• DMRG phase transition

The aim of my master thesis is the study of a strongly anisotropic fermionic ladder system 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 (real dimension), but with only few sites on the y direction (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.
Since our lattice is not strictly unidimensional, it is possible to introduce a synthetic gauge field. The presence of a synthetic magnetic flux allows us to study geometric frustration effects depending both on the magnitude of the field and on the density of particles in the chain. These effects causes interesting outcomes such as fermionic localization.
This kind of structure can be encoded in a 1D system with an effective local hamiltonian dimension 2^n where n is the number of sites on the synthetic dimension. It was proven that through the presence of nontrivial complex phases in the hopping term along the y axis, which mimic the effect of a magnetic field, the behavior of this system shows similarities with the physics of Quantum Hall Effect (QHE).
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 fractional density of particles.
So our purpose is to understand the properties of this model in the strongly interacting regime via density matrix renormalization group (DMRG) simulations if an interaction Hubbard term is added to the Hamiltonian. This will be achieved by finding a transition between a crystalline and a liquid phase[3] in dependence of the Hubbard coupling. The analysis is performed with a matrix product states (MPS) algorithm, 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. This approach enables to easily measure quantities like local magnetization, local density, correlation functions and bipartite von Neumann Entropy. In conclusion the results of this study will be useful to attain a better understanding of the possible fermionic phases in low dimensional structures with the presence of a magnetic gauge field.
Our investigation on the quantum phase diagram of the system has shed light on interesting phenomena. We work both on the 2-leg and on the 3-leg ladder configuration, the former has only two sites on the synthetic dimension, the latter has three sites on this dimension. Although the difference seems irrelevant, the two configurations appear to behave very differently. This model is SU(N)-symmetric where N is the number of legs chosen, because we work with periodic boundary conditions on the synthetic dimension. Notice that also the interaction terms that we will consider respect this group of symmetry.
An important quantity to take in account is the filling number ν which is defined as ν=n/[(M + 1)φ]
where n is the density of particles along the real direction, M depends on the kind of system, M = 1 for the 2-leg and M = 2 for the 3-leg, and φ is the parameter of the complex phase that simulate the magnetic field. 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 function of the site, from a gapless one, where the entanglement entropy has a bell-shape.
By the variation of both ν and φ the MPS algorithm permits us to observe either a crystalline ordered phase, where the local density of fermions is periodic, or a liquid disordered one, in which the local density of particles is flat.
In the 2-leg lattice we see an ordered phase only when φ = 1/2 and ν ≥ 1/2 if U is sufficiently large, while for ν < 1/2 a crystalline phase appears only if a repulsive nearest neighbors interaction is switched on. The order parameter for the phase transition is the charge gap ∆E which is defined as ∆E = E(k+1)+E(k−1)−2E(k) where E(k) is the energy of the system with k particles. In contrast in the 3-leg lattice crystal phases arise even if φ ̸= 1/2, and an ordered phase is stabilized for proper U in the case ν = 1/3.
We also notice a peculiar behavior of the correlations of the magnetization in our chain, because they appear to be periodic even under the transition point, so the system shows magnetic order and structural disorder at the same time, both in the 2-leg and in the 3-leg case.