FROM FEW TO MANY An analytical study on the crossover of a one-dimensional Fermi gas, from a few-particle system to the many-body limit

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Report Bachelor Project Physics and Astronomy

FROM FEW TO MANY

An analytical study on the crossover of a one-dimensional Fermi gas, from a

few-particle system to the many-body limit

by

David Gunneweg

6361862

July 3, 2014

15 ECTS

April 2013 - June 2013

Supervisor:

Prof.dr. C.J.M. Schoutens

Second assessor:

dhr. dr. P.R.Corboz

Institute for Theoretical Physics

University of Amsterdam

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Abstract

How many particles are needed for a system to describe its macroscopic properties by many-body theory? For this generally hard question, we examine the crossover from few to many-body physics by analytically studying a one-dimensional infinite square well potential with ultracold spin-1/2 fermions, consisting of a single spin-_{" particle} interact-ing with an increasinteract-ing number of spin-_{# particles. With a Bethe Ansatz we solve the} interaction energy of such a system, as a function of the number of spin-# particles, in the limits of weak and strong interaction. In the strong interaction limit, already for a small number of particles we find a fast convergence towards the many-body solution. In the weak interaction limit, further research with a larger number of particles is needed

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Populaire samenvatting

Het bestuderen van systemen met veel deeltjes is een belangrijk gebied binnen de natu-urkunde. Bijvoorbeeld bij onderzoek naar de eigenschappen van een klein stukje metaal, heb je als onderzoeker al te maken met vele miljarden atomen. Deze atomen zijn samen verantwoordelijk voor de kenmerken van het metaal. Bijvoorbeeld warmte- en stroomgelei-ding en de kleur van het materiaal zijn het gevolg van de eigenschappen van individuele atomen en de manier waarop deze gerangschikt zijn. Hierdoor kun je met behulp van de eigenschappen van bijvoorbeeld een enkel koperatoom, voorspellingen doen over de stroomweerstand van een koperdraad van miljarden atomen. Zulke berekeningen aan sys-temen met veel deeltjes worden ingewikkelder naarmate het aantal deeltjes toeneemt. Om hiermee om te gaan, wordt bij grote systemen daarom vaak de aanname gemaakt dat het aantal deeltjes naar oneindig gaat. Dat klinkt abstract maar dit komt er op neer dat het systeem niet verandert als er een deeltje wordt toegevoegd of er uit wordt gehaald. In een koperdraad met miljarden atomen maakt het voor de stroomweerstand niet uit of er ´e´en atoom meer of minder in zit. Deze aanname vereenvoudigt de berekening en zorgt ervoor dat er op deze manier eigenschappen van materialen berekend kunnen worden.

Bij deze methode is het van belang dat het systeem waar je een uitspraak over wilt doen genoeg deeltjes bevat om aan te nemen dat dit aantal naar oneindig gaat. Zijn hiervoor miljarden atomen nodig of zijn enkele tientallen of zelf een aantal atomen al genoeg?

In dit onderzoek hebben we deze vraag analytisch onderzocht voor een systeem van fermionen in een eendimensionale potentiaalput. Fermionen zijn een bepaald type deelt-jes waar elektronen bijvoorbeeld ook onder vallen. Een eendimensionale potentiaalput wil zeggen dat de deeltjes kunnen bewegen over een rechte lijn met een beperkte lengte. Denk aan een flinterdun geleidende draad dat met de uiteinden vast is gemaakt aan twee sterke isolatoren, waardoor elektronen kunnen bewegen. Verder bevat het systeem één minderheidsdeeltje, dat is een fermion dat alle andere fermionen, de meerderheidsdeeltjes, afstoot als het er tegenaan botst. De meerderheidsdeeltjes reageren niet op elkaar maar worden alleen afgestoten door het enkele minderheidsdeeltje. Een dergelijk systeem bevat meer energie dan eenzelfde systeem met alleen maar meerderheidsdeeltjes die elkaar niet afstoten. Deze extra energie wordt de interactie-energie van het systeem genoemd. In dit onderzoek is de interactie-energie berekend van een systeem met één minderheids-deeltje en 1, 2 tm 5 meerderheidsminderheids-deeltjes, voor hele sterke en een hele zwakke interacties. Deze interactie-energien zijn vergeleken met de interactie-energie van een systeem met één minderheidsdeeltje dat reageert met oneindig veel meerderheidsdeeltjes. De totale interactie-energie stijgt met het aantal meerderheidsdeeltjes en gaat voor oneindig veel deeltjes naar oneindig. Daarom normaliseren we de interactie-energie met de Fermi en-ergie, de natuurlijke energie maat van het systeem.

We vinden dat de genormaliseerde interactie energie voor een toenemend aantal meerder-heidsdeeltjes naar de oplossing voor een oneindig aantal meerdermeerder-heidsdeeltjes gaat. Voor

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sterke interacties wordt de genormaliseerde interactie-energie al vanaf drie `a vier meerder-heidsdeeltjes grotendeels beschreven door de oplossing voor het systeem met oneindig meerderheidsdeeltjes. Hieruit volgt dat voor sterke interacties al voor een klein aantal meerderheidsdeeltjes bepaalde eigenschappen van het systeem kunnen worden berekend door aan te nemen dat het aantal deeltjes naar oneindig gaat. Voor zwakke interac-ties zien we een minder sterke overgang van de oplossing en is er onderzoek met meer meerderheidsdeeltjes nodig om te bepalen wanneer deze aanname gemaakt kan worden.

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1 Introduction

Deriving the macroscopic properties of a many-body system from the microscopic prop-erties of the individual particles is one of the great accomplishments of physics. For this derivation usually the assumption is made that the number of bodies tends to infinity. In this limit, the complexity of the system can be reduced by using continuous in stead of discrete variables. This method raises the question how many particles are needed for this many-body approximation to be valid. This is a particular hard question to answer since for most systems their microscopic description becomes extremely complex already for a small number of particles.

This question has been addressed experimentally by Wenz et al. (2013) in a quasi one-dimensional harmonic trapping potential with ultra colt atoms consisting of a single impurity interacting with an increasing number of identical fermions. In their research Wenz et al. measure the interaction energy of such a system as a function of the number of majority atoms for di↵erent interaction strengths. Comparing their measurements to the analytical solutions for a system with one majority particle and a Fermi sea of majority particles, they find a fast convergence to the many-body limit.

In this research we will address this question analytically for a one-dimensional infinite square well potential with spin-1₂ fermions consisting of a single spin-" particle interacting with an increasing number of spin-_{# particles, see Figure 1. Specifically, we examine the} system size needed to describe the interaction energy, normalized to the Fermi energy, with the solution for the many-body limit of a single spin-" fermion interacting with a Fermi sea of spin-_{# fermions.}

First we will derive a Bethe Ansatz solution for the ground state energies of the few particles systems. Using the (anti)-symmetry properties of the system and the boundary specifications, we will derive the first and second level Bethe equations for the given configuration. Next we will approximate the solutions of the obtained Bethe equations in the limits of weak and strong interaction in order to find the ground state energies. Then we will determine the corresponding normalized interaction energies and compare those to the analytical solution for a system with one spin-_{" particle and a Fermi sea of spin-#} particles in the obtained interaction limits.

Figure 1: From few to many. A single spin-_{" fermion (blue) interacting with, from left to right,} one, few and many spin-_{# fermions (red) in a infinite square well potential. In the many-body body} limit the assumption is made that N tends to infinity.

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2 A Bethe Ansatz solution for few-particle systems

2.1 The Bethe equations

First we look at an infinite square well of length L with arbitrary N interacting spin-1 2

fermions. The fermions interact through a delta potential. In units where ~ = 2m = 1, this system is described by the Hamiltonian

H = N X i=1 @2 @x2 i + 2c N X i<j (xi xj), (1)

in which xirepresents the position of the i-th particle, and c 0 the repulsive interaction

strength.

The solution of this system is described by wavefunctions of the form

(x1, . . . , xN, 1, . . . , N), with xi 2 [0, L] and the spin coordinates i 2 {", #}, for i =

1, . . . , N . The eigenfunctions of the Hamiltonian (1) are constraint to the following additional properties:

1. Anti-symmetry under particle exchange: = for (xi, i)$ (xj, j)

2. Continuity of for{xi} 2 [0, L]

3. Infinite wall boundary conditions, (. . . xi= 0 . . .) = (. . . xi= L . . .) = 0

The solutions of this system can be derived using the Bethe Ansatz (Gaudin 1983). The essential idea is as follows: the contact interaction (xi xj) only plays a role ’near’

the points where the two coordinates are the same: xi ! xj. Furthermore, it has been

shown, by integrating over the delta potential, that the wave function is continuous, but has a finite jump in its first derivative at the points xi ! xj (Fl¨ugge 1971). Therefore,

this problem can be seen as a free particle problem away from the points xi ! xj, with

plane wave solutions. As a result, the exact eigenfunctions of the Hamiltonian (1) can be obtained by patching together the free plane wave solutions, constrained under the appropriate ’patching conditions’ - accounting for the jump in the first derivative as a result of the delta potential.

To find the form of the free wave functions, we treat the Hamiltonian as describing the interaction of a single particle with a fixed potential in a N -dimensional space (Yang 1967). The single particle coordinates are given by the positions of each of the particles, {x1, . . . , xN}. The fixed potential is given by a series of intersecting (N 1)-dimensional

delta functions in the N dimensional space. These intersecting delta functions form the boundaries of N ! di↵erent regions in the N -dimensional space. Each of those regions corresponds to one of the N ! permutations of the particles along the line. Therefore we call each of the permutations of the particle coordinates a region. For example 0_{ x}1<

x2< . . . < xN  L is a region.

Now, for a set of wave numbers kj, j = 1, . . . , N , in one region of space, we have

2N

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{nj} = {1, . . . , N} (Oelkers et al. 2006). The ±-sign for the ki-s corresponds to incident

and reflected waves.

Now let Q =_{Q1, . . . , QN} and P = {P1, . . . , PN} be two permutations of the numbers

{1,. . . ,N} and let ✏ = ±1. In the region 0  xQ1 < xQ2 < . . . < xQN  L the Bethe

Ansatz wavefunction is of the form:

(x1, . . . , xN, 1, . . . , N) = X P X ✏_Pi=± A 1,..., N(✏P1, . . . , ✏PN|Q1, . . . , QN)· exp 0 @i N X j=1 ✏PjkPjxQj 1 A (2) for xi6= xj,8i 6= j, with i, j = 1, . . . , N.

The sums run over all N ! permutations P , over all N signs ✏ =_{± and each} i can be

either " or #. For each argument (x1, . . . , xN) there is exactly one permutation Q, such

that 0_{ x}Q1 < xQ2< . . . < xQN  L. In total there are N! di↵erent regions. So in total

there are N !_⇥N!⇥2N

⇥2N _{complex coefficients A}

1,..., N(✏P1, . . . , ✏PN|Q). The problem of

finding the eigenfunctions of the Hamiltonian (1) got reduced to finding the appropriate coefficients A and the corresponding quasi-momenta _{k1, . . . , kN}. Because this wave

function is a combination of plane waves, the energy eigenvalue of the Hamiltonian (1) is determined by the quasi-momenta through:

E =

N

X

i=1

ki2 (3)

Since we are only interested in the energy eigenvalues of the Hamiltonian (1), we will proceed by reducing the coefficients in (2) by imposing restrictions on the Bethe Ansatz wavefunctions.

Oelkers et al. (2006) describe a method for reducing the Bethe Ansatz wavefunctions of the form (2) to a system of equations from which the wave numbers _{ki} can be derived.

The Bethe Ansatz wavefunctions in (2) are first reduced by imposing restriction on the coefficients A 1,..., N(✏P1, . . . , ✏PN|Q), using the key properties of the wavefunction.

First of all, applying the anti-symmetry condition under particle exchange ( = for (xi, i)$ (xj, j)) to the wavefunction (2) gives the following property, with a clear

abbreviation in the notation:

A i j(✏P1. . . , ✏PN|QaQb) = A j i(✏P1. . . , ✏PN|QbQa), (4)

where Qa and Qb refer to the positions of xi and xj for a given argument of , where

we assume xi < xj. Next we define i = Pa and j = Pb and introduce the permutation

operator (Tij₎ 01... N0

1... N = i, 0j j, 0i

Q

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be written as

(Tij) 10... 0N

1... NA ₁0... _N0 (✏P1. . . , ✏PN|QaQb) = A 1... N(✏P1. . . , ✏PN|QbQa). (5)

Furthermore, using the continuity of at the intersections of two regions requires

(x1, . . . , xN, 1, . . . , N)xxii<x!xjj = (x1, . . . , xN, 1, . . . , N) xi>xj xi!xj (6) and therefore, A 1... N(✏Pa✏Pb|QaQb) + A 1... N(✏Pb✏Pa|QaQb) = A 1... N(✏Pa✏Pb|QbQa) + A 1... N(✏Pb✏Pa|QbQa). (7)

Since we consider the delta function potential as the barrier between two regions of the N -dimensional space, the delta potential results in a two-sided boundary condition. By integrating the Schr¨odinger equation over an infinitesimal region, which spans the delta-function potential between the two coordinates in the argument, xQa= xiand xQb= xj,

the delta interaction results in (Fl¨ugge 1971) "✓ @ @xQa @ @xQb ◆ x_Qa x_Qb=0+ ✓ _@ @xQa @ @xQb ◆ x_Qa x_Qb=0 # = c _|_x Qa=xQb. (8)

Applying this condition to the wave function, gives the following relation on the coefficients

Using equation (7) this relation can be reduced to the following equation relating A coefficients with identical spin values i j and di↵erent permutations Q and P

[ikPb ikPa]· ⇥ A i j(✏Pa✏Pb|QaQb) A i j(✏Pb✏Pa|QbQa) ⇤ = c⇥A i j(✏Pa✏Pb|QaQb) + A i j(✏Pb✏Pa|QaQb) ⇤ . (10)

By using the T -operator from equation (5), Yang (1967) transformed this equation into a relation between coefficients A with the same permutation P and a di↵erent permutation Q for a similar system with periodic boundary conditions. Oelkers et al. (2006) found the corresponding relation for the infinite square well, in the form of a matrix equation,

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saying A i j(✏Pa✏Pb|QaQb) =  i(✏PbkPb ✏PakPa)T ij_{+ cId} i(✏PbkPb ✏PakPa) c 0 1... N0 1... N A 0 i 0j(✏Pb✏Pa|QaQb). (11)

We found the equations relating the the coefficients of di↵erent permutations P and Q and di↵erent configurations of 1, . . . , N. By imposing the infinite square well boundary

conditions and using the obtained relations for the coefficients A, Oelkers et al. (2006) found a restriction on the values of the ki in the form of the Bethe equations. For the

spin-1

2 fermions, the first level Bethe equations reduce to N equations

ei2kiL₌ M Y j=1 ki ⇤j+1₂ic ki ⇤j 1₂ic ki+ ⇤j+1₂ic ki+ ⇤j 1₂ic (12)

for (i = 1, . . . , N ), M is the number of spin-" particles and {⇤1, . . . , ⇤M} are the so-called

spin roots. These equations satisfy M second level Bethe equations

N Y j=1 ⇤i kj 1₂ic ⇤i kj+1₂ic ⇤i+ kj 1₂ic ⇤i+ kj+1₂ic = M Y j=1 j6=i ⇤i ⇤j 1₂ic ⇤i ⇤j+1₂ic ⇤i+ ⇤j 1₂ic ⇤i+ ⇤j+1₂ic . (13)

2.2 Ground state energies for N

_"

= 1 configuration

In the rest of this section we look at the special case of the system described in the subsection 2.1, with N = 2, . . . , 6 spin-1

2 fermions, trapped in a one-dimensional infinite

square well of length L. We will derive the ground state energies of configurations with one spin-_{" fermion, which we call the impurity particle and (N 1) spin-# fermions, which} we refer to as the majority particles. The particles interact via a delta function potential with interaction strength c. This configuration with c = 0 is described in Figure 2.

The Hamiltonian of this system is given by equation (1), so the energy eigenvalues are entirely determined by the quasi-momenta {k1, . . . , kN} through equation (3). In order

to find the quasi-momenta we reduce the Bethe equations (12) and (13) for the specific system.

We have M = N_" = 1 and N M = N_# = N 1, so the first level Bethe equations (12) reduce to ei2kiL ₌ki ⇤ + 1 2ic ki ⇤ 1₂ic ki+ ⇤ +1₂ic ki+ ⇤ 1₂ic . (14)

The second level Bethe equations (13) reduce to

N Y j=1 ⇤ kj 1₂ic ⇤ kj+1₂ic ⇤ + kj 1₂ic ⇤ + kj+1₂ic = 1. (15)

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Figure 2: One-dimensional infinite square well with spin-1₂ fermions. We consider systems with one spin-" fermion, indicated with a blue ball, and N 1 = 1 . . . 5 spin-# fermions, indicated with red balls. This figure displays particle states for non-interaction fermions. For each state, the energy level of the Femi energy is indicated with EF

In this research we attempted to find numerical solutions to these Bethe equations for general c > 0, using Wolfram Mathematica. However, even for the simple configuration with N = 2, we did not manage to find a solution. Furthermore, for this configuration we tried to analytically solve the Hamiltonian, using centre of mass co¨ordinates. Also these attempts did not result in any insightful solutions. Therefore, in this research we focus on the weak interaction limit, where c_⌧ _L1, and the strong interaction limit, where c_{! 1.} This allows us to solve the Bethe equations analytically in each of these limits.

2.2.1 Weak interaction limit

First we discuss the solutions of the N" = 1 configuration, as described above, in the

weak interaction limit, where c⌧ 1

L. From numerical analysis of the Bethe equations, it

is known that the root distribution evolves continuously from the free fermion case where there is no interaction (c = 0) (Oelkers et al. 2006). For c = 0 the Bethe equations (14) and (15) collapse to one equation

ei2kiL_{= 1,} ₍₁₆₎

with as solutions the free momenta ki = ⇡_Lni, ni = 1, 2, 3, . . .. As we are considering

spin-1

2 fermions, every root can be occupied by at most two fermions, a pair of a spin-#

and a spin-" fermion. So each lattice side can be occupied by one ki or two ki’s, in which

case the quasi-momenta are interspaced by a spin root ⇤ (Oelkers et al. 2006). Since we are considering the situation of the ground state with only one spin-_{" minority particle} and (N 1) spin-# majority particles (see Figure 2), we will find a pair spin-# and spin-" in the lowest attice side, interspaced by a spin root, and single spin-_{# majority particles} in the next (N 2) lattice sides (Figure 3).

For nonzero small c, the roots can be obtained by expanding the Bethe equations in c. For these solutions to be valid it is important that c⌧ ki ⇤ for i = 1, . . . , N 1 (see

the reduced Bethe equations (14) and (15)). This can be ensured by separately treating spin roots and quasi-momenta in the same lattice side. Numerical analysis by Oelkers et

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Figure 3: Bethe equation root distribution for the ground state for c = 0.1, with M = 1 spin-_{" particle} and N M = 5 spin-_{# particles (top) and N} M = 1 spin-_{# particle (bottom). The blue} squares indicate the quasi-momenta kiand the red square indicates the spin root ⇤. The roots develop

continuously from the case c = 0, where they take values at integer multiples of ⇡_L. The roots are obtained by solving the Bethe equations (14) and (15) in the weak interaction limit.

al. (2006) shows that the Bethe roots are of the form

k1= ⇡ L+ c p c k2= ⇡ L + c + p c (17) ⇤ = ⇡ L+ c (18) ki= ⇡ L· i + ic i = 3, . . . , (N 1). (19) The roots in (17) correspond to the paired roots in the first lattice side. ⇤ is the spin root belonging to the first lattice side. The ki’s in (19) correspond to the unpaired roots in the

higher lattice sides. , , and the i’s are real constants that are left to be determined.

For N = 2 we have two quasi-momenta k1 and k2, so , and are left to be

deter-mined. First we substitute the roots in the form of equations (17) and (18) into the Bethe equations (14) and (15) for N = 2. We obtain

ei2L(⇡L+c p c)₌ c p_c _{c +}1 2ic c pc c 1₂ic· 2⇡ L + c p_{c + c +}1 2ic 2⇡ L + c p_{c + c} 1 2ic (20)

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and ei2L(⇡L+c + p c)₌ c + p_c _{c +}1 2ic c + pc c 1₂ic· 2⇡ L + c + p_{c + c +}1 2ic 2⇡ L + c + p c + c 1₂ic (21) for the first level Bethe equations (14) and

1 = c c + p_c 1 2ic c c + pc + 1 2ic · 2⇡ L + c + p_{c + c} 1 2ic 2⇡ L + c + p_{c + c +}1 2ic ⇥ c c p_c 1 2ic c c pc + 1 2ic · 2⇡ L + c p_{c + c} 1 2ic 2⇡ L + c p_{c + c +}1 2ic (22)

for the second level Bethe equation (15). In order to solve this system for , and , we expand the equations (20),(21) and (22) in powers ofpc up to the first order of c. After simplifying the result, we obtain

1 2i pc + 2iL c 2 2L2c = 1 ipc + i( 2_{L + ⇡(i} _{2 + 2 ))} 2 2_⇡ c (23) and 1 + 2i pc + 2iL c 2 2L2c = 1 + ipc + i( 2_{L + ⇡(i} _{2 + 2 ))} 2 2_⇡ c (24)

for the first level Bethe equations and

1 = 1 2i( ₂ )c iL

⇡c (25)

for the second level Bethe equation. Next we solve this system for , and and we find

= 1 2⇡, = 1 p 2L and = 3 4⇡. (26)

So we have the quasi-momenta

k1= ⇡ L+ 1 p 2L p c + 1 2⇡c and k2= ⇡ L 1 p 2L p c + 1 2⇡c (27) And the ground state energy is now given by equation (3), which gives

EGR =2⇡ 2 L2 + 3 Lc + 1 2⇡2c 2_. ₍₂₈₎

Following the same method of first approximating the roots of the Bethe equations by roots of the form in equations (17),(18) and (19), and then expanding the Bethe equations in powers of pc, we obtain the ground state energies given in Table 1. For more details on the computations see appendix A.

2.2.2 Strong interaction limit

In this section we look at the solutions of the N"= 1 configuration in the strong interaction

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Table 1: Approximate ground state energies of a system of interacting fermions in an one-dimensional infinite square well of length L, in the weak interaction limit, where c⌧ 1

L. The system consists

of a single spin-" minority fermion and (N 1) = 1, . . . , 5 spin-# majority particles.

Number of

c

2

the free values ki = _L⇡ · ni, ni = 1, 2,· · · when c = 0, in the limit of c ! 1 the roots

come back to those values but now without the existence of a paired root. In this limit the ground-state roots are given by

ki = ⇡ Lni+ ic 1_, _{i = 1,} · · · , N {ni} = {1, . . . , N} (29) ⇤ = c (30)

(Oelkers et al. 2006; Takahasi 1999), where the ki’s are the quasi-momenta, ⇤ is the spin

root and and the i’s are real constants left to be determined. We substitute these roots

in the second level Bethe equation (15) and we obtain

1 = N Y j=1 c _L⇡nj jc 1 1₂ic c _L⇡nj jc 1+1₂ic c +_L⇡nj+ jc 1 1₂ic c +_L⇡nj+ jc 1+1₂ic {n j} 2 {1, . . . , N}. (31) Next we expand in p1 c to the power c 1 _{and we get} N Y j=1 ( i 2) 2 ( + i 2)2 = i 2 + i 2 !2N = 1. (32)

And it follows directly that = 0. Now we substitute = 0 and the roots of the form from equation (29) into the first level Bethe equations (14) and we obtain

exp ✓ 2i(L i+ cni⇡ c ◆ =( ni⇡ L + d c+ ic 2)2 (ni⇡ L + d c ic 2)2 . (33)

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Again, we expand in p1 c to the power c 1_{and we find} e2ini⇡₊2i iL c e 2ini⇡_{= 1} 8ini⇡ cL . (34) {ni} 2 {1, . . . , N} so 1 +2i iL c = 1 8ini⇡ cL . (35) and i= 4⇡ cLni. (36)

So the quasi-momenta (29) are of the form

ki= ⇡ Lni· ✓ 1 4 cL ◆ where _{ni} 2 {1, . . . , N}. (37)

Using equation (3) we find the ground state energies for a system of N particles

EGR= ⇡2 6L2 ✓ 1 8L c + 16 c2 ◆ (2N3_{+ 3N}2_{+ N )} ₍₃₈₎

2.3 Interaction energies

Having found the ground state energies of the systems in both limits, we can now examine the interaction energies. To find the interaction energy, we subtract the ground state energy of the same system but without interaction. The ground state energy of a one-dimensional infinite square well with a single minority particle and N 1 majority particles is obtained by finding the eigenvalues of the Hamiltonian( 1) for c = 0. We find the energy levels

En=

n2_⇡2

L2 n = 1, 2, , . . . (39)

in units where _{~ = 2m = 1. And for a system of a single minority particle in the lowest} energy state with (N 1) majority particles in the lowest possible states (see Figure 2), we find the ground state energy

EGR|c=0= 2⇡2 L2 + N_X1 i=2 ⇡2 L2i 2₌ ⇡2 L2 ✓ 1 +2N 3 _3N2_{+ N} 6 ◆ (40)

The interaction energy is now given by

E = EGR EGR|c=0. (41)

The addition of each majority atom increases the number of particles the minority atom interacts with, so the interaction energy increases with N and diverges for N _{! 1.} Therefore, we rescale the interaction energy with the Fermi energy of the majority atoms. The Fermi energy is given by the lowest single particle state not occupied by a majority

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particle, see Figure 2. For N 1 majority particles that is the N th level energy state

EF = N 2_⇡2

L2 (42)

Furthermore, to compensate for the increasing density, caused by the increase of the number of majority particles, we rescale c with the line density of majority particles, (N 1)/L. We obtain the new dimensionless interaction parameter

= 2m ~2 L N 1c = cL N 1. (43)

2.3.1 Weak interaction limit

Using the approximate ground state energies for a di↵erent number of majority particles in the weak interaction limit, given in table 1, we determine the interaction energy (41). Neglecting the c2 _{terms we find for N = 2, . . . , 6 particles the following relation for the}

interaction energy

E = 2N 1

L c +O(c

2_). ₍₄₄₎

Rescaling with the Fermi energy gives

E = _EE F = L ⇡2 2N 1 N2 c. (45)

In terms of the interaction parameter (43),

E = _⇡12

2N2 _{3N + 1}

N2 (46)

2.3.2 Strong interaction limit

The approximate ground state energy for a given interaction strength c are given by equation (38). First we determine the interaction energy by equation (41) and neglecting the c 2 _{terms, which gives for a system of N particles}

E = ⇡ 2 L2 ✓ N2 8N3 3L 1 c 4N2 L 1 c 4N 3L 1 c ◆ +_O(c 2₎ ₍₄₇₎

Next we rescale by the Fermi energy (42), which result in

E = _EE F = 1 ✓_8N 3L + 4 L+ 4 3N L ◆₁ c. (48)

And in terms of the interaction parameter

E = 1 ✓₈ 3 N N 1+ 4 N 1 + 4 3 1 N (N 1) ◆₁ (49)

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3 Crossover from few to many particles

In this section we look at the crossover of the normalized interaction energy as a function of the interaction parameter from few to many particles. The normalized interaction energy of a system of one impurity atom in an finite square well, interacting with a large number N 1 1 of majority atoms, as a function of the interaction strength, has analytically been calculated by McGuire (1965) and is given by

E|N 1= E EF N 1 = 2 ⇡ " c 2kF + tan 1 ✓ _c 2kF ◆ ✓ _c 2kF ◆2✓ ⇡ 2 tan 1✓ c 2kF ◆◆# , (50)

where kF is the Fermi momentum kF = N ⇡/L in units of _L1p~_2m. In terms of the

interaction parameter we find

E|N 1= E EF _N ₁ = ⇡2  1 4 + ✓ 2⇡ + 2⇡◆ tan 1⇣ 2⇡ ⌘ . (51)

To observe the crossover from few to many particle system we plot the normalized inter-action energy for N = 2, . . . , 6 and N ! 1 as a function of the dimensionless interaction parameter .

In Figure 4A the normalized interaction energies in the weak (left) and the strong (right) interaction limit are plotted as a function of the interaction parameter . To show the crossover in more detail, in Figure 4B we subtract the interaction energy of the N = 2 configuration. In these plots, the di↵erent colors indicate the number of majority particles.

In order to quantify the crossover we look at the ratio of the interaction energy of the few-body systems to the interaction energy of the many-body limit. In Table 2 the minimum values of these ratios on the plotted intervals are given for the di↵erent number of majority particles in weak and strong interaction limit.

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Figure 4: (A) Normalized interaction energy as function of the the interaction parameter , of inter-acting spin-1

2 fermions in a one dimensional infinite square well of length L, with interaction strength

c. The interacting energy is given in the weak interaction limit where c_⌧ _L1 (left) and in the strong interaction limit where c_{! 1 (right). The system consist of one spin-" minority particle and N} 1 spin-# majority particles. The di↵erent colors indicate the total number of particles. For N = 2, . . . , 6 the plots are obtained by analytically solving the Bethe equations in the limit c ! 1. The plot for N ! 1 is analytically obtained by McGuire (1965). (B) Di↵erence between the normalized interaction energies and the analytical solution for the N = 2 configuration.

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Table 2: Minimum values of the ratios of the interaction energy of few-body systems to the interaction energy of the many-body limit, on the indicated intervals of . For the weak interaction limit (left), these ratios are approximately constant on the given interval. In the strong interaction limit (right) these ratios slowly approach 1, because all normalized interaction energies approach 1 in this limit.

Minimum values ratios

Weak interaction limit

Strong interaction limit

0 <

< 0.1

0.764

0.985

4 Discusion

In Figure 4 we see that the interaction energies coincide for the limits of vanishing and diverging interactions. For = 0 the interaction energy is 0. For _{! 1 the limit of} fermionization is reached, where the energy of the spin-# fermion interacting with N 1 spin-" fermions equals the the energy of N identical fermions (Z¨urn et al. 2012). So the interaction energy in this limit equals the Fermi energy EF. In the regions close to these

limits the normalized interaction energy increases for a increasing number of particles. In the strong interaction limit we find a fast convergence towards the many-body limit. Already from three particles the interaction energy takes values very close to the many-body limit.

In the weak interaction limit for an increasing number of particles, we observe a drift towards the many-body solution. However, for the system sizes we studied, the interaction energy is does not take values close to the many-body solution. In order to get more insight in the crossover in this limit, larger systems need to be studied. We find that in this limit, in the given interval, the ratios take approximately constant values, close to the minimum ratios. If we reduce the interval of in the weak interaction limit to 0 < < 0.01, we still find approximately the same minimum ratio. This indicates that the slower convergence in the weak limit is not due to a lower accuracy in the approximation. If we look at analytical solutions for a similar system in a harmonic trapping potential, we also find that the interaction energy of three interacting fermions in the strong interaction limit is relatively closer to the many-body solution then in the weak limit (Bush et al. 1998; Gharasi et al. 2012; McGuire 1965).

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5 Conclusion

In the strong interaction limit, already for a few particles, the normalized interaction energy is described by the many-body solution. These finding correspond to theoretically obtained results for a similar system with a harmonic potential and smaller interaction strengths (Wenz et al. 2013). In the weak interaction limit, the drift towards the many-body solution for an increasing number of particles suggests the existence of a similar crossover for a larger number of particles then studied in this research. To gain further insight into the crossover in this region, research on larger systems is needed. Finally, since theoretical studies suggest a few-particle crossover for general interaction strength (Wenz et al. 2013), it would be interesting to study the interactiong energy in this system for general c > 0.

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6 Acknowledgements

Around the time that most of my fellow students were knocking doors of potential su-pervisors to get the bachelor project of their choice, I was living a rough ten hour flight away, studying for one term at the University of British Columbia, Vancouver. Since I got accepted to a Master programme starting from September, I was challenged to pick up a bachelor project from Vancouver, in order to be able to finish on time. Emailing didn’t turn out to be as e↵ective as knocking doors and most of my requests for projects got the reply that the project was already taken by a student that was in Amsterdam. When I explained this situation to prof. Kareljan Schoutens he replied that although all his projects also were already taken, he recently came across an interesting article and he was willing to turn this into a bachelor project with me. It is thanks to his help and flexibility that I was able to do this project in this unusual situation. I would like to thank him for his guidance and help in understanding and solving the Bethe Ansanz.

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A

Calculations of the ground state energy in the weak

interaction limit

In this appendix we give an overview of the Wolfram Mathematica (.nb) code, that we used for calculating the ground state energy in the weak interaction limit.

A.1 N=2

Calculation of Bethe roots for M = 1, N = 2 in the weak interaction limit (c < 1/L). Defining the approximation of the Bethe roots

lambda[c ]:=Pi/L + gamma_{⇤ c} lambda[c ]:=Pi/L + gamma⇤ c lambda[c ]:=Pi/L + gamma_{⇤ c} k1[c ]:=Pi/L + d1⇤ c beta⇤ Sqrt[c] k1[c ]:=Pi/L + d1_{⇤ c} beta_{⇤ Sqrt[c]} k1[c ]:=Pi/L + d1⇤ c beta⇤ Sqrt[c] k2[c ]:=Pi/L + d1_{⇤ c + beta ⇤ Sqrt[c]} k2[c ]:=Pi/L + d1_{⇤ c + beta ⇤ Sqrt[c]} k2[c ]:=Pi/L + d1_{⇤ c + beta ⇤ Sqrt[c]} The Bethe equations

f [x , y , c ]:=Exp[I_{⇤ 2 ⇤ x ⇤ L]} (x y + I_{⇤ c/2)/(x} y I_{⇤ c/2)} f [x , y , c ]:=Exp[I_{⇤ 2 ⇤ x ⇤ L]} (x y + I_{⇤ c/2)/(x} y I_{⇤ c/2)} f [x , y , c ]:=Exp[I_{⇤ 2 ⇤ x ⇤ L]} (x y + I_{⇤ c/2)/(x} y I_{⇤ c/2)} ⇤(x + y + I ⇤ c/2)/(x + y I⇤ c/2) ⇤(x + y + I ⇤ c/2)/(x + y_{⇤(x + y + I ⇤ c/2)/(x + y} II_{⇤ c/2)}⇤ c/2)

g[x1 ]:=(y x1 I_{⇤ c/2)/(y} x1 + I_{⇤ c/2) ⇤ (y + x1} I_{⇤ c/2)/(y + x1 + I ⇤ c/2)} g[x1 ]:=(y x1 I⇤ c/2)/(y x1 + I⇤ c/2) ⇤ (y + x1 I⇤ c/2)/(y + x1 + I ⇤ c/2) g[x1 ]:=(y x1 I_{⇤ c/2)/(y} x1 + I_{⇤ c/2) ⇤ (y + x1} I_{⇤ c/2)/(y + x1 + I ⇤ c/2)} G[x1 , x2 , y , c ] = Apply[Times,_{{g[x1], g[x2]}]}

G[x1 , x2 , y , c ] = Apply[Times,{g[x1], g[x2]}] G[x1 , x2 , y , c ] = Apply[Times,_{{g[x1], g[x2]}]}

( ic

2 x1+y)( ic2+x1+y)( ic2 x2+y)( ic2+x2+y)

(ic 2 x1+y)( ic 2+x1+y)( ic 2 x2+y)( ic 2+x2+y)

The Bethe equations expressed in the approximate roots G1[c , gamma , d1 , beta ]:=G[k1[c], k2[c], lambda[c], c] G1[c , gamma , d1 , beta ]:=G[k1[c], k2[c], lambda[c], c] G1[c , gamma , d1 , beta ]:=G[k1[c], k2[c], lambda[c], c] F1[c , gamma , d1 , beta ]:=f [k1[c], lambda[c], c] F1[c , gamma , d1 , beta ]:=f [k1[c], lambda[c], c] F1[c , gamma , d1 , beta ]:=f [k1[c], lambda[c], c] F2[c , gamma , d1 , beta ]:=f [k2[c], lambda[c], c] F2[c , gamma , d1 , beta ]:=f [k2[c], lambda[c], c] F2[c , gamma , d1 , beta ]:=f [k2[c], lambda[c], c]

Now we replace c for b^2, in order to expand in Sqrt[c] G1b[b , gamma , d1 , beta ]:=G1[b^2, gamma, d1, beta] G1b[b , gamma , d1 , beta ]:=G1[b^_{2, gamma, d1, beta]}

G1b[b , gamma , d1 , beta ]:=G1[b^2, gamma, d1, beta] F1b[b , gamma , d1 , beta ]:=F1[b^_{2, gamma, d1, beta]}

F1b[b , gamma , d1 , beta ]:=F1[b^2, gamma, d1, beta] F1b[b , gamma , d1 , beta ]:=F1[b^_{2, gamma, d1, beta]}

F2b[b , gamma , d1 , beta ]:=F2[b^_{2, gamma, d1, beta]}

F2b[b , gamma , d1 , beta ]:=F2[b^2, gamma, d1, beta] F2b[b , gamma , d1 , beta ]:=F2[b^_{2, gamma, d1, beta]}

Expansion in b (!Sqrt[c])

EG1b[b , gamma , d1 , beta ] = Normal[Series[G1b[b, gamma, d1, beta],{b, 0, 2}]] EG1b[b , gamma , d1 , beta ] = Normal[Series[G1b[b, gamma, d1, beta],_{{b, 0, 2}]]} EG1b[b , gamma , d1 , beta ] = Normal[Series[G1b[b, gamma, d1, beta],{b, 0, 2}]]

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1 ib

2₍

beta2L+2d1⇡ 2gamma⇡) beta2_⇡

EF1b[b , gamma , d1 , beta ] = Normal[Series[F1b[b, gamma, d1, beta],_{{b, 0, 2}]]} EF1b[b , gamma , d1 , beta ] = Normal[Series[F1b[b, gamma, d1, beta],_{{b, 0, 2}]]} EF1b[b , gamma , d1 , beta ] = Normal[Series[F1b[b, gamma, d1, beta],_{{b, 0, 2}]]} b i

beta 2ibetaL + b 2 1

2beta2 +_betaid12

igamma

beta2 + 2id1L 2beta

2_L2 iL 2⇡

EF2b[b , gamma , d1 , beta ] = Normal[Series[F2b[b, gamma, d1, beta],{b, 0, 2}]] EF2b[b , gamma , d1 , beta ] = Normal[Series[F2b[b, gamma, d1, beta],_{{b, 0, 2}]]} EF2b[b , gamma , d1 , beta ] = Normal[Series[F2b[b, gamma, d1, beta],{b, 0, 2}]] b _betai + 2ibetaL + b2 _2beta1 2 +_betaid12

igamma

beta2 + 2id1L 2beta

2

L2 iL_2⇡

Replace b by Sqrt[c]

EG1c[c , gamma , d1 , beta ]:=EG1b[Sqrt[c], gamma, d1, beta] EG1c[c , gamma , d1 , beta ]:=EG1b[Sqrt[c], gamma, d1, beta] EG1c[c , gamma , d1 , beta ]:=EG1b[Sqrt[c], gamma, d1, beta] EF1c[c , gamma , d1 , beta ]:=EF1b[Sqrt[c], gamma, d1, beta] EF1c[c , gamma , d1 , beta ]:=EF1b[Sqrt[c], gamma, d1, beta] EF1c[c , gamma , d1 , beta ]:=EF1b[Sqrt[c], gamma, d1, beta] EF2c[c , gamma , d1 , beta ]:=EF2b[Sqrt[c], gamma, d1, beta] EF2c[c , gamma , d1 , beta ]:=EF2b[Sqrt[c], gamma, d1, beta] EF2c[c , gamma , d1 , beta ]:=EF2b[Sqrt[c], gamma, d1, beta]

Solving the expanded equations

f [x , y , c ]:=Exp[I_{⇤ 2 ⇤ x ⇤ L]} (x y + I_{⇤ c/2)/(x} y I_{⇤ c/2)} f [x , y , c ]:=Exp[I_{⇤ 2 ⇤ x ⇤ L]} (x y + I_{⇤ c/2)/(x} y I_{⇤ c/2)} f [x , y , c ]:=Exp[I_{⇤ 2 ⇤ x ⇤ L]} (x y + I_{⇤ c/2)/(x} y I_{⇤ c/2)} ⇤(x + y + I ⇤ c/2)/(x + y I_{⇤ c/2)} ⇤(x + y + I ⇤ c/2)/(x + y_{⇤(x + y + I ⇤ c/2)/(x + y} II_{⇤ c/2)}_{⇤ c/2)} nn gamma_! 3 4⇡, d1! 1 2⇡, beta! 1 p 2pL o ,ngamma_! 3 4⇡, d1! 1 2⇡, beta! 1 p 2pL oo

Without loss of generality we define beta to be positive list:=sol[[2]]

list:=sol[[2]] list:=sol[[2]]

Now the quasi momenta are:

K1[c ]:=Pi/L + (d1/.list)_{⇤ c} (beta/.list)_{⇤ Sqrt[c]} K1[c ]:=Pi/L + (d1/.list)⇤ c (beta/.list)⇤ Sqrt[c] K1[c ]:=Pi/L + (d1/.list)_{⇤ c} (beta/.list)_{⇤ Sqrt[c]} K2[c ]:=Pi/L + (d1/.list)_{⇤ c + (beta/.list) ⇤ Sqrt[c]} K2[c ]:=Pi/L + (d1/.list)⇤ c + (beta/.list) ⇤ Sqrt[c] K2[c ]:=Pi/L + (d1/.list)_{⇤ c + (beta/.list) ⇤ Sqrt[c]} And the ground state energy is given by:

Eg[c ] = Simplify[K1[c]^_{2 + K2[c]}^_2] Eg[c ] = Simplify[K1[c]^2 + K2[c]^2] Eg[c ] = Simplify[K1[c]^_{2 + K2[c]}^_2] 3c L + c2 2⇡2 +2⇡ 2 L2

A.2 N=3

Calculation of Bethe roots for M=1, N=3. Defining approximation of the Bethe roots lambda[c ]:=Pi/L + gamma_{⇤ c}

lambda[c ]:=Pi/L + gamma⇤ c lambda[c ]:=Pi/L + gamma_{⇤ c} k1[c ]:=Pi/L + d1⇤ c beta⇤ Sqrt[c] k1[c ]:=Pi/L + d1_{⇤ c} beta_{⇤ Sqrt[c]} k1[c ]:=Pi/L + d1⇤ c beta⇤ Sqrt[c]

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k2[c ]:=Pi/L + d1⇤ c + beta ⇤ Sqrt[c] k2[c ]:=Pi/L + d1_{⇤ c + beta ⇤ Sqrt[c]} k2[c ]:=Pi/L + d1⇤ c + beta ⇤ Sqrt[c] k3[c ]:=2_{⇤ Pi/L + d2 ⇤ c} k3[c ]:=2⇤ Pi/L + d2 ⇤ c k3[c ]:=2_{⇤ Pi/L + d2 ⇤ c} The Bethe equations

f [x , y , c ]:=Exp[I_{⇤ 2 ⇤ x ⇤ L]} (x y + I_{⇤ c/2)/(x} y I_{⇤ c/2)} f [x , y , c ]:=Exp[I⇤ 2 ⇤ x ⇤ L] (x y + I⇤ c/2)/(x y I⇤ c/2) f [x , y , c ]:=Exp[I_{⇤ 2 ⇤ x ⇤ L]} (x y + I_{⇤ c/2)/(x} y I_{⇤ c/2)} ⇤(x + y + I ⇤ c/2)/(x + y I_{⇤ c/2)} ⇤(x + y + I ⇤ c/2)/(x + y⇤(x + y + I ⇤ c/2)/(x + y II⇤ c/2)_{⇤ c/2)}

g[x1 ]:=(y x1 I_{⇤ c/2)/(y} x1 + I_{⇤ c/2) ⇤ (y + x1} I_{⇤ c/2)/(y + x1 + I ⇤ c/2)} g[x1 ]:=(y x1 I⇤ c/2)/(y x1 + I⇤ c/2) ⇤ (y + x1 I⇤ c/2)/(y + x1 + I ⇤ c/2) g[x1 ]:=(y x1 I_{⇤ c/2)/(y} x1 + I_{⇤ c/2) ⇤ (y + x1} I_{⇤ c/2)/(y + x1 + I ⇤ c/2)} G[x1 ,x2 ,x3 ,y ,c ]:= Apply[Times,{g[x1],g[x2],g[x3]}]

G[x1 ,x2 ,x3 ,y ,c ]:= Apply[Times,_{{g[x1],g[x2],g[x3]}]} G[x1 ,x2 ,x3 ,y ,c ]:= Apply[Times,{g[x1],g[x2],g[x3]}]

The Bethe equations expressed in the approximate roots G1[c , gamma , d1 , d2 , beta ]:=G[k1[c], k2[c], k3[c], lambda[c], c] G1[c , gamma , d1 , d2 , beta ]:=G[k1[c], k2[c], k3[c], lambda[c], c] G1[c , gamma , d1 , d2 , beta ]:=G[k1[c], k2[c], k3[c], lambda[c], c] F1[c , gamma , d1 , d2 , beta ]:=f [k1[c], lambda[c], c]

F1[c , gamma , d1 , d2 , beta ]:=f [k1[c], lambda[c], c] F1[c , gamma , d1 , d2 , beta ]:=f [k1[c], lambda[c], c] F2[c , gamma , d1 , d2 , beta ]:=f [k2[c], lambda[c], c] F2[c , gamma , d1 , d2 , beta ]:=f [k2[c], lambda[c], c] F2[c , gamma , d1 , d2 , beta ]:=f [k2[c], lambda[c], c] F3[c , gamma , d1 , d2 , beta ]:=f [k3[c], lambda[c], c] F3[c , gamma , d1 , d2 , beta ]:=f [k3[c], lambda[c], c] F3[c , gamma , d1 , d2 , beta ]:=f [k3[c], lambda[c], c]

Now we replace c for b^_{2, in order to expand in Sqrt[c]}

G1b[b , gamma , d1 , d2 , beta ]:=G1[b^_{2, gamma, d1, d2, beta]}

F1b[b , gamma , d1 , d2 , beta ]:=F1[b^_{2, gamma, d1, d2, beta]}

F1b[b , gamma , d1 , d2 , beta ]:=F1[b^2, gamma, d1, d2, beta] F1b[b , gamma , d1 , d2 , beta ]:=F1[b^_{2, gamma, d1, d2, beta]}

F2b[b , gamma , d1 , d2 , beta ]:=F2[b^2, gamma, d1, d2, beta] F2b[b , gamma , d1 , d2 , beta ]:=F2[b^_{2, gamma, d1, d2, beta]}

F2b[b , gamma , d1 , d2 , beta ]:=F2[b^2, gamma, d1, d2, beta] F3b[b , gamma , d1 , d2 , beta ]:=F3[b^2, gamma, d1, d2, beta] F3b[b , gamma , d1 , d2 , beta ]:=F3[b^_{2, gamma, d1, d2, beta]}

F3b[b , gamma , d1 , d2 , beta ]:=F3[b^2, gamma, d1, d2, beta]

Expansion in b

EG1b[b , gamma , d1 , d2 , beta ] = Normal[Series[G1b[b, gamma, d1, d2, beta],_{{b, 0, 2}]]} EG1b[b , gamma , d1 , d2 , beta ] = Normal[Series[G1b[b, gamma, d1, d2, beta],{b, 0, 2}]] EG1b[b , gamma , d1 , d2 , beta ] = Normal[Series[G1b[b, gamma, d1, d2, beta],_{{b, 0, 2}]]} 1 ib

2₍_beta2_{L+6d1⇡ 6gamma⇡}₎

3beta2_⇡

EF1b[b , gamma , d1 , d2 , beta ] = Normal[Series[F1b[b, gamma, d1, d2, beta],{b, 0, 2}]] EF1b[b , gamma , d1 , d2 , beta ] = Normal[Series[F1b[b, gamma, d1, d2, beta],_{{b, 0, 2}]]} EF1b[b , gamma , d1 , d2 , beta ] = Normal[Series[F1b[b, gamma, d1, d2, beta],{b, 0, 2}]] b i

2beta2 +_betaid12

igamma

beta2 + 2id1L 2beta

2_L2 iL 2⇡

EF2b[b , gamma , d1 , d2 , beta ] = Normal[Series[F2b[b, gamma, d1, d2, beta],_{{b, 0, 2}]]} EF2b[b , gamma , d1 , d2 , beta ] = Normal[Series[F2b[b, gamma, d1, d2, beta],{b, 0, 2}]] EF2b[b , gamma , d1 , d2 , beta ] = Normal[Series[F2b[b, gamma, d1, d2, beta],_{{b, 0, 2}]]} b _betai + 2ibetaL + b2 1

2beta2 +_betaid12 igamma_beta2 + 2id1L 2beta2L2 iL_2⇡

EF3b[b , gamma , d1 , d2 , beta ] = Normal[Series[F3b[b, gamma, d1, d2, beta],{b, 0, 2}]] EF3b[b , gamma , d1 , d2 , beta ] = Normal[Series[F3b[b, gamma, d1, d2, beta],_{{b, 0, 2}]]} EF3b[b , gamma , d1 , d2 , beta ] = Normal[Series[F3b[b, gamma, d1, d2, beta],{b, 0, 2}]] b2 _2id2L 4iL

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EG1c[c , gamma , d1 , d2 , beta ]:=EG1b[Sqrt[c], gamma, d1, d2, beta] EG1c[c , gamma , d1 , d2 , beta ]:=EG1b[Sqrt[c], gamma, d1, d2, beta] EG1c[c , gamma , d1 , d2 , beta ]:=EG1b[Sqrt[c], gamma, d1, d2, beta] EF1c[c , gamma , d1 , d2 , beta ]:=EF1b[Sqrt[c], gamma, d1, d2, beta] EF1c[c , gamma , d1 , d2 , beta ]:=EF1b[Sqrt[c], gamma, d1, d2, beta] EF1c[c , gamma , d1 , d2 , beta ]:=EF1b[Sqrt[c], gamma, d1, d2, beta] EF2c[c , gamma , d1 , d2 , beta ]:=EF2b[Sqrt[c], gamma, d1, d2, beta] EF2c[c , gamma , d1 , d2 , beta ]:=EF2b[Sqrt[c], gamma, d1, d2, beta] EF2c[c , gamma , d1 , d2 , beta ]:=EF2b[Sqrt[c], gamma, d1, d2, beta] EF3c[c , gamma , d1 , d2 , beta ]:=EF3b[Sqrt[c], gamma, d1, d2, beta] EF3c[c , gamma , d1 , d2 , beta ]:=EF3b[Sqrt[c], gamma, d1, d2, beta] EF3c[c , gamma , d1 , d2 , beta ]:=EF3b[Sqrt[c], gamma, d1, d2, beta]

sol = Solve[_{{EG1c[c, gamma, d1, d2, beta] == 1, EF1c[c, gamma, d1, d2, beta] == 0,} sol = Solve[_{{EG1c[c, gamma, d1, d2, beta] == 1, EF1c[c, gamma, d1, d2, beta] == 0,} sol = Solve[_{{EG1c[c, gamma, d1, d2, beta] == 1, EF1c[c, gamma, d1, d2, beta] == 0,} EF2c[c, gamma, d1, d2, beta] == 0,

EF2c[c, gamma, d1, d2, beta] == 0,

EF2c[c, gamma, d1, d2, beta] == 0,EF3c[c, gamma, d1, d2, beta] == 0EF3c[c, gamma, d1, d2, beta] == 0EF3c[c, gamma, d1, d2, beta] == 0_{}, {gamma, d1, d2, beta}]}_{}, {gamma, d1, d2, beta}]}_{}, {gamma, d1, d2, beta}]} nn gamma_! 5 12⇡, d1! 1 3⇡, d2! 2 3⇡, beta! 1 p 2pL o , n gamma_! 5 12⇡, d1! 1 3⇡, d2! 2 3⇡, beta! 1 p 2pL o

K1[c ]:=Pi/L + (d1/.list)_{⇤ c} (beta/.list)_{⇤ Sqrt[c]} K1[c ]:=Pi/L + (d1/.list)⇤ c (beta/.list)⇤ Sqrt[c] K1[c ]:=Pi/L + (d1/.list)_{⇤ c} (beta/.list)_{⇤ Sqrt[c]} K2[c ]:=Pi/L + (d1/.list)⇤ c + (beta/.list) ⇤ Sqrt[c] K2[c ]:=Pi/L + (d1/.list)_{⇤ c + (beta/.list) ⇤ Sqrt[c]} K2[c ]:=Pi/L + (d1/.list)⇤ c + (beta/.list) ⇤ Sqrt[c] K3[c ]:=2_{⇤ Pi/L + (d2/.list) ⇤ c}

K3[c ]:=2_{⇤ Pi/L + (d2/.list) ⇤ c} K3[c ]:=2_{⇤ Pi/L + (d2/.list) ⇤ c}

And the ground state energy is given by: Eg[c ] = Simplify[K1[c]^_{2 + K2[c]}^_{2 + K3[c]}^_2] Eg[c ] = Simplify[K1[c]^_{2 + K2[c]}^_{2 + K3[c]}^_2] Eg[c ] = Simplify[K1[c]^_{2 + K2[c]}^_{2 + K3[c]}^_2] 5c L + 2c2 3⇡2 +6⇡ 2 L2

A.3 N=4

Calculation of Bethe roots for M=1, N=4. Defining approximation of the Bethe roots lambda[c ]:=Pi/L + gamma⇤ c

lambda[c ]:=Pi/L + gamma_{⇤ c} lambda[c ]:=Pi/L + gamma⇤ c k1[c ]:=Pi/L + d1_{⇤ c} beta_{⇤ Sqrt[c]} k1[c ]:=Pi/L + d1⇤ c beta⇤ Sqrt[c] k1[c ]:=Pi/L + d1_{⇤ c} beta_{⇤ Sqrt[c]} k2[c ]:=Pi/L + d1_{⇤ c + beta ⇤ Sqrt[c]} k2[c ]:=Pi/L + d1⇤ c + beta ⇤ Sqrt[c] k2[c ]:=Pi/L + d1_{⇤ c + beta ⇤ Sqrt[c]} k3[c ]:=2⇤ Pi/L + d2 ⇤ c k3[c ]:=2_{⇤ Pi/L + d2 ⇤ c} k3[c ]:=2⇤ Pi/L + d2 ⇤ c k4[c ]:=3_{⇤ Pi/L + d3 ⇤ c} k4[c ]:=3_{⇤ Pi/L + d3 ⇤ c} k4[c ]:=3_{⇤ Pi/L + d3 ⇤ c}

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The Bethe equations f [x , y , c ]:=Exp[If [x , y , c ]:=Exp[If [x , y , c ]:=Exp[I_{⇤ 2 ⇤ x ⇤ L]}_{⇤ 2 ⇤ x ⇤ L]}⇤ 2 ⇤ x ⇤ L] (x(x(x y + Iy + Iy + I⇤ c/2)/(x_{⇤ c/2)/(x}_{⇤ c/2)/(x} yyy III_{⇤ c/2)}⇤ c/2)_{⇤ c/2)} ⇤(x + y + I ⇤ c/2)/(x + y I_{⇤ c/2)}

⇤(x + y + I ⇤ c/2)/(x + y⇤(x + y + I ⇤ c/2)/(x + y II⇤ c/2)_{⇤ c/2)}

g[x ]:=(y x I_{⇤ c/2)/(y} x + I_{⇤ c/2) ⇤ (y + x} I_{⇤ c/2)/(y + x + I ⇤ c/2)} g[x ]:=(y x I⇤ c/2)/(y x + I⇤ c/2) ⇤ (y + x I⇤ c/2)/(y + x + I ⇤ c/2) g[x ]:=(y x I_{⇤ c/2)/(y} x + I_{⇤ c/2) ⇤ (y + x} I_{⇤ c/2)/(y + x + I ⇤ c/2)} G[x1 , x2 , x3 , x4 , y , c ] := Apply[Times,{g[x1], g[x2], g[x3], g[x4]}]

G[x1 , x2 , x3 , x4 , y , c ] := Apply[Times,_{{g[x1], g[x2], g[x3], g[x4]}]} G[x1 , x2 , x3 , x4 , y , c ] := Apply[Times,{g[x1], g[x2], g[x3], g[x4]}] The Bethe equations expressed in the approximate roots

G1[c , gamma , d1 , d2 , d3 , beta ] := G[k1[c], k2[c], k3[c], k4[c], lambda[c], c] G1[c , gamma , d1 , d2 , d3 , beta ] := G[k1[c], k2[c], k3[c], k4[c], lambda[c], c] G1[c , gamma , d1 , d2 , d3 , beta ] := G[k1[c], k2[c], k3[c], k4[c], lambda[c], c] F1[c , gamma , d1 , d2 , d3 , beta ]:=f [k1[c], lambda[c], c]

F1[c , gamma , d1 , d2 , d3 , beta ]:=f [k1[c], lambda[c], c] F1[c , gamma , d1 , d2 , d3 , beta ]:=f [k1[c], lambda[c], c] F2[c , gamma , d1 , d2 , d3 , beta ]:=f [k2[c], lambda[c], c] F2[c , gamma , d1 , d2 , d3 , beta ]:=f [k2[c], lambda[c], c] F2[c , gamma , d1 , d2 , d3 , beta ]:=f [k2[c], lambda[c], c] F3[c , gamma , d1 , d2 , d3 , beta ]:=f [k3[c], lambda[c], c] F3[c , gamma , d1 , d2 , d3 , beta ]:=f [k3[c], lambda[c], c] F3[c , gamma , d1 , d2 , d3 , beta ]:=f [k3[c], lambda[c], c] F4[c , gamma , d1 , d2 , d3 , beta ]:=f [k4[c], lambda[c], c] F4[c , gamma , d1 , d2 , d3 , beta ]:=f [k4[c], lambda[c], c] F4[c , gamma , d1 , d2 , d3 , beta ]:=f [k4[c], lambda[c], c]

G1b[b , gamma , d1 , d2 , d3 , beta ]:=G1[b^_{2, gamma, d1, d2, d3, beta]}

F1b[b , gamma , d1 , d2 , d3 , beta ]:=F1[b^_{2, gamma, d1, d2, d3, beta]}

F1b[b , gamma , d1 , d2 , d3 , beta ]:=F1[b^2, gamma, d1, d2, d3, beta] F1b[b , gamma , d1 , d2 , d3 , beta ]:=F1[b^_{2, gamma, d1, d2, d3, beta]}

Expansion in b

EG1b[b , gamma , d1 , d2 , d3 , beta ] = Normal[Series[G1b[b, gamma, d1, d2, d3, beta],_{{b, 0, 2}]]} EG1b[b , gamma , d1 , d2 , d3 , beta ] = Normal[Series[G1b[b, gamma, d1, d2, d3, beta],{b, 0, 2}]] EG1b[b , gamma , d1 , d2 , d3 , beta ] = Normal[Series[G1b[b, gamma, d1, d2, d3, beta],_{{b, 0, 2}]]} 1 ib

2₍

beta2L+24d1⇡ 24gamma⇡) 12beta2_⇡

EF1b[b , gamma , d1 , d2 , d3 , beta ] = Normal[Series[F1b[b, gamma, d1, d2, d3, beta],_{{b, 0, 2}]]} EF1b[b , gamma , d1 , d2 , d3 , beta ] = Normal[Series[F1b[b, gamma, d1, d2, d3, beta],_{{b, 0, 2}]]} EF1b[b , gamma , d1 , d2 , d3 , beta ] = Normal[Series[F1b[b, gamma, d1, d2, d3, beta],_{{b, 0, 2}]]} b i

2beta2 +_betaid12

igamma

beta2 + 2id1L 2beta

2_L2 iL 2⇡

EF2b[b , gamma , d1 , d2 , d3 , beta ] = Normal[Series[F2b[b, gamma, d1, d2, d3, beta],{b, 0, 2}]] EF2b[b , gamma , d1 , d2 , d3 , beta ] = Normal[Series[F2b[b, gamma, d1, d2, d3, beta],_{{b, 0, 2}]]} EF2b[b , gamma , d1 , d2 , d3 , beta ] = Normal[Series[F2b[b, gamma, d1, d2, d3, beta],{b, 0, 2}]] b _betai + 2ibetaL + b2 _2beta1 2 +_betaid12

igamma

beta2 + 2id1L 2beta

2

L2 iL_2⇡

EF3b[b , gamma , d1 , d2 , d3 , beta ] = Normal[Series[F3b[b, gamma, d1, d2, d3, beta],_{{b, 0, 2}]]} EF3b[b , gamma , d1 , d2 , d3 , beta ] = Normal[Series[F3b[b, gamma, d1, d2, d3, beta],{b, 0, 2}]] EF3b[b , gamma , d1 , d2 , d3 , beta ] = Normal[Series[F3b[b, gamma, d1, d2, d3, beta],_{{b, 0, 2}]]} b2 _2id2L 4iL

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EF4b[b , gamma , d1 , d2 , d3 , beta ] EF4b[b , gamma , d1 , d2 , d3 , beta ] EF4b[b , gamma , d1 , d2 , d3 , beta ]

= Normal[Series[F4b[b, gamma, d1, d2, d3, beta],{b, 0, 2}]] = Normal[Series[F4b[b, gamma, d1, d2, d3, beta],_{{b, 0, 2}]]} = Normal[Series[F4b[b, gamma, d1, d2, d3, beta],{b, 0, 2}]] b2 _2id3L 3iL

4⇡

(*ReplacebbySqrt[c]*) (*ReplacebbySqrt[c]*) (*ReplacebbySqrt[c]*)

EG1c[c , gamma , d1 , d2 , d3 , beta ]:=EG1b[Sqrt[c], gamma, d1, d2, d3, beta] EG1c[c , gamma , d1 , d2 , d3 , beta ]:=EG1b[Sqrt[c], gamma, d1, d2, d3, beta] EG1c[c , gamma , d1 , d2 , d3 , beta ]:=EG1b[Sqrt[c], gamma, d1, d2, d3, beta] EF1c[c , gamma , d1 , d2 , d3 , beta ]:=EF1b[Sqrt[c], gamma, d1, d2, d3, beta] EF1c[c , gamma , d1 , d2 , d3 , beta ]:=EF1b[Sqrt[c], gamma, d1, d2, d3, beta] EF1c[c , gamma , d1 , d2 , d3 , beta ]:=EF1b[Sqrt[c], gamma, d1, d2, d3, beta] EF2c[c , gamma , d1 , d2 , d3 , beta ]:=EF2b[Sqrt[c], gamma, d1, d2, d3, beta] EF2c[c , gamma , d1 , d2 , d3 , beta ]:=EF2b[Sqrt[c], gamma, d1, d2, d3, beta] EF2c[c , gamma , d1 , d2 , d3 , beta ]:=EF2b[Sqrt[c], gamma, d1, d2, d3, beta] EF3c[c , gamma , d1 , d2 , d3 , beta ]:=EF3b[Sqrt[c], gamma, d1, d2, d3, beta] EF3c[c , gamma , d1 , d2 , d3 , beta ]:=EF3b[Sqrt[c], gamma, d1, d2, d3, beta] EF3c[c , gamma , d1 , d2 , d3 , beta ]:=EF3b[Sqrt[c], gamma, d1, d2, d3, beta] EF4c[c , gamma , d1 , d2 , d3 , beta ]:=EF4b[Sqrt[c], gamma, d1, d2, d3, beta] EF4c[c , gamma , d1 , d2 , d3 , beta ]:=EF4b[Sqrt[c], gamma, d1, d2, d3, beta] EF4c[c , gamma , d1 , d2 , d3 , beta ]:=EF4b[Sqrt[c], gamma, d1, d2, d3, beta] Solving the expanded equations

sol = Solve[{EG1c[c, gamma, d1, d2, d3, beta] == 1, EF1c[c, gamma, d1, d2, d3, beta] == 0, sol = Solve[_{{EG1c[c, gamma, d1, d2, d3, beta] == 1, EF1c[c, gamma, d1, d2, d3, beta] == 0,} sol = Solve[{EG1c[c, gamma, d1, d2, d3, beta] == 1, EF1c[c, gamma, d1, d2, d3, beta] == 0, EF2c[c, gamma, d1, d2, d3, beta] == 0,EF2c[c, gamma, d1, d2, d3, beta] == 0,EF2c[c, gamma, d1, d2, d3, beta] == 0,EF3c[c, gamma, d1, d2, d3, beta] == 0,EF3c[c, gamma, d1, d2, d3, beta] == 0,EF3c[c, gamma, d1, d2, d3, beta] == 0, EF4c[c, gamma, d1, d2, d3, beta] == 0EF4c[c, gamma, d1, d2, d3, beta] == 0EF4c[c, gamma, d1, d2, d3, beta] == 0}, {gamma, d1, d2, d3, beta}]_{}, {gamma, d1, d2, d3, beta}]}}, {gamma, d1, d2, d3, beta}]

nn gamma_! 7 24⇡, d1! 13 48⇡, d2! 2 3⇡, d3! 3 8⇡, beta! 1 p 2pL o , n gamma_! 7 24⇡, d1! 13 48⇡, d2! 2 3⇡, d3! 3 8⇡, beta! 1 p 2pL o

Now the quasi momenta are: Clear[K1, K2, K3, K4] Clear[K1, K2, K3, K4] Clear[K1, K2, K3, K4]

K1[c ]:=Pi/L + (d1/.list)_{⇤ c} (beta/.list)_{⇤ Sqrt[c]} K1[c ]:=Pi/L + (d1/.list)⇤ c (beta/.list)⇤ Sqrt[c] K1[c ]:=Pi/L + (d1/.list)_{⇤ c} (beta/.list)_{⇤ Sqrt[c]} K2[c ]:=Pi/L + (d1/.list)⇤ c + (beta/.list) ⇤ Sqrt[c] K2[c ]:=Pi/L + (d1/.list)_{⇤ c + (beta/.list) ⇤ Sqrt[c]} K2[c ]:=Pi/L + (d1/.list)⇤ c + (beta/.list) ⇤ Sqrt[c] K3[c ]:=2⇤ Pi/L + (d2/.list) ⇤ c K3[c ]:=2_{⇤ Pi/L + (d2/.list) ⇤ c} K3[c ]:=2⇤ Pi/L + (d2/.list) ⇤ c K4[c ]:=3_{⇤ Pi/L + (d3/.list) ⇤ c} K4[c ]:=3⇤ Pi/L + (d3/.list) ⇤ c K4[c ]:=3_{⇤ Pi/L + (d3/.list) ⇤ c}

And the ground state energy is given by:

Eg[c ] = Simplify[K1[c]^_{2 + K2[c]}^_{2 + K3[c]}^_{2 + K4[c]}^_2] Eg[c ] = Simplify[K1[c]^2 + K2[c]^2 + K3[c]^2 + K4[c]^2] Eg[c ] = Simplify[K1[c]^_{2 + K2[c]}^_{2 + K3[c]}^_{2 + K4[c]}^_2] 7c L + 281c2 384⇡2 + 15⇡2 L2

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A.4 N=5

Calculation of Bethe roots for M=1, N=5. Defining approximation of the Bethe roots lambda[c ]:=Pi/L + gamma⇤ c

lambda[c ]:=Pi/L + gamma_{⇤ c} lambda[c ]:=Pi/L + gamma⇤ c k1[c ]:=Pi/L + d1_{⇤ c} beta_{⇤ Sqrt[c]} k1[c ]:=Pi/L + d1⇤ c beta⇤ Sqrt[c] k1[c ]:=Pi/L + d1_{⇤ c} beta_{⇤ Sqrt[c]} k2[c ]:=Pi/L + d1_{⇤ c + beta ⇤ Sqrt[c]} k2[c ]:=Pi/L + d1⇤ c + beta ⇤ Sqrt[c] k2[c ]:=Pi/L + d1_{⇤ c + beta ⇤ Sqrt[c]} k3[c ]:=2⇤ Pi/L + d2 ⇤ c k3[c ]:=2_{⇤ Pi/L + d2 ⇤ c} k3[c ]:=2⇤ Pi/L + d2 ⇤ c k4[c ]:=3_{⇤ Pi/L + d3 ⇤ c} k4[c ]:=3_{⇤ Pi/L + d3 ⇤ c} k4[c ]:=3_{⇤ Pi/L + d3 ⇤ c} k5[c ]:=4_{⇤ Pi/L + d4 ⇤ c} k5[c ]:=4⇤ Pi/L + d4 ⇤ c k5[c ]:=4_{⇤ Pi/L + d4 ⇤ c}

The Bethe equations

f [x , y , c ]:=Exp[I⇤ 2 ⇤ x ⇤ L] (x y + I⇤ c/2)/(x y I⇤ c/2) f [x , y , c ]:=Exp[I_{⇤ 2 ⇤ x ⇤ L]} (x y + I_{⇤ c/2)/(x} y I_{⇤ c/2)} f [x , y , c ]:=Exp[I⇤ 2 ⇤ x ⇤ L] (x y + I⇤ c/2)/(x y I⇤ c/2) ⇤(x + y + I ⇤ c/2)/(x + y I⇤ c/2) ⇤(x + y + I ⇤ c/2)/(x + y_{⇤(x + y + I ⇤ c/2)/(x + y} II_{⇤ c/2)}⇤ c/2)

g[x ]:=(y x I⇤ c/2)/(y x + I⇤ c/2) ⇤ (y + x I⇤ c/2)/(y + x + I ⇤ c/2) g[x ]:=(y x I_{⇤ c/2)/(y} x + I_{⇤ c/2) ⇤ (y + x} I_{⇤ c/2)/(y + x + I ⇤ c/2)} g[x ]:=(y x I⇤ c/2)/(y x + I⇤ c/2) ⇤ (y + x I⇤ c/2)/(y + x + I ⇤ c/2) G[x1 , x2 , x3 , x4 , x5 , y , c ] := Apply[Times,_{{g[x1], g[x2], g[x3], g[x4], g[x5]}]} G[x1 , x2 , x3 , x4 , x5 , y , c ] := Apply[Times,{g[x1], g[x2], g[x3], g[x4], g[x5]}] G[x1 , x2 , x3 , x4 , x5 , y , c ] := Apply[Times,_{{g[x1], g[x2], g[x3], g[x4], g[x5]}]}

The Bethe equations expressed in the approximate roots

G1[c , gamma , d1 , d2 , d3 , d4 , beta ]:=G[k1[c], k2[c], k3[c], k4[c], k5[c], lambda[c], c] G1[c , gamma , d1 , d2 , d3 , d4 , beta ]:=G[k1[c], k2[c], k3[c], k4[c], k5[c], lambda[c], c] G1[c , gamma , d1 , d2 , d3 , d4 , beta ]:=G[k1[c], k2[c], k3[c], k4[c], k5[c], lambda[c], c] F1[c , gamma , d1 , d2 , d3 , d4 , beta ]:=f [k1[c], lambda[c], c]

F1[c , gamma , d1 , d2 , d3 , d4 , beta ]:=f [k1[c], lambda[c], c] F1[c , gamma , d1 , d2 , d3 , d4 , beta ]:=f [k1[c], lambda[c], c] F2[c , gamma , d1 , d2 , d3 , d4 , beta ]:=f [k2[c], lambda[c], c] F2[c , gamma , d1 , d2 , d3 , d4 , beta ]:=f [k2[c], lambda[c], c] F2[c , gamma , d1 , d2 , d3 , d4 , beta ]:=f [k2[c], lambda[c], c] F3[c , gamma , d1 , d2 , d3 , d4 , beta ]:=f [k3[c], lambda[c], c] F3[c , gamma , d1 , d2 , d3 , d4 , beta ]:=f [k3[c], lambda[c], c] F3[c , gamma , d1 , d2 , d3 , d4 , beta ]:=f [k3[c], lambda[c], c] F4[c , gamma , d1 , d2 , d3 , d4 , beta ]:=f [k4[c], lambda[c], c] F4[c , gamma , d1 , d2 , d3 , d4 , beta ]:=f [k4[c], lambda[c], c] F4[c , gamma , d1 , d2 , d3 , d4 , beta ]:=f [k4[c], lambda[c], c] F5[c , gamma , d1 , d2 , d3 , d4 , beta ]:=f [k5[c], lambda[c], c] F5[c , gamma , d1 , d2 , d3 , d4 , beta ]:=f [k5[c], lambda[c], c] F5[c , gamma , d1 , d2 , d3 , d4 , beta ]:=f [k5[c], lambda[c], c]

G1b[b , gamma , d1 , d2 , d3 , d4 , beta ]:=G1[b^_{2, gamma, d1, d2, d3, d4, beta]}

F1b[b , gamma , d1 , d2 , d3 , d4 , beta ]:=F1[b^_{2, gamma, d1, d2, d3, d4, beta]}

F1b[b , gamma , d1 , d2 , d3 , d4 , beta ]:=F1[b^2, gamma, d1, d2, d3, d4, beta] F1b[b , gamma , d1 , d2 , d3 , d4 , beta ]:=F1[b^_{2, gamma, d1, d2, d3, d4, beta]}

F2b[b , gamma , d1 , d2 , d3 , d4 , beta ]:=F2[b^2, gamma, d1, d2, d3, d4, beta] F3b[b , gamma , d1 , d2 , d3 , d4 , beta ]:=F3[b^2, gamma, d1, d2, d3, d4, beta] F3b[b , gamma , d1 , d2 , d3 , d4 , beta ]:=F3[b^_{2, gamma, d1, d2, d3, d4, beta]}

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F5b[b , gamma , d1 , d2 , d3 , d4 , beta ]:=F5[b^2, gamma, d1, d2, d3, d4, beta]

Expansion in b (So expansion in Sqrt[c]) EG1b[b , gamma , d1 , d2 , d3 , d4 , beta ] EG1b[b , gamma , d1 , d2 , d3 , d4 , beta ] EG1b[b , gamma , d1 , d2 , d3 , d4 , beta ]

= Normal[Series[G1b[b, gamma, d1, d2, d3, d4, beta],_{{b, 0, 2}]]} = Normal[Series[G1b[b, gamma, d1, d2, d3, d4, beta],{b, 0, 2}]] = Normal[Series[G1b[b, gamma, d1, d2, d3, d4, beta],_{{b, 0, 2}]]} 1 + ib

2₍_beta2_{L 40d1⇡+40gamma⇡}₎

20beta2_⇡

EF1b[b , gamma , d1 , d2 , d3 , d4 , beta ] EF1b[b , gamma , d1 , d2 , d3 , d4 , beta ] EF1b[b , gamma , d1 , d2 , d3 , d4 , beta ]

= Normal[Series[F1b[b, gamma, d1, d2, d3, d4, beta],{b, 0, 2}]] = Normal[Series[F1b[b, gamma, d1, d2, d3, d4, beta],_{{b, 0, 2}]]} = Normal[Series[F1b[b, gamma, d1, d2, d3, d4, beta],{b, 0, 2}]] b i

2beta2 +_betaid12

igamma

beta2 + 2id1L 2beta

2_L2 iL 2⇡

= Normal[Series[F2b[b, gamma, d1, d2, d3, d4, beta],_{{b, 0, 2}]]} = Normal[Series[F2b[b, gamma, d1, d2, d3, d4, beta],{b, 0, 2}]] = Normal[Series[F2b[b, gamma, d1, d2, d3, d4, beta],_{{b, 0, 2}]]} b _betai + 2ibetaL + b2 1

2beta2 +_betaid12 igamma_beta2 + 2id1L 2beta2L2 iL_2⇡

= Normal[Series[F3b[b, gamma, d1, d2, d3, d4, beta],{b, 0, 2}]] = Normal[Series[F3b[b, gamma, d1, d2, d3, d4, beta],_{{b, 0, 2}]]} = Normal[Series[F3b[b, gamma, d1, d2, d3, d4, beta],{b, 0, 2}]] b2 _2id2L 4iL

3⇡

= Normal[Series[F4b[b, gamma, d1, d2, d3, d4, beta],_{{b, 0, 2}]]} = Normal[Series[F4b[b, gamma, d1, d2, d3, d4, beta],{b, 0, 2}]] = Normal[Series[F4b[b, gamma, d1, d2, d3, d4, beta],_{{b, 0, 2}]]} b2 _2id3L 3iL

4⇡

= Normal[Series[F5b[b, gamma, d1, d2, d3, d4, beta],{b, 0, 2}]] = Normal[Series[F5b[b, gamma, d1, d2, d3, d4, beta],_{{b, 0, 2}]]} = Normal[Series[F5b[b, gamma, d1, d2, d3, d4, beta],{b, 0, 2}]] b2 _2id4L 8iL

15⇡

EG1c[c , gamma , d1 , d2 , d3 , d4 , beta ]:=EG1b[Sqrt[c], gamma, d1, d2, d3, d4, beta] EG1c[c , gamma , d1 , d2 , d3 , d4 , beta ]:=EG1b[Sqrt[c], gamma, d1, d2, d3, d4, beta] EG1c[c , gamma , d1 , d2 , d3 , d4 , beta ]:=EG1b[Sqrt[c], gamma, d1, d2, d3, d4, beta] EF1c[c , gamma , d1 , d2 , d3 , d4 , beta ]:=EF1b[Sqrt[c], gamma, d1, d2, d3, d4, beta] EF1c[c , gamma , d1 , d2 , d3 , d4 , beta ]:=EF1b[Sqrt[c], gamma, d1, d2, d3, d4, beta] EF1c[c , gamma , d1 , d2 , d3 , d4 , beta ]:=EF1b[Sqrt[c], gamma, d1, d2, d3, d4, beta] EF2c[c , gamma , d1 , d2 , d3 , d4 , beta ]:=EF2b[Sqrt[c], gamma, d1, d2, d3, d4, beta] EF2c[c , gamma , d1 , d2 , d3 , d4 , beta ]:=EF2b[Sqrt[c], gamma, d1, d2, d3, d4, beta] EF2c[c , gamma , d1 , d2 , d3 , d4 , beta ]:=EF2b[Sqrt[c], gamma, d1, d2, d3, d4, beta] EF3c[c , gamma , d1 , d2 , d3 , d4 , beta ]:=EF3b[Sqrt[c], gamma, d1, d2, d3, d4, beta] EF3c[c , gamma , d1 , d2 , d3 , d4 , beta ]:=EF3b[Sqrt[c], gamma, d1, d2, d3, d4, beta] EF3c[c , gamma , d1 , d2 , d3 , d4 , beta ]:=EF3b[Sqrt[c], gamma, d1, d2, d3, d4, beta] EF4c[c , gamma , d1 , d2 , d3 , d4 , beta ]:=EF4b[Sqrt[c], gamma, d1, d2, d3, d4, beta] EF4c[c , gamma , d1 , d2 , d3 , d4 , beta ]:=EF4b[Sqrt[c], gamma, d1, d2, d3, d4, beta] EF4c[c , gamma , d1 , d2 , d3 , d4 , beta ]:=EF4b[Sqrt[c], gamma, d1, d2, d3, d4, beta] EF5c[c , gamma , d1 , d2 , d3 , d4 , beta ]:=EF5b[Sqrt[c], gamma, d1, d2, d3, d4, beta] EF5c[c , gamma , d1 , d2 , d3 , d4 , beta ]:=EF5b[Sqrt[c], gamma, d1, d2, d3, d4, beta] EF5c[c , gamma , d1 , d2 , d3 , d4 , beta ]:=EF5b[Sqrt[c], gamma, d1, d2, d3, d4, beta]

(30)

sol = Solve[{EG1c[c, gamma, d1, d2, d3, d4, beta] == 1, EF1c[c, gamma, d1, d2, d3, d4, beta] == 0, sol = Solve[_{{EG1c[c, gamma, d1, d2, d3, d4, beta] == 1, EF1c[c, gamma, d1, d2, d3, d4, beta] == 0,} sol = Solve[{EG1c[c, gamma, d1, d2, d3, d4, beta] == 1, EF1c[c, gamma, d1, d2, d3, d4, beta] == 0, EF2c[c, gamma, d1, d2, d3, d4, beta] == 0,EF2c[c, gamma, d1, d2, d3, d4, beta] == 0,EF2c[c, gamma, d1, d2, d3, d4, beta] == 0,EF3c[c, gamma, d1, d2, d3, d4, beta] == 0,EF3c[c, gamma, d1, d2, d3, d4, beta] == 0,EF3c[c, gamma, d1, d2, d3, d4, beta] == 0, EF4c[c, gamma, d1, d2, d3, d4, beta] == 0, EF5c[c, gamma, d1, d2, d3, d4, beta] == 0EF4c[c, gamma, d1, d2, d3, d4, beta] == 0, EF5c[c, gamma, d1, d2, d3, d4, beta] == 0EF4c[c, gamma, d1, d2, d3, d4, beta] == 0, EF5c[c, gamma, d1, d2, d3, d4, beta] == 0},_},}, {gamma, d1, d2, d3, d4, beta}]{gamma, d1, d2, d3, d4, beta}]{gamma, d1, d2, d3, d4, beta}]

nn gamma_! 9 40⇡, d1! 19 80⇡, d2! 2 3⇡, d3! 3 8⇡, d4! 4 15⇡, beta! 1 p 2pL o , n gamma_! 9 40⇡, d1! 19 80⇡, d2! 2 3⇡, d3! 3 8⇡, d4! 4 15⇡, beta! 1 p 2pL o

K1:=Pi/L + (d1/.list)_{⇤ c} (beta/.list)_{⇤ Sqrt[c]} K1:=Pi/L + (d1/.list)⇤ c (beta/.list)⇤ Sqrt[c] K1:=Pi/L + (d1/.list)_{⇤ c} (beta/.list)_{⇤ Sqrt[c]} K2:=Pi/L + (d1/.list)⇤ c + (beta/.list) ⇤ Sqrt[c] K2:=Pi/L + (d1/.list)_{⇤ c + (beta/.list) ⇤ Sqrt[c]} K2:=Pi/L + (d1/.list)⇤ c + (beta/.list) ⇤ Sqrt[c] K3:=2⇤ Pi/L + (d2/.list) ⇤ c K3:=2_{⇤ Pi/L + (d2/.list) ⇤ c} K3:=2⇤ Pi/L + (d2/.list) ⇤ c K4:=3_{⇤ Pi/L + (d3/.list) ⇤ c} K4:=3⇤ Pi/L + (d3/.list) ⇤ c K4:=3_{⇤ Pi/L + (d3/.list) ⇤ c} K5:=4_{⇤ Pi/L + (d4/.list) ⇤ c} K5:=4⇤ Pi/L + (d4/.list) ⇤ c K5:=4_{⇤ Pi/L + (d4/.list) ⇤ c}

And the ground state energy is given by:

Eg = Simplify[K1^2 + K2^2 + K3^2 + K4^2 + K5^2] Eg = Simplify[K1^_{2 + K2}^_{2 + K3}^_{2 + K4}^_{2 + K5}^_2] Eg = Simplify[K1^2 + K2^2 + K3^2 + K4^2 + K5^2] 9c L + 22147c2 28800⇡2 +31⇡ 2 L2

A.5 N=6

Calculation of Bethe roots for M=1, N=6 in the weak interaction limit (c<1/L) Defining approximation of the Bethe roots

lambda[c ]:=Pi/L + gamma⇤ c lambda[c ]:=Pi/L + gamma_{⇤ c} lambda[c ]:=Pi/L + gamma⇤ c k1[c ]:=Pi/L + d1⇤ c beta⇤ Sqrt[c] k1[c ]:=Pi/L + d1_{⇤ c} beta_{⇤ Sqrt[c]} k1[c ]:=Pi/L + d1⇤ c beta⇤ Sqrt[c] k2[c ]:=Pi/L + d1_{⇤ c + beta ⇤ Sqrt[c]} k2[c ]:=Pi/L + d1_{⇤ c + beta ⇤ Sqrt[c]} k2[c ]:=Pi/L + d1_{⇤ c + beta ⇤ Sqrt[c]} k3[c ]:=2_{⇤ Pi/L + d2 ⇤ c} k3[c ]:=2⇤ Pi/L + d2 ⇤ c k3[c ]:=2_{⇤ Pi/L + d2 ⇤ c} k4[c ]:=3⇤ Pi/L + d3 ⇤ c k4[c ]:=3_{⇤ Pi/L + d3 ⇤ c} k4[c ]:=3⇤ Pi/L + d3 ⇤ c

FROM FEW TO MANY An analytical study on the crossover of a one-dimensional Fermi gas, from a few-particle system to the many-body limit

Report Bachelor Project Physics and Astronomy

FROM FEW TO MANY

An analytical study on the crossover of a one-dimensional Fermi gas, from a

few-particle system to the many-body limit

by

David Gunneweg

6361862

July 3, 2014

15 ECTS

April 2013 - June 2013

Supervisor:

Prof.dr. C.J.M. Schoutens

Second assessor:

dhr. dr. P.R.Corboz

Institute for Theoretical Physics

University of Amsterdam

Populaire samenvatting

Contents

1

Introduction

2

A Bethe Ansatz solution for few-particle systems

2.1

The Bethe equations

2.2

Ground state energies for N

= 1 configuration

2.2.1

Weak interaction limit

2.2.2

Strong interaction limit

Number of

Ground state energy

majority particles

2

E

=

+

c +

c

3

E

=

+

c +

c

4

E

=

+

c +

c

5

E

=

+

c +

c

6

E

=

+

c +

c

2.3

Interaction energies

2.3.1

Weak interaction limit

2.3.2

Strong interaction limit

3

Crossover from few to many particles

Minimum values ratios

Weak interaction limit

Strong interaction limit

0 <

< 0.1

100



_