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arXiv:2310.11263 · Physical review. B./Physical review. B

Magnetic correlations of a doped and frustrated Hubbard model: Benchmarking the two-particle self-consistent theory against a quantum simulator

Guan-Hua Huang, Zhigang Wu

Published Sep 13, 2024• 3 citations
arXivHTMLPDF

Abstract

Recently, a quantum simulator for the two-dimensional Fermi-Hubbard model on an anisotropic triangular lattice has been realized, where both geometrical frustration and doping can be continuously tuned. Here, we provide a comprehensive comparison between the magnetic correlations calculated by the two-particle self-consistent (TPSC) theory and those measured in this quantum simulator, at temperatures comparable to the spin-exchange energy and Hubbard interactions comparable to the bandwidth. We find overall excellent agreements between the TPSC calculations and the measurements from the quantum simulator at all levels of frustration and doping. This is quite remarkable considering that the Hubbard model is already in the intermediate- to strong-coupling regime, for which very few methods yield reliable results. Our work showcases the potential of TPSC as a theoretical approach capable of providing reasonably accurate descriptions of the Hubbard model even at fairly strong interactions and when the frustration is present.

DOI
10.1103/physrevb.110.l100406

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Recently, a quantum simulator for the two-dimensional Fermi-Hubbard model on an anisotropic triangular lattice has been realized, where both geometrical frustration and doping can be continuously tuned. Here, we provide a comprehensive comparison between the magnetic correlations calculated by the two-particle self-consistent (TPSC) theory and those measured in this quantum simulator, at temperatures comparable to the spin-exchange energy and Hubbard interactions comparable to the bandwidth. We find overall excellent agreements between the TPSC calculations and the measurements from the quantum simulator at all levels of frustration and doping. This is quite remarkable considering that the Hubbard model is already in the intermediate- to strong-coupling regime, for which very few methods yield reliable results. Our work showcases the potential of TPSC as a theoretical approach capable of providing reasonably accurate descriptions of the Hubbard model even at fairly strong interactions and when the frustration is present.
Kinetic-to-magnetic frustration crossover and linear confinement in the doped triangular t−Jt-Jt−J model
Microscopically understanding competing orders in strongly correlated systems is a key challenge in modern quantum many-body physics. For example, the study of magnetic polarons and their relation to pairing in the Fermi-Hubbard model in different geometries remains one of the central questions, and may help to understand the mechanism underlying unconventional superconductivity in cuprates or transition metal dichalcogenides. With recent advances in analog quantum simulation of the Fermi-Hubbard model on non-bipartite lattices, frustrated physics can now be explored experimentally in systems lacking particle-hole symmetry. Here, we study the singly doped t−Jt-Jt−J model on the triangular lattice, focusing on the competition between kinetic and magnetic frustration as a function of temperature. In doublon doped systems, we uncover a crossover between Nagaoka-type ferromagnetic (FM) correlations at high temperature and exchange mediated antiferromagnetic (AFM) order around the doublon at low temperature. For hole doped systems, kinetic Haerter-Shastry-type AFM at high temperature as well as exchange interactions at low temperature favor 120∘120^{\circ}120∘ order, strengthening magnetic correlations compared to the undoped system. In the ground state, the presence of AFM correlations throughout a wide range of interactions indicates confinement of both types of dopants. In this regime we firmly establish the presence of linear confining potentials via energy scaling arguments, supporting the picture of geometric strings in the frustrated triangular t−Jt-Jt−J model.
The two-particle self-consistent approach (TPSC) is a method for the one-band Hubbard model that can be both numerically efficient and reliable. However, TPSC fails to yield physical results deep in the renormalized classical regime of the bidimensional Hubbard model where the spin correlation length becomes exponentially large. We address the limitations of TPSC with improved approaches that we call TPSC+ and TPSC+SFM (spin fluctuation mediated). In this work, we show that these improved methods satisfy the Mermin-Wagner theorem and the Pauli principle. We also show that they are valid in the renormalized classical regime of the two-dimensional Hubbard model, where they recover a generalized Stoner criterion at zero temperature in the antiferromagnetic phase. We discuss some limitations of the TPSC+ approach with regards to the violation of the fff-sum rule and conservation laws, which are solved within the TPSC+SFM framework. Finally, we benchmark the TPSC+ and TPSC+SFM approaches for the one-band Hubbard model in two dimensions and show how they have an overall better agreement with available diagrammatic Monte Carlo results than the original TPSC approach.
Microscopically understanding competing orders in strongly correlated systems is a key challenge in modern quantum many-body physics. For example, the study of magnetic polarons and their relation to pairing in the Fermi-Hubbard model in different geometries remains one of the central questions, and may help to understand the mechanism underlying unconventional superconductivity in cuprates or transition metal dichalcogenides. With recent advances in analog quantum simulation of the Fermi-Hubbard model on nonbipartite lattices, frustrated physics can now be explored experimentally in systems lacking particle-hole symmetry. Here, we study the singly doped $t\ensuremath{-}J$ model on the triangular lattice, focusing on the competition between kinetic and magnetic frustration as a function of temperature. In doublon doped systems, we uncover a crossover between Nagaoka-type ferromagnetic (FM) correlations at high temperature and exchange mediated antiferromagnetic (AFM) order around the doublon at low temperature. For hole-doped systems, kinetic Haerter-Shastry-type AFM at high temperature as well as exchange interactions at low temperature favor ${120}^{\ensuremath{\circ}}$ order, strengthening magnetic correlations compared to the undoped system. In the ground state, the presence of AFM correlations throughout a wide range of interactions indicates confinement of both types of dopants. In this regime we firmly establish the presence of linear confining potentials via energy scaling arguments, supporting the picture of geometric strings in the frustrated triangular $t\ensuremath{-}J$ model.
The semimetal to antiferromagnet quantum phase transition of the Hubbard model on the honeycomb lattice has come to the forefront in the context of the proposal that a semimetal to spin liquid transition can occur before the transition to the antiferromagnetic phase. To study the semimetal to antiferromagnet transition, we generalize the two-particle self-consistent (TPSC) approach to the honeycomb lattice (a structure that can be realized in graphene for example). We show that the critical interaction strength where the transition occurs is ${U}_{c}/t=3.79\ifmmode\pm\else\textpm\fi{}0.01$, quite close to the value ${U}_{c}/t=3.869\ifmmode\pm\else\textpm\fi{}0.013$ reported using large-scale quantum Monte Carlo simulations. This reinforces the conclusion that the semimetal to spin-liquid transition is preempted by the transition to the antiferromagnet. Since TPSC satisfies the Mermin-Wagner theorem, we find temperature-dependent results for the antiferromagnetic and ferromagnetic correlation lengths as well as the dependence of double occupancy and of the renormalized spin and charge interactions on the bare interaction strength. We also estimate the value of the crossover temperature to the renormalized classical regime as a function of interaction strength.
We present the nonequilibrium implementation of the two-particle self-consistent (TPSC) approach, which has been shown to provide a reliable equilibrium description of interacting lattice systems in the weak- and intermediate-correlation regime. This method captures the effects of local and nonlocal correlations in two- and higher-dimensional systems and satisfies the Mermin-Wagner theorem. We demonstrate the versatility of nonequilibrium TPSC with calculations of the time-dependent spin and charge response functions and the evolution of effective temperatures extracted from different correlation functions, after interaction ramps in the two-dimensional Hubbard model.
Analog quantum simulation provides a unique toolkit to investigate quantum many-body problems of whose solutions may be incredibly challenging. In particular, using ultracold fermionic atoms in optical lattices it realizes the Fermi Hubbard model, which is a fundamental model in condensed matter physics exhibiting properties relevant to many intriguing strongly-correlated systems. However, most quantum simulators of the Hubbard model have so far restricted to the square geometries. In addition, access to low temperatures where the exotic quantum phases are predicted to develop has still remained elusive. In this thesis, we engineer a novel optical lattice with dynamically tunable geometries and significantly reduced technical noises. We extend for the first time the study of quantum magnetism into the exotic phases in a Fermi Hubbard system with tunable geometric frustration. As we continuously tune our system from a square to a triangular lattice geometry, we observe a transition from a Néel antiferromagnet to a 120-degree spin spiral state. We then study, again for the first time, how doping affects a frustrated quantum magnet. To our surprise, we found an emergent ferromagnetic state with strong particle doping, which is absent on the hole-doped side and is in stark contrast with the particle-hole symmetric doping dependence in a square lattice. To shed light on the role of itinerant dopants in the emergence of these new magnetic properties, we furthermore leverage the unique single-particle resolution capabilities of our platform to probe higher-order density and spin correlations. These measurements hint at a new type of magnetism induced by kinetic frustration, which manifests as local antiferromagnetic correlations around hole dopants and and ferromagnetic correlations around particle dopants. Unlike magnetism induced by exchange or superexchange interactions, the energy associated with this mechanism is the kinetic energy, closely related to the famous rigorous results by Nagaoka. Furthermore, the dynamical tunability of the lattice allows adibatic engineering of quantum states in optical lattices. We discuss the preliminery efforts to prepare strongly-correlated states with significantly reduced temperatures, which may pave the way towards low temperature states in the Hubbard model.