Ground State Energy of Dilute Fermi Gases in 1D

TL;DR

This paper analyzes the ground state energy of a dilute spin-J Fermi gas with repulsive interactions in one dimension, linking it to the energy of a related spin chain model.

Contribution

It establishes the asymptotic behavior of the ground state energy for the dilute Fermi gas in 1D, connecting it to the Heisenberg antiferromagnet for spin-1/2.

Findings

Derived asymptotics of the ground state energy in the dilute limit.

Linked the Fermi gas energy to the energy of a spin chain model.

Provided a rigorous mathematical framework for the asymptotic analysis.

Abstract

We study the spin-J Fermi gas, interacting through a general repulsive 2-body potential, and prove asymptotics of the ground state energy in the dilute limit. The asymptotic behaviour is given in terms of the ground state energy of a spin chain, which is the Heisenberg antiferromagnet in the case of spin-1/2 fermions.

Equations

$$H_N=-\sum _{i=1}^N{\partial }_{x_i}^2+\sum _{1\le i<j\le N}v(x_i-x_j),$$

$$N\frac{{\pi }^2}{3}{\rho }^2\left(1+2\rho a_e+\dots \right),$$

$$\prod _{1\le i<j\le N}|\sin (\pi \frac{x_j-x_i}{L})|,$$

$$\underset{\begin{array}{c}\psi \in H^1[-R,R]\\ \psi (R)=\psi (-R)=1\end{array}}{\inf }{\int }_{-R}^{R}2|{\partial }_x\psi {|}^2+v(x)|\psi (x){|}^{2}dx={\int }_{-R}^{R}2|{\partial }_x{\psi }_{\text{even}}{|}^2+v(x)|{\psi }_{\text{even}}(x){|}^{2}dx=\frac{4}{R-a_e}.$$

$$\prod _{1\le i<j\le N}{f}_{i,j}(x_i,x_j),$$

$$f_{i,j}(x_i,x_j)=\{\begin{array}{ll}\sin (\pi \frac{x_j-x_i}{L})& j>i+1\text{ or }|x_j-x_i|>R\\ \sin (\pi \frac{R}{L}){\psi }_{\text{even}}(x_{i+1}-x_i)& j=i+1\text{ and }|x_{i+1}-x_i|\le R.\end{array}$$

$$\underset{\begin{array}{c}\psi \in H^1[-R,R]\\ \psi (R)=\psi (-R)=-1\end{array}}{\inf }{\int }_{-R}^{R}2|{\partial }_x\psi {|}^2+v(x)|\psi (x){|}^{2}dx={\int }_{-R}^{R}2|{\partial }_x{\psi }_{\text{odd}}{|}^2+v(x)|{\psi }_{\text{odd}}(x){|}^{2}dx=\frac{4}{R-a_o},$$

$$N\frac{{\pi }^2}{3}{\rho }^2\left(1+2\rho a_o+\dots \right),$$

$$N\frac{{\pi }^2}{3}{\rho }^2(1+2\ln (2)\rho a_e+2(1-\ln (2))\rho a_o+\dots ),$$

$$\left(\prod _{1\le i<j\le N}{f}_{i,j}(x_i,x_j)\right){\varphi }_{\text{spin}}({\sigma }_1,\dots ,{\sigma }_N),$$

$$f_{i,j}(x_i,x_j)=$$

$$2\rho a_e{⟨\frac{1}{N}\sum _i{P}_A^{i,i+1}⟩}_{{\varphi }_{\text{spin}}}+2\rho a_o{⟨\frac{1}{N}\sum _i{P}_S^{i,i+1}⟩}_{{\varphi }_{\text{spin}}}.$$

$$2P_S^{i,i+1}=(S_i+S_{i+1}{)}^2=\frac{3}{2}+2S_i\cdot S_{i+1},$$

$${⟨\frac{1}{N}\sum _i{P}_S^{i,i+1}⟩}_{{\varphi }_{\text{Heis}}}=\frac{3}{4}+{⟨\frac{1}{N}\sum _i{S}_i\cdot S_{i+1}⟩}_{{\varphi }_{\text{Heis}}}=1-\ln (2).$$

$$H_{\text{LS}}=\sum _{i=1}^N{P}_S^{i,i+1},$$

$$H_{\text{LL}}=-\sum _{i=1}^N{\partial }_{x_i}^2+\sum _{1\le i<j\le N}2c\delta (x_i-x_j).$$

$$H=-\sum _{i=1}^N{\partial }_i^2+\sum _{i<j}v(x_i-x_j),$$

$$\mathcal{E}(\Psi )={\int }_{[0,L{]}^N}\sum _{i=1}^N⟨{\partial }_i\Psi |{\partial }_i\Psi ⟩+\sum _{i<j}⟨\Psi |v(x_i-x_j)\Psi ⟩,$$

$$\left\{\Psi \in {\wedge }^N{L}^2([0,L];{\mathbb{C}}^{2J+1})|\Psi \in H^1([0,L{]}^N;({\mathbb{C}}^{2J+1}{)}^{\otimes N})\text{ and }\mathcal{E}(\Psi )<\infty \right\}.$$

$$E_J(N,L)=N\frac{{\pi }^2}{3}{\rho }^2\left(1+2\rho \left[a_e+(a_o-a_e){ϵ}_J^{\text{LS}}\right]+\mathcal{O}((\rho \max (R_0,a_o,\left|a_e\right|){)}^{6/5}+N^{-2/3})\right),$$

$$h_{LS}=\frac{1}{N}\sum _{i=1}^N{P}_S^{i,i+1}.$$

$$\underset{N\to \infty }{\lim }{ϵ}_J^{\text{LS}}=1-\frac{1}{2J+1}\left[\psi (1)-\psi (1/(2J+1))\right],$$

$${ϵ}_J^{\text{LS}}=1-\frac{1}{2J+1}\left[\psi (1)-\psi (1/(2J+1))\right]+\mathcal{O}(1/N).$$

$$N\frac{{\pi }^2}{3}{\rho }^2\left(1+2\rho \ln (2)a_e+2\rho (1-\ln (2))a_o+\mathcal{O}\left((\rho \max (R_0,a_o,\left|a_e\right|){)}^{6/5}+N^{-2/3}\right)\right),$$

$$N\frac{{\pi }^2}{3}{\left(\frac{N}{L-Na}\right)}^2=N\frac{{\pi }^2}{3}{\rho }^2{\left(1-\rho a\right)}^{-2}=N\frac{{\pi }^2}{3}{\rho }^2\left(1+2\rho a+\mathcal{O}\left((\rho a{)}^2\right)\right).$$

$$\frac{{\pi }^2}{3}{\rho }^3\left(1+2\ln (2)\rho a_e+\mathcal{O}\left((\rho /c{)}^2\right)\right),$$

$${\int }_{-R}^{R}2|{\partial }_x\psi {|}^2+v(x)|\psi (x){|}^{2}dx\ge \frac{2}{R-a_e}\int ({\delta }_R(x)+{\delta }_{-R}(x))|\psi (x){|}^{2}dx,$$

$$N\frac{{\pi }^2}{3}{\rho }^2(1+2\rho R+\dots )(1-2\rho (R-a)+\dots )=N\frac{{\pi }^2}{3}{\rho }^2(1+2\rho a+\dots ).$$

$${\mathcal{E}}^{\text{LLH}}(\Psi )={\int }_{[0,L{]}^N}\sum _{i=1}^N⟨{\partial }_i\Psi |{\partial }_i\Psi ⟩+\sum _{i<j}⟨\Psi |2c^{\prime }{P}_A^{i,j}\delta (x_i-x_j)+2c{P}_S^{i,j}\delta (x_i-x_j)|\Psi ⟩,$$

$$\left\{\Psi \in (L^2[0,L]{)}^{\vee N}\otimes ({\mathbb{C}}^2{)}^{\otimes N}|\Psi \in H^1([0,L{]}^N;({\mathbb{C}}^2{)}^{\otimes N})\text{ and }\mathcal{E}(\Psi )<\infty \right\}.$$

Full text

1 Introduction

A well-known feature of ground state energies of interacting quantum gases is that they show a kind of ‘universality’ in the interaction potential when the density of the gas is low. The best known example is perhaps the validity of the Lee–Huang–Yang energy expansion for 3D dilute Bose gases with general potentials [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], but other set-ups have been studied extensively in the mathematics literature in recent years, such as 2D dilute Bose gases [13, 14, 15, 16, 17, 18] and 3D dilute Fermi gases [19, 20, 21, 22, 23, 24] (see the introductions of the papers cited for further references).

In contrast, in one dimension, dilute Bose and Fermi gases interacting with general potentials have not been investigated as much111Note that weakly-interacting gases in 1D are not well described by the dilute limit, see Section 1.1 for a discussion.. To address this, we previously studied the 1D Bose gas [25], and considered a gas of $N$ spinless bosons in the interval $[0,L]$ (with density $\rho=N/L$), interacting via the $N$-body Hamiltonian

[TABLE]

with $v$ a potential with even-wave scattering length $a_{e}$. We showed that for a wide class of potentials $v$, the ground state energy in the low-density limit $\rho|a_{e}|\ll 1$ can be expanded as

[TABLE]

(where error terms have been omitted to have an informal discussion; see [25, Theorem 1] for precise statements). This expression is in some sense ‘universal’ because only the even wave scattering length appears to first order in the low-density ground state energy expansion, independent of the exact shape of the potential $v$. As indicated, a stronger ‘universality’ (continuing the expansion to higher orders and finding that it depends only on $a$) has been demonstrated in 2 and 3D, but a crucial difference with higher dimensions is that in 1D, the leading-order term in the energy expansion is the ground state of the free Fermi gas. This means that the techniques used to prove (1.2) differ significantly from those used in 2 and 3D (see [25, Section 1.2] for a summary).

To better understand (1.2), we can study relevant trial wave functions. The leading-order term in (1.2) is the ground state energy of the free Fermi gas in 1D. For spinless bosons, it can be found by using the wave function

[TABLE]

which is the (unnormalized, periodic b.c.) 1D free Fermi ground state on the sector $0\leq x_{1}<\dots<x_{N}<L$, extended symmetrically to other orderings of the particles to render the wave function bosonic. This is the exact ground state of the Tonks–Girardeau gas [26], which corresponds to the low-density limit ($\rho/c\to 0$) of the Lieb–Liniger gas ($v=2c\delta$ in (1.1) [27]). This is effectively the state around which we expand to find the expansion (1.2), while the expansions for the dilute Bose gas in 2 and 3D use the Bose condensate as a point of departure.

The next term in (1.2)—the correction involving the scattering length $a_{e}$—can be made intuitive by studying the trial state that was used to obtain the upper bound in [25]. The idea is that (1.3) needs to be modified in those areas of the configuration space where at least two bosons are close. In those regions, the potential should impact the wave function, and this needs to be accounted for. A 2-body energy minimization problem helps to define a good modification: for potentials with compact support contained in $[-R_{0},R_{0}]$, we minimize the 2-body energy on some suitable scale $[-R,R]$ with $R\geq R_{0}$,

[TABLE]

Here, one should think of $x$ as the distance between the particles, so that the boundary condition $\psi(R)=\psi(-R)=1$ is truly a bosonic one. The minimum is attained by the even-wave scattering solution $\psi_{\text{even}}$, while the minimal energy defines the even-wave scattering length $a_{e}$. It turns out that $\psi_{\text{even}}$ is a good tool to modify (1.3). If we focus on the sector $0\leq x_{1}<\dots<x_{N}<L$, the trial state producing (1.2) is roughly

[TABLE]

with

[TABLE]

That is, we introduce the scattering solution for neighbouring particles, on a suitable scale $R$. Note the wave function is continuous. It suffices to consider neighbouring particles because we restricted to $0\leq x_{1}<\dots<x_{N}<L$; the full trial state is found by symmetrically extending to other orderings of the particles on the line.

In [25], we used a slightly more complicated trial state of the type (1.5) to obtain the upper bound needed for (1.2), and we also proved a matching lower bound. Intuitively speaking, the appearance of the scattering length $a_{e}$ in the energy expansion (1.3) in the dilute limit $\rho|a_{e}|\ll 1$ comes from the importance of the problem (1.4)—the gas is so dilute that, to first order, interactions are successfully described by 2-body scattering only.

A similar result was found for the spinless (or spin-polarized) Fermi gas with Hamiltonian (1.1) [25] (see also an improved upper bound [20] that followed our preprint). This requires only a small modification to (1.4):

[TABLE]

where the boundary condition $\psi(R)=\psi(-R)=-1$ is now fermionic. This defines the odd-wave scattering solution $\psi_{\text{odd}}$ and the odd-wave scattering length $a_{o}$ (we can see that $a_{e}\leq a_{o}$ by comparing the energies (1.4) and (1.6) and even/odd boundary conditions). Replacing $\psi_{\text{even}}$ by $\psi_{\text{odd}}$ in the trial state (1.5), and extending antisymmetrically to get a fermionic wave function gives the upper bounded needed to prove a ground state energy expansion

[TABLE]

again for a general class of potentials. The lower bound could also be adapted to this case. Note the leading order is again the free Fermi ground state—this is independent of the exact symmetry of the particles.

The logical next question is: what happens for a dilute 1D spin-1/2 Fermi gas with Hamiltonian (1.1)? We can expect a free Fermi leading-order term like before, but this time, spin is present, and the wave function should be antisymmetric under the combined exchange of space and spin. If we again want to proceed by modifying the free Fermi state with 2-body scattering solutions, this means both even-wave and odd-wave scattering are needed: depending on the combined spin of the two particles involved in the scattering, the spatial wave function has to be symmetric or antisymmetric under exchange of the spatial coordinates. Indeed, in [25, Remark 5], we conjecture that the correct ground state energy is

[TABLE]

which is based on the exactly-solvable cases of the hard-core gas and the Lieb–Liniger gas (in analogy with the original conjecture for bosons [28, 25]). Demonstrating (1.8) rigorously is the main motivation for this article (though the results extend beyond this case, see Section 2).

How should the trial state (1.5) be modified? Again restricting to $0\leq x_{1}<\dots<x_{N}<L$, and adding a spin wave function $\phi_{\text{spin}}$, we can study

[TABLE]

with

[TABLE]

where $P^{i,i+1}_{A}$ is the antisymmetric spin projection (spin zero) for particles $i$ and $i+1$ and $P^{i,i+1}_{S}$ is the symmetric spin projection (spin one) (again note that we only get these operators for neighbouring particles by construction). Thus, antisymmetric spin gets coupled to even-wave scattering and vice versa, which means the trial state can be extended fermionically to other orderings of the particles. Note that spin state $\phi_{\text{spin}}$ still needs to be chosen. The even-wave scattering function $\psi_{\text{even}}$ has a lower energy than $\psi_{\text{odd}}$ because even boundary conditions give a lower energy, which means that the spin singlet is preferred for neighbouring spins. Indeed, calculations similar to the ones that gave (1.2) and (1.7) for spinless particles now produce a first-order correction that is

[TABLE]

Given $a_{e}\leq a_{o}$ as pointed out before, to achieve the lowest energy we need to maximize the first expectation value (and hence minimize the second, given $P^{i,i+1}_{S}+P^{i,i+1}_{A}=\mathds{1}$). This is exactly the Heisenberg spin chain since

[TABLE]

and the minimal expectation value is attained by the ground state $\phi_{\text{Heis}}$ of the Heisenberg chain [29, 30],

[TABLE]

Formally, this gives the upper bound needed to prove (1.8). In other words, the ground state problem for the dilute, interacting spin-$1/2$ Fermi gas in 1D effectively reduces to the antiferromagnetic Heisenberg spin chain. More generally, ground state problems of 1D, dilute spin-$J$ Fermi gases give rise to an effective spin chain that is known as the Lai–Sutherland model [31, 32, 33, 34, 35, 36],

[TABLE]

where $P^{i,i+1}_{S}$ should now be understood as the projection onto all symmetric spin combinations of spins $i$ and $i+1$ (for example, if $J=5/2$, these would be combined spin $1$, $3$ and $5$, whereas spin [math], $2$ and $4$ are antisymmetric). This is not difficult to understand: as indicated above, in a dilute gas, the first order in the low-density energy expansion is effectively decided by 2-body scattering, which can be either even or odd wave. This means that the effective spin chain should distinguish even and odd spin combinations, and that is exactly what (1.11) does. We again find (1.10) with the spin-$J$ interpretation of $P^{i,i+1}_{S}$ and $P^{i,i+1}_{A}=\mathds{1}-P^{i,i+1}_{S}$. Note that for spin-$J$ bosons, $a_{e}$ and $a_{o}$ switch places because the wave function needs to be symmetric upon combined exchange of space and spin coordinates. This means the correct Hamiltonian to study is now $-H_{LS}$. In this article, we demonstrate the relevance of the effective spin chains rigorously, for a general class of potentials $v$.

In the physics literature, interacting 1D gases with spin have been studied a lot (see [37, 38, 39, 40] for reviews), particularly with the contact interaction that first appeared in the Lieb–Liniger Hamiltonian [27]

[TABLE]

Following the introduction of the bosonic model and the exact Bethe ansatz solution by Lieb and Liniger in 1963, Gaudin and Yang independently found an exact solution for particles with spin in 1967 [41, 42], with Sutherland extending the result to general symmetry types given by Young diagrams in 1968 [43]. The Hamiltonian (1.12) for particles with spin is now known as the Yang–Gaudin model. Despite the exact solution, the connection with effective spin chains such as the Heisenberg model was not made at the time.

This appears to have been done for the first time in the discrete set-up (the 1D Fermi–Hubbard model) by Ogata and Shiba in 1990 [44]. Interestingly, they were inspired by the Bethe ansatz solution by Lieb and Wu [45], who in turn adapted Yang and Gaudin’s solution of the continuum model to the discrete case. Eventually, the spin chain observation made it back to the continuous setting [46, 47] for spin-$1/2$ bosons with contact interactions. The possibility to study fermions and more general interactions was hinted at as well. For contact interactions between fermions, the phenomenon was studied in more detail with perturbation theory methods starting from 2014 [33, 34, 35, 36]. It was found that the relevant effective spin chain is the Lai–Sutherland model (1.11).

After 2014, it was noticed that the effective spin chain can be engineered to differ from the Lai–Sutherland model, for example with external trapping potentials, spin-dependent interactions, or by considering several species of particles [48, 49, 50, 51]. Effective spin chains have also been observed experimentally [52].

Our main result is a rigorous derivation of the appearance of the Lai–Sutherland model for the ground state energy, in the absence of a trapping potential. It applies to general interaction potentials (1.1), and therefore extends what has been done for contact interactions in the literature.

This paper is organized as follows. Section 2.1 describes our main result for spin-$J$ fermions, while Section 2.2 describes some key ideas used in the proofs. Section 2.3 contains results about the Lieb–Liniger–Heisenberg model and Section 2.4 suggests a few open problems. It turns out the upper bounds can be formulated in slightly greater generality using matrix-valued potentials. Such potentials are introduced in Section 3, while the upper bound is stated and proved in Section 4. Finally, Section 5 describes the lower bound and its proof.

1.1 Luttinger theory and weakly-interacting 1D gases

This paper studies the ground state energy of dilute gases of particles with spin in 1D, and shows that the first correction term in the expansion only depends on the even and odd wave scattering lengths. This follows work that showed that for dilute gases in 2D and 3D, the ground state energy is even ‘universal’ for higher terms in the expansion (see the references at the start of the introduction).

Of course, the study of universal properties of 1D gases has a long history, in particular through Tomonaga–Luttinger theory, bosonization, and the Luttinger liquid universality class [53, 54, 55, 56, 57, 58, 59]. An important conclusion is that the theory of excitations of a large class of 1D models has a universal shape, independent of the exact microscopic model. However, this fact does not directly seem to comment on the ground state energy of the microscopic model, and how it depends on the shape of the interaction potential, which is the topic of interest here.

In the mathematical physics literature, many results have been obtained by Benfatto, Gallavotti, Mastropietro and others for weakly-interacting Fermi gases [60, 61, 62, 63, 64]. Using renormalization group techniques, these papers demonstrate a notion of universality, commonly used in the physics literature, that concerns the ground state energy, but also goes a great deal beyond that.

However, while in 2D and 3D the dilute and weakly-interacting limits are related, this is not the case in 1D. This is most easily seen in the Lieb–Liniger and Yang–Gaudin models, whose solution can entirely be expressed in terms of the parameter $\gamma=c/\rho$ (with $2c$ the parameter in front of the $\delta$-potential, see (1.12)) [27]. This means that the case of “fixed potential and low density” considered in this article corresponds to $\gamma\gg 1$, while the case of “fixed density and weak interactions” that is the focus of the references above corresponds to $\gamma\ll 1$. Hence, for the models with a $\delta$-interaction, the weakly-interacting regime and the low-density regime are opposites.

It is easy to carry out a similar scaling for potentials that are not concentrated at the origin like the $\delta$-function. In that case, the scaling that is needed to go between the two set-ups will also affect the range (or concentration) of the potential, but the conclusion remains the same: the regime considered in this article corresponds to strong interactions in the set-up used in the works by Benfatto, Gallavotti, Mastropietro and others, and so the results do not overlap.

2 Main Results

2.1 The ground state energy of the dilute spin-$J$ Fermi gas

We consider a general even, compactly-supported, repulsive (i.e. positive) potential $v$ that is a measure. We the study the Hamiltonian

[TABLE]

with energy functional given by

[TABLE]

and with form domain

[TABLE]

We will not assume $v$ to be absolutely continuous with respect to the Lebesgue measure, but we abuse notation and write $v(x)dx$ when integrating with respect to $v$.

We denote by $P_{S}$ is the projection onto the symmetric subspace $\mathbb{C}^{2J+1}\vee\mathbb{C}^{2J+1}$, while $P_{A}$ is the projection onto the antisymmetric subspace $\mathbb{C}^{2J+1}\wedge\mathbb{C}^{2J+1}$. We will add indices $i,j$ to indicate the spin projection is acting on particles $i$ and $j$, e.g. $P_{A}^{i,j}$. The even and odd-wave scattering lengths $a_{e}$ and $a_{o}$ were mentioned in (1.4) and (1.6) (see [25, Lemma 2 and Appendix A] for a precise definition, and Definition 8 below for a more general definition in the case of matrix-valued potentials).

Theorem 1** (Main result for fermions).**

Let $v$ be an even, repulsive (i.e. positive) interaction compactly supported in $[-R_{0},R_{0}]$, with $a_{e}$ and $a_{o}$ its even and odd wave scattering lengths. Consider a spin-$J$ Fermi gas with energy functional (2.2) and domain (2.3) restricted by any Robin boundary condition. Let $\rho=N/L$. Then, for $\rho|a_{e}|$, $\rho a_{o}$ and $\rho R_{0}$ sufficiently small, the ground state energy $E_{J}(N,L)$ satisfies

[TABLE]

where $\epsilon^{\textnormal{LS}}_{J}$ is the minimal energy per site of the spin-$J$ Lai–Sutherland model, i.e. the minimal expectation value of 222We can assume periodic boundary conditions for the spin chain, but it does not matter for the theorem: other boundary conditions change $\epsilon^{\textnormal{LS}}_{J}$ by at most $N^{-1}$, which can be absorbed in the error term.

[TABLE]

The theorem follows from an upper bound proved for Dirichlet boundary conditions (Corollary 15, stated and proved in Section 4) and a lower bound proved for Neumann boundary conditions (Theorem 25 in Section 5). The upper bound is a corollary of a more general result that also covers matrix-valued potentials (Theorem 13 in Section 4), which has the advantage that it allows us to disprove a conjecture by Girardeau (see Section 2.3). The lower bound is only proved for scalar potentials.

Remark 2**.**

For bosons, we can prove a similar theorem, with an analogous proof. The only difference is that for bosons, symmetric spin ($P_{S}$) gets coupled to symmetric spatial behaviour ($a_{e})$, while $P_{A}$ gets coupled to $a_{o}$, so that $a_{e}$ and $a_{o}$ swap roles. However, since the total ground state is known to be spatially symmetric [65], this means that it was already covered by our result for the spinless Bose gas [25]: a fully polarized state has $\langle h_{\textnormal{LS}}\rangle=1$, so that $a_{o}+(a_{e}-a_{o})\langle h_{\textnormal{LS}}\rangle=a_{e}$, and this gives the regular bosonic energy expansion.

Remark 3**.**

The assumption that the potential is compactly supported is not necessary: we may allow for potentials decaying fast enough to satisfy $\int_{b}^{\infty}v(x)x^{6+\epsilon}dx<\infty$ for some $b\geq 0$ and some $\epsilon>0$. In that case, the error term in (2.4) is slightly more complicated. For simplicity, we only discuss the case of compactly-supported potentials in this paper, and refer to [25] for details on how to include potentials of non-compact support.

Remark 4**.**

Note that the notion of diluteness needed for the theorem is that $\rho|a_{e}|$, $\rho a_{o}$ and $\rho R_{0}$ are all small (where the absolute values are needed for $a_{e}$ because it can be negative, and the error term involving $R_{0}$ cannot be avoided, see [25, Remarks 2 and 3]). Since the free case $v=0$ has $a_{e}=-\infty$ (see [25, Remark 4]), it cannot be considered dilute and $\rho|a_{e}|$ in the theorem is infinite.

Remark 5**.**

The theorem covers scalar potentials $v\geq 0$ with even and odd wave scattering lengths $a_{e}$ and $a_{o}$. These satisfy $a_{e}\leq a_{o}$ and $a_{o}\geq 0$. The full range of scattering lengths defined by those two equations can be attained by varying the potential, see Remark 38 in Appendix A.

Note that the Lai–Sutherland model is the antiferromagnetic Heisenberg model for spin $1/2$. It is solved exactly by the Bethe ansatz for any spin $J$ [31, 32].333This means that the Bethe ansatz gives exact eigenstates of the Hamiltonian. Whether the Bethe ansatz states form a complete set is a more delicate problem, see [66, 67]. Yang and Yang proved the validity of the Bethe ansatz for the ground state for the spin-$1/2$ Heisenberg chain in [68]. The thermodynamic ground state energy per site is [32]

[TABLE]

where $\psi(x)=\frac{d}{dx}\ln\left(\Gamma(x)\right)$ is Euler’s digamma-function. Hence, the thermodynamic energy per site of the spin-$1/2$ antiferromagnetic Heisenberg chain is $\lim\limits_{N\to\infty}\epsilon^{\textnormal{LS}}_{1/2}=1-\ln(2)$ (also see [69, 30]).

For a finite chain of length $N$, Lemma 39 in Appendix B shows that the ground state energy per site satisfies

[TABLE]

Noting that $a_{o}\geq a_{e}$ and combining Theorem 1 and (2.7) shows that the ground state energy of the dilute spin-$1/2$ Fermi gas is

[TABLE]

which proves a conjecture that we made in [25, Remark 5].

Remark 6**.**

*Here are two exactly-solvable models that are covered by (2.8).

The hard core potential of radius $a>0$ has $a_{e}=a_{o}=a$, and the ground state is $(2J+1)^{N}$-fold degenerate, as there is one ground state for each spin state. The ground state energy in the dilute limit $\rho a\ll 1$ is given by*

[TABLE]

The (fermionic) Yang–Gaudin model is a Fermi gas with point interactions, $v(x)=2c\delta(x)$. It was solved with the Bethe ansatz by Gaudin and Yang in [42, 41], and the solution was generalized to any spin by Sutherland in [43]. The ground state energy in the dilute limit $\rho/c\ll 1$ is

[TABLE]

where $a_{e}=-2/c$ is the even-wave scattering length of the interaction $v$ (note $a_{o}=0$).

2.2 Ideas used in the proof

The most important ingredient in the proof are generalizations of the minimizations (1.4) for symmetric wave functions and (1.6) for antisymmetric wave functions. Those were used to prove the first-order ground state energy expansions for dilute spinless Bose and Fermi gases in [25]. For clarity, we first summarize the strategies used in the spinless case (see [25, Section 1.2] for a more extensive summary), followed by the adaptations needed for the case with spin.

The upper bound uses the trial state argument outlined in Section 1: for spinless bosons, the trial state departs from Girardeau’s solution of the impenetrable Bose gas (1.3), which is the free Fermi solution with absolute values to create a symmetric wave function. This produces the leading order in the expansion. The Girardeau wave function is then patched with the scattering solution (1.4) on some suitable small scale, and for the closest pair of particles only, which, up to negligible error terms in the limits $\rho a_{e}\to 0$ and $\rho R_{0}\to 0$ (with $R_{0}$ the range of a compactly supported potential), gives our first-order result.

The case of spin-$J$ fermions is similar, but it requires the spin-dependent patching discussed above in (1.9) and (1.10), leading to a first-order correction related to the Lai–Sutherland model (1.11). Given that we will actually prove the upper bound for the more general case of matrix-valued potentials, the proof requires a few additional ideas. The most important one is the matrix generalization of (1.4) and (1.6): the matrix-valued scattering solution discussed in Section 3. The precise trial state is introduced in Definition 8 in Section 4.2. The upper bound is found by estimating the energy of the trial state, and the proof involves calculations that reduce to the free Fermi ground state on which the trial state is based, as well as the matrix scattering solution of Definition 8.

The lower bound for spinless bosons in [25] again uses the minimization (1.4). An important idea (see [2, 3] in the 3D case) is that (1.4) can be written as

[TABLE]

for all $\psi\in H^{1}[-R,R]$ (note the boundary condition $\psi(R)=\psi(-R)=1$ in the minimization in (1.4)). This effectively lower bounds the kinetic and potential energy on $[-R,R]$ by a symmetric delta potential at radius $R$. To generalize this to an $N$-boson wave function, we repeatedly apply (2.10) to obtain the symmetric delta potential for any neighbouring pairs of bosons. Inspired by [70], we then throw away the regions where $|x_{i+1}-x_{i}|\leq R$ (which gives a lower bound since $v$ is positive) and as a consequence, the two delta functions at radius $|x_{i+1}-x_{i}|=R$ collapse into a single delta at $|x_{i+1}-x_{i}|=0$, with value $4/(R-a_{e})$. This corresponds to the Lieb–Liniger energy of some wave function on the reduced interval, which is of course lower bounded by the explicitly known Lieb–Liniger ground state energy [27]. Heuristically, this corresponds to the following calculation: starting with an interval of length $L$, cut it back to length $L-(N-1)R$, so that the Lieb–Liniger ground state energy (see e.g. Lemma 27) with $c=2/(R-a_{e})$ and new density $N/(L-(N-1)R)=\rho(1+\rho R+\dots)$ gives a lower bound

[TABLE]

As is done in [70], we also show that the ground state wave function has little weight in the regions that get thrown out, so that (2.11) is accurate.

For spin-$J$ fermions, the strategy is the same, but some additional ideas are needed. The most important step is to combine symmetric scattering (1.4) and antisymmetric scattering (1.6) into a single (Dyson’s) lemma involving the spin projections $P_{S}$ and $P_{A}$, see Lemma 26 in Section 5.1. A summary of the additional ideas needed to solve the case with spin $J$ is given at the start of Section 5.2.

2.3 Upper bound on the ground state energy of the Lieb–Liniger–Heisenberg model

The Lieb–Liniger–Heisenberg model is an effective model for spin-$1/2$ fermions in a tight waveguide introduced by Girardeau [71, 72]. Is has energy functional

[TABLE]

with $c^{\prime},c>0$, and its spatially symmetric domain is

[TABLE]

Note that the energy (2.12) depends explicitly on spin, and this is the main motivation for studying the upper bound in the context of matrix-valued potentials in Section 4. The trial state argument is as sketched in the previous section, but of course we need to patch with a scattering solution that allows for matrix-valued potentials and ensure correct boundary conditions (see Definition 8 and Remarks 9 and 14).

Theorem 7** (Upper bound for the LLH model).**

Consider a gas of spin-1/2 particles with energy functional (2.12) with $c,c^{\prime}>0$ and domain (2.13) with Dirichlet boundary conditions. Let $\rho=N/L$. Then, for $\rho/c$, $\rho/c^{\prime}$ sufficiently small, there exists a constant $C_{U}>0$ such that the ground state energy $E^{\textnormal{LLH}}(N,L)$ satisfies

[TABLE]

where $\epsilon^{\textnormal{LLH}}(c,c^{\prime})$ is the ground state energy of the spin chain Hamiltonian

[TABLE]

and $H_{\textnormal{Heis}}=\sum_{i=1}^{N}P_{S}^{i,i+1}$ is the Heisenberg antiferromagnet.

In [72], Girardeau used a trial state argument to obtain an upper bound on the Lieb–Liniger–Heisenberg ground state energy in the case $c>c^{\prime}>0$,

[TABLE]

and he conjectured this might be sharp. However, $\epsilon^{\textnormal{LLH}}(c,c^{\prime})=-2\left(\frac{\ln(2)}{c^{\prime}}+\frac{1-\ln(2)}{c}\right)+\mathcal{O}(1/N)$ for $c>c^{\prime}>0$ by the ground state energy of the Heisenberg model ((2.7) with $J=1/2$), so that it follows from the weighted harmonic-arithmetic mean inequality that Theorem 7 improves (2.14), disproving Girardeau’s conjecture. The upper bound of Theorem 7 is presumably sharp, but we cannot prove this as our lower bound only covers scalar potentials.

2.4 Open problems

Besides extending the lower bound to matrix-valued potentials and thus demonstrating that the expansion for the LLH model in Theorem 7 is correct, there are several other questions one could study.

For example, in [25, Section 1.5], we conjectured that the energy expansions for dilute, spinless bosons and fermions (1.3) and (1.7) depend only on the scattering length to second order. In particular, for dilute bosons, we expect that the ground state energy is

[TABLE]

for a wide class of potentials. We can ask if this is also the case for the dilute spin-$1/2$ fermion result (2.8). Can we simply follow the bosonic analogue and conjecture that the right expression is

[TABLE]

In general, does $a_{e}$ get replaced by $a_{e}+(a_{o}-a_{e})\epsilon^{\textnormal{LS}}_{J}$ like in Theorem 1,

[TABLE]

and likewise for the matrix-valued case?

Of course, the other open problems for dilute, spinless 1D bosons mentioned in [25, Section 1.5]—studying momentum distributions and positive temperature for general potentials—extend to dilute gases with spin as well (see e.g. [73, 74, 75] in the physics literature). It would be particularly interesting to see to what extent the physics is still described by the Lai–Sutherland model (2.5).

3 Matrix-valued potentials

In the previous sections, we presented our results for scalar-valued potentials, but the upper bound we prove in the next section also applies to matrix-valued potentials. To state that result, we need a definition of the scattering length that is more general than the approach taken in (1.4) and (1.6) in the introduction (explained in more detail in [25, Lemma 2 and Appendix A], also see [76, Appendix C] for a 3D gas). Background facts needed for the following definition have been collected in Appendix A.

Definition 8** (The scattering length matrix for spin-$J$ fermions).**

Consider a symmetric, positive-semidefinite measure $V$ that takes values in the space of complex $(2J+1)^{2}\times(2J+1)^{2}$ matrices. Assume that $V$ commutes with $P_{S}$ and $P_{A}$ (meaning $V$ is block diagonal, leaving the symmetric and antisymmetric spin spaces invariant), and assume that $V$ is supported in $[-R_{0},R_{0}]$ for some $R_{0}\geq 0$.

Given $R\geq R_{0}$, let $F_{0}\in H^{1}([-R,R];\operatorname{Mat}((2J+1)^{2},(2J+1)^{2}))$ be the unique solution (in the distributional sense) of the matrix equation

[TABLE]

on $[-R,R]$ with boundary conditions $F_{0}(R)=I$ and $F_{0}(-R)=P_{A}-P_{S}$ (see Proposition 36 in Appendix A). The matrix-valued function $F_{0}$ is called the scattering solution of $V$, and it satisfies (see Proposition 36 for proofs)

[TABLE]

for some matrix $\mathsf{A}$ called the scattering length matrix of $V$, and

[TABLE]

It follows from the above that that $\mathsf{A}$ is independent of $R\geq R_{0}$, that it is Hermitian and (by letting $R\to R_{0}$) that $\mathsf{A}\leq R_{0}$ (since $V$ is positive-semidefinite).

Remark 9**.**

For spin-$J$ bosons, all we have to change in the definition above is the boundary condition. This becomes $F_{0}(R)=I$ and $F_{0}(-R)=P_{S}-P_{A}$, so that there is symmetry upon mapping $R\to-R$ and exchanging the spins, rather than antisymmetry.

In the Lieb–Liniger–Heisenberg model (Section 2.3), the (spin-1/2) particles are spatially symmetric and the correct boundary condition to define the matrix scattering solution is $F_{0}(R)=F_{0}(-R)=I$. Since the interaction is $V=2c^{\prime}P_{A}+2cP_{S}$, this will mean that $F_{0}$ is constant on the spin singlet and triplet subspaces, with the spatial behaviour of the scattering solution of the Lieb–Liniger model (with constant $c^{\prime}$ for the singlet and $c$ for the triplet).

For the scalar potentials $V=vI$ that were discussed in the previous sections, it is easily seen that the scattering length matrix $\mathsf{A}$ splits in two diagonal blocks and may be written

[TABLE]

where $a_{e}$ and $a_{o}$ are the even and odd-wave scattering lengths discussed in (1.4) and (1.6) in the introduction (see [25, Lemma 2 and Appendix A] for more properties).

In the general case, the scattering matrix $\mathsf{A}$ has two blocks corresponding to the symmetric and antisymmetric spin spaces. These can be diagonalized to give specific scattering lengths with corresponding (anti)symmetric eigenvectors. These might be linear combinations of different total spin. For example, if $J=1$, total spin 0 and 2 are both symmetric, and a linear combination of these might give an eigenvector of $\mathsf{A}$.

Remark 10**.**

Generalizing the case $v=0$ discussed in Remark 4, whenever $F_{0}$ has a constant eigenvector $\chi(x)=\chi$ (necessarily spin antisymmetric since it is spatially symmetric), we make this fit with (3.2) using the convention $\mathsf{A}\chi=-\infty\chi$. In that case, $\left\lVert\mathsf{A}\right\rVert=\infty$ and the error terms in Theorems 1, 13, and 25 are infinite.

4 Upper Bound for matrix-valued potentials

To state our upper bound in the most general way, we consider an even (repulsive) potential $V$ that is a positive semidefinite matrix-valued measure acting on $\mathbb{C}^{2J+1}\otimes\mathbb{C}^{2J+1}$. We study the Hamiltonian

[TABLE]

with corresponding energy functional

[TABLE]

where $V_{ij}$ denotes the matrix potential $V(x_{i}-x_{j})$ acting on the spins of particles $i$ and $j$ (the indices of the matrix will not be used much and these are not $i$ and $j$). The associated domain of $\mathcal{E}$ is

[TABLE]

We will decompose the potential as $V=vI+\tilde{V}$, where $v$ is a positive scalar potential as in the previous sections. For technical reasons, $\tilde{V}$ is assumed to be of bounded variation, for which we recall the following definition.

Definition 11**.**

*We say that an $M\times M$ Hermitian matrix-valued measure $\tilde{V}$ has bounded variation if $\langle\chi|\tilde{V}|\chi\rangle$ is a finite signed measure for any $\chi\in\mathbb{C}^{M}$. In this case, we define the total variation of $\tilde{V}$ to be the finite444Let $\{e_{\sigma}\}_{\sigma=1}^{M}$ be the standard basis of $\mathbb{C}^{M}$. Note that $\|\tilde{V}\|(X)$ is finite by writing out $\xi_{n}$ in this basis

$\sup_{\begin{subarray}{c}\xi_{n}\\ |\xi_{n}|=1\end{subarray}}|\langle\xi_{n}|\tilde{V}(X_{n})\xi_{n}\rangle|\leq\sum_{1\leq\sigma,\sigma^{\prime}\leq M}|\langle e_{\sigma}|\tilde{V}(X_{n})|e_{\sigma^{\prime}}\rangle|.$

and using polarization identities to reduce to the diagonal entries. (positive) measure*

[TABLE]

where $X$ is a Borel-measurable set and the first supremum is over partitions of the set $X$ into a countable number of disjoint Borel-measurable subsets $X_{n}$.

Recall that we abuse notation and write $V(x)dx$ when integrating with respect to the measure $V$, even if the measure does not allow for this. If we let $X$ be all of $\mathbb{R}$, the above definition says that $\int\|\tilde{V}\|<\infty$, and we will use this constant in some of our estimates. The precise assumptions on $V$ are as follows.

Assumption 12**.**

$V$* is assumed to be an even, positive semidefinite, $(2J+1)\times(2J+1)$ matrix-valued measure that commutes with $P_{S}$ (and hence $P_{A}$), compactly supported in $[-R_{0},R_{0}]$. It is also assumed that $V=vI+\tilde{V}$, where $v$ is any (positive) Borel measure and $\tilde{V}$ is a Hermitian matrix-valued Borel measure of bounded variation (with $\tilde{V}\geq-vI$). Denote by $\mathsf{A}$ the scattering length matrix of $V$ (see Definition 8).*

Our upper bound is given by the following theorem.

Theorem 13**.**

Consider a spin-$J$ Fermi gas with Hamiltonian (4.1) and potential $V$ satisfying Assumption 12. Let $\rho=N/L$. Then, for $\rho R_{0}$ and $\rho\|\mathsf{A}\|$ sufficiently small (where $\|\mathsf{A}\|$ denotes the operator norm of $\mathsf{A}$), there exist a constant $C_{U}>0$ such that the ground state energy with Dirichlet boundary conditions satisfies

[TABLE]

where $\epsilon_{J}(\mathsf{A})$ is the ground state energy of the spin chain Hamiltonian 555As indicated in Footnote 2, the boundary conditions of (4.5) are irrelevant for the statement. We will work with periodic boundary conditions for the spin chain in the proof.

[TABLE]

with $\mathsf{A}^{i,i+1}$ acting as the scattering length matrix $\mathsf{A}$ on spins $i$ and $i+1$ and as the identity on the rest.

Remark 14**.**

A key feature of the proof of Theorem 13 is that the trial state is initially defined on the ordered sector $\{0\leq x_{1}<x_{2}<\dots<x_{N}<L\}$, and then extended to other sectors by symmetry. Hence, the proof can easily be adapted to other symmetries—such as spin-$J$ bosons or the set-up studied in the Lieb–Liniger–Heisenberg model (Section 2.3)—by extending in a different way to other orderings of the particles. The only change needed (to guarantee continuity of the trial state) is the choice of boundary conditions in the Definition 8 of $\mathsf{A}$—also see Remark 9.

As a special case, for $V=vI$, we find the following corollary to Theorem 1 by (3.4).

Corollary 15** (Upper bound in Theorem 1).**

Let $v$ be an even, positive interaction compactly supported in $[-R_{0},R_{0}]$, with $a_{e}$ and $a_{o}$ its even and odd wave scattering lengths. Consider a spin-$J$ Fermi gas with energy functional (2.2) and domain (2.3) with Dirichlet boundary conditions. Then, for $\rho R_{0}$, $\rho a_{e}$ and $\rho a_{o}$ sufficiently small, there exists a constant $C_{U}>0$ such that the ground state energy $E^{\textnormal{Dirichlet}}_{J}(N,L)$ satisfies

[TABLE]

where $\epsilon^{\textnormal{LS}}_{J}$ is the minimal energy per site of the spin-$J$ Lai–Sutherland model, i.e. the minimal expectation value of

[TABLE]

4.1 Preparatory facts about the free Fermi ground state

Let $\Psi_{F}\in L^{2}([0,L]^{N})$ be the spinless Dirichlet free Fermi ground state. Its $k$-particle reduced density matrix is given by (note the choice of normalization)

[TABLE]

Similarly, we define the $k$-particle reduced density of $\psi_{F}$ by

[TABLE]

We shall, given a function $f(x)$, in the following use the notation $f|_{x_{1}}=f(x)|_{x_{1}}=f(x_{1})$ and $f|_{x_{1}}^{x_{2}}=f(x)|_{x_{1}}^{x_{2}}=f(x_{2})-f(x_{1})$.

We will need some facts about $\gamma^{(k)}$ and $\rho^{(k)}$ from [25], and we add a new, very similar lemma.

Lemma 16** (Lemma 9 in [25]).**

[TABLE]

with $f$ a suitable function that satisfies $\int_{[0,L]}\left\lvert f(x_{2})\right\rvert dx_{2}\leq\textnormal{ const. }\rho^{3}\ln(N)$.

Lemma 17** (from Lemma 10 in [25]).**

[TABLE]

Proof.

(See [25] for more details.) Wick’s theorem says that all free Fermi reduced densities matrices can be expressed in terms of $\gamma^{(1)}$. Also, we can verify that all derivatives of $\gamma^{(1)}$ can be bounded uniformly in the coordinates in terms of powers of $\rho$ (where the power can be obtained from dimensional analysis)—and this fact extends to all higher-order reduced density matrices by Wick’s theorem. The estimates above then follow from using the quotient rule for the fractions (if present) and Taylor expansion, simplified by symmetry and dimensional analysis to determine the correct power of $\rho$ (see [25, Lemma 10] for examples). ∎

Similarly, we obtain the following estimates.

Lemma 18** (New estimates, similar to Lemma 17).**

[TABLE]

4.2 Constructing a trial state

4.2.1 Notation

We will use $|.|$ for absolute values and the vector 2-norm in $\otimes^{N}\mathbb{C}^{2J+1}$, and $\langle.|.\rangle$ for its inner product. Let $\|\mathsf{A}\|$ denote the corresponding matrix norm of the scattering length matrix $\mathsf{A}$, which is equal to the spectral radius since $\mathsf{A}$ is Hermitian. Definition 11 and the remark below it discuss the meaning of $\int\|\tilde{V}\|$.

As explained in the introduction and Section 2.2, the ground state energy of a spinless gas in the dilute limit can be approximated by making use the free Fermi gas when particles are far apart, adjusted with the 2-body scattering solution when they are close [25, 77]. We denote by $b>R_{0}$ the range within which we will alter the free Fermi state to account for scattering effects. This will be chosen appropriately in the proof of Theorem 13 in Section 4.3.3. Let

[TABLE]

denote the closest and second closest distance between two particles, respectively.

For particles $i$ and $j$, we define666Note the absolute values: this is convenient for our method, and it makes no difference since we will define the trial state on $0\leq x_{1}<\dots<x_{N}<L$ only and extend antisymmetrically. We will often use the scattering energy (3.3), but this is unaffected by the sign differences created by the absolute values.

[TABLE]

where $F_{ij}$ is the matrix scattering solution from Definition 8 with $R$ replaced by $b\geq R_{0}$, acting on the spin indices of the particles with coordinates $i$ and $j$. We would also like to have this definition for the closest pair: say particles $i$ and $j$ are the closest pair in a given configuration $x$, i.e. $\mathcal{R}(x)=\left\lvert x_{i}-x_{j}\right\rvert$, define

[TABLE]

Similarly, let

[TABLE]

be related to the scattering solution from Definition 8 for the hard-core potential with radius $R_{0}$. We again extend this definition to the closest pair,777For our proof to work, we could also choose the hard-core radius to be the spectral radius of the positive part of $\mathsf{A}$, $\|\mathsf{A}_{+}\|$. This possibility is important to apply the proof to non-compactly supported potentials.

[TABLE]

Note that there is an important difference between (4.8) and (4.10): the latter matrix acts as the identity on the spins, which means that (4.10) is a continuous function in $x$ (given that $\mathcal{R}(x)$ is). However, (4.8) acts non-trivially on the closest pair of spins, and so it may not be continuous whenever $\mathcal{R}(x)=\mathcal{R}_{2}(x)$, because which pair of particles is the closest may change at this point and the action of the scattering solution matrix will jump with it.

All this means that our preferred strategy—patching with the scattering solution (4.8) for the closest pair, which was done for spinless bosons in [25]—leaves discontinuities. To solve this technical challenge, we introduce the continuous function (4.10), and the continuous and almost everywhere differentiable interpolation function

[TABLE]

We use $\eta$ to define a continuous trial state (4.12) below, interpolating between the preferred, but discontinuous (4.8) and the continuous (4.10) at a negligible length scale (i.e. at the cost of an acceptable error).

4.2.2 The trial state

Let $\Psi_{F}$ be the spinless free Fermi state with Dirichlet boundary conditions (depending only on spatial coordinates). Using the notation introduced in the previous subsection, we define the (unnormalized) trial state to be as follows on the set $\{0\leq x_{1}<x_{2}<\ldots<x_{N}<L\}$,

[TABLE]

where $\chi\in\bigotimes^{N}\mathbb{C}^{2J+1}$ is a fixed spin state to be chosen below in Section 4.3.3. Note that $W^{\mathcal{R}}$ and $W_{\text{hc}}^{\mathcal{R}}$ are matrices acting on $\chi$. The extension of $\Psi_{\chi}$ to other orderings of the coordinates $x_{i}$ is defined by antisymmetry in the combined space-spin variables, and we denote the extension by $\Psi_{\chi}$ as well. We will assume that $\chi$ is translation symmetric (with periodic boundary conditions), meaning that it is an eigenvector of the spin translation operator.

This trial state is in the relevant domain (4.3): note that $P_{S}^{i,j}\Psi_{\chi}\rvert_{x_{i}=x_{j}}=0$ due to the boundary condition at [math] satisfied by $W^{\mathcal{R}}$ and $W_{\text{hc}}^{\mathcal{R}}$, so that the antisymmetric extension of the part with symmetric spin is not discontinuous at $x_{i}=x_{j}$; also note that the discontinuity in $W^{\mathcal{R}}$ at $\mathcal{R}(x)=\mathcal{R}_{2}(x)$ discussed in the previous subsection is avoided because of the interpolation with $\eta$ and the continuity of $W_{\text{hc}}^{\mathcal{R}}$.

As was the case in [25, 77], the trial state given in (4.12) produces an error that grows with the particle number. This is not sufficient to prove Theorem 13. However, we can construct a full trial state on the set $\{0\leq x_{1}<x_{2}<\ldots<x_{N}<L\}$ by taking a product of wave functions of the form (4.12) defined on small, non-overlapping intervals (leaving some distance to avoid interactions between the intervals), and extending to $[0,L]^{N}$ by antisymmetry. The details are worked out in Section 4.3.3.

4.3 Proof of Theorem 13

4.3.1 Ingredients in the estimates

Given the fact that the definition (4.12) of $\Psi_{\chi}$ involves $\min_{i<j}\left\lvert x_{i}-x_{j}\right\rvert$, it is useful to define

[TABLE]

where $B_{12}$ will be convenient to use because of antisymmetry of $\Psi_{\chi}$. The following domains of integration (all subsets of $[0,L]^{N}$) will be used in the proof.

[TABLE]

The distinction between the $B$’s is made because of the interpolation with $\eta$ in (4.12). The $A$’s are convenient for calculations because they relax the requirements about being the (next) closest pair of particles. The energy density is positive so we often extend e.g. $B_{12}\subset A_{12}$ at the cost of an acceptable error.

Using the definition (4.7) of $W_{12}$, we also define

[TABLE]

where $(\Psi)_{12}$ is now a matrix-valued function, related to $F_{12}$ (the matrix scattering solution of the potential acting on the spin indices of the particles with coordinates $1$ and $2$). At this point, it may be good to contrast this with the spinless free Fermi ground state $\Psi_{F}$, which is scalar valued, and our trial state $\Psi_{\chi}$, which is vector valued.

Given that $\Psi_{\chi}$ was only made explicit on the set $\{0\leq x_{1}<x_{2}<\ldots<x_{N}<L\}$ in (4.12), we introduce notation for other ‘sectors’—sets that order the coordinates in a different way,

[TABLE]

with $\sigma=(\sigma_{1},\dots,\sigma_{N})$ a permutation that sends $(1,...,N)\mapsto(\sigma_{1},...,\sigma_{N})$. It will be useful to find an explicit expression for $\Psi_{\chi}$ for other particle orders as well. We work out the case of $B_{12}^{\geq}\cap\{\sigma\}$ here, and refer to these steps in other cases later.

Note that by the definition (4.12) of $\Psi_{\chi}$ and (4.8) and (4.11), we have that $\Psi_{\chi}=(\Psi)_{12}\chi$ on the set $B_{12}^{\geq}\cap\{0\leq x_{1}<x_{2}<\ldots<x_{N}<L\}$ (this is the reason for using the symbol $(\Psi)_{12}$). Can we find a similar expression when the positions are ordered in a different way? It turns out that, on $B_{12}^{\geq}\cap\{\sigma\}$ (assuming $\sigma$ is such that the intersection is not empty888Note that since particles 1 and 2 are closest on $B_{12}^{\geq}$, they they must be next to each other on the line, and only permutations that respect this will give a non-zero intersection $B_{12}^{\geq}\cap\{\sigma\}$.), we have that

[TABLE]

where $\chi_{\sigma}$ denotes the spin state $\chi$ with spins permuted by $\sigma$. To understand this, it is useful to consider an example for $N=4$. On $B_{12}^{\geq}\cap\{0\leq x_{3}\leq x_{4}\leq x_{2}\leq x_{1}<L\}$, the trial state is defined by ordering the spins and positions such that the positions are ascending (up to a sign that is $-$ in this case). Denoting by $\left\langle s_{1},s_{2},s_{3},s_{4}|\cdot\right\rangle$ the spin coordinate function $\left\langle s_{1}\otimes s_{2}\otimes s_{3}\otimes s_{4}|\cdot\right\rangle$ (where $s_{1},...,s_{4}\in\mathbb{C}^{2J+1}$ are any set of basis states of the spin space), we find that, given that the particles with coordinates $x_{1}$ and $x_{2}$ form the closest pair (which, for the purpose of the second equality and using the definition (4.12) of $\Psi_{\chi}$, are the third and fourth spins),

[TABLE]

A similar, but general argument gives (4.15). From the third line, we can obtain an additional fact about inner products: on $B_{12}^{\geq}$, the particles with coordinates $x_{1}$ and $x_{2}$ are always next to each other because they form the closest pair. This means that in an inner product in $(\mathbb{C}^{2J+1})^{\otimes N}$, we can translate the corresponding spins (3 and 4 in the example) back to 1 and 2 (without doing anything with the spatial coordinates). Of course, we also need to translate the spins in $\chi$, but $\chi$ was assumed to be translation invariant. Thus, for example, a general version of this argument gives for all $x\in B^{\geq}_{12}\cap\{\sigma\}$ and any $1\leq i\leq N$,

[TABLE]

Since the latter is independent of $\sigma$, the equality between the first and last expression actually holds on all of $B^{\geq}_{12}$. We will apply this idea to various inner products in the proof, such as the expression above. A similar argument can for example be used to show that for all $x\in B^{\geq}_{12}$,

[TABLE]

4.3.2 Splitting the energy into parts and estimating them

We can now start estimating. The quantity of interest is the energy (4.2) of the trial state $\Psi_{\chi}$ (we will later account for the norm of $\Psi_{\chi}$),

[TABLE]

Let $E_{F}=N\frac{\pi^{2}}{3}\rho^{2}(1+\mathcal{O}(1/N))$ denote the ground state energy of the spinless Dirichlet free Fermi gas. By adding and subtracting $\int_{B}\sum_{i}\left\lvert\partial_{i}\Psi_{F}\right\rvert^{2}$ and using antisymmetry in the second step, we find that

[TABLE]

where we used that there are no interactions outside of $B$ because $b$ is bigger than the range of the potential $R_{0}$ by assumption, as well as the fact that $|\partial_{i}\Psi_{\chi}(x)|^{2}=|\partial_{i}\Psi_{F}(x)|^{2}$ outside of $B$ by the definition (4.12).

We wish to integrate the last term in (4.19) over $A_{12}$ rather than $B_{12}$: note that $x\in A_{12}\setminus B_{12}$ implies $x\in A_{ij}$ for some $(i,j)\neq(1,2)$, so we can estimate

[TABLE]

We use this estimate on (4.19) to rewrite the third term as integration over $A_{12}$. In the other two terms, we write the terms involving $\partial_{1}$, $\partial_{2}$ and $V_{12}$ separately and split some of the integrals, with equality, using the integration domains (4.13) and antisymmetry to collect similar terms. This gives

[TABLE]

The following lemmas show that the last four lines all end up in the error term. The first three are proved in Section 4.3.4, the last (concerning the free Fermi gas) was proved in [25, Lemma 12].

Lemma 19**.**

[TABLE]

Lemma 20**.**

Assuming $b>2\max(\left\lVert\mathsf{A}\right\rVert,R_{0})$,

[TABLE]

Lemma 21**.**

Assuming $N(\rho b)^{3}<1$ and $b>2\max(\left\lVert\mathsf{A}\right\rVert,R_{0})$,

[TABLE]

Lemma 22** (Lemma 12 in [25]).**

Assuming $N(\rho b)^{3}<1$,

[TABLE]

For the integral over $B^{\geq}_{12}$ in the first line of (4.20), we use (4.17) and (4.18), and extend to $A_{12}\supset B^{\geq}_{12}$ by positivity. The four lemmas above then say that

[TABLE]

Besides the proof of Lemmas 19, 20 and 21 (postponed to Section 4.3.4) there are now two steps left in the proof of Theorem 13: 1. Estimating the remaining terms to find the spin chain. We will do this now in Lemma 23, and it gives the upper bound on $\mathcal{E}(\Psi_{\chi})$ formulated in Proposition 24. 2. Formulating a full trial state built from copies of $\Psi_{\chi}$ and finishing the proof in Section 4.3.3.

Lemma 23**.**

Assuming $\rho b<1$ and $b>2\max(\left\lVert\mathsf{A}\right\rVert,R_{0})$,

[TABLE]

with $\mathsf{A}^{i,i+1}$ acting as the scattering length matrix $\mathsf{A}$ from Definition 8 on spins $i$ and $i+1$ and as the identity on the rest, and $\mathsf{A}^{N,N+1}=\mathsf{A}^{N,1}$

Proof.

Inserting the expression for $(\Psi)_{12}$ from (4.14), we find that

[TABLE]

Applying partial integration to the second $\partial_{1}$ in the following integral, using notation introduced above Lemma 17, gives

[TABLE]

Since $\Psi_{F}:[0,L]^{N}\to\mathbb{R}$ is real valued, this equation allows us to treat the second term in (4.23) and cancel the third. If we also apply the product rule to the final term in (4.24), integrate out the variables $x_{3},\dots,x_{N}$ and use that $\|W_{12}(\pm b)\chi\|^{2}=b^{2}\|F_{12}(b)\chi\|^{2}=b^{2}$ by the boundary condition of $F_{12}$, we find that (4.23) equals

[TABLE]

Also, by antisymmetry of $\Psi_{F}$ and partial integration (notice only $x_{1}$ gives a boundary term), and using that $\Psi_{F}$ is an eigenfunction of $-\sum_{i=1}^{N}\partial_{i}^{2}$ with eigenvalue $E_{F}$,

[TABLE]

The first term is negative and can be thrown out for an upper bound, while the second cancels a similar term in the previous equation. Putting these equations together and integrating out the coordinates $x_{3},...,x_{N}$ for the first equality, we find that the quantity of interest satisfies

[TABLE]

where, for the first inequality, we use Lemma 16 in the first line, Lemma 16 and (3.3) from Definition 8 in the second line (with $R=b$), and Lemma 17 and the uniform bound $|W_{12}(x_{1}-x_{2})\chi|^{2}\leq b^{2}$ in the last line.999This bound follows from convexity of $|W_{12}(x_{1}-x_{2})\chi|^{2}$ with boundary conditions $|W_{12}(\pm b)\chi|^{2}=b^{2}$, which in turn follows from the fact that $W_{12}(x_{1}-x_{2})=bF_{12}(|x_{1}-x_{2}|)$ and $F_{12}$ solves (3.1) with positive definite $V$ (see (A.4) for the straightforward calculation of the second derivative). For the second inequality, we use that $b>2\left\lVert\mathsf{A}\right\rVert$ by assumption. Finally, to find the desired result (4.22), we rewrite the operator in a translation-independent way (with periodic boundary conditions) using the translation symmetry of $\chi$. ∎

Combining (4.21) with Lemma 23, we arrive at the following estimate of $\mathcal{E}(\Psi_{\chi})$, which will be used to estimate the energy of the full trial state in the next section.

Proposition 24**.**

Assuming $N(\rho b)^{3}<1$ and $b>2\max(\left\lVert\mathsf{A}\right\rVert,R_{0})$, the energy of the (unnormalized) trial state $\Psi_{\chi}$ defined in (4.12) is bounded by

[TABLE]

4.3.3 A full trial state (Proof of Theorem 13)

This proof is similar to the strategy used for bosons [25]: concatenating copies of the (Dirichlet) trial state $\Psi_{\chi}$ on smaller intervals, or ‘boxes’.

Proof of Theorem 13.

Let $\rho\left\lVert\mathsf{A}\right\rVert<\frac{1}{32}$ and fix $b$ to satisfy

[TABLE]

Then, $(\rho\left\lVert\mathsf{A}\right\rVert)^{4/5}<\frac{1}{16}$, which implies $\frac{1}{16}\geq\rho b>(\rho\left\lVert\mathsf{A}\right\rVert)^{-1/5}\rho\left\lVert\mathsf{A}\right\rVert\geq 2\rho\left\lVert\mathsf{A}\right\rVert$. Therefore, we find

[TABLE]

Notice first that one box suffices for $N<\frac{1}{2}(\rho b)^{-3/2}$: the result follows directly from Proposition 24 and Lemma 17 with the observation that

[TABLE]

as well as $\frac{16\pi^{2}}{3}N(\rho b)^{3}\leq\frac{8\pi^{2}}{3}(\rho b)^{3/2}<1/2$, so that $\frac{1}{1-N(\rho b)^{3}}\leq 1+2N(\rho b)^{3}$. The error term of order $N(\rho b)^{3}\left(\rho b^{2}\int\|\tilde{V}\|\right)$ is lower order in $\rho b$: for $\rho b$ sufficiently small (or $\rho\left\lVert A\right\rVert$ and $\rho R_{0}$ sufficiently small as in the assumptions of Theorem 13) we have $\rho b^{2}\int\|\tilde{V}\|\leq 1$.

For $N\geq\frac{1}{2}(\rho b)^{-3/2}$, we consider $M$ disjoint ‘boxes’ (intervals) $I_{1},I_{2},\dots,I_{M}$, with $\bigcup_{k=1}^{M}I_{M}\subset[0,L]$. We then set the number of particles in each box to be either $\left\lceil N/M\right\rceil$ or $\left\lceil N/M\right\rceil-1$, adding up to $N$ in total. We choose

[TABLE]

so that

[TABLE]

Hence, the number of particles in each box lies between $\frac{1}{4}(\rho b)^{-3/2}$ and $\frac{33}{64}(\rho b)^{-3/2}$. We write $\tilde{N}=\left\lceil N/M\right\rceil$ and fix the length of each box so that the density is $\tilde{\rho}=\frac{N}{L-(M-1)b}$. In that way, the sum of the lengths is $L-(M-1)b$, leaving a distance $b$ between each box. Also,

[TABLE]

which follows from $b(M-1)/L\leq 2(\rho b)^{5/2}<1/2$.

We now consider the glued trial state $\Psi_{\chi,\text{full}}=\prod_{k=1}^{M}\Psi_{\chi,k}$, where $\Psi_{\chi,k}$ denotes the state defined in (4.12), on the box $I_{k}$. Note that there is no interaction energy between the boxes, as the distance $b$ between the boxes (4.25) is greater than the range of the potential $R_{0}$.

To study the energy within each box, let $e_{0,I_{k}}=\frac{\pi^{2}}{3}\left\lvert I_{k}\right\rvert\tilde{\rho}^{3}(1+\text{const. }\tilde{N}^{-1})$ be the free Fermi energy in box $I_{k}$ (where $\left\lvert I_{k}\right\rvert$ denotes the length of the box). From Proposition 24, the trial state constructed using the small boxes gives the upper bound

[TABLE]

Notice that $\tilde{\rho}\left\lVert\mathsf{A}\right\rVert\frac{\ln(\tilde{N})}{\tilde{N}}\leq\frac{2}{e}\max(\tilde{N}^{-1},(\tilde{\rho}\left\lVert\mathsf{A}\right\rVert)^{3/2})$ (this can be seen by considering the cases $\tilde{N}\leq(\tilde{\rho}\left\lVert\mathsf{A}\right\rVert)^{-1}$ and $\tilde{N}>(\tilde{\rho}\left\lVert\mathsf{A}\right\rVert)^{-1}$ separately). Hence, this term is subleading, and we absorb it into other error terms. We now choose $\chi$ to be a translation-invariant ground state of $\sum_{i=1}^{\tilde{N}}\mathsf{A}^{i,i+1}$, and using a simple trial state argument (cutting up a length $N$ spin chain in $M$ parts and adding periodic terms to chains of length $\tilde{N}$), we find

[TABLE]

where $\epsilon_{J}(\mathsf{A})$ was defined to be the ground state energy of $\frac{1}{N}\sum_{i=1}^{N}\mathsf{A}^{i,i+1}$. Therefore, using (4.26),

[TABLE]

The second term is of order $(\rho b)^{3/2}\rho\left\lVert\mathsf{A}\right\rVert$ by (4.28) and is therefore subleading. Using (4.27) and (4.29) and the observations above, we see that (4.30) is upper bounded by

[TABLE]

Using $M/N\leq(\rho b)^{3/2}+\frac{1}{N}$ and the fact that $\frac{16\pi^{2}}{3}\tilde{N}(\rho b)^{3}\leq 1/2$, so that

[TABLE]

as well as $\tilde{N}(\rho b)^{3}\leq\textnormal{const. }(\rho b)^{3/2}$, the desired result (4.4) follows (recall the remark about $\int\|\tilde{V}\|$ below (4.27)). Note the error term $N\frac{\pi^{2}}{3}\rho^{2}\mathcal{O}(N^{-1})$ in (4.4) is only important when $N<(\rho b)^{-3/2}$, so it does not appear in (4.31). ∎

This concludes the proof of Theorem 13. It remains to prove the lemmas that we used.

4.3.4 Proof of Lemmas 19, 20 and 21

Proof of Lemma 19.

Given that the integration domain is $B^{\geq}_{12}$, we use (4.17), and extend to $A_{12}\supset B^{\geq}_{12}$ by positivity. As $\Psi_{F}$ is an eigenfunction of $-\sum_{i=1}^{N}\partial_{i}^{2}$ with eigenvalue $E_{F}$, by partial integration, we see that

[TABLE]

where we used $|W_{12}(x_{1}-x_{2})\chi|^{2}\leq b^{2}$ (see Footnote 9) and Lemma 17 in the last two steps. ∎

Proof of Lemma 20.

Recalling the definitions (4.7), (4.9) and (4.11), define

[TABLE]

with $\tilde{W}_{12}(x)=\eta(x)W_{12}(x_{2}-x_{1})+(1-\eta(x))W_{\textnormal{hc},12}(x_{2}-x_{1})$. Note that $\Psi_{\chi}(x)=(\tilde{\Psi})_{12}(x)\chi$ on the set $B_{12}\cap\{0\leq x_{1}<\dots<x_{N}<L\}$ by the definition (4.12). Similar to (4.15), we find that on $B_{12}\cap\{\sigma\}$ (assuming the intersection is not empty),

[TABLE]

Note that we can repeat the argument that led to (4.17) and (4.18) to find that on all of $B_{12}$,

[TABLE]

Before we start estimating, it is useful to note that

[TABLE]

The first inequality follows from the convexity of squaring and the fact that $\left\langle W_{\textnormal{hc},12}\chi|W_{\textnormal{hc},12}\chi\right\rangle\leq\left\langle W_{12}\chi|W_{12}\chi\right\rangle$ pointwise (see Lemma 37 in Appendix A for a proof). The second follows from convexity of $\left\langle W_{12}\chi|W_{12}\chi\right\rangle$ (see Footnote 9 and the proof just referenced) and the boundary condition $|W_{12}(\pm b)\chi|^{2}=b^{2}$.

Recall that by Assumption 12, we can write $V_{ij}=v_{ij}+\tilde{V}_{ij}$, with $v_{ij}$ positive and scalar, and $\tilde{V}_{ij}$ a Hermitian matrix-valued measure of bounded variation, both with support contained in $[-b,b]$. A useful fact for the estimates is that, for a function $g\in L^{2}(\mathbb{R}^{N};(\mathbb{C}^{2J+1})^{\otimes N})$, 101010To see this, it suffices to study a simple function $g(x)=\sum_{k=1}^{n}\chi_{k}\mathbbm{1}_{X_{k}}$. Then $\int\langle g(x)|\tilde{V}(x)g(x)\rangle dx=\sum_{k=1}^{n}\int_{X_{k}}\langle\xi_{k}|\tilde{V}\xi_{k}\rangle\leq\sum_{k=1}^{n}\int_{X_{k}}|\xi_{k}|^{2}\|\tilde{V}\|=\int|g(x)|^{2}\|\tilde{V}\|(x)dx$. We can then generalize to $L^{2}$-functions by approximating with simple functions in the usual way.

[TABLE]

This gives

[TABLE]

By (4.33) and antisymmetry to collect similar terms, we find that the $\tilde{V}_{ij}$-part of (4.36) satisfies

[TABLE]

where we used Lemma 17 for the estimates. For the $v_{ij}$-part of (4.36), we note that for fixed $\left\lvert x_{i}-x_{j}\right\rvert$ and $(i,j)\neq(1,2)$, by positivity of $v_{ij}$,

[TABLE]

since the function $|W_{12}(x_{1}-x_{2})\chi|^{2}$ is increasing in $|x_{1}-x_{2}|$ because it is convex and even, and $\left\lvert x_{i}-x_{j}\right\rvert\geq\left\lvert x_{1}-x_{2}\right\rvert$ on $B_{12}$. Thus, using (4.33), followed by (4.34) and (4.38) and antisymmetry, and integrating out certain coordinates, we find that

[TABLE]

Hence, combining (4.36), (4.37) and (4.39), and by adding and subtracting interaction terms with $\tilde{V}_{12}(x_{2}-x_{3})$ and $\tilde{V}_{12}(x_{3}-x_{4})$ instead of $v(|x_{3}-x_{4}|)$ and $v(|x_{2}-x_{3}|)$ and estimating the subtraction as in (4.36) and (4.37), and using Lemma 17, we find that

[TABLE]

where we used the scattering energy (3.3) and the assumption $b>2\|\mathsf{A}\|$ in the last steps. ∎

Proof of Lemma 21.

Consider the trial state $\Psi_{\chi}$ (4.12) on the set $B_{12}^{34}$ (4.13), which means that the particles with positions $x_{1}$ and $x_{2}$ are the closest pair, while those with positions $x_{3}$ and $x_{4}$ are the next closest pair. Similar to (4.32) (also see (4.15) and (4.16)), we see that on $B_{12}^{34}\cap\{\sigma\}$,

[TABLE]

where we defined $W_{12}^{34}(x)=W_{12}^{34}(x_{1},x_{2},x_{3},x_{4})$ in terms of $W_{12}$ (4.7) and $W_{hc,12}$ (4.9),

[TABLE]

Also, because of reasoning similar to (4.33), $B_{12}^{34}\subset A_{12}^{34}$, and partial integration in $x_{5},\dots,x_{N}$,

[TABLE]

Using that $\Psi_{F}$ is an eigenfunction of $-\Delta$ with eigenvalue $E_{F}$, we find that

[TABLE]

By convexity of squaring, the scattering energy (3.3) and the assumption $b>2\|\mathsf{A}\|$, we find that

[TABLE]

Also, similar to (4.34), we find that $\left\langle W_{12}^{34}\chi|W_{12}^{34}\chi\right\rangle\leq\left\langle W_{12}\chi|W_{12}\chi\right\rangle\leq b^{2}$. Using the dependence on $x_{3}$ in (4.40) for the first inequality and Cauchy–Schwarz for the other two, we find that

[TABLE]

Thus, using Lemmas 17 and 18 and the assumption $N(\rho b)^{3}<1$ (and the notation from Section 4.1), we find that (4.41) (the kinetic energy part of the object we wish to estimate) is upper bounded by

[TABLE]

We continue with the potential energy part. The strategy will be similar to what was done in Lemma 20, but simplified because we are only dealing with the $V_{12}$-term. By Assumption 12, $V_{ij}=v_{ij}+\tilde{V}_{ij}$, with $v_{ij}$ positive and scalar, and $\tilde{V}_{ij}$ a Hermitian matrix-valued measure of bounded variation, both with support contained in $[-b,b]$. Applying (4.35) and using an argument similar to the one that gave (4.33) (also see (4.17) and (4.18)), followed by $\left\langle W_{12}^{34}\chi|W_{12}^{34}\chi\right\rangle\leq\left\langle W_{12}\chi|W_{12}\chi\right\rangle\leq b^{2}$ already used before, we find that

[TABLE]

Adding and subtracting $\frac{\left\lvert\Psi_{F}\right\rvert^{2}}{\left\lvert x_{2}-x_{1}\right\rvert^{2}}\langle W_{12}\chi|\tilde{V}_{12}W_{12}\chi\rangle$ and estimating the subtraction as in the previous step, followed by $B^{34}_{12}\subset A^{34}_{12}$ and integrating out the coordinates $x_{5},...,x_{N}$, and using Lemma 17 and the scattering energy (3.3) and the assumption $b>2\|\mathsf{A}\|$ gives

[TABLE]

This concludes the proof of the first estimate of the lemma. The second estimate can be found by straightforwardly adapting all the steps above to the $B^{23}_{12}$ case. ∎

5 Lower Bound for scalar potentials

The following lower bound is used to prove Theorem 1. Our method does not allow us to extend to the matrix-valued potentials that we considered for the upper bound—see Footnote 12 for the technical obstruction.

Theorem 25** (Lower bound in Theorem 1).**

Let $v$ be an even, positive interaction compactly supported in $[-R_{0},R_{0}]$, with $a_{e}$ and $a_{o}$ its even and odd wave scattering lengths. Consider a spin-$J$ Fermi gas with energy functional (2.2) and domain (2.3) with Neumann boundary conditions. Then, for $\rho R_{0}$, $\rho a_{e}$ and $\rho a_{o}$ sufficiently small, there exists a constant $C_{L}>0$ such that the ground state energy $E_{J}^{\textnormal{Neumann}}(N,L)$ satisfies

[TABLE]

where $\epsilon^{\textnormal{LS}}_{J}$ is the minimal energy per site of the spin-$J$ Lai–Sutherland model, i.e. the minimal expectation value of 111111We will prove the bound for a chain with open boundary conditions, but this is irrelevant, see Footnote 2.

[TABLE]

5.1 Preparatory Facts: Dyson’s Lemma and the Lieb–Liniger Model

For fermions with spin, we will use the following generalization of Dyson’s lemma, which was proved for spinless bosons (and fermions) in [25].

Lemma 26** (Dyson for spin-$J$ fermions, see Lemmas 2 and 21 in [25] and Appendix C in [76]).**

Let $R>R_{0}=\textnormal{range}(v)$ and let

[TABLE]

Then, for any interval $\mathcal{I}\ni 0$,

[TABLE]

where $a_{e}$ and $a_{o}$ are the even and odd-wave scattering lengths of $v$.

The proof is simply that of the bosonic case, or alternatively the spin-polarized fermionic case, both discussed in [25], together with the fact that two spin-$J$ fermions have a symmetric spatial wave function only when their spin is antisymmetric and vice versa. The second inequality in (5.3) follows from positivity and $a_{o}\geq a_{e}$.

5.1.1 The Lieb–Liniger model

The Lieb–Liniger model describes bosons in one dimension interacting through a contact potential. The formal Hamiltonian (to be understood in the quadratic form sense) is

[TABLE]

acting on the Hilbert space $L^{2}_{\text{sym}}([0,L]^{N})$. We write $E_{\textnormal{LL}}^{\textnormal{Neumann}}(N,L,c)$ for its ground state energy at interaction strength $c$ and interval length $L$. In [25, Lemma 17], we proved the following lower bound. The proof uses the exact solution for periodic boundary conditions and a method of Robinson [78] to adapt to Neumann boundary conditions.

Lemma 27** (Lemma 17 in [25]).**

The ground state energy of the Lieb–Liniger model (5.4) with Neumann boundary conditions is bounded below by

[TABLE]

5.2 Proof of the lower bound (Theorem 25)

We will adapt the strategy used in the bosonic case [25], outlined heuristically in Section 2.2. Summarizing, this strategy used the following steps.

• •

Let’s say the particle configuration is such that $x_{i+1}$ and $x_{i}$ are neighbouring particles. We fix some radius $R$ bigger than the support of the potential $R_{0}$, to be decided later, and apply Dyson’s lemma to lower bound the energy contained in the region $|x_{i+1}-x_{i}|\leq R$ by a Hamiltonian involving $\delta$-potentials at $x_{i+1}-x_{i}=\pm R$.

• •

We throw away the regions with $|x_{i+1}-x_{i}|<R$ for a lower bound (note the energy density is positive), which causes the $\delta$-potentials to collapse into a single delta (of twice the strength) at $x_{i+1}=x_{i}$. This gives the Lieb–Liniger model on a smaller interval. Using the known ground state energy, we find that it produces the desired expression.

• •

To really obtain the result of this calculation as a lower bound, we demonstrate that the norm of the wave function in the region that was thrown out is negligible.

• •

However, this strategy only works for a limited number of particles, and so an argument to extend to any particle number is needed. We use boxes and superadditivity caused by the positive potential as in [3].

What is different for particles with spin? As we can see from Lemma 26 (Dyson’s lemma for spin-$J$ fermions), the strength of the delta function that we obtain at radius $R$ depends on the spin symmetry of the two particles: antisymmetric spin goes with the even-wave scattering length $a_{e}$, whereas symmetric spin goes with the odd-wave $a_{o}$. This means we do not find Lieb and Liniger’s spin-independent delta-function by using the above strategy and Lemma 26. This is a problem because the use of the exact expression for the ground state energy of the Lieb–Liniger model was crucial to the proof.

However, there is something we can try. Note that in Lemma 26, we could use two different radii for the spin singlet and triplet, say $R>R_{0}$ and $R^{\prime}>R_{0}$, such that $R-a_{o}=R^{\prime}-a_{e}$. This would mean that we again obtain a spin-independent delta-interaction. Of course, there is a catch: if we were to follow the steps above, we should then throw out two different regions $|x_{i+1}-x_{i}|<R$ and $|x_{i+1}-x_{i}|<R^{\prime}$ depending on whether the combined spin of the two particles is symmetric or antisymmetric!

This seems very inconvenient, perhaps even hopeless, but it is not. If we keep track of the spin-symmetry of each pair of neighbouring particles using projection operators, we know how much space we need to remove. This depends on the spin symmetry, but in all cases, we end up with the Lieb–Liniger model—what differs is the length of the interval. Summing the exact formula for the ground state energy over all possible spin symmetries, we obtain the Lai–Sutherland spin chain (5.2). These ideas are made precise in Section 5.2.1.

The final two steps of the strategy above are essentially unchanged: we discuss these briefly in Sections 5.2.2 and 5.2.3, referencing the bosonic proof in [25].

5.2.1 Lower bound for small particle numbers

Recall that $v$ is an even, repulsive (positive) interaction compactly supported in $[-R_{0},R_{0}]$, and that $a_{e}$ and $a_{o}$ are its even and odd wave scattering lengths. For now, let $\Psi$ be some (normalized) wave function describing $N$ fermions (in the domain (2.3) of the energy functional (2.2)), which means that $\Psi(y_{1},\dots,y_{N})$ takes values in $(\mathbb{C}^{2J+1})^{\otimes N}$. We fix a radius $R>R_{0}+a_{o}-a_{e}$ that we will use throughout the proof.

To apply Dyson’s lemma (Lemma 26) to the many-body case, define the set

[TABLE]

and the distances

[TABLE]

with $R>R_{0}+a_{o}-a_{e}$ so that both distances are strictly bigger than the support of the potential $R_{0}$ (note $R_{0}\geq a_{o}\geq a_{e}$ and $a_{o}\geq 0$, given the energies in (1.4) and (1.6)). These are the radii that we will use to apply Lemma 26: the bigger radius $R>R_{0}$ for symmetric spin combinations, and the smaller radius $R-(a_{o}-a_{e})>R_{0}$ for antisymmetric spin combinations. The fact that these radii are different will mean that the result of the lemma depends on the spin symmetry defined by $T\in\Theta$. Therefore, we write

[TABLE]

which is defined on the set

[TABLE]

This corresponds to selecting a particular spin symmetry for $\Psi$ with projection operators defined by $T$, and cutting away the regions $\{x_{i}-x_{i-1}<R_{T_{i}}\}$ from the domain, where $1\leq i\leq N-1$ (that is, those regions in which neighbouring particles are within distance $R_{T_{i}}$), which we will do after applying Dyson’s lemma. The length of the remaining interval is $L-\sum_{i=1}^{N-1}R_{T_{i}}$. The order of the projections $T_{i}$ matters for the definition: projections appearing within $\Pi^{T}_{e}$ and $\Pi^{T}_{o}$ commute, and their definition is unambiguous, but $\Pi^{T}_{e}$ and $\Pi^{T}_{o}$ themselves do not commute. The following lemma gives the details on applying Dyson’s lemma.

Lemma 28**.**

With the definitions above, for any $\Psi$ in the domain (2.3) of the energy functional (2.2), we have that

[TABLE]

where the last line contains the ground state energy of the Lieb–Liniger model with Neumann boundary conditions discussed in Section 5.1.1.

Proof.

We start by proving the first inequality in (5.7). Recalling that $|.|^{2}$ relates to the inner product in $\otimes^{N}\mathbb{C}^{2J+1}$, and decomposing into orthonormal projections $P_{A}^{1,2}+P_{S}^{1,2}=I$, we have121212This step is the obstruction in proving the lower bound for the matrix-valued potentials that we considered for the upper bound. For our method, it is crucial that we can split the energy in a sum of terms that contain either $P_{A}^{1,2}\Psi$ or $P_{S}^{1,2}\Psi$, but not both. However, for e.g. a matrix-valued interaction term $\langle\Psi|V_{23}\Psi\rangle$, this split is not possible since $V_{23}$ may not commute with $P_{A}^{1,2}$ and $P_{S}^{1,2}$ (such as in the LLH model (2.12), for which $V_{23}=(c^{\prime}P_{A}^{2,3}+cP_{S}^{2,3})\delta_{23}$). For a scalar term $v_{23}|\Psi|^{2}$, this is not a problem.

[TABLE]

Since $P_{A}^{1,2}\Psi$ and $P_{S}^{1,2}\Psi$ are symmetric and antisymmetric as functions of $x_{1}$ and $x_{2}$, respectively, we can apply the version of Dyson’s lemma from Lemma 26 to

[TABLE]

From (5.8), this gives

[TABLE]

where we used positivity of $v_{1j}$ for $j\geq 3$, and we discarded the integral over the region $\{0<x_{2}-x_{1}<R_{T_{1}}\}$ by positivity. Note that the amount of space removed differs between the symmetric and antisymmetric cases, but that the delta function ends up with coupling $2/(R-a_{o})$ in both cases.

We now repeat the above steps to insert $T_{2},T_{3},\dots T_{N-1}$, but we need to do this in a particular order to arrive at the claim. Given the order of the projections in the definition (5.6) of $\psi_{T}$, we first repeat the steps above for odd $k$, followed by the even values of $k$.

After making the $T$-dependent change of coordinates $(y_{1},y_{2},\dots,y_{N})=(x_{1},x_{2}-R_{T_{1}},\dots,x_{N}-\sum^{N-1}_{i=1}R_{T_{i}})$, we find the first inequality in (5.7) as desired. The second inequality follows by noting that the second line of (5.7) is the Lieb–Liniger energy of the state $\psi_{T}$ in a box of length $L-\sum_{i\in T}R_{i}$ with Neumann boundary conditions and coupling $c=2/(R_{0}-a_{o})$, restricted to the particle ordering defined by $M_{T}$. ∎

The terms of the final sum in (5.7) have a $T$-dependence, and we will have to deal with this to continue the lower bound. In particular, the functions $\psi_{T}$ are defined on an interval of length

[TABLE]

with $n_{P_{S}}(T)$ denoting the number of times $P_{S}$ appears in $T\in\Theta$ (see (5.5)). It turns out to be convenient to cut back the functions $\psi_{T}$ to an interval of a fixed length $L-(N-1)R$ by defining $\tilde{\psi}\in L^{2}([0,L-(N-1)R]^{N},\otimes^{N}\mathbb{C}^{2J+1})$, again starting from a given wave function $\Psi$, as

[TABLE]

extended symmetrically in the $z_{i}$ to other orderings of the particles (this ensures continuity so that we can later use $\tilde{\psi}$ as a trial state for the bosonic Lieb–Liniger model). We have that the integral in the final line of (5.7) can be lower bounded by

[TABLE]

where we introduced an inner product notation for the integral over the ordered sector $\{0\leq x_{1}<\dots<x_{N}<L-(N-1)R\}$ that will be used in the rest of the proof. We will write $\langle.|.\rangle_{L^{2}}$ for the $L^{2}([0,L-(N-1)R]^{N},\otimes^{N}\mathbb{C}^{2J+1})$ inner product on the full domain (recall $\langle.|.\rangle$ is used for the $\otimes^{N}\mathbb{C}^{2J+1}$ inner product). The following lemma is proved in Section 5.2.2.

Lemma 29**.**

Let $\Psi$ be the ground state corresponding to $E_{J}^{\textnormal{Neumann}}(N,L)$ considered in Theorem 25. With the definition of $\tilde{\psi}$ above and assuming that $N(\rho R)^{2}\leq\frac{3}{16^{2}\pi^{2}}\frac{1}{8}$, and $\rho R\leq\frac{1}{2}$,

[TABLE]

The purpose of this lemma is to explain why it was not a problem to cut the ground state $\Psi$ (and $\psi_{T}$) back to $\tilde{\psi}$: the wave function remains normalized up to a small error. With this fact, we are ready to prove a lower bound on the energy with an error that is small when the particle number is not too large.

Proposition 30** (Lower bound for small particle number).**

Assuming $N(\rho R)^{2}\leq\frac{3}{16^{2}\pi^{2}}\frac{1}{8}$, and $\rho R\leq\frac{1}{2}$, it holds that

[TABLE]

Proof.

Combining (5.12) with (5.7), we have that for any $\Psi$,

[TABLE]

where we used positivity of $E^{\textnormal{Neumann}}_{\textnormal{LL}}$. Given $R_{T_{i}}\leq R$, and using the assumptions on $R$ made in the statement of the lemma, note that we can write out expansions like

[TABLE]

Using estimates of this type, the expression (5.10) for $\sum_{i}R_{T_{i}}$, Lemma 27, and $a_{o}\geq 0$, we can prove that

[TABLE]

To continue, we introduce a notation to be able to choose even and odd projections in $T\in\Theta$ independently:

[TABLE]

where the last indices are simply the highest odd and even numbers $\leq N-1$. We will slightly abuse the notation introduced in (5.6) and note that the projection

[TABLE]

is fully defined by the projections $T_{i}$ with even indices $i$ collected in $T_{e}\in\Theta_{e}$. Now note that

[TABLE]

which follows directly from $P_{A}+P_{S}=I$. We also have that

[TABLE]

where we used (5.15) and the fact that for $T_{o}\in\Theta_{o}$,

[TABLE]

as well as the relation

[TABLE]

which again uses $P_{A}+P_{S}=I$. Combining (5.13), (5.14), (5.15) and (5.16), we find that

[TABLE]

For the relevant operator in this equation, we know that for all $\tilde{\psi}$,

[TABLE]

To complete the proof, we take $\Psi$ to be the (normalized) Neumann ground state, combine (5.17) with $P_{A}+P_{S}=I$ and (5.18), and use that $\Pi^{T}_{o}$ are projections with $\sum_{T_{o}\in\Theta_{0}}\Pi^{T}_{o}=I$ so that we obtain $\langle\tilde{\psi}|\tilde{\psi}\rangle^{\textnormal{sec}}_{L^{2}}$. All the integrals are now only on a sector, but we find the full energy $E_{\mathcal{J}}^{\textnormal{Neumann}}(N,L)$ by multiplying by the number of sectors $N!$ and using antisymmetry of $\Psi$. All this gives

[TABLE]

since $N!\langle\tilde{\psi}|\tilde{\psi}\rangle^{\textnormal{sec}}_{L^{2}}=\langle\tilde{\psi}|\tilde{\psi}\rangle_{L^{2}}$ by the symmetric extension of $\tilde{\psi}$ discussed below (5.11). The proposition now follows from Lemma 29 stated above (proved in the next section), as well as $a_{e}\leq a_{o}\leq R_{0}\leq R$ and boundedness of the spin chain ground state energy. ∎

5.2.2 Estimating $\langle\tilde{\psi}|\tilde{\psi}\rangle_{L^{2}}$ (proof of Lemma 29)

The purpose of this section is to prove the missing ingredient in the proof of Proposition 30: the bound on the $L^{2}$-norm of $\tilde{\psi}$ in Lemma 29. A proof of this result was given for spinless bosons (or scalar $\Psi$ and $\tilde{\psi}$) in [25, Section 3.2]. We adapt it to the case in which $\Psi$ and $\tilde{\psi}$ take values in $\otimes^{N}\mathbb{C}^{2J+1}$ in this section. To make it easier for the reader to see that the same strategy applies, we restate the lemmas used in [25] and remark on the minor adaptations in the proofs. Note that the only place where we use the assumption that $\Psi$ is the ground state corresponding to $E_{J}^{\textnormal{Neumann}}(N,L)$ is in applying the upper bound Theorem 13 in the last step.

Note that the set that gets removed from the domain in going from $\Psi$ to $\tilde{\psi}$ in (5.11) is $\{x\in\mathbb{R}^{n}|\min_{i,j}\left\lvert x_{i}-x_{j}\right\rvert<R\}$. We then see that since $\Psi$ is normalized,

[TABLE]

where $D_{i}^{j}:=\{x\in\mathbb{R}^{n}|\mathfrak{r}_{i}(x)=\left\lvert x_{i}-x_{j}\right\rvert<R\}$ with $\mathfrak{r}_{i}(x):=\min_{j\neq i}(\left\lvert x_{i}-x_{j}\right\rvert)$ (note that $D_{i}^{j}$ is not symmetric in $i$ and $j$; that for $j\neq j^{\prime}$, $D_{i}^{j}\cap D_{i}^{j^{\prime}}=\emptyset$ up to sets of measure zero; and that $\{x\in\mathbb{R}^{n}|\min_{i,j}\left\lvert x_{i}-x_{j}\right\rvert<R\}=\cup_{i<j}D_{i}^{j}$). The first relevant lemma in [25] is the following.

Lemma 31** (Lemma 20 in [25]).**

For general $\Psi$ and $\tilde{\psi}$ defined in (5.11), we have

[TABLE]

Regarding the proof: we have already verified the first inequality in (5.19), and the second inequality immediately generalizes from the scalar to the multi-component case by linearity. The second relevant lemma in [25] is the following.

Lemma 32** (Lemma 22 in [25]).**

Let $\epsilon\in[0,1]$. For general $\Psi$, and $\tilde{\psi}$ defined in (5.11) and $E_{\textnormal{LL}}^{\textnormal{Neumann}}$ introduced in Section 5.1.1,

[TABLE]

The proof is the same as before, replacing the use of Dyson’s lemma for spinless bosons [25, Lemma 21] by the analogue for spin-$J$ from Lemma 26 (using the bound that only involves $a_{e}$). (Note that the ground state energy of the multicomponent Lieb–Liniger model—for which $\tilde{\psi}$ is a trial state—is simply the bosonic ground state energy. Also note that $R>R_{0}+a_{o}-a_{e}>-a_{e}$ so that $R>|a_{e}|$ by the assumption on $R$ made at the start of Section 5.2.1.)

With Lemma 32, we can prove Lemma 29, which was the missing ingredient in the previous section—we again simply follow our earlier work [25, Proof of Lemma 23]. In the proof, the use of the upper bound [25, Proposition 6] gets replaced the corresponding Theorem 13 (or rather the version for scalar-valued potentials in Corollary 15) and the estimates $a_{e}+(a_{o}-a_{e})\epsilon^{\textnormal{LS}}_{J}\leq a_{o}<R$ and $|a_{e}|<R$.

Note that we have now fully proved the lower bound from Proposition 30 that works well for limited particle numbers. It still needs to be extended to arbitrary particle numbers, and we discuss this in the next subsection.

5.2.3 Lower bound for arbitrary particle numbers (proof of Theorem 25)

In this subsection, we extend the lower bound in Proposition 30 to large particle numbers to prove Theorem 13. We again use the strategy from [25], which consisted in noticing that it suffices to restrict to (boxes with) particle numbers of order $(\rho R)^{-9/5}$, then performing a Legendre transformation in the particle number and studying the grand canonical ensemble, while using superadditivity caused by the positive potential. Indeed, for particle numbers of order $(\rho R)^{-9/5}$, we get the desired lower bound as a corollary of Proposition 30.

Corollary 33**.**

For $\frac{\tau}{2}(\rho R)^{-9/5}\leq N\leq\tau(\rho R)^{-9/5}$ with $\tau=\frac{3}{16^{2}\pi^{2}}\frac{1}{8}$ and $\rho R\leq\frac{1}{2}$, we have

[TABLE]

This replaces [25, Corollary 25]. Starting from this corollary and Proposition 30, we now follow the strategy used in [25], where $a$ gets replaced by the infimum from the previous equation. The proof is then completely identical, with the proof of the lower bound Theorem 25 mimicking the proof of [25, Proposition 14], and use of the exact analogue of [25, Lemma 26].

Acknowledgments

JA and JPS were partially supported by the Villum Centre of Excellence for the Mathematics of Quantum Theory (QMATH, Grant No. 10059). JA was partially supported by the Novo Nordisk Foundation, Grant number NNF22SA0081175, NNF Quantum Computing Programme. RR was partially supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (ERC CoG UniCoSM, Grant Agreement No. 724939). RR thanks the University of Copenhagen and the University of Cambridge for hospitality during visits, and the Royal Society for supporting the research visit to Cambridge (AL$\backslash$24100022). RR is a member of the Gruppo Nazionale per la Fisica Matematica (GNFM) of the Istituto Nazionale di Alta Matematica Francesco Severi (INdAM).

Appendix A Facts about the scattering length matrix

Consider a symmetric, positive-semidefinite measure $V$ that takes values in the space of complex $(2J+1)^{2}\times(2J+1)^{2}$ matrices. Assume that $V$ commutes with $P_{S}$ and $P_{A}$ (meaning $V$ leaves the symmetric and antisymmetric spin spaces invariant), and assume that $V$ is supported in $[-R_{0},R_{0}]$ for some $R_{0}\geq 0$.

Let $R\geq R_{0}$ and define the scattering energy functional $\mathcal{E}_{\textnormal{scattering}}:H^{1}([-R,R];\mathbb{C}^{2J+1}\otimes\mathbb{C}^{2J+1})\to[0,\infty]$ by

[TABLE]

To state Definition 8, we need the following facts.

Lemma 34**.**

For $\chi\in\mathbb{C}^{2J+1}\otimes\mathbb{C}^{2J+1}$, denote by $\mathcal{E}_{\chi}$ the functional $\mathcal{E}_{\textnormal{scattering}}$ restricted to functions $\psi$ with boundary conditions $\psi(R)=\chi$ and $\psi(-R)=(P_{A}-P_{S})\chi$. Then, $\mathcal{E}_{\chi}$ is strictly convex.

Proof.

Note that $\mathcal{E}_{\chi}$ is quadratic, and that it is non-negative by the assumptions on $V$. Hence, given $\psi_{1},\psi_{2}$ in the domain of $\mathcal{E}_{\chi}$ and $\alpha\in[0,1]$, we can write

[TABLE]

Now assume that $\psi_{1}\neq\psi_{2}$. Since they satisfy the same boundary condition, they cannot differ by a constant, and thus the kinetic energy in $\mathcal{E}_{\chi}(\psi_{1}-\psi_{2})$ is strictly positive. By $V$ being positive semi-definite, we conclude that $\mathcal{E}_{\chi}$ is strictly convex,

[TABLE]

∎

This directly implies the uniqueness of the scattering solution claimed in Definition 8.

Corollary 35**.**

There is a unique minimizer of $\mathcal{E}_{\textnormal{scattering}}$ with boundary condition $\psi(R)=\chi$ and $\psi(-R)=(P_{A}-P_{S})\chi$.

Given this uniqueness, we can define a map $F_{0}:\mathbb{C}^{2J+1}\otimes\mathbb{C}^{2J+1}\to H^{1}([-R,R];\mathbb{C}^{2J+1}\otimes\mathbb{C}^{2J+1})$ that maps a boundary spin-state $\chi$ to the corresponding minimizer of $\mathcal{E}_{\chi}$. Note that any minimizer is the unique solution of the Euler-Lagrange equation

[TABLE]

with boundary conditions $\psi(R)=\chi$ and $\psi(-R)=(P_{A}-P_{S})\chi$. Thus, linearity of (A.3) and the boundary condition imply linearity of $F_{0}$. This means that $F_{0}$ can be identified with an element in $H^{1}([-R,R];\operatorname{Mat}((2J+1)^{2},(2J+1)^{2}))$. This proves the first part of the following proposition.

Proposition 36**.**

The scattering solution $F_{0}\in H^{1}([-R,R];\operatorname{Mat}((2J+1)^{2},(2J+1)^{2}))$ is the unique solution of (3.1) on $[-R,R]$ with boundary conditions $F(R)=I$ and $F(-R)=P_{A}-P_{S}$. Moreover, $F_{0}$ satisfies (3.2) and (3.3).

Proof.

This first part of the proposition follows from the analysis above. To check (3.2), notice that for $\left\lvert x\right\rvert>R_{0}$ (outside the range of $V$), $F_{0}$ is harmonic by (3.1). Hence, there exists $\mathsf{A}\in\operatorname{Mat}((2J+1)^{2},(2J+1)^{2})$ such that (3.2) holds. Integration by parts and use of (3.1) gives (3.3). ∎

The following result is used in the proof of the upper bound.

Lemma 37**.**

Consider the set-up in Definition 8—that is, let $F$ be the scattering solution and $\mathsf{A}$ the scattering length matrix of some potential $V$. Furthermore, let $F_{\rm hc}$ be the scattering solution for the hard-core potential with radius $R_{0}$. Then, for all $\xi\in\mathbb{C}^{2J+1}$ and $x\in[-R,R]$,

[TABLE]

Proof.

Note that $\left\langle F(x)\xi|F(x)\xi\right\rangle$ is convex in $x$: by the scattering equation (3.1),

[TABLE]

and this goes for $\left\langle F_{\rm hc}(x)\xi|F_{\rm hc}(x)\xi\right\rangle$ as well. To prove the claim, note that for $\left\lvert x\right\rvert\leq R_{\rm hc}$, we have $\left\langle F_{\rm hc}(x)\xi|F_{\rm hc}(x)\xi\right\rangle=0$ and the statement is trivially true. For $\left\lvert x\right\rvert>R_{\rm hc}$, we use this fact at $x=R_{0}$, convexity, and that by (3.2) and $\mathsf{A}\leq R_{0}$, the derivatives at $R$ satisfy

[TABLE]

while the functions are equal at $R$ by the boundary condition. ∎

We add a remark about scalar potentials, to prove a claim made in Remark 5.

Remark 38**.**

In Remark 5, we claimed that all scattering lengths $a_{e}$ and $a_{o}$ with $a_{e}\leq a_{o}$ and $a_{o}\geq 0$ can be attained by varying the potential. To see this, consider the double delta potential, $v=2c(\delta_{-R_{0}}+\delta_{R_{0}})$, with strength $c>0$ and range $R_{0}\geq 0$. This gives $a_{e}=R_{0}-\frac{1}{c}$ and $a_{o}=R_{0}-\frac{R_{0}}{1-R_{0}c}$. For $a_{o}\geq 0$ fixed, we find $R_{0}=\frac{a_{o}+\sqrt{a_{o}^{2}+4a_{o}/c}}{2}$. Thus, $a_{e}(c)=\frac{a_{o}+\sqrt{a_{o}^{2}+4a_{o}/c}}{2}-\frac{1}{c}$. Since $a_{e}(c)$ is a continuous function of $c>0$ with $\lim_{c\to\infty}a_{e}(c)=a_{o}$ and $\lim_{c\to 0^{+}}a_{e}(c)=-\infty$, we see that any $a_{e}\leq a_{o}$ can be achieved.

Appendix B Remarks about the Lai–Sutherland and Yang–Gaudin models

B.1 Yang–Gaudin

Consider the spin-$1/2$ Yang–Gaudin model, which is the spin-$1/2$ Fermi gas with point interactions, $v(x)=2c\delta(x)$. If the total spin-$z$ is fixed to $(N-2M)/2$, the Bethe ansatz gives the following equations for the thermodynamic limit in which $N,M,L\to\infty$ proportionally [41],

[TABLE]

Here, $e_{\textnormal{YG}}$ denotes the energy density of a given solution. By the Lieb–Mattis result [79], the ground state is known to have total spin 0, so that it satisfies $M=N/2$. This is achieved for $B\to\infty$, which follows from (B.1) and (B.3). It is not hard to show (see e.g. [77]) that, given a solution to (B.1)–(B.4) with $B=\infty$, one finds

[TABLE]

where $a_{e}=-2/c$ is the even-wave scattering length of the interaction $v(x)=2c\delta(x)$ (note $a_{o}=0$ for a point interaction).

While the technique used to solve the Yang–Gaudin model [41, 42, 80] is similar to that for the Lieb–Liniger model [27], the mathematical rigour is somewhat less clear. More specifically, while the algebraic equations obtained with the Bethe ansatz for the Lieb–Liniger model are known to have unique solutions [81], this is unclear for the ones found for the Yang–Gaudin model. Furthermore, even if existence and uniqueness of solutions to the relevant set of algebraic equations is assumed, one needs to argue that the ground state is among the Bethe ansatz states to begin with. For the Lieb–Liniger model, this follows from uniqueness of the bosonic ground state, and the fact that Girardeau’s $c=\infty$ ground state is of the Bethe ansatz shape. More generally, it was proved by Dorlas in [67] that the Lieb–Liniger Bethe ansatz states form a complete orthogonal set, but none of these arguments are directly valid for the spin-$J$ (or even spin-$1/2$) Fermi gas. Hence, there does not seem to be a proof in the literature that the Bethe ansatz is valid for the ground state. However, Theorem 1 and the known expansion of the Heisenberg ground state energy (see (2.7)) prove that first two terms of (2.9) are correct in the dilute limit.

B.2 Lai–Sutherland

The following lemma was used in (2.7).

Lemma 39**.**

Let $E_{\textnormal{LS}}^{N}$ denote the ground state energy of the $N$-spin Lai–Sutherland model $H_{\textnormal{LS}}=\sum_{i=1}^{N}P_{S}^{i,i+1}$ with periodic boundary conditions (for some fixed spin $J$), and let $e_{\textnormal{LS}}=\lim\limits_{N\to\infty}\frac{1}{N}E_{\textnormal{LS}}^{N}$ be the thermodynamic ground state energy per site. Then,

[TABLE]

Proof.

For $M>N$, we have

[TABLE]

This can be seen by taking the ground state of a chain of length $M$, truncating it at length $N$ to obtain a trial state for the chain of length $N$, and using that $N-1$ terms in the Hamiltonian all have the same expectation value by translation invariance of the chain of length $M$, while the last term is estimated by the maximal eigenvalue 1 of a projection operator.

On the other hand, for $m\in\mathbb{N}$, take a translation-invariant ground state of the chain of length $N$, concatenate $m$ of them to obtain a trial state for the chain of length $mN$, and again use that all terms in the Hamiltonian corresponding to spins in the interior of the chain of length $N$ give the same energy contribution. This gives

[TABLE]

Taking the limits $m\to\infty$ and $M\to\infty$ in (B.6) and (B.7) gives the desired result. ∎