arXiv subjects

Pierre Nataf

Publications and source records attributed to Pierre Nataf.

At least 19 recordsLinked to original sources

arXiv2026

$\mathbb{Z}_L$ symmetry breaking in SU(N) Fermi-Hubbard dots at zero and finite temperature

We address the SU(N) Fermi-Hubbard model on a chain, with $N$ the number of degenerate orbitals, or colors, for each fermion. In the limit of both large number of colors $N$ and particles, and small number of sites $L \geq 2$, the model is proved to undergo a $\mathbb{Z}_L$ symmetry breaking for attractive local interaction amplitude $U$. Using a combination of Exact Diagonalization with full SU(N) symmetry, generalized L-levels Holstein-Primakoff transformation, Hartree-Fock method and large-N saddle point approximation of the partition function, we extend the results obtained in [PRA 111, L020201 (2025)] to $L \geq 3$ and finite temperature $T>0$. In particular, we show that at $T=0$ for $U<U_c\sim -1/N$, the ground state is L-fold degenerate, while for positive temperatures, the critical temperature is both proportional to $N$ and $U$, i.e. $T_c \propto -U N$, making this phase transition particularly suitable for large-N fermions.

cond-mat.str-el↗

arXiv2024

The SU(N) Fermi-Hubbard Model on two sites: Bethe Ansatz solution and Quantum Phase Transition of the Lipkin-Meshkov-Glick Model in the large-N limit

We show that the SU(N) Fermi-Hubbard model (FHM) on two sites, where N is the number of flavors of each fermion, corresponds to an exactly solvable two-level many-boson model that Richardson [J. Math. Phys. 9, 1327 (1968)] analytically solved long ago. We express the Bethe ansatz solutions as a function of the physical parameters of the SU(N) FHM, and recast its eigenvalues and eigenstates in terms of the Richardson pair energies and creation operators. In this context, the connection with the well-studied Lipkin-Meshkov-Glick (LMG) model, known as equivalent to the Richardson model, is established and serves as a guideline to the prediction of some N-body physics phenomena in the two-site SU(N) FHM with N particles. In particular, the LMG second-order quantum phase transition (QPT) is shown to occur in the SU(N) FHM for an attractive density-density interaction U equal to U_c=-1/(2N), in units of the (absolute value of the) tunneling amplitude between the two sites. We show the finite-size energies, the gap, and the kinetic energy, which all reveal the transition, as a function of U for values of N from N = 3 to N = 36, suggesting that the QPT could be experimentally achieved with current technologies involving SU(N) ultracold atoms or molecules. Finally, we show the entanglement entropy of the first site with respect to the second, and it scales like N at the transition, in contrast with several two-mode models.

cond-mat.quant-gas↗

arXiv2024

Density Matrix Renormalization Group simulations of the SU(N) Fermi-Hubbard chain implementing the full SU(N) symmetry via Semi-Standard Young Tableaux and Unitary Group Subduction Coefficients

We have developed an efficient method for performing density matrix renormalization group (DMRG) simulations of the SU(N) Fermi-Hubbard chain with open boundary conditions, fully leveraging the SU(N) symmetry of the problem. This method extends a previously developed approach for the SU(N) Heisenberg model and relies on the systematic use of the semi-standard Young tableaux (SSYT) basis in a DMRG algorithm `a la White. Specifically, the method aligns the site-by-site growth process of the infinite-size part of the DMRG, in its original formulation, with the site-by-site construction of the SSYT (or Gelfand-like) basis, based on the chain of unitary subgroups $U(1)\subset U(2) \subset U(3) \subset U(4)\cdots $. We give special emphasis to the calculation of the symmetry-resolved reduced matrix elements of the hopping terms between the left and the right block, which makes direct use of the basis of SSYT and of the Gelfand-Tsetlin coefficients, offering a computational advantage in scaling with N compared to alternative methods that rely on summing over Clebsch-Gordan coefficients. Focusing on the model with homogeneous hopping between nearest neighbors, we have calculated the ground state energy as a function of U, i.e the atom-atom interaction amplitude, up to N=6 for filling 1/N (one particle per site in average), and for one atom (resp. hole) away from filling 1/N, alllowing us to compute the charge gaps, and to estimate in the thermodynamical limit, the critical value $U_c$, separating the Mott insulator from the metallic phase. Central charges c are extracted from the entanglement entropy using the Calabrese-Cardy formula, and are consistent with the theoretical predictions: c=N-1, expected from the $SU(N)_1$ Wess-Zumino-Witten CFTs in the spin sector for the Mott phase, and c=N in the metallic phase, reflecting the presence of one additional (charge) gapless critical mode.

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arXiv2024

Numerical observation of $\mathrm{SU}(N)$ Nagaoka ferromagnetism

We provide numerical evidence of the Nagaoka's theorem in the $\mathrm{SU}(N)$ Fermi-Hubbard model on various cluster geometries, such as the square, the honeycomb and the triangular lattices. In particular, by diagonalizing several finite-size clusters, we show that for one hole away from filling $1/N$, the itinerant ferromagnetism arises for $U$ (the positive on-site interaction) larger than $U_c$ (the value at the transition), which strongly depends on the coordination number $z$ and on $N$, the number of degenerate orbitals, that we vary from $N=2$ to $N=6$ in our simulations. We prove that $U_c$ is a non decreasing function of $N$. In addition, we find that the lattice dependency is rooted in the kinetic energy of the hole. We find that large coordination numbers $z$ lower the value of $U_c$. Complementary, we explore the effect of long-range hopping on the appearance of itinerant ferromagnetism and demonstrate that it acts as an increased coordination number, protecting the ferromagnetic phase at small $U$. Finally, both the effects of the presence of some additional holes and of the finite size of the clusters are briefly discussed.

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arXiv2023

Exact Diagonalization of $\mathrm{SU}(N)$ Fermi-Hubbard Models

We show how to perform exact diagonalizations of $\mathrm{SU}(N)$ Fermi-Hubbard models on $L$-site clusters separately in each irreducible representation ({irrep}) of $\mathrm{SU}(N)$. Using the representation theory of the unitary group $\mathrm{U}(L)$, we demonstrate that a convenient orthonormal basis, on which matrix elements of the Hamiltonian are very simple, is given by the set of {\it semistandard Young tableaux} (or, equivalently the Gelfand-Tsetlin patterns) corresponding to the targeted irrep. As an application of this color factorization, we study the robustness of some $\mathrm{SU}(N)$ phases predicted in the Heisenberg limit upon decreasing the on-site interaction $U$ on various lattices of size $L \leq 12$ and for $2 \leq N \leq 6$. In particular, we show that a long-range color ordered phase emerges for intermediate $U$ for $N=4$ at filling $1/4$ on the triangular lattice.

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arXiv2022

Superradiant Quantum Phase transition for Landau Polaritons with Rashba and Zeeman couplings

We develop a theory of cavity quantum electrodynamics for a two-dimensional electron gas in the presence of Rashba spin-orbit and Zeeman couplings and perpendicular magnetic field, coupled to a spatially nonuniform quantum photon field. We show that the superradiant quantum phase transition (SQPT), also known as photon condensation, can in principle occur through a pure in-plane Zeeman coupling, but it requires extremely small (unrealistic) quantum well widths or extremely fine tuning of the effective Landé factor which makes two Landau levels coincide. Landau level crossings can also be induced by the Rashba spin-orbit coupling and they promote the SQPT which can be obtained for certain values of the effective Landé factor and filling factors.

cond-mat.mes-hall↗

arXiv2021

Abelian SU$(N)_1$ Chiral Spin Liquids on the Square Lattice

In the physics of the Fractional Quantum Hall (FQH) effect, a zoo of Abelian topological phases can be obtained by varying the magnetic field. Aiming to reach the same phenomenology in spin-like systems, we propose a family of SU($N$)-symmetric models in the fundamental representation, on the square lattice with short-range interactions restricted to triangular units, a natural generalization for arbitrary $N$ of an SU($3$) model studied previously where time-reversal symmetry is broken explicitly. Guided by the recent discovery of SU($2$)$_1$ and SU($3$)$_1$ chiral spin liquids (CSL) on similar models we search for topological SU($N$)$_1$ CSL in some range of the Hamiltonian parameters via a combination of complementary numerical methods such as exact diagonalizations (ED), infinite density matrix renormalization group (iDMRG) and infinite Projected Entangled Pair State (iPEPS). Extensive ED on small (periodic and open) clusters up to $N=10$ and an innovative SU($N$)-symmetric version of iDMRG to compute entanglement spectra on (infinitely-long) cylinders in all topological sectors provide unambiguous signatures of the SU($N$)$_1$ character of the chiral liquids. An SU($4$)-symmetric chiral PEPS, constructed in a manner similar to its SU($2$) and SU($3$) analogs, is shown to give a good variational ansatz of the $N=4$ ground state, with chiral edge modes originating from the PEPS holographic bulk-edge correspondence. Finally, we discuss the possible observation of such Abelian CSL in ultracold atom setups where the possibility of varying $N$ provides a tuning parameter similar to the magnetic field in the physics of the FQH effect.

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arXiv2021

Edge states and universality class of the critical two-box symmetric SU(3) chain

We numerically demonstrate that, although it is critical, the two-box symmetric $\mathrm{SU}(3)$ chain possesses edge states in the adjoint representation whose excitation energy scales with the number of sites $N_s$ as $1/(N_s \log N_s)$, in close analogy to those found in half-integer $\mathrm{SU}(2)$ chains with spin $S\ge 3/2$. We further show that these edge states dominate the entanglement entropy of finite chains, explaining why it has been impossible so far to verify with DMRG simulations the field theory prediction that this model is in the $\mathrm{SU}(3)_1$ universality class. Finally, we show that these edge states are very efficiently screened by attaching adjoint representations at the ends of the chain, leading to an estimate of the central charge consistent within 1\% with the prediction $c=2$ for $\mathrm{SU}(3)_1$.

cond-mat.str-el↗

arXiv2020

Haldane Gap of the Three-Box Symmetric $\mathrm{SU}(3)$ Chain

Motivated by the recent generalization of the Haldane conjecture to $\mathrm{SU}(3)$ chains [M. Lajkó et al., Nucl. Phys. B924, 508 (2017)] according to which a Haldane gap should be present for symmetric representations if the number of boxes in the Young diagram is a multiple of three, we develop a density matrix renormalization group algorithm based on standard Young tableaus to study the model with three boxes directly in the representations of the global $\mathrm{SU}(3)$ symmetry. We show that there is a finite gap between the singlet and the symmetric $[3\,0\,0]$ sector $Δ_{[3\,0\,0]}/J = 0.040\pm0.006$ where $J$ is the antiferromagnetic Heisenberg coupling, and we argue on the basis of the structure of the low energy states that this is sufficient to conclude that the spectrum is gapped.

cond-mat.str-el↗

arXiv2019

Landau polaritons in highly non-parabolic 2D gases in the ultra-strong coupling regime

We probe ultra-strong light matter coupling between metallic terahertz metasurfaces and Landau-level transitions in high mobility 2D electron and hole gases. We utilize heavy-hole cyclotron resonances in strained Ge and electron cyclotron resonances in InSb quantum wells, both within highly non-parabolic bands, and compare our results to well known parabolic AlGaAs/GaAs quantum well (QW) systems. Tuning the coupling strength of the system by two methods, lithographically and by optical pumping, we observe a novel behavior clearly deviating from the standard Hopfield model previously verified in cavity quantum electrodynamics: an opening of a lower polaritonic gap.

cond-mat.mes-hall↗

arXiv2019

Rashba cavity QED: a route towards the superradiant quantum phase transition

We develop a theory of cavity quantum electrodynamics for a 2D electron gas in the presence of Rashba spin-orbit coupling and perpendicular static magnetic field, coupled to spatially nonuniform multimode quantum cavity photon fields. We demonstrate that the lowest polaritonic frequency of the full Hamiltonian can vanish for realistic parameters, achieving the Dicke superradiant quantum phase transition. This singular behaviour originates from soft spin-flip transitions possessing a non-vanishing dipole moment at non-zero wave vectors and can be viewed as a magnetostatic instability.

cond-mat.str-el↗

arXiv2018

DMRG simulations of SU(N) Heisenberg chains using standard Young tableaux: fundamental representation and comparison with finite-size Bethe ansatz

We develop an efficient method to perform density matrix renormalization group simulations of the SU(N) Heisenberg chain with open boundary conditions taking full advantage of the SU(N) symmetry of the problem. This method is an extension of the method previously developed for exact diagonalizations and relies on a systematic use of the basis of standard Young tableaux. Concentrating on the model with the fundamental representation at each site (i.e. one particle per site in the fermionic formulation), we have benchmarked our results for the ground state energy up to N = 8 and up to 420 sites by comparing them with Bethe ansatz results on open chains, for which we have derived and solved the Bethe ansatz equations. The agreement for the ground state energy is excellent for SU(3) (12 digits). It decreases with N , but it is still satisfactory for N = 8 (6 digits). Central charges c are also extracted from the entanglement entropy using the Calabrese-Cardy formula, and agree with the theoretical values expected from the SU(N)1 Wess-Zumino-Witten CFTs.

cond-mat.str-el↗

arXiv2017

Linear Flavor-Wave Theory for Fully Antisymmetric $\mathrm{SU}(N)$ Irreducible Representations

The extension of the linear flavor-wave theory (LFWT) to fully antisymmetric irreducible representations (irreps) of $\mathrm{SU}(N)$ is presented in order to investigate the color order of $\mathrm{SU}(N)$ antiferromagnetic Heisenberg models in several two-dimensional geometries. The square, triangular and honeycomb lattices are considered with $m$ fermionic particles per site. We present two different methods: the first method is the generalization of the multiboson spin-wave approach to $\mathrm{SU}(N)$ which consists of associating a Schwinger boson to each state on a site. The second method adopts the Read and Sachdev bosons which are an extension of the Schwinger bosons that introduces one boson for each color and each line of the Young tableau. The two methods yield the same dispersing modes, a good indication that they properly capture the semi-classical fluctuations, but the first one leads to spurious flat modes of finite frequency not present in the second one. Both methods lead to the same physical conclusions otherwise: long-range Néel-type order is likely for the square lattice for $\mathrm{SU}(4)$ with two particles per site, but quantum fluctuations probably destroy order for more than two particles per site, with $N=2m$. By contrast, quantum fluctuations always lead to corrections larger than the classical order parameter for the tripartite triangular lattice (with $N=3m$) or the bipartite honeycomb lattice (with $N=2m$) for more than one particle per site, $m>1$, making the presence of color very unlikely except maybe for $m=2$ on the honeycomb lattice, for which the correction is only marginally larger than the classical order parameter.

cond-mat.str-el↗

arXiv2017

Exact diagonalization of $\mathrm{SU}(N)$ Heisenberg and AKLT chains using the full $\mathrm{SU}(N)$ symmetry

We present a method for the exact diagonalization of the $\mathrm{SU}(N)$ Heisenberg interaction Hamiltonian, using Young tableaux to work directly in each irreducible representation of the global $\mathrm{SU}(N)$ group. This generalized scheme is applicable to chains consisting of several particles per site, with any $\mathrm{SU}(N)$ symmetry at each site. Extending some of the key results of substitutional analysis, we demonstrate how basis states can be efficiently constructed for the relevant $\mathrm{SU}(N)$ subsector, which, especially with increasing values of $N$ or numbers of sites, has a much smaller dimension than the full Hilbert space. This allows us to analyze systems of larger sizes than can be handled by existing techniques. We apply this method to investigate the presence of edge states in $\mathrm{SU}(N)$ Heisenberg and AKLT Hamiltonians.

cond-mat.str-el↗

arXiv2016

Exact diagonalization of Heisenberg $SU(N)$ chains in the fully symmetric and antisymmetric representations

Motivated by recent experimental progress in the context of ultra-cold multi-color fermionic atoms in optical lattices, we have developed a method to exactly diagonalize the Heisenberg $SU(N)$ Hamiltonian with several particles per site living in a fully symmetric or antisymmetric representation of $SU(N)$. The method, based on the use of standard Young tableaux, takes advantage of the full $SU(N)$ symmetry, allowing one to work directly in each irreducible representations of the global $SU(N)$ group. Since the $SU(N)$ singlet sector is often much smaller than the full Hilbert space, this enables one to reach much larger system sizes than with conventional exact diagonalizations. The method is applied to the study of Heisenberg chains in the symmetric representation with two and three particles per site up to $N=10$ and up to 20 sites. For the length scales accessible to this approach, all systems except the Haldane chain ($SU(2)$ with two particles per site) appear to be gapless, and the central charge and scaling dimensions extracted from the results are consistent with a critical behaviour in the $SU(N)$ level $k$ Wess-Zumino-Witten universality class, where $k$ is the number of particles per site. These results point to the existence of a cross-over between this universality class and the asymptotic low-energy behavior with a gapped spectrum or a critical behavior in the $SU(N)$ level $1$ WZW universality class.

cond-mat.str-el↗

arXiv2016

SU(6) Heisenberg model on the honeycomb lattice: competition between plaquette and chiral order

We revisit the SU(6) Heisenberg model on the honeycomb lattice, which has been predicted to be a chiral spin liquid by mean-field theory [G. Szirmai et al., Phys. Rev. A 84, 011611 (2011)]. Using exact diagonalizations of finite clusters, infinite projected entangled pair states simulations, and variational Monte Carlo simulations based on Gutzwiller projected wave functions, we provide strong evidence in favour of the competing plaquette state, which was reported to be higher but close by in energy according to mean-field theory. This is further confirmed by the investigation of the model with a ring exchange term, which shows that there is a transition between the plaquette state and the chiral state at a finite value of the ring exchange term.

cond-mat.quant-gas↗

arXiv2016

Chiral spin liquids in triangular lattice SU(N) fermionic Mott insulators with artificial gauge fields

We show that, in the presence of a $π/2$ artificial gauge field per plaquette, Mott insulating phases of ultra-cold fermions with $SU(N)$ symmetry and one particle per site generically possess an extended chiral phase with intrinsic topological order characterized by a multiplet of $N$ low-lying singlet excitations for periodic boundary conditions, and by chiral edge states described by the $SU(N)_1$ Wess-Zumino-Novikov-Witten conformal field theory for open boundary conditions. This has been achieved by extensive exact diagonalizations for $N$ between $3$ and $9$, and by a parton construction based on a set of $N$ Gutzwiller projected fermionic wave-functions with flux $π/N$ per triangular plaquette. Experimental implications are briefly discussed.

cond-mat.quant-gas↗

arXiv2015

Variational Monte-Carlo investigation of SU($N$) Heisenberg chains

Motivated by recent experimental progress in the context of ultra-cold multi-color fermionic atoms in optical lattices, we have investigated the properties of the SU($N$) Heisenberg chain with totally antisymmetric irreducible representations, the effective model of Mott phases with $m < N$ particles per site. These models have been studied for arbitrary $N$ and $m$ with non-abelian bosonization [I. Affleck, Nuclear Physics B 265, 409 (1986); 305, 582 (1988)], leading to predictions about the nature of the ground state (gapped or critical) in most but not all cases. Using exact diagonalization and variational Monte-Carlo based on Gutzwiller projected fermionic wave functions, we have been able to verify these predictions for a representative number of cases with $N \leq 10$ and $m \leq N/2$, and we have shown that the opening of a gap is associated to a spontaneous dimerization or trimerization depending on the value of m and N. We have also investigated the marginal cases where abelian bosonization did not lead to any prediction. In these cases, variational Monte-Carlo predicts that the ground state is critical with exponents consistent with conformal field theory.

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