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arXiv:2503.15344 · Journal of Physics A Mathematical and Theoretical

Higher-order tacnodes in a free fermionic model

Andrea Maroncelli, Jean-Marie Stéphan

Published Oct 28, 2025• 0 citations
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Abstract

Abstract We investigate a one-dimensional free fermion model with nearest and next-nearest neighbor hopping, evolving in imaginary time from a product state with N consecutive fermions, and conditioned to go back to the same state after a given time. Such types of models are quantum reformulations of well-studied two-dimensional classical lattice models, which are known to give rise to limit shapes where expectation values of simple local observables, such as density, depend on position in an appropriate scaling limit. In the case of only nearest neighbor hopping, this model is known to have two fluctuating regions which can be tuned to merge depending on ratio between time and N . Correlations near the merger are governed by a so-called tacnode kernel. Here we show that another universal higher-order tacnode process can appear upon including the next-nearest neighbor term. We also discuss the limit shapes, and compute analytically the corresponding density profile.

DOI
10.1088/1751-8121/ae1879

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Abstract We investigate a one-dimensional free fermion model with nearest and next-nearest neighbor hopping, evolving in imaginary time from a product state with N consecutive fermions, and conditioned to go back to the same state after a given time. Such types of models are quantum reformulations of well-studied two-dimensional classical lattice models, which are known to give rise to limit shapes where expectation values of simple local observables, such as density, depend on position in an appropriate scaling limit. In the case of only nearest neighbor hopping, this model is known to have two fluctuating regions which can be tuned to merge depending on ratio between time and N . Correlations near the merger are governed by a so-called tacnode kernel. Here we show that another universal higher-order tacnode process can appear upon including the next-nearest neighbor term. We also discuss the limit shapes, and compute analytically the corresponding density profile.
In a recent contribution [P. Corboz, R. Or\'us, B. Bauer, and G. Vidal, Phys. Rev. B 81, 165104 (2010)] fermionic projected entangled-pair states (PEPSs) were used to approximate the ground state of free and interacting spinless fermion models, as well as the $t\text{\ensuremath{-}}J$ model. This paper revisits these three models in the presence of an additional next-nearest hopping amplitude in the Hamiltonian. First we explain how to account for next-nearest neighbor Hamiltonian terms in the context of fermionic PEPS algorithms based on simulating time evolution. Then we present benchmark calculations for the three models of fermions and compare our results against analytical, mean-field, and variational Monte Carlo results, respectively. Consistent with previous computations restricted to nearest neighbor Hamiltonians, we systematically obtain more accurate (or better converged) results for gapped phases than for gapless ones.
We discuss a reaction–diffusion model in one dimension subjected to an external driving force. Each lattice site may be occupied by at most one particle. The particles hop with rates (1±η)/2 to the right or left nearest neighbor site if it is vacant, and annihilate with rate one if it is occupied. The representation of an uncorrelated random initial state in terms of free fermions allows the calculation of multiple time-dependent higher order correlation functions of the local density. We outline the procedure using a field theoretic approach. We also compute the long time behavior of the density profile if the initial density profile is a step function. The fermion representation of the step function state also allows the calculation of multiple time-dependent correlation functions. Experimental applications of these results are pointed out.
We consider an extended Hubbard model of interacting fermions on a lattice. The fermion kinetic energy corresponds to a tight-binding Hamiltonian with nearest-neighbor (-t) and next-nearest-neighbor (t∣IH{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}t∣IH) hopping matrix elements. In addition to the on-site Hubbard interaction (U) we also consider a nearest-neighbor repulsion (V). We obtain the zero-temperature phase diagram of our model within the Hartree-Fock approximation. We consider ground states having charge- and spin-density-wave ordering as well as states with orbital antiferromagnetism or spin nematic order. The latter two states correspond to particle-hole binding with ${\mathrm{d}}_{{\mathrm{x}}^{2}\mathrm{\ensuremath{-}}{\mathrm{y}}^{2}}$ symmetry in the charge and spin channels, respectively. For t∣IH{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}t∣IH=0, only the charge-density-wave and spin-density-wave states are energetically stable. For nonzero t∣IH{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}t∣IH, we find that orbital antiferromagnetism (or spin nematic) order is stable over a finite portion of the phase diagram at weak coupling. This region of stability is seen to grow with increasing values of t∣IH{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}t∣IH.
We study a one-dimensional lattice model of generalized statistics in which particles have next-nearest-neighbor hopping between sites which depends on the occupation number at the intermediate site and a statistical parameter $\ensuremath{\phi}$. The model breaks parity and time-reversal symmetries and has four-fermion interactions if $\ensuremath{\phi}\ensuremath{\ne}0$. We first analyze the model using mean field theory and find that there are four Fermi points whose locations depend on $\ensuremath{\phi}$ and the filling $\ensuremath{\eta}$. We then study the modes near the Fermi points using the technique of bosonization. Based on the quadratic terms in the bosonized Hamiltonian, we find that the low-energy modes form two decoupled Tomonaga-Luttinger liquids with different values of the Luttinger parameters which depend on $\ensuremath{\phi}$ and $\ensuremath{\eta}$; further, the right- and left-moving modes of each system have different velocities. A study of the scaling dimensions of the cosine terms in the Hamiltonian indicates that the terms appearing in one of the Tomonaga-Luttinger liquids will flow under the renormalization group and the system may reach a nontrivial fixed point in the long distance limit. We examine the scaling dimensions of various charge density and superconducting order parameters to find which of them is the most relevant for different values of $\ensuremath{\phi}$ and $\ensuremath{\eta}$. Finally, we look at two-particle bound states that appear in this system and discuss their possible relevance to the properties of the system in the thermodynamic limit. Our work shows that the low-energy properties of this model of generalized statistics have a rich structure as a function of $\ensuremath{\phi}$ and $\ensuremath{\eta}$.
We study the magnetic properties of the two-dimensional Hubbard model with nearest-neighbor (t) and next-nearest-neighbor (t2{t}_{2}t2​) hopping and on-site repulsion U. We first calculate the mean-field phase diagram as a function of band filling and U. Because of the Van Hove singularity in the density of states, we find a ferromagnetic phase extending to zero U for certain band filling. For the half-filled band case, antiferromagnetism sets in at a finite value of U if t2{t}_{2}t2​\ensuremath{\ne}0. We study the behavior of spin-spin correlation functions for small lattices of up to N=64 atoms using Monte Carlo simulations, as well as exact diagonalization for N=4. Our results show enhanced ferromagnetic correlations in some regions, but apparently no ferromagnetic long-range order. In the half-filled case, our numerical results are consistent with a nonzero critical U. For a non-half-filled band our results suggest that there is no long-range order.
For a class of translation-invariant free-fermion systems including those with uniform nearest-neighbor hopping) on a d-dimensional L×⋯×L hypercubic lattice, we prove that, starting from an arbitrary pure initial state, the system equilibrates with respect to the coarse-grained density within a timescale of order L. This scaling is optimal, since there exist initial states whose equilibration requires time of order L. Our result establishes O(L) as the equilibration timescale, as is expected in normal macroscopic systems with a conserved quantity, such as total number of particles.