epsilon }_s\left(c_0\right)$ (solid line, dashed line, dotted line, dashed-dotted line) and ${\eta }_1\left(c_0\right)$ (white triangle, white circle, white diamond, ×), calculated for $H_0$ with $\gamma =\frac{U_0}{t}=1,2,4,8$, as said in the text.

Furthermore it is in order to compare ${\epsilon }_s$ with the electronic energy ${\eta }_2$, obtained with help of the correlated Fermi gas (CFG) model[7] in normal metals, the characteristic features of which will be recalled now for self-containedness. Each independent-electron band of dispersion $ϵ(k\in \left[-\pi ,\pi \right])$, as given in Eq.(1), accommodating at most 2 electrons of opposite spin direction per site, splits into one singlet and one triplet band, each of them accommodating at most 1 electron per site. The corresponding dispersion curves ${ϵ}_s\left(k\right),{ϵ}_t\left(k\right)$ read for the general Hamiltonian H as follows

$$\begin{array}{c}{ϵ}_t\left(k\right)=\sum _{k^{\prime }}〈0\left|c_{k,\sigma }{c}_{k^{\prime },\sigma }H{c}_{k^{\prime },\sigma }^{+}{c}_{k,\sigma }^{+}\right|0〉\hfill \\ {ϵ}_s\left(k\right)=\sum _{k^{\prime }}〈0\left|c_{k,\sigma }{c}_{k^{\prime },-\sigma }H{c}_{k^{\prime },-\sigma }^{+}{c}_{k,\sigma }^{+}\right|0〉\hfill \end{array},$$

(29)

for which $〈0\left|c_{k,\sigma }{c}_{k,\sigma }^{+}\right|0〉=〈0\left|c_{k^{\prime },\pm \sigma }{c}_{k^{\prime },\pm \sigma }^{+}\right|0〉=1$ has been assumed and $c_{k,\sigma }^{+},c_{k,\sigma }$ are one-electron creation and annihilation operators on the Bloch[9] state $\left|k,\sigma \right.〉$

$$\left|k,\sigma \right.〉=c_{k,\sigma }^{+}\left|0\right.〉,\left|0\right.〉=c_{k,\sigma }\left|k,\sigma \right.〉.$$

Besides, the calculation requires the expression of H in momentum space, which will be taken from a previous work[8] for illustration in case of $H_0,H_1$. Note that ${ϵ}_t\left(k\right),{ϵ}_s\left(k\right)$ are found in general to depend not only on $h_{e-e}$ but also on the concentration of singlet or triplet electrons $c_s,c_t$ and there is ${ϵ}_t\left(k\right)={ϵ}_t(-k),{ϵ}_s\left(k\right)={ϵ}_s(-k)$. Then both singlet and triplet bands are populated in accordance with Fermi-Dirac statistics[9], which yields in the $d=1$ case at $T=0K$

$$\begin{array}{c}{c}_s=\frac{k_s}{\pi },{\eta }_s={\int }_0^{k_s}{ϵ}_s\left(k\right)\frac{dk}{\pi },{ϵ}_s\left(k_s\right)=E_F\hfill \\ c_t=\frac{k_t}{\pi },{\eta }_t={\int }_0^{k_t}{ϵ}_t\left(k\right)\frac{dk}{\pi },{ϵ}_t\left(k_t\right)=E_F\hfill \\ c_0=c_s+c_t,{\eta }_2\left(c_0\right)={\eta }_s\left(c_s\right)+{\eta }_t\left(c_t\right)\hfill \end{array},$$

(30)

with $E_F,{\eta }_s,{\eta }_t$ standing for the Fermi energy[9] and the partial singlet and triplet energy, respectively.

The expressions of ${ϵ}_t,{\eta }_t,{ϵ}_s,{\eta }_s$ are recalled[7] to read for the Hubbard model

$$\begin{array}{c}{ϵ}_t\left(k\right)=ϵ\left(k\right),{\eta }_t=-\frac{2t}{\pi }sin{k}_t\hfill \\ {ϵ}_s\left(k\right)=ϵ\left(k\right)+U_0\frac{c_s}{2},{\eta }_s=-\frac{2t}{\pi }sin{k}_s+U_0\frac{c_s^2}{2}\hfill \end{array}.$$

Unlike ${\epsilon }_s\ne {\eta }_1$, the energies ${\eta }_1,{\eta }_2$ have been found[7] quite close to each other, namely there is $\left|1-\frac{{\eta }_1(c_0,\frac{U_0}{t})}{{\eta }_2(c_0,\frac{U_0}{t})}\right|<.01$ for all $c_0,\frac{U_0}{t}$-values. As a matter of fact this agreement is all the more baffling, since ${\eta }_1$ is a true eigenvalue, whereas ${\eta }_2$ comes out of a variational calculation, and furthermore the associated eigen- and variational many-electron states belong in quite different vector spaces. Hence this unexpected feature is likely to stem from both states comprising singlet and triplet electrons.

6. 5-Groundstate energy for $H_1$

$\frac{\partial {\theta }_s}{\partial k}\left(k,u\right)=-.5$ is inferred from Eq.(21) to be constant, which entails owing to Eq.(26) that ${\rho }_s\left(k\right)=\frac{1}{2\left(k_s+\pi \right)}$ is constant too, and eventually thanks to Eq.(27)

$$\begin{array}{c}{c}_s=\frac{k_s}{k_s+\pi }\Rightarrow \hfill \\ {\epsilon }_s=-\frac{2t}{k_s+\pi }sin{k}_s=-\frac{2t}{\pi }\left(1-c_s\right)sin\left(\frac{\pi c_s}{1-c_s}\right)\hfill \end{array}.$$

(31)

Figure 4. Plots of ${\epsilon }_s\left(c_0\right)$, as given in Eq.(31) (white square), ${\epsilon }_t\left(c_0\right)$ (solid line, dashed line, dotted line, dashed-dotted line) and ${\eta }_2\left(c_0\right)$ (white triangle, white circle, white diamond, ×), calculated for $H_1$ with $\gamma =\frac{U_1-J}{t}=1,.75,.5,.25$ and $J=.1U_1$, as said in the text.

Figure 4. Plots of ${\epsilon }_s\left(c_0\right)$, as given in Eq.(31) (white square), ${\epsilon }_t\left(c_0\right)$ (solid line, dashed line, dotted line, dashed-dotted line) and ${\eta }_2\left(c_0\right)$ (white triangle, white circle, white diamond, ×), calculated for $H_1$ with $\gamma =\frac{U_1-J}{t}=1,.75,.5,.25$ and $J=.1U_1$, as said in the text.

It results from Eq.(31) that $c_s\le .5$ and ${\epsilon }_s\left(c_s\right)$ shows up independent from $U_1-J$. As a matter of fact, there is no meaningful ${\epsilon }_t$ solution either, i.e. with ${\rho }_t\left(k\right)>0,\forall k$, for $c_t>.5$. The ${\epsilon }_t,{\epsilon }_s$ data, plotted in Fig.4, show that the inequality ${\epsilon }_t\left(c_0\right)<{\epsilon }_s\left(c_0\right)$ holds for all $\gamma$-values.

Implementing Eqs.(29,30) for $H_1$ yields

$$\begin{array}{c}{ϵ}_t\left(k\right)=ϵ\left(k\right)+\left(U_1-J\right)\left(c_t+\frac{sin\left(k-k_t\right)-sin\left(k+k_t\right)}{2\pi }\right)\hfill \\ {\eta }_t=-\frac{2t}{\pi }sin{k}_t+\left(U_1-J\right)\left(c_t^2+\frac{cos\left(2k_t\right)-1}{2{\pi }^2}\right)\hfill \\ {ϵ}_s\left(k\right)=ϵ\left(k\right)+U_1{c}_s+J\frac{sin\left(k+k_s\right)-sin\left(k-k_s\right)}{\pi }\hfill \\ {\eta }_s=-\frac{2t}{\pi }sin{k}_s+U_1{c}_s^2+J\frac{1-cos\left(2k_s\right)}{{\pi }^2}\hfill \end{array}.$$

The ${\eta }_2$ data have been plotted in Fig.4. The inequality ${\eta }_2\left(c_0\right)<{\epsilon }_t\left(c_0\right)$ is seen to hold for all $c_0<.5$ values and is likely to ensue again from the $CFG$ state, including triplet and singlet electrons with $c_t\approx c_s\approx \frac{c_0}{2},\forall c_0$, by contrast with Bethe’s wave-function of eigenvalue ${\epsilon }_t\left(c_0\right)$ comprising only triplet electrons.

7. 6-Discussion

Since the groundstate is widely believed to describe the properties of any physical system at $T=0K$, it is of significance to sort, out of the various many electron states, discussed hereabove, that one, likely to account at best for the observed properties of interacting electrons. To that end, it should be noticed that all of them share a common property, namely the total energy $\epsilon$ consists in a sum over one-fermion energy either $ϵ\left(k\right)$ or ${ϵ}_{\beta =s,t}\left(k\right)$. Thus, the groundstate can be built[9] by populating every one-electron state from the bottom of the one-electron band, corresponding to $ϵ(k=0)$, up to a $c_0$ dependent upper bound, designated as the Fermi energy $E_F$. The reader can check that this requirement is met by all kinds of many-electron states of concern hereabove, i.e. Bethe’s wave-function and $CFG$ states. However Fermi-Dirac statistics requires in addition at $T=0K$ that the relationship $E_F=\frac{\partial \epsilon }{\partial c_0}$ hold[9,10]. Obviously solely the CFG solution meets successfully this constraint, because it obeys Fermi-Dirac statistics by definition (see $E_F={ϵ}_t\left(k_t\right)={ϵ}_s\left(k_s\right)$ in Eq.(30)), whereas there is $E_F\ne \frac{\partial \epsilon }{\partial c_0}$ for all kinds of Bethe’s wave-functions, studied here and elsewhere[3,4,6]. Yet, as seen above, the CFG solution is not the groundstate for $H_0$ in case of $\frac{U_0}{t}<4$.

Buttressing the claim that the CFG state is observable would help validate this analysis. To that end, let us discuss electron spin resonance (ESR) in case of $H_1$. Applying an external magnetic field H lifts[11] the degeneracy between the respective energies of one-electron states ${ϵ}_{\pm }\left(k\right)$ of $\left.c_{k,+}^{+}\right|0〉,\left.c_{k,-}^{+}\right|0〉$ for triplet electrons as follows

$$\begin{array}{c}{ϵ}_{\pm }\left(k\right)=f(k,k_{\pm }\left(H\right))\pm \frac{\mu H}{2}\hfill \\ f(k_{\pm }\left(H\right),k_{\pm }\left(H\right))\pm \frac{\mu H}{2}=E_F\left(H\right)\hfill \end{array},$$

for which $\mu$ stands for the electron magnetic moment and there is $f(x,y)={ϵ}_t\left(x\right)$, taken from Eq.(29) with $x=k$ and $y=k_{\pm }\left(H\right)$ instead of $k_t$ in case of $H\ne 0$, which thence implies $f(k_t,k_t)=E_F(H=0)$. Then the experiment consists of measuring the absorption of a resonant electromagnetic field of frequency $\omega$, such that $\hslash \omega =f(k_{+},k_{+})-f(k_{+},k_{-})$, which implies

$$\omega =\left(1+\frac{\frac{\partial f}{\partial y}(k_t,k_t)}{\frac{\partial f}{\partial x}(k_t,k_t)+\frac{\partial f}{\partial y}(k_t,k_t)}\right)\frac{\mu H}{\hslash }.$$

(32)

Remarkably the singlet electrons are seen not to contribute to the ESR signal, because their associated many-electron state is not degenerate at $H=0$, due to the electrons of spin $\pm \sigma$ keeping same concentration $=\frac{c_s}{2}$ even for $H\ne 0$. Actually the ESR signal has been observed[11] but in a paucity of cases, namely alcali, noble metals and Al at $\frac{\omega }{H}\approx 3$GHz/KG. This might stem from $\frac{\partial f}{\partial x}(k_t,k_t)+\frac{\partial f}{\partial y}(k_t,k_t)\approx 0$ in Eq.(32), which would shift $\omega$ upward at fixed H. Hopefully this work might kindle attempts at seeking the ESR signal in the IF rather than microwave range, as done usually[11] with $H<10KG$.

8. Conclusion

Bethe’s wave-function has been shown to subtend the subspace of scattering eigenstates of the $d=1$ Hubbard Hamiltonian. Though this analysis is independent of a previous one[3,4], both pieces of work lead to the same conclusions, summarised by Eqs.(3,7). The two-body scattering plays a key role. Likewise, though the Coulomb force does not depend on the spin of both electrons, involved in the scattering process, its outcome does indeed (see Eqs.(15,17,19,21)), as a consequence of Pauli’s principle. This analysis can be applied to any realistic $d=1$ Hamiltonian, which has been exemplified on $H_1$. Nevertheless Bethe’s wave-function turns out to be of limited significance in condensed matter physics, because it has been shown not to be observable. Actually, the groundstate can be observed at $T=0K$ solely for atoms, molecules and insulators. Conversely it cannot in superconducting and magnetic compounds and in normal metals, because, as argued elsewhere[7], there are two kinds of conduction electrons at thermal equilibrium with one another in each case, namely normal versus superconducting, normal versus magnetic and triplet versus singlet electrons, respectively.