Different phases of interacting fermions in low dimensional lattices with synthetic gauge fields

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Anno accademico 2017/2018

Different phases of interacting

fermions in low dimensional

lattices with synthetic gauge field

Supervisor: Davide Rossini

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Introduction

The aim of my master thesis is the study of a strongly anisotropic ladder system with interacting fermions through numerical variational methods. The system considered in our work consists in a quasi-1D lattice filled with spinless fermions, which is very long on the x direction, the real dimension, but with only few sites on the y direction, the synthetic dimension.

The motivation for studying this model arises from various ultacold-atom experiments, in which scientists have been able to realize through magneto-optical confining techniques these kind of lattices pierced by a synthetic gauge field [1, 2, 3].

Since our lattice is not strictly unidimensional, it is possible to introduce a synthetic gauge field as we will show in chapter 2. The presence of a synthetic magnetic flux allows us to study geometric frustration effects de-pending both on the magnitude of the field and on the density of particles in the chain. These effects cause interesting outcomes such as fermionic lo-calization [4].

Strongly correlated systems are extremely hard, if not impossible, to solve analytically. Moreover, in our case, perturbation techniques typically fail, because the ratio between the magnitude of the interacting terms and that of the hopping term is r ≥ 1. In fact, if there were no interactions, the model would be exactly solvable by mapping it in a free fermions system. However an interaction term is necessary to unveil a much richer physics, which can encompass peculiar phenomena including fermionic crystallization at frac-tional density of particles, as observed in chapters 4 and 5.

So our purpose is to understand the properties of this model in the strongly interacting regime via density matrix renormalization group (DMRG), that will be discussed in chapter 3, simulations if an interaction Hubbard term

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is added to the Hamiltonian. This will be achieved by finding a transition between a crystalline and a liquid phase [4] in dependence of the Hubbard coupling. The analysis is performed with a matrix product states (MPS) algorithm [5], which is known to be the most accurate method to deal with 1D quantum chains. MPS algorithm is a more flexible yet equally powerful transposition of DMRG, a concise overview on the MPS formalism will be given in chapter 3. This approach enables to easily measure quantities like local magnetization, local density, correlation functions and bipartite von Neumann entropy.

In order to characterize the various phases we will introduce various quan-tities, that can be found by our algorithm, in chapter 2.

A way to identify the phase of our system is to measure the bipartite von Neumann entropy, or entanglement entropy: if the system is in a pure state and it is split in two parts, the entanglement entropy is defined as the von Neumann entropy of one of the two subsystems. This quantity allows to discriminate a gapped system, in which entanglement entropy is a flat func-tion of the site, from a gapless one, where the entanglement entropy has a bell-shape [6], as we will see in chapter 2.

A theoretical treatment of this problem can be found in [4, 1], but in none of the papers currently present in literature there is a study on the critical transition for the 2-leg model, nor it is well understood the connection be-tween the magnetic flux and the crystalline phase.

There are several motivations that guided our work. First the experimental research of this area proved that these ultracold-atom techniques can simu-late accurately the physics of the quantum Hall effect with arbitrary value of the magnetic field [2]. Moreover very recently systems like our 3-leg have been experimentally reproduced [7].

In the models considered the geometric frustration leads to peculiar prop-erties of the matter that can only be studied via numerical techniques. Our efforts have been mainly driven by the idea of characterizing precisely for the first time in literature the phases of this system and their relation in the strong coupling regime where only numerical simulations are be able to make predictions, in order to both understand better the new and rich phenomenology that has been discovering in the last years, and to propose new ideas to experimentalists in this exciting field.

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1.1 Quantum Phase Transitions

In the following we want to briefly introduce to the reader the principal facts of the Quantum Phase Transitions (QPT), they are a phenomenon that dif-fers strongly from the classical ones and that has been studied in depth only recently. Our discussion can be found with much greater details in [8]. Let us take into account an Hamiltonian ˆH(g), whose degrees of freedom reside on the sites of a lattice, and which varies as a function of a dimen-sionless coupling g. If we follow the evolution of the ground state energy, for the case of a finite lattice, this ground state energy will generally be an analytic function of g, in such kind of system no QPT occurs. The main possibility of an exception comes from the case when g couples only to a conserved quantity, i.e. H(g) = ˆˆ H0 + g ˆH1, where ˆH0 and ˆH1 commute. This means that ˆH0 and ˆH1 can be simultaneously diagonalized and so the eigenfunctions are independent of g even though the eigenvalues vary with g; then there can be a level-crossing where an excited level reaches the same energy of the ground state at g = gc, and then it becomes the new ground state, creating a point of nonanalyticity of the ground state energy as a function of g. Avoided level-crossing occurs when the energy of the excited state between the ground state and an excited state come close but they do not exchange; the distance between the excited state and the ground state in a finite lattice may become thinner as the lattice size increases, leading to a nonanalyticity at g = gc in the thermodynamic limit. We shall identify any point of nonanalyticity in the ground state energy of the infinite lattice system as a quantum phase transition: The nonanalyticity could be either the limiting case of an avoided level-crossing or an actual level-crossing. The phase transition is usually accompanied by a qualitative change in the na-ture of the correlations in the ground state.

An important class of phase transitions are the continuous transition, where the energy of the ground state is a continuous function of g. These are transitions at which the characteristic energy scale of fluctuations above the ground state vanishes as g approaches gc. Let the energy ∆ represent a scale characterizing some significant spectral density of fluctuations at zero temperature for g 6= gc . Thus ∆ could be the energy of the lowest excita-tion above the ground state, if this is nonzero, i.e. there is an energy gap, or if there are excitations at arbitrarily low energies in the infinite lattice limit, i.e. the energy spectrum is gapless, ∆ is the energy scale at which the spectrum of the Hamiltonian changes dramatically. In most cases, we will

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find that as g approaches gc, ∆ vanishes as

∆ ∼ J |g − gc|zν. (1.1)

Here J is the energy scale of a characteristic microscopic coupling, and zν is a critical exponent. The value of zν is usually universal, that is, it is in-dependent of most of the microscopic details of the Hamiltonian ˆH(g). The behavior (1.1) holds both for g > gc and for g < gc with the same value of the exponent zν, but with different nonuniversal constants of proportional-ity.

In addition to a vanishing energy scale, continuous quantum phase transi-tions invariably have a diverging characteristic length scale ξ . This could be the length scale determining the exponential decay of equal-time corre-lations in the ground state or the length scale at which some characteristic crossover occurs to the correlations at the longest distances. This length diverges as

ξ−1∼ Λ|g − gc|ν (1.2)

where Λ is an inverse length scale of order the lattice spacing and ν is a critical exponent. The characteristic energy scale vanishes as the zth power of the characteristic inverse length scale

∆ ∼ ξ−z (1.3)

It is important to note that the discussion above refers to singularities in the ground state of the system. So strictly speaking, quantum phase tran-sitions occur only at zero temperature, T = 0. Because all experiments are necessarily at some nonzero, though possibly very small, temperature, it is crucial to understand how the singularity at T = 0 affect the observables quantities at T > 0. It turns out that working outward from the quantum critical point at g = gc and T = 0 is a powerful way of understanding and describing the thermodynamic and dynamic properties of numerous systems over a broad range of values of |g − gc| and T . Indeed, it is not even neces-sary that, for the system of interest, the microscopic couplings reach a value such that g = gc: it can still be very useful to argue that there is a quantum critical point at a physically inaccessible coupling g = gc and to develop a description in the deviation |g − gc|.

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1.1.1 A paradigmatic example: the quantum Ising chain

The most straightforward yet very instructive example of QPT is the Quan-tum Ising Model, the Hamiltonian defined on a spin 1₂ chain is

ˆ HI = −J X i ˆ σ_izσˆ_i+1z − J gX i ˆ σ_ix. (1.4)

In the usual notation J is an exchange constant, which sets the microscopic scale of energy, and g is a dimensionless coupling constant, whose variation determines the phase transition. The spin operators are the Pauli matrices acting locally on a single site, the Pauli matrices are

ˆ σx=0 1 1 0 ˆ σy =0 −i i 0 ˆ σz =1 0 0 −1 . (1.5)

This model can be solved exactly thanks to the application of a Jordan-Wigner transformation and subsequently of Bogoliubov-Valatin transforma-tion, through this method the Hamiltonian can be mapped to a free massless spinless fermions model [8], that is easily tractable. It can be shown that there is a QPT at gc= 1, for g = 0 the ground state is either

|↑i =O i

|↑i_i or |↓i =O

i

|↓i_i, (1.6)

here |↑i_i and |↓i_i are the eigenvectors of ˆσz_i, the former has 1 as eigenvalue, the latter has −1.

For 0 < g < gcthe ground state is not so simple but qualitatively the system is in a ferromagnetic phase, where can be observed a nonzero value of the magnetization along the direction of ˆσz.

On the other hand for g → ∞ the ground state is

|→i =O

i

|→i_i (1.7)

where

|→i_i = |↑ii√+ |↓ii 2 |←i_i = |↑ii√− |↓ii

2

(1.8)

are the eigenstates of ˆσx_i with eigenvalues ±1 respectively.

For g > gc the system undergoes a paramagnetic phase with a zero global magnetization.

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1.2 Quantum phases

In the last years a lot of new models have been studied in this framework, and their interest is not limited to theoretical reasoning, an example to that are the Hubbard and the Bose-Hubbard [11, 12] models, the Hamiltonian of the first one in one dimension can be written as [13, 14]

ˆ H = −tX j,s (ˆc†_j,scˆj+1,s+ h.c.) + U X j ˆ nj,↑nˆj,↓, (1.9)

where j = 1, .., L is the site index, s =↑, ↓ is the spin index, ˆc†_j,s is the local fermionic creation operator, ˆcj,s is the local fermionic annihilation operator and ˆnj,s= ˆc†j,scˆj,s is the local number operator. The first term is commonly the hopping term to nearest neighbor sites, while the second term in taken into account to describe the Coulomb interaction between electrons. This type of system cannot be mapped to free models and they cannot be easily diagonalized, unlike the quantum Ising chain; while the Bose-Hubbard sys-tem is not exactly solvable even in one dimension. Their study is of crucial interest for many fields, as an example the 3D Hubbard model can predict the metal-insulator transition, the so called Mott transition, that is fun-damental in solid-state physics, this transition is nothing but a QPT that occurs to a certain value of U/t and makes the properties of the considered metal change drastically.

An entire book on this argument cannot cover all the possibilities discovered so far, even so in our introduction we tried to show how this relatively new field has led to new groundbreaking insights.

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The system

2.1 Experimental setup

Recently intense effort has been devoted to the creation of gauge fields for electrically neutral atoms, synthetic magnetic fields have been engineered in periodic optical lattices.

To achieve this goal experimentalists have exploited ultracold-atoms tech-niques [15]: they trap a chain of neutral atoms in a 1D optical lattice in the x direction called real dimension through counter-propagating laser beams, and then they use the hyperfine structure spin manifold of the atom to real-ize the y direction called synthetic dimension, so practically the spin levels of the internal degrees of freedom of the atom become the sites of a fictitious 2D lattice as shown in figure 2.1.

Moreover the synthetic gauge field is crafted by optical Raman coupling be-tween the spin level of the atom [16], this imprints a site dependent Peierls phase factor onto the wave function on the synthetic dimension [2, 3], which is equivalent to the effect of a piercing magnetic field on the lattice.

Since this method permits to obtain large magnetic fluxes, that are impos-sible in solid state physics experiments [2], this framework is very useful to investigate various interesting phenomena such as Quantum Hall Effect and chiral edge state [2, 3].

Experiments have been conducted with alkaline [2] and alkaline-earth(-like) atoms [3] all in tight binding approximation.

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Figure 2.1: A pictorial representation of the experimental setup, the spin manifold of the neutral atoms in the optical lattice constitute the synthetic dimension [1].

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2.2 Theoretical model

We can theoretically model the previously described physical system as a quasi-1D lattice with a ladder geometry: it is very long on the real dimension, infinite in the thermodynamic limit, but with only few sites on the synthetic dimension: the Hamiltonian can be written as:

ˆ H = ˆH0+ ˆH1+ ˆHint (2.1) ˆ H0= −t X j,m (ˆc†_j,mˆcj+1,m+ h.c.) (2.2) ˆ H1 = Ω X j M −1 X m=0 (e−2πiϕjˆc†_j,mcˆj,m+1+h.c)+Ω0 X j (e2πiϕjˆc†_j,0ˆcj,M+h.c.) (2.3) ˆ Hint= U X j,m<m0 ˆ nj,mnˆj,m0+ V X j,m,m0 ˆ nj,mnˆj+1,m0 (2.4) where j = 1, .., L m = 0, .., M. (2.5)

Here j is the index of the real dimension, while m is the index of the synthetic dimension ˆc†_j,m and ˆcj,m are the fermionic creation and annihilation local operators respectively, acting on the jth site in the x direction and mth site on the y direction, ϕ is the parameter that defines the piercing magnetic field applied on the lattice and ˆnj,m = ˆc†j,mˆcj,mis the local number operator or local density operator.

These relations hold:

{ˆcj,m, ˆcj0_,m0} = {ˆc†

j,m, ˆc †

j0_,m0} = 0 (2.6)

{ˆcj,m, ˆc†_j0_,m0} = δj,j0δ_m,m0 (2.7)

∀j, j0, m, m0, where {,} denotes the anticommutator.

We immediately notice that experimentally j corresponds to the site of the optical lattice, and L is its size, while m depends on the nuclear spin of the used atomic species, so, if the spin is I, the spin manifold has dimension 2I + 1 and M = 2I.

ˆ

H0 represents the hopping term on the real dimension, physically it is the tunneling of the neutral atom between the sites of the optical lattice. The Hamiltonian written above is SU(N)-invariant where N is the number of possible legs in the model [17, 18].

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CHAPTER 2. THE SYSTEM

1D atomic gas

Raman

Lattice

(c) Concept

Figure 2.2: The figure, found in [2], illustrates the action of the synthetic magnetic flux on the quasi 1D lattice, here γ = 2πϕ.

ˆ

H1 is the hopping term that describes the hopping on the synthetic di-mension, the nontrivial complex site-dependent phase, described in 2.2, is the Peierls phase which mimics the action of a magnetic flux of amplitude Φ = ϕΦ0 in the tight binding approximation with Φ0 = h/e where h is the Planck constant and e is the elementary charge. Experimentally it is real-ized with the Raman coupling between the levels of the spin manifold of the atom. The difference between Ω and Ω0 arises from the fact that experimen-tally they are couplings realized through different beams so, in principle, they could not have the same value. We deal with the case t = Ω = Ω0. Finally, the first term of ˆHint is the Hubbard term responsible for the on-site repulsion, which is caused by the Coulomb repulsion; while the second term introduces a nearest neighbor repulsive interaction, this is an extension of the model, that can be crated in the laboratory via Rydberg atoms or molecules [19], these experimental setups are currently really hard to realize, but hopefully in few years this kind of interaction terms will be engineered. We are interested in the properties of the system with Open Boundary Con-ditions (OBC) in the real dimension and with Periodic Boundary ConCon-ditions (PBC) in the synthetic dimension.

Our aim is to stay the behavior of spinless fermions embedded in this syn-thetic manifold, the peculiar hopping term on the synsyn-thetic dimension mim-ics the effect of a piercing magnetic field in fact if the fermion travels through a plaquette its wave function acquires a complex phase e±2πiϕ, here the sign depends on the direction of the closed path.

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2.2.1 Tight binding

The tight binding approximation is a method used to describe the behavior of electrons inside a lattice, its main application is the calculation of the electronic band structure.

The main ansatz is that the electron wave function is a sum of wave func-tions localized near the lattice site, i.e. the Wannier funcfunc-tions. The hopping term allowed for these function is only a nearest neighbor term.

The sum of Wannier function is in the form described by the Bloch’s theo-rem, and it is called Bloch sum.

The coupling constant in the hopping term is nothing but the hopping in-tegral between two near Wannier functions.

A much deeper discussion can be found in [20].

2.2.2 Peierls phase factor

In this part we will show that a complex phase in ˆH1 is equivalent to the action of a gauge field.

The general Hamiltonian of an electron in a crystalline lattice is given by H = |p|_2m2 + U (r) where U (r) is the periodic potential of the crystal, in the tight binding approximation we know we can write the electronic wave function as a Bloch sum of Wannier functions φR(r)

Ψk(r) = 1 √ N X R eik·RφR(r) (2.8)

where N is the total number of lattice sites. The hopping integral is tRR’= −

Z

φ∗_R(r) ˆHφR’(r)dr (2.9)

if a magnetic field is applied the Hamiltonian becomes ˆ

˜

H(t) = |ˆp − q ˆA(r, t)| 2

2m + ˆU (r) (2.10)

where ˆA(r, t) is the vector potential operator and q is the charge of the particle. In order to amend this difference we can consider a change in the Wannier functions ˜ φR= ei q ~ Rr RA(r’,t)dr’_φ_R _(2.11) so that ˜ Ψk(r) = 1 √ N X R eik·Rφ˜R(r) (2.12)

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We see that these corrected wave functions are eigenstate of the new Hamil-tonian, using ˆp = −i~∇

ˆ ˜ H(t) ˜φk(r) =ei q ~ Rr RA(r’,t)dr’ |ˆp − q ˆA(r, t) + q∇GR’(r, t)| 2 2m + ˆU (r) φk(r) = =eiq~ Rr RA(r’,t)dr’Hφˆ _k(r) (2.13) Where GR(r, t) = Rr

RA(r’, t)dr’ and ∇GR(r, t) = ˆA(r, t) holds for the tight binding condition and if the magnetic field is invariant at the scale of the crystal lattice. Then finally, assuming that the vector potential changes slowly, we can compute the new hopping integral

˜ tRR’ = − Z ˜ φ∗_R(r)H ˜ˆ˜φR’(r)dr = = − eiq~ RR R’A(r’,t)dr’ Z φ∗_R(r)eiq~ΦR,r,R’Hφˆ _R’(r)dr (2.14)

where we have defined ΦR,r,R’= H

R→r→R’A(r, t) · dr but, since we assume A(r, t) approximately uniform at the lattice scale, we obtain ΦR,r,R’ ≈ 0 so that ˜tRR’ ≈ ei

q ~

RR

R’A(r’,t)dr’t_RR’ [21, 22, 23, 24] In our system the vector

potential is both uniform on the lattice scale and stationary in time, so all the above calculation is valid.

2.2.3 Fermionic liquid

The liquid phase is common in fermionic systems, usually interacting fermions do not possess any particular spatial structure, important exceptions can be found in the realm of the quantum Hall effect. The wave function in this phase is delocalized and appears to be energetically gapless, thus the phase is disordered.

In our system it is the dominant phase for the majority of cases that we investigated.

2.2.4 Wigner crystal

The ordered phase, stabilized by large enough values of the interaction, is a realization of the Wigner crystal.

Typically a Wigner crystal appears in fermionic system, when the density becomes low and the Coulomb interaction begins to dominate with respect of the mean kinetic energy, so the ground state has a crystalline structure. A detailed discussion on this state of matter can be found in [25].

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2.3 Relevant quantities

Below we define the quantities that are relevant to our investigation, the filling number is a parameter of the model, while the following are our mea-surable observables.

2.3.1 Filling number

A fundamental quantity to consider is the filling number

ν = n

(M + 1)ϕ. (2.15)

Here n, regarded as density of particles, is the ratio between the number of particles and the number of sites in the real dimension, ν represents the ratio between the number of particles and the number of magnetic flux quanta piercing the synthetic lattice [4].

In our system it is possible to have crystalline a phase stabilized by certain values of U and V even if ν is a rational non integer number.

2.3.2 Density

An important used quantity to characterize the phase of a physical system is the local number operator

ˆ Nj = X m ˆ c†_jmˆcjm, (2.16)

it clearly counts the number of particles in the real site j summing all the fermions on the synthetic dimension. This operator gives us clear infor-mation on the charge density distribution of the lattice, so if we see a flat shape in function of j we observe a superfluid phase where all the fermions are delocalized, while a periodic behavior spots a structurally ordered phase.

2.3.3 Magnetization

We can introduce in our framework the magnetizations operators through the definition ˆ M_jα ≡ X m,m0 ˆ c†_j,m(Tα)mm0cˆ_j,m0 (2.17)

where Tα is the (2I + 1)-dimensional representation of the SU(2) spin oper-ator (α = x, y, z) [4].

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2.3.4 Charge gap

If we want to find the critical value of U , which can discriminates between two different phases, we first have to find an order parameter.

The compressibility is κ = ∂n_∂µ where n is the density of particles and µ is the chemical potential, we call the ground state energy in the system with N fermions at fixed size EN, EN +1 is the ground state energy with one more fermion and EN −1 is that without a particle, i.e. with the addition of an hole, the chemical potentials µp and µh as

µp = EN +1− EN µh = EN − EN −1. (2.18) The gapless phase is superfluid thus compressible κ = ∞ so µp = µh, while in the gapped phase µp 6= µh [11].

Our discussion implies that

∆E = EN +1+ EN −1− 2EN = µp− µh (2.19) is zero in the liquid phase, plus terms of order of one over the lattice size, while it is positive in the crystalline phase.

2.4 Quantum correlations

We want to introduce some basic notion on quantum correlations. They arise solely in quantum mechanics and give birth to amazing new effects. We will use in the following concepts deriving form quantum entanglement and quantum information theory [26] such as the bipartite von Neumann entropy, in order to characterize both the efficiency of our algorithm and the presence of different phases of our system.

2.4.1 Entanglement

In quantum mechanics a non classical phenomenon appears: the quantum entanglement. An entangled system is defined to be one whose quantum state cannot be factored as a product of states of its local constituents; that is to say, they are not individual particles but are an inseparable whole. In an entangled state, one constituent cannot be fully described without con-sidering the others. Note that the state of a composite system is always expressible as a sum, or superposition, of products of states of local con-stituents; it is entangled if this sum necessarily has more than one term. A typical example of this particular effect is the combined system of two

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identical particles with spin 1₂, if we prepare the state with zero total spin, the only possible wave function written as a combination of the two particles is

|ψi_Bell= |↑i1⊗ |↓i2√− |↓i1⊗ |↑i2

2 (2.20)

except for a global phase, so if we separate the particles and the measure the spin of the first particle and we find that it is the up state, the second particle must be in the down state and, conversely, if we measure a down state on the first spin, the second particle must be in the up state.

This interesting aspect has risen a lot of controversies in time, as pointed out by Einstein; Podolski and Rosen in their famous paradox, it is a purely non local effect, the non locality of the effect has been proven by the experimental violation of the Bell inequalities [27].

2.4.2 Bipartite von Neumann entropy

In quantum mechanics the von Neumann entropy is defined as

S = −tr[ ˆρ log ( ˆρ)] (2.21)

where ˆρ is the density matrix of the considered quantum mechanical system. von Neumann Entropy represents the departure from a pure state, in fact if the density matrix is a projector, i.e. ˆρ2 = ˆρ, S vanishes, while it is maximal if the system is in a maximally mixed state, where ˆρ = P

ipi|ψii hψi| and all the pi are equal.

If we split our system in two subsystems called A and B we can define the bipartite von Neumann entropy, or entanglement entropy: if ˆρAB is the density matrix of the whole system we can define ˆρA = trB[ ˆρAB] and conversely ˆρB = trA[ρAB], where we used the definition of partial trace on B and A respectively, thus we can define SA = −tr[ ˆρAlog ( ˆρA)] and SB = −tr[ ˆρBlog ( ˆρB)]. It can be proven that if the system is in a pure state, then SA = SB, so, because we are solely interested in the ground state of the system, there will not be misunderstandings if we do not specify the subsystem from which we calculate the entanglement entropy.

Entanglement entropy can be used to measure the degree of entanglement in a system, in fact if we take the simple product state |↑i ⊗ |↑i we see that the reduced density matrix on first subsystem, say, A is trivially

ˆ ρA= 1 0 0 0 , (2.22)

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therefore its bipartite von Neumann entropy SA= SB= 0, while if we take the state |ψi_Bell as in (2.20) we can compute

ˆ ˜ ρA= 1 2 1 0 0 1 (2.23) whose entanglement entropy is SA = SB = log(2), which is the maximum possible value, indeed we can recognize ˆρ as the density matrix of the maxi-˜ mally mixed state in two dimension. So we see that if the state is maximally entangled the bipartite von Neumann entropy is also maximal, and the re-sult is independent of the choice of the subsystem.

Moreover another useful fact we used in our work is that if we have a quan-tum system in a gapped phase the entanglement entropy scales as the bound-ary of the configuration, so it remains constant in one dimension, while if the system is in a gapless phase the bipartite entropy has logarithmic corrections in one dimension, there are rigorous proofs only in the 1D case [28], these results are called area laws. In our system the open boundary conditions impose a functional form of the entanglement entropy in a gapless phase: if we divide the chain in two part, the first of length l and the second of length L − l the bipartite entropy as a function of l becomes

S(l) = A + c 6log 2L π sin πl L , (2.24)

where c is constant called central charge defined in the context of 1+1 di-mensional conformal field theory and A is a constant [6].

Thanks to the previous arguments we can analyze the behavior of the en-tanglement entropy in order to discriminate between different phases, if S(l) is flat, then we are in the presence of a gapped phase, while if it has a bell shape the phase is gapless.

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Algorithm

In this chapter we explain the numerical methods that have been used in our research and the mathematical transformation required to implement the system in our algorithm.

3.1 Density matrix renormalization group

Low-dimensional strongly correlated quantum systems are really hard to study both analytically and numerically; the most powerful numerical method to deal with those kind of problems in 1D is density matrix renormalization group (DMRG) [29].

DMRG variational algorithms consist of two fundamental steps: infinite-system DMRG and finite-infinite-system DMRG.

The following discussion can be found with much greater detail in [5].

3.1.1 Infinite-system DMRG

Infinite-system DMRG deals with this problem by considering a chain of increasing length, usually L = 2, 4, 6, ..., and discarding a sufficient number of states to keep Hilbert space size manageable. This decimation procedure is key to the success of the algorithm: we assume that there exists a reduced state space which can describe the relevant physics and that we can develop a procedure to identify it.

To explain this procedure we have to introduce left and right blocks A and B, which in a first step could be one site, or a spin, such that the total chain length is 2. Longer chains are built iteratively by inserting pairs of

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sites between the two blocks, such that the chain grows to 4, 6, and so on; at each step previous sites are absorbed into the left and right blocks, such that block sizes grow as 1, 2, 3, and so on, leading to exponential growth of the dimension of the full block state space as 2l, where l is the current block size. Our chains then always have a block-site-site-block structure, A••B. We are interested in finding the ground state of the above constructed chain also referred as superblock, but we want to reduce the Hilbert space dimen-sion to D, which depends on our numerical resources. Any state of the superblock A••B can be described by

|ψi = X

aAσAaBσB

ψaAσAaBσB|aiA|σiA|aiB|σiB ≡ X

iAjB

ψiAjB|iiA|jiB (3.1)

where the states of the site next to A are in the set {|σAi} of local state dimension d, and analogously those of the site next to B. By numerical diagonalization we find the |ψi that minimizes the energy

E = hψ| ˆHA••B|ψi

hψ|ψi (3.2)

with respect to the Hamiltonian of the superblock. To this purpose, we need some iterative sparse matrix eigensolver such as provided by the Lanczos or Jacobi-Davidson methods. We now have to explain how we choose a certain reduced Hilbert space after we add a site to the block A the total dimension of A• is dD, analogously for •B, if we want to cut the dimension to D we have to find the right procedure to do so, we define the reduced density operator for A•

ˆ

ρA•= tr•B[|ψi hψ|] (ρA•)ij = X

k

ψikψ∗jk. (3.3)

We now find the eigensystem of ˆρA• and we retain only the D eigenvectors with the highest eigenvalues {wa}a<D. If we call them |biA the vector en-tries are simply the expansion coefficients in the previous block-site basis, Ahaσ|bi_A.

After an approximate transformation of all desired operators on A• in the new basis, the system size can be increased again, until the final desired size is reached. B is grown at the same time, for reflection-symmetric systems by simple mirroring.

It can be proven that the previous prescription follows both from statisti-cal physics arguments or from demanding that the 2-norm distance || |ψi −

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|ψi_trunc||2between the current ground state |ψi and its projection onto trun-cated block bases of dimension D, |ψi_trunc, is minimal. The overall success of DMRG rests on the observation that even for moderate D, often only a few hundreds, the total weight of the truncated eigenvalues, given by the truncation error ε = 1 −P

a<Dwa if we assume descending order of the wa, is extremely close to 0, say 10−10 or less.

Operators acting on blocks

Now we show how to express operators acting on blocks in the current block bases, in order to construct the Hamiltonian and observables of interest. Let us assume a general operator ˆO acting on site l with matrix element O_σ_l_,σ0

l = hσl| ˆO|σ

0

li. We imagine that site l is on the left-hand side and is added to block A, its construction is then initialized when block A grows from l − 1 → l, as ha_l| ˆO|a0_li = X al−1,σl,σ0l ha_l|a_l−1σli hσl| ˆO|σ0li hal−1σ0l|a 0 li . (3.4)

Here |ali and |a0_li are the effective basis states of blocks of length l and l − 1 respectively. The necessary update after a further growth of A is

ha_l+1| ˆO|a0_l+1i = X al,a0l,σl+1

ha_l+1|a_lσl+1i hal| ˆO|al0i ha0lσl+1|a0l+1i . (3.5)

In Hamiltonians product operator ˆO ˆP occur, the correct way is to update the first operator, counting from the left for a left block A, until the site of the second operator is reached, and to incorporate it as

hψ| ˆO|ψi = X

aA,a0_A,σA,σB,aB

hψ|aAσAσBaBi haA| ˆO|a0Ai ha 0

AσAσBaB|ψi (3.7)

where suitable bracketing turns this into an operation of order O(D3d2). An important special case is given if we are looking for the expectation value of a local operator that acts on one of the free sites • in the final block-site configuration. Then

hψ| ˆO|ψi = X

aA,σA,σA0,σB,aB

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which is an expression of order O(D2d3), for many operators, even O(D2d2). If we stop infinite-system DMRG at some superblock size L, we can take it as an approximation to the exact state for the superblock of size L and evaluate expectation values. The accuracy is limited not only by the truncations, but also by the fact that the first truncations were carried out for extremely small superblocks: the choice of relevant short block states is likely to be a not too good approximation to those one would have chosen for these short blocks embedded in the final system of length L.

3.1.2 Finite-system DMRG

Once the desired final size system size is reached, it is important to follow up on it by the so-called finite-system DMRG procedure. This will not merely lead to some slight quantitative improvements of our results, but may change them completely.

The finite-system algorithm corrects the choices made for reduced bases in the context of a superblock that was not the system of interest of final length L, but some sort of smaller proxy for it. What the finite-system algorithm does is the following: it continues the growth process of, for example, block B following the same prescription as before: finding the ground state of the superblock system, determining the reduced density operator, finding the eigensystem, retaining the D highest weight eigenstates for the next larger block. But it does so at the expense of block A, which shrinks. This is continued until A is so small as to have a complete Hilbert space, i.e. of dimension not exceeding D, one may also continue until A is merely one site long; results are not affected. Then the growth direction is reversed: A grows at the expense of B, including new ground state determinations and basis choices for A, until B is small enough to have a complete Hilbert space, which leads to yet another reversal of growth direction. This sweeping through the system is continued until energy or, more precisely, the wave function converges. The intuitive motivation for this, in practice highly suc-cessful, procedure is that after each sweep, blocks A or B are determined in the presence of an ever improved embedding.

The whole algorithm, that is graphically shown in figure 3.1, selects cleverly the right subspace of the global hilbert space of the problem to find the ground state energy.

As a final remark we want to remind that we cannot be completely sure that the final state we obtain is the ground state, because DMRG is a deter-ministic variational algorithm on a local set of parameters, i.e. it optimizes variationally few parameters at a time. This could lead to a local minimum

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of the energy, that is not the ground state. superblock block B 2 sites block A subsystem A subsystem B

new block A new block B

block B 2 sites block A block B growth block A growth end of infinite DMRG block A size minimal end of finite DMRG (retrieved) (retrieved) (re pe ate d s w ee ps )

Figure 3.1: The left half of the figure describes schematically the infinite-system DMRG, while the right part illustrates the finite-system DMRG procedure, the picture can be found in [5].

3.1.3 DMRG and entanglement

In the previous part we discussed a numerical procedure to approximate efficiently a quantum state, the key is retaining only the D density matrix eigenvectors with the highest eigenvalues wa, so knowing the eigenspectra of the reduced density operators on |ψi would allow us to easily assess the quality of the DMRG approximation; it simply depends on how quickly the eigenvalues wa decrease.

The DMRG algorithm simply leads to consideration on bipartite systems, where the parts are A• and •B, so the entanglement entropy, that we defined in 2.4.2, can come to help.

SA= SB = − X

a

walog(wa). (3.9)

In a mathematically non-rigorous way one can make contact between DMRG and the area laws of quantum entanglement, with two D-dimensional spaces the maximal entanglement entropy is SA = log D, where ρA is maximally mixed, so if we ant to encode properly the entropy S of the system we need at least of a state of dimension D ∼ 2S, so in a unidimensional gapped system one is not forced to increase D dramatically when the system size L goes up, while in the two-dimensional case the DMRG is doomed to fail because S scales as the square of the size. In a gapped phase even in on dimension the situation might be problematic due to the logarithmic correction of the entanglement entropy[28, 6], but simulations are usually numerically affordable for up to L ∼ 103.

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3.2 Matrix Product States

DMRG algorithm finds a more flexible implementation on a class of states called matrix product states (MPS).

We start our brief discussion noting that every rectangular matrix M of dimension N × M can be decomposed as

M = U SV†, (3.10)

here U is a matrix of dimension N × min(N, M ) and U†U = I, with I the identity.

S is a diagonal matrix of dimension min(N, M ) × min(N, M ) with non negative entries, the so called singular values.

V† is of dimension min(N, M ) × M and V†V = I. This is the so called singular value decomposition (SVD).

For a lattice of size L with d-dimensional local space states {σi} the most general pure quantum state reads

|ψi = X

σ1,...,σL

cσ1,...,σL|σ1...σLi . (3.11)

Through SVD we can write the coefficient as cσ1,...,σL= X a1,...,aL−1 Aσ1 a1A σ2 a1,a2...A σL aL−1= A σ1_...AσL_, _(3.12)

where Aσi _{are matrices of varying dimension for i 6= 1, L, while for i = 1, L}

they are column vector and it holds X

σl

Aσl†_Aσl_{= I.} _(3.13)

So every quantum state can be written as an MPS: |ψi = Aσ1_...AσL_|σ

1...σLi . (3.14)

Analogously we can write

|ψi = Bσ1_...BσL_|σ 1...σLi , (3.15) with X σl Bσl_Bσl†_{= I,} _(3.16)

and finally we can also express a state as

|ψi = Aσ1_...Aσl_SBσl+1_...BσL_|σ

1...σLi . (3.17)

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3.3 Our algorithm

In this part we want to describe briefly the algorithm that we used to carry out our analysis.

The algorithm exploits the MPS formalism, it generates a chain of the de-sired size and obtain a quantum state |ψi written as (3.17). For the varia-tional step it takes one of the matrices Aσl _{or B}σl _{and fixes all the others,}

then it minimize the energy as a function of the chosen matrix. After that it takes the successive matrix and repeat the procedure until a suitable con-vergence is obtained.

3.4 Implementation

In order to simulate numerically our system via a DMRG algorithm, which is a strictly 1D technique, we had to perform two transformation before the actual implementation: first we have grouped all the sites on the same leg, so that every site in the synthetic dimension has been considered as a unique block, since every site has local Hilbert space dimension equal to 2, with base vector |0i = 1

0

and |1i = 0 1

, which represent respectively the absence and the presence of an electron in the site, the corresponding block has local dimension 2n_{where n is the length of the synthetic dimension;} mathematically we have a tensor product of local Hilbert spaces lying on the same leg. After this operation we obtain a 1D chain made by blocks, due to the tricky nature of the canonical anti commutation relations, we may want to transform the previously defined fermionic operators into spin operators, to carry out this task we perform a Jordan-Wigner transformation.

3.5 Jordan-Wigner transformation

In this section we give the details of the Jordan-Wigner transformation [8] we carried out to simulate our system. Given the three Pauli matrices as in (1.5) we define ˆ σ_k+= 1 2(ˆσ x k− iˆσ y k) σˆ − k = 1 2(ˆσ x k+ iˆσ y k), (3.18)

where the subscript refers to the fact that they are local operators acting on the kth site. So we can define ˆ σ+_k = ˆf_k† σˆ_k−= ˆfk ˆσzk= 1 − 2 ˆf † kfˆk, (3.19)

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thanks to these new operators we can construct fermionic operators with the right anti commutation relations

ˆ c†_k= eiπPk−1l=1 fˆ † lfˆl_fˆ† k ˆ ck= e−iπ Pk−1 l=1 fˆ † lfˆl_fˆ k. (3.20)

These relations define the Jordan-Wigner transformation, that is evidently a bijection between spin and fermionic operators. We note that ˆf_k†fˆk ∈ {0, 1}, so e±iπPk−1l=1 fˆ † lfˆl₌ k−1 Y l=1 (1 − 2 ˆf_l†fˆl) = k−1 Y l=1 ˆ σz_l. (3.21)

So, in order to implement the model into the code, we have to write all the operators in the Hamiltonian in terms of spin operators, i.e. matrices, acting on the previously defined block, so it is important to notice that the only terms which are now non local are ˆH1 and the nearest neighbor interaction. We put in appendix A.1 all the operators used in our calculation obtained after the application transformations described above for the 2-leg system, and in appendix A.2 for the three leg-system.

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Results for the 2-leg

configuration

In this section we show the results of our analysis for the 2-leg system varying the parameters of the model.

First we show the presence of Wigner crystals only for a particular value of the parameter ϕ, then we examine the behavior of the charge gap, and we find evidence of a phase transition governed by the parameter U/t. At the end we show that under the transition point the phase is liquid and we highlight the connection of the ordered and disordered phases with the entanglement entropy.

In order to give the right estimate of the error for our observables in appendix B we show the analysis we have put forward. Moreover in appendix C we give all the details about our simulations.

4.1 Filling number ν =

1₂

Here we discuss the properties of the model in the case ν = 1₂.

We notice that rich physics occurs when only the Hubbard term is consid-ered. Therefore, in order to analyze the different phases in this particular condition, we will take V = 0, so we will not consider the effect of a nearest neighbor interaction in the following.

4.1.1 Local density

First of all it is important to examine the charge distribution of the wave function when ϕ and U vary. We start by analyzing the situation in the

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limit U/t = ∞, in order to spot a Wigner crystal.

We observe a Wigner crystal in the Hubbard dominating limit, i.e. U/t → ∞ only for ϕ = 1/2, while an ordered phase cannot be stabilized for different values of the magnetic flux, as shown in 4.1.

With this value of filling the number of particles N , with lattice-size L = 80, is N = 40 for ϕ = 1/2, N = 20 for ϕ = 1/4 and N = 10 for ϕ = 1/8.

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Figure 4.1: Here we see the charge distribution in the limit U/t → ∞, the first one is simulated with ϕ = 1/2 and appears to be a crystal due to its manifest structural order. The others have ϕ = 1/4 and ϕ = 1/8 respectively, their wave function is delocalized signaling a liquid phase. The error bars are too small to be drawn at this scale.

We see in the picture above a kink in the plot with ϕ = 1/2, this can be explained as a numerical symmetry breaking in our algorithm. The meaning of this point is that the ground state is degenerate, so the state variationally found is a linear combination of two states with the same energy. We will see that both magnetizations and entanglement entropy have a singular point corresponding to the one of the local density.

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4.1.2 Magnetization

It is very interesting to analyze the behavior of the magnetization operators that we defined in 2.3.3, we see an ordered pattern of h ˆMxi for the Wigner crystal, while for the liquid phase we see an analogous behavior as charge density for h ˆMxi and h ˆMyi in figure 4.2, we do not show data for ˆMzbecause it is always zero as expected by the symmetry of the Hamiltonian.

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Figure 4.2: Here we observe the behavior of h ˆMx

ji with the blue circles, and h ˆM y

ji marked

by green triangles, for ϕ = 1/2, 1/4, 1/8 respectively. The errors are too small to lessen in figure

4.1.3 Charge gap

We now exhibit the behavior of the charge gap ∆E for ϕ = 1/2 defined in 2.3.4, which serves as the order parameter for our transition.

We see in figure 4.3 a distinct jump by varying U , so this is a strong evidence that a phase transition occurred. We have conducted our analysis with three different sizes L = 40, 60, 80, we were not be able to go further because of the computational cost of our simulations.

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Moreover a crude estimate of the transition point Uc can be extrapolated by our data.

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L = 60

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Figure 4.3: We see the charge gap as a function of the Hubbard coupling U/t, for different sizes of the system. A jump is clearly visible in the central part of the figure, signaling a phase transition.

We distinctly see a jump in the charge gap for the region 30 < Uc/t < 40, so we can roughly estimate the critical point as Uc/t = 35 ± 5

4.1.4 Behavior at the critical point

We want to examine the local density and the magnetization in the transition region in order to understand better our phase transition.

We see in figure 4.4a the quantities in the middle of the critical region, we observe a distinct crystalline behavior and the coexistence of two different group states in agree with the analysis conducted above the transition point. While in 4.4b we see that the crystalline pattern is broken at the end of the region, in fact the quantities shown begin to have a disordered trend.

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(a) Local density and magnetization at U/t = 34 for L = 80.

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4.1.5 Relevant quantities below the transition point

It is also interesting to investigate how the density and magnetizations be-have under the transition point, in order to see if the system really underwent a transition from a crystal phase to a liquid one.

We have already shown previously the local density and the magnetizations in the limit U/t → ∞, so we now focus on their value under the critical point for ϕ = 1/2.

We can notice in figure 4.5 that the local density evidences a delocalized wave function apart from border effects, h ˆM_jxi shows a noisy behavior.

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Figure 4.5: Here we show the density profile in the first graph and the average magneti-zation on the x axis in second one. In this simulation we have ϕ = 1/2, U/t = 10 and L = 80, so we are below the transition point.

4.1.6 Correlations

In order to further explain the trend of h ˆMxi, we have also measured the correlation functions h ˆM_ixMˆ_jxi, and we calculated the covariance, seen in

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figure 4.6:

Cov( ˆM_ixMˆ_jx) = h ˆM_ixMˆ_jxi − h ˆM_ixi h ˆM_jxi . (4.1) This enables us to better understand the magnetic structure of the liquid phase. The second plot in figure 4.6 shows a distinct power law behavior of the connected correlation functions.

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Figure 4.6: In this graph we see a decaying of the covariance in magnetization on the x direction, signaling the magnetic disorder of the liquid phase. The simulation has been conducted with ϕ = 1/2 and U/t = 10. The second graph shows the absolute value of the same data in a logarithmic scale.

4.1.7 Entanglement entropy

Here we analyze the trend of the bipartite von Neumann entropy defined in section 2.4.2, in order to confirm the previous analysis. As we see in figure 4.7a, only for ϕ = 1/2 we can observe a flat behavior, the other two plots exhibit a bell-shape trend, sign of a disordered phase, in the limit of high U . We observe in figure 4.7b the form of the entanglement entropy below the transition point, and we distinctly see the sign of a liquid phase.

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We note in the third plot of figure 4.7a ten peaks. This kind of oscillation, studied in [30], represents the ten fermions present in the system.

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(a) Here we see the entanglement entropy for U/t → ∞, for ϕ = 1/2, 1/4, 1/8 for lattice size L = 80.

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(

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(b) In figure there is the bipartite von Neumann entropy of the system at U/t = 5, for L = 80, we see a bell-shape and a value of the fitted central charge c = 1.85 ± 0.05. Figure 4.7: Bipartite von Neumann entropy of the system for different values of ϕ and U/t.

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4.2 Filling number ν =

1₃

We now consider the case in which ν = 1/3, we will see a lot of analogies with the last case but also we will notice an important difference between the two cases.

As pointed out in [4], a crystalline phase cannot be reproduced with the Hubbard term alone, so we have to turn on the nearest neighbor interactions. In fact bosonization arguments find that if ν diminishes we need to increase the rage of interaction in order to form a crystalline phase.

4.2.1 Local density

As stated above in order to stabilize a crystal phase at ν = 1/3 it is necessary to turn on the nearest neighbor interaction, so unlike in the previous sections we will deal with V > 0.

In the last part we dealt with the phases of the system at ν = 1/2 and we discovered a transition between two different phases by only tuning the Hubbard coupling U , in the following we will always work in the limit U/t → ∞ and we will search for different phases by varying V .

In this part of the analysis we have N = 28 for ϕ = 1/2, N = 14 for ϕ = 1/4 and N = 7 for ϕ = 1/8.

If we set a large value of V , in our case V /t = 100 we can unveil a trend similar to the one we have already analyzed seen, as shown in figure 4.8. We see a similar picture as the previous case, for ϕ = 1/2 we can stabilize a crystalline phase for large values of the nearest neighbor interaction, while in the other cases the liquid phase is dominant.

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Figure 4.8: Here we see the average number operator for L = 84, the first graph shows the data at ϕ = 1/2, while the others have a magnetic flux given by ϕ = 1/4, 1/8 respectively.

4.2.2 Magnetization

We also have a behavior of the magnetization operators similar to the case with filling number ν = 1/2, in the limit U/t → ∞. We see in figure 4.9 an ordered behavior for h ˆMxi in the case with ϕ = 1/2. The value of h ˆMzi is zero in every simulation analogously to the previous case.

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Figure 4.9: Here we have different values of h ˆMjxi in blue, and of h ˆMyi in green, for

ϕ = 1/2, 1/4, 1/8 respectively and for L = 84.

4.2.3 Charge gap

Now we discuss the presence of a phase transition in the case ϕ = 1/2, we maintained the system in the limit U/t → ∞ and we varied the parameter that regulates the nearest neighbor interaction V , we used the charge gap as an order parameter and in figure 4.10 we distinctly see a critical region for 10 < V /t < 20. We performed the analysis varying the lattice size and using L = 42, 60, 84.

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∆ E /t

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Figure 4.10: We see above the charge gap for ν = 1/3, we clearly observe a jump in all of the three sizes sampled.

We can crudely extract from our data a value for Vc/t = 15 ± 5. Due to limited time and resources we are not be able to perform a finite-size scaling analysis to improve our result.

4.2.4 Behavior at the critical point

Analogously to the previous case we show the local density and magnetiza-tion in the critical region.

In figure 4.11a we see that the crystalline phase persist for V /t = 12, so in a neighborhood of the critical point we can distinctly see an ordered phase. On the other hand in figure 4.11b we see that slightly under the transition region the crystalline pattern is replaced by a disordered one.

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(a) Local density and magnetization at V /t = 12 for L = 84.

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4.2.5 Relavant quantities below the transition point

We now are interested in the measurable quantities of our configuration below the transition point.

We see in figure 4.12 a noisy behavior of both density and h ˆMxi.

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Figure 4.12: Density and magnetization along x axis for ϕ = 1/2, V /t = 2 and L = 84.

4.2.6 Correlations

We want to see quantitively if the quantities obtained below the transitions are in a disordered phase, so we proceed in the same fashion as in the previous section calculating the covariance of the density as we show in figure 4.13. In the second plot of the same figure we see a distinct power law behavior of the connected correlation functions.

As regard the magnetization the result is analogous, so we do not need to report the graph.

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0.025

0.020

0.015

0.010

0.005

0.000

0.005

C ov (ˆ Ni ˆN)j

10

0

₁₀

1

₁₀

2 |i−j|

10

-6

10

-5

10

-4

10

-3

10

-2

10

-1 | C ov (ˆ Ni ˆN)j |

Figure 4.13: Here we observe that both the density is uncorrelated on the lattice under the transition point. There ϕ = 1/2, U/t → ∞ and V /t = 2. In the second image there is the absolute value of the same data of the previous graph in a logarithmic scale.

So we see that under the critical point both density and magnetization are uncorrelated, this confirms that under Vcthe system is in a liquid phase.

Now we analyze the behavior of the bipartite von Neumann entropy. At first we show in figure 4.14a the trend for ϕ = 1/2, 1/4, 1/8 in the case with V /t = 100, we only see shape compatible with a Wigner crystal in the first plot, analogously to the previous case.

In order to corroborate the fact that we are in a disordered phase under the phase transition we show in figure 4.14b the entanglement entropy for ϕ = 1/2 simulated under Vc. We clearly see that below the critical point we are in a disordered phase.

As discussed in the previous sections the seven peaks of the third plot in figure 4.14a are a sign of the number of particles in the system, seven in this case.

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1.0

0.8

0.6

0.4

0.2

0.0

0.2

0.4

0.6

0.8

1.0

0.5

0.0

0.5

1.0

S ( j )

10

20

30

40

50

60

70

80

Sitej

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

(a) Here we have the values of entanglement entropies in function of the lattice site, they are obtained at ϕ = 1/2, 1/4, 1/8 respectively with lattice size L = 84.

0

10

20

30

40

50

60

70

80

90 Site j

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0 S

(

j

)

(b) Here is depicted the bipartite von Neumann entropy for V /t = 2, ϕ = 1/2 and L = 84. We can fit the data with the function (2.24) and we obtain a central charge c = 0.88±0.06. Figure 4.14: Bipartite von Neumann entropy of the system for different values of ϕ and V /t.

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4.3 Summary of the results for the 2-leg ladder

In the last chapter we examined the behavior of the 2-leg ladder system. On the one hand for filling number ν = 1/2 we have seen that in order to form Wigner crystal it is not required a nearest neighbor interaction in the case ϕ = 1/2, as shown in [4], but we showed that for different values of the synthetic magnetic flux the system is in a liquid state even for the limit U/t → ∞. We analyzed the model by varying the Hubbard parameter U , and we individued a transition region, we saw that by crossing the region the crystal becomes progressively a liquid, and it loses both structural and magnetic ordering. All the analysis have been reinforced by the study of the bipartite von Neumann entropy of the model.

On the other hand for filling number ν = 1/3, we need to turn on the nearest neighbor parameter V , if we want stabilize an ordered phase, as explained in [4], but analogously to the previous case a structural and magnetic crystal is formed only for ϕ = 1/2, even for large V and in the limit of infinite U . In the same limit we varied V and we found a critical region, where the crystal dissolves into a fermionic liquid. As in the previous study all of the phases uncovered are been confirmed by the shape of the entanglement entropy.

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Results for the 3-leg

configuration

We now examine a slightly different case, the synthetic dimension in this configuration has three sites simulated with periodic boundary conditions. We see a different behavior form the previous case although the two systems appear similar.

As stated in 3.1, the leading term of the computational cost is of order O(D2d3) where d is the local hamiltonian dimension. Since d is doubled with respect to the 2-leg case, we now have to to exploit four times more of our computational resources than before, for this reason it has not been possible to analyze the emergence of a phase transition.

We have conducted an analysis, shown in appendix B, to estimate the error on our observables. On the other hand in appendix C we show the details of our simulations.

5.1 Filling number ν =

1₂

5.1.1 Density

Thanks to the addition of a new row of sites we are able to see a new phe-nomenon: Wigner crystallizations happens for new values of the synthetic gauge fields as we see in figure 5.1 in the limit of U/t → ∞.

The results have been obtained with ϕ = 1/2, 1/4, 1/8 and L = 80, for ϕ = 1/2 the number of particle N = 60, for ϕ = 1/4 N = 30 and for ϕ = 1/8 N = 15.

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0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

< ˆNj >

0

10

20

30

40

50

60

70

80

Sitej

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Figure 5.1: We see spatially periodic structures for ϕ = 1/4, 1/8, i.e. the last two graphs, while we see a particular periodic structure for ϕ = 1/2 in the first figure

5.1.2 Magnetization

We show in figure 5.2 in figure that the structural order observed in the density profiles extend to magnetic order for new values of the Peierls phase in the infinite U limit.

We see that in the 3-leg configuration the degeneracy problem of the ground state is even more severe than in the previous case. In fact the algorithm cannot find a unique ground state, but it reaches a combination of two or more states with minimal energy.

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1.0

0.5

0.0

0.5

1.0

0.5

0.0

0.5

1.0

< ˆ_M x >j

,

< ˆ_M y >j

0

10

20

30

40

50

60

70

80

Sitej

1.0

0.5

0.0

0.5

1.0

Figure 5.2: Here are reported h ˆMjxi and h ˆM y

ji, with values of ϕ = 1/2, 1/4, 1/8

respec-tively, the lattice size is L = 80

The entanglement entropy in figure 5.3 confirms the crystalline phase we have seen thanks to the other observables.

We also notice a phenomenon of dimerization for ϕ = 1/2, in fact the bi-partite von Neumann entropy periodically jumps from zero to a finite value, that does not depend on the size of the system, as it is in a liquid phase, where the entanglement entropy maximum depends on the lattice size.

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0.00

0.05

0.10

0.15

0.20

0.25

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

S ( j )

0

10

20

30

40

50

60

70

80

Sitej

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Figure 5.3: Here we see the bipartite von Neumann entropy for ϕ = 1/2, 1/4, 1/8, L = 80 and U/t → ∞.

5.2 Filling number ν =

1₃

5.2.1 Density

We see immediately in figure 5.4 a striking difference between the 2-leg and the 3-leg configuration: we can stabilize a crystal phase without the nearest neighbor interaction in the ν = 1/3 case, moreover we see new flux amplitudes in which an ordered phase can be formed in the limit U/t → ∞. These facts prove e great change in the physics of the system with only one site added in the synthetic dimension.

Here for ϕ = 1/2 the number of particle N = 40, for ϕ = 1/4 N = 20 and for ϕ = 1/8 N = 10.

Different phases of interacting fermions in low dimensional lattices with synthetic gauge fields

Anno accademico 2017/2018