Effective Low-energy Hamiltonian Research Articles

LiHoF4 as a spin-half non-standard quantum Ising system

LiHoF_{4}4 is a magnetic material known for its Ising-type anisotropy, making it a model system for studying quantum magnetism. However, the theoretical description of LiHoF_{4}4 using the quantum Ising model has shown discrepancies in its phase diagram, particularly in the regime dominated by thermal fluctuations. In this study, we investigate the role of off-diagonal dipolar terms in LiHoF_{4}4, previously neglected, in determining its properties. We analytically derive the low-energy effective Hamiltonian of LiHoF_{4}4, including the off-diagonal dipolar terms perturbatively, both in the absence and presence of a transverse field. Our results encompass the full phase diagram, confirming the significance of the off-diagonal dipolar terms in reducing the zero-field critical temperature and determining the critical temperature’s dependence on the transverse field. We also highlight the sensitivity of this mechanism to the crystal structure by comparing our calculations with the Fe_{8}8 system.

Dome structure in pressure dependence of superconducting transition temperature for HgBa2Ca2Cu3O8 : Studies by ab initio low-energy effective Hamiltonian

The superconducting (SC) cuprate HgBa2Ca2Cu3O8 (Hg1223) has the highest SC transition temperature Tc among cuprates at ambient pressure Pamb, namely, Tcopt≃138 K experimentally at the optimal hole doping concentration. Tcopt further increases under pressure P and reaches 164 K at optimal pressure Popt≃30 GPa, then Tcopt decreases with increasing P>Popt generating a dome structure [Gao , ]. This nontrivial and nonmonotonic P dependence of Tcopt calls for a theoretical understanding and mechanism. To answer this open question, we consider the low-energy effective Hamiltonian (LEH) for the antibonding (AB) Cu3dx2−y2/O2pσ band derived generally for the cuprates. In the AB LEH for cuprates with Nℓ≤2 laminated CuO2 planes between block layers, it was proposed that Tcopt is determined by a universal scaling Tcopt≃0.16|t1|FSC [Schmid , ], where t1 is the nearest-neighbor hopping, and the SC order parameter at optimal hole doping FSC mainly depends on the ratio u=U/|t1| where U is the onsite effective Coulomb repulsion: The u dependence of FSC has a peak at uopt≃8.5 and a steep decrease with decreasing u in the region u<uopt irrespective of materials dependent on other parameters. In this paper, we show that |t1| increases with P, whereas u decreases with P in the Hamiltonian of Hg1223. Based on these facts, we show that the domelike P dependence of Tcopt can emerge at least qualitatively if we assume Hg1223 with Nℓ=3 follows the same universal scaling for Tcopt, and Hg1223 is located at the slightly strong coupling region u≳uopt at Pamb and u≃uopt at Popt by taking account of expected corrections to our calculation. The consequence of these assumptions is the following: With increasing P within the range P<Popt, the increase in Tcopt is accounted for by the increase in |t1|, whereas FSC is insensitive to the decrease in u around ≃uopt and hence to P as well. At P>Popt, the decrease in Tcopt is accounted for by the decrease in u below uopt, which causes a rapid decrease in FSC dominating over the increase in |t1|. We further argue the appropriateness of these assumptions based on the insight from studies on other cuprate compounds in the literature. In addition, we discuss the dependencies of u and |t1| on each crystal parameter (CP), which provides hints for design of even higher Tcopt materials. Published by the American Physical Society 2024

Topological nano-switches in higher-order topological insulators

We consider multi-terminal transport through a flake of rectangular shape of a two-dimensional topological insulator in the presence of an in-plane magnetic field. This system has been shown to be a second-order topological insulator, thus exhibiting corner states at its boundaries. The position of the corner states and their decay length can be controlled by the direction of the magnetic field. In the leads we assume that the magnetic field is absent and therefore we have helical one-dimensional propagating states characteristic of the spin-Hall effect. Using a low-energy effective Hamiltonian we show analytically that, in a two-terminal setup, transport can be turned on and off by a rotation of the in-plane magnetic field. Similarly, in a three terminal configuration, the in-plane magnetic field can be used to turn on and off the transmission between neighbouring contacts, thus realising a directional switch. Analytical calculations are supplemented by a numerical finite-difference method. For small values of the Fermi energy and field strength, the analytical results agree exceptionally well with the numerics. The effect of disorder is also addressed in the numerical approach. We find that the switching functionality is remarkably robust to the presence of strong disorder stemming from the topological nature of the states contributing to the electron transport.

Excitonic enhancement of cavity-mediated interactions in a two-band Hubbard model

We study cavity-mediated interactions that are generated in a two-dimensional two-band Hubbard model coupled to an optical cavity, when it is driven in-gap by a strong laser. Starting from a Floquet description of the driven system, we derive effective low-energy Hamiltonians by projecting out the high-energy degrees of freedom and treating intrinsic interactions on a mean-field level. We then investigate how the emergence of high-energy Frenkel excitons from the electronic interband coupling, which form near the upper electronic band, affects the interactions as well as the laser-induced Floquet renormalization of the electronic band structure. Cavity-mediated interactions are enhanced strongly when the light couples to an excitonic transition. Additionally, the interaction as well as the Floquet renormalization are strongly broadened in reciprocal space, which could further boost the impact of cavity-mediated interactions on the driven-dissipative steady state. Published by the American Physical Society 2024

DFT2kp: Effective kp models from ab-initio data

The k\cdot pk⋅p method, combined with group theory, is an efficient approach to obtain the low energy effective Hamiltonians of crystalline materials. Although the Hamiltonian coefficients are written as matrix elements of the generalized momentum operator \pi= p+ p_{SOC}π=p+pSOC (including spin-orbit coupling corrections), their numerical values must be determined from outside sources, such as experiments or ab initio methods. Here, we develop a code to explicitly calculate the Kane (linear in crystal momentum) and Luttinger (quadratic in crystal momentum) parameters of k\cdot pk⋅p effective Hamiltonians directly from ab initio wavefunctions provided by Quantum ESPRESSO. Additionally, the code analyzes the symmetry transformations of the wavefunctions to optimize the final Hamiltonian. This is an optional step in the code, where it numerically finds the unitary transformation UU that rotates the basis towards an optimal symmetry-adapted representation informed by the user. Throughout the paper, we present the methodology in detail and illustrate the capabilities of the code applying it to a selection of relevant materials. Particularly, we show a “hands-on” example of how to run the code for graphene (with and without spin-orbit coupling). The code is open source and available at https://gitlab.com/dft2kp/dft2kp.

Codebase release 0.0 for DFT2kp

The k\cdot pk⋅p method, combined with group theory, is an efficient approach to obtain the low energy effective Hamiltonians of crystalline materials. Although the Hamiltonian coefficients are written as matrix elements of the generalized momentum operator \pi= p+ p_{SOC}π=p+pSOC (including spin-orbit coupling corrections), their numerical values must be determined from outside sources, such as experiments or ab initio methods. Here, we develop a code to explicitly calculate the Kane (linear in crystal momentum) and Luttinger (quadratic in crystal momentum) parameters of k\cdot pk⋅p effective Hamiltonians directly from ab initio wavefunctions provided by Quantum ESPRESSO. Additionally, the code analyzes the symmetry transformations of the wavefunctions to optimize the final Hamiltonian. This is an optional step in the code, where it numerically finds the unitary transformation UU that rotates the basis towards an optimal symmetry-adapted representation informed by the user. Throughout the paper, we present the methodology in detail and illustrate the capabilities of the code applying it to a selection of relevant materials. Particularly, we show a “hands-on” example of how to run the code for graphene (with and without spin-orbit coupling). The code is open source and available at https://gitlab.com/dft2kp/dft2kp.

Update of [formula omitted]: Newly added functions and methods in versions 2 and 3

HΦ [aitch-phi] is an open-source software package of numerically exact and stochastic calculations for a wide range of quantum many-body systems. In this paper, we present the newly added functions and the implemented methods in vers. 2 and 3. In ver. 2, we implement spectrum calculations by the shifted Krylov method, and low-energy excited state calculations by the locally optimal block preconditioned conjugate gradient (LOBPCG) method. In ver. 3, we implement the full diagonalization method using ScaLAPACK and GPGPU computing via MAGMA. We also implement a real-time evolution method and the canonical thermal pure quantum (cTPQ) state method for finite-temperature calculations. The Wannier90 format for specifying the Hamiltonians is also implemented. Using the Wannier90 format, it is possible to perform the calculations for the abinitio low-energy effective Hamiltonians of solids obtained by the open-source software RESPACK. We also update Standard mode—simplified input format in HΦ—to use these functions and methods. We explain the basics of the implemented methods and how to use them. Program summaryProgram Title:HΦ [aitch-phi]CPC Library link to program files:https://doi.org/10.17632/vnfthtyctm.2Developer's repository link:https://github.com/issp-center-dev/HPhiLicensing provisions: GPLv3Programming language: C, FortranJournal reference of previous version: Comput. Phys. Commun. 217 (2017) 180–192Does the new version supersede the previous version?: Yes. The latest version has compatibility with the old versions. Although most functions are available in newer versions, some redundant/unnecessary functions were abolished in newer versions.Reasons for the new version: Implementation of new functions and methods, development of utilities, and bug fixes.Summary of revisions: We added new functions to obtain excited spectrum, low-energy excited states, and time-dependent physical quantities. We also implemented the full diagonalization method by using MAGMA on GPGPUs and finite-temperature simulations by using the canonical thermal pure quantum state method. In addition, we developed utilities for connection with RESPACK and a submodule for a generator of input files.Nature of problem: Physical properties in quantum lattice models with finite system sizesSolution method: In HΦ, we implemented several numerical methods such as the Lanczos method, the full diagonalization method, the locally optimal block preconditioned conjugate gradient (LOBPCG) method, the real-time evolution method based on the Taylor expansion, the shifted Krylov method, and the microcanonical/canonical thermal pure quantum state method.

Revisiting electromagnetic response of superconductors in mean-field approximation

In the standard mean-field treatment of superconductors, the electron-electron interactions are assumed to be written in terms of local density operators. However, more general interactions, such as pair-hopping interactions, may exist or may be generated in a low-energy effective Hamiltonian. In this work, we study the effect of correlated hopping interactions toward the electromagnetic response of superconductors. When only the Hamiltonian after the mean-field approximation is provided, one cannot unambiguously determine its electromagnetic response whenever such interactions are allowed. This work demonstrates that such interactions induce additional terms in the current operator, leading to modifications in the Meissner weight and optical conductivities that deviate from conventional expectations. These results underscore the need for caution when incorporating gauge fields into the Bogoliubov–de Gennes Hamiltonian. Published by the American Physical Society 2024

Photogalvanic response in multi-Weyl semimetals

We investigate the dependence of the photogalvanic response of a twofold degenerate multi-Weyl semimetal on its topological charge, tilt, and chemical potential. We derive analytical expressions for the shift and injection conductivities for tilted charge-n Weyl points using a low-energy two-band effective Hamiltonian. We compute the response for more realistic tight-binding models of a double-Weyl semimetal with broken time-reversal symmetry to find significant deviations from the effective low-energy continuum model predictions. We analyze several different limits of these models, describe the nature of these deviations, and provide estimates of their dependence on the frequency and other model parameters. Our analysis provides a simple explanation for the first-principle calculation based frequency dependence of the injection current in SrSi2. We also obtain analytical results for the charge-4 Weyl semimetal using a new approach, providing all relevant information about the nature of its second-order dc response and the precise condition for observing quantized circular photogalvanic effect. This approach can easily be extended to a systematic study of second harmonic generation and first-order optical conductivity in charge-4 Weyl semimetals. Published by the American Physical Society 2024

Ground-state phase diagram, symmetries, excitation spectra and finite-frequency scaling of the two-mode quantum Rabi model

We investigate the two-mode quantum Rabi model (QRM) describing the interaction between a two-level atom and a two-mode cavity field. The quantum phase transitions are found when the ratio η of transition frequency of atom to frequency of cavity field approaches infinity. We apply the Schrieffer–Wolff (SW) transformation to derive the low-energy effective Hamiltonian of the two-mode QRM, thus yielding the critical point and rich phase diagram of quantum phase transitions. The phase diagram consists of four regions: a normal phase, an electric superradiant phase, a magnetic superradiant phase and an electromagnetic superradiant phase. The quantum phase transition between the normal phase and the electric (magnetic) superradiant phase is of second order and associates with the breaking of the discrete Z 2 symmetry. On the other hand, the phase transition between the electric superradiant phase and the magnetic superradiant phase is of first order and relates to the breaking of the continuous U(1) symmetry. Several important physical quantities, for example the excitation energy and average photon number in the four phases, are derived. We find that the excitation spectra exhibit the Nambu–Goldstone mode. We calculate analytically the higher-order correction and finite-frequency exponents of relevant quantities. To confirm the validity of the low-energy effective Hamiltonians analytically derived by us, the finite-frequency scaling relation of the averaged photon numbers is calculated by numerically diagonalizing the two-mode quantum Rabi Hamiltonian.

Interface tool from Wannier90 to RESPACK: wan2respack

We develop the interface tool wan2respack, which connects RESPACK (software that derives the low-energy effective Hamiltonians of solids) with Wannier90 (software that constructs Wannier functions). wan2respack converts the Wannier functions obtained by Wannier90 into those used in RESPACK, which is then used to derive the low-energy effective Hamiltonians of solids. In this paper, we explain the basic usage of wan2respack and show its application to standard compounds of correlated materials, namely, the correlated metal SrVO3 and the high-Tc superconductor La2CuO4. Furthermore, we compare the low-energy effective Hamiltonians of these compounds using Wannier functions obtained by Wannier90 and those obtained by RESPACK. We confirm that both types of Wannier functions give the same Hamiltonians. This benchmark comparison demonstrates that wan2respack correctly converts Wannier functions in the Wannier90 format into those in the RESPACK format. Program summaryProgram title:wan2respackCPC Library link to program files:https://doi.org/10.17632/6zfj2dkv5b.1Licensing provisions: GNU General Public License version 3Programming language:Fortran and python3External routines/libraries:Quantum ESPRESSO (version 6.6), Wannier90 (version 3.0.0), RESPACK (version 20200113), tomli.Nature of problem: Using RESPACK, one can derive low-energy effective Hamiltonians of solids from maximally localized Wannier functions. However, due to the differences in the representation of Wannier functions, the Wannier functions obtained by Wannier90 cannot be directly used in RESPACK.Solution method:wan2respack converts the Wannier functions in the Wannier90 format into those in the RESPACK format. Using the converted Wannier functions, one can derive the low-energy effective Hamiltonians using RESPACK.

Superconductivity, Charge Density Wave, and Supersolidity in Flat Bands with a Tunable Quantum Metric.

Predicting the fate of an interacting system in the limit where the electronic bandwidth is quenched is often highly nontrivial. The complex interplay between interactions and quantum fluctuations driven by the band geometry can drive competition between various ground states, such as charge density wave order and superconductivity. In this work, we study an electronic model of topologically trivial flat bands with a continuously tunable Fubini-Study metric in the presence of on-site attraction and nearest-neighbor repulsion, using numerically exact quantum MonteCarlo simulations. By varying the electron filling and the minimal spatial extent of the localized flat-band Wannier wave functions, we obtain a number of intertwined orders. These include a phase with coexisting charge density wave order and superconductivity, i.e., a supersolid. In spite of the nonperturbative nature of the problem, we identify an analytically tractable limit associated with a "small" spatial extent of the Wannier functions and derive a low-energy effective Hamiltonian that can well describe our numerical results. We also provide unambiguous evidence for the violation of any putative lower bound on the zero-temperature superfluid stiffness in geometrically nontrivial flat bands.

Disorder-induced anomalous Hall effect in type-I Weyl metals: Connection between the Kubo-Streda formula in the spin and chiral basis

We study the anomalous Hall effect (AHE) in tilted Weyl metals with weak Gaussian disorder under the Kubo-Streda formalism in this work. To separate the three different contributions, namely the intrinsic, side jump and skew scattering contribution, it is usually considered necessary to go to the eigenstate (chiral) basis of the Kubo-Streda formula. However, it is more straight-forward to compute the total Hall current in the spin basis. For the reason, we develop a systematic and transparent scheme to separate the three different contributions in the spin basis for relativistic systems by building a one-to-one correspondence between the Feynman diagrams of the different mechanisms in the chiral basis and the products of the symmetric and anti-symmetric part of the polarization operator in the spin basis. We obtain the three contributions of the AHE in tilted Weyl metals by this scheme and found that the side jump contribution exceeds both the intrinsic and skew scattering contribution for the low-energy effective Hamiltonian. We compared the anomalous Hall current obtained from our scheme with the results from the semi-classical Boltzmann equation approach under the relaxation time approximation and found that the results from the two approaches agree with each other in the leading order of the tilting velocity.

Two-band description of the strong ‘spin’-orbit coupled one-dimensional hole gas in a cylindrical Ge nanowire

The low-energy effective Hamiltonian of the strong ‘spin’-orbit coupled one-dimensional hole gas in a cylindrical Ge nanowire in the presence of a strong magnetic field is studied both numerically and analytically. Basing on the Luttinger–Kohn Hamiltonian in the spherical approximation, we show this strong ‘spin’-orbit coupled one-dimensional hole gas can be accurately described by an effective two-band Hamiltonian , as long as the magnetic field is purely longitudinal or purely transverse. The explicit magnetic field dependent expressions of the ‘spin’-orbit coupling and the effective g-factor are given. When the magnetic field is applied in an arbitrary direction, the two-band Hamiltonian description is still a good approximation.

Gap opening mechanism for correlated Dirac electrons in organic compounds α−(BEDT−TTF)2I3 and α−(BEDT−TSeF)2I3

To determine how electron correlations open a gap in two-dimensional massless Dirac electrons in the organic compounds $\alpha$-(BEDT-TTF)$_2$I$_3$ [$\alpha$-(ET)$_2$I$_3$] and $\alpha$-(BEDT-TSeF)$_2$I$_3$ [$\alpha$-(BETS)$_2$I$_3$], we derive and analyze $ab$ $initio$ low-energy effective Hamiltonians for these two compounds. We find that the horizontal stripe charge ordering opens a gap in the massless Dirac electrons in $\alpha$-(ET)$_2$I$_3$, while an insulating phase without explicit symmetry breaking appears in $\alpha$-(BETS)$_2$I$_3$. We clarify that the combination of the anisotropic transfer integrals and the electron correlations induces a dimensional reduction in the spin correlations, i.e., one-dimensional spin correlations develop in $\alpha$-(BETS)$_2$I$_3$. We show that the one-dimensional spin correlations open a gap in the massless Dirac electrons. Our finding paves the way for opening gaps for massless Dirac electrons using strong electronic correlations.

Ab initio low-energy effective Hamiltonians for the high-temperature superconducting cuprates Bi2Sr2CuO6, Bi2Sr2CaCu2O8, HgBa2CuO4, and CaCuO2

We derive $ab$ $initio$ low-energy effective Hamiltonians (LEH) for high-temperature superconducting (SC) copper oxides Bi$_2$Sr$_2$CuO$_6$ (Bi2201, $N_{\ell}=1$, $T_c^{\rm exp} \sim 10$ K), Bi$_2$Sr$_2$CaCu$_2$O$_8$ (Bi2212, $N_{\ell}=2$, $T_c^{\rm exp} \sim 84$ K), HgBa$_2$CuO$_4$ (Hg1201, $N_{\ell}=1$, $T_c^{\rm exp} \sim 90$ K) and CaCuO$_2$ (Ca11, $N_{\ell}=\infty$, $T_c^{\rm exp} \sim 110$ K), with different experimental optimal SC transition temperature $T_c^{\rm exp}$ and number $N_{\ell}$ of laminated CuO$_2$ planes between the two neighboring block layers. We apply the latest methodology of the multiscale $ab$ $initio$ scheme for correlated electron systems (MACE), and focus on the LEH consisting of one antibonding (AB) Cu$3d_{x^2-y^2}$/O$2p_{\sigma}$ orbital centered on each Cu atom. We discuss prominent features of this LEH: (1) The ratio $U/|t_1|$ between the onsite effective Coulomb repulsion (ECR) $U$ and amplitude of nearest neighbour hopping $t_1$ increases with $T^{\rm exp}_c$ and $N_{\ell}$, consistently with the expected increase in $d$-wave SC correlation function $P_{dd}$ with $U/|t_1|$. One possible cause of the increase of $U/|t_1|$ is the replacement of apical O atoms by Cu atoms from neighbouring CuO$_2$ planes when $N_{\ell}$ increases. Furthermore, we show that the increase in distance between Cu and apical O atoms decreases the effective screening (ES) by electrons outside of the LEH and increases $U/|t_1|$. (2) For Hg1201 and Ca11, we show that $U/|t_1|$ decreases when hole doping per AB orbital $\delta$ increases, which may partly account for the disappearance of SC when $\delta$ exceeds the optimal value in experiment. (3) For $N_{\ell} \geq 2$, off-site inter-CuO$_2$ plane ECR is comparable to off-site intra-CuO$_2$ plane ECR. We discuss contributions of inter-CuO$_2$ plane ECR to both $P_{dd}$ and the stability of the SC state.

Effective Low-Energy Hamiltonians and Unconventional Landau-Level Spectrum of Monolayer C3N.

We derive low-energy effective k·p Hamiltonians for monolayer C3N at the Γ and M points of the Brillouin zone, where the band edge in the conduction and valence band can be found. Our analysis of the electronic band symmetries helps to better understand several results of recent ab initio calculations for the optical properties of this material. We also calculate the Landau-level spectrum. We find that the Landau-level spectrum in the degenerate conduction bands at the Γ point acquires properties that are reminiscent of the corresponding results in bilayer graphene, but there are important differences as well. Moreover, because of the heavy effective mass, n-doped samples may host interesting electron-electron interaction effects.

Josephson junctions of two-dimensional time-reversal invariant superconductors: Signatures of the topological phase

We determine the current-phase relation (CPR) of two-terminal configurations of Josephson junctions containing two-dimensional (2D) time-reversal invariant topological superconductors (TRITOPS), including TRITOPS-TRITOPS, as well as junctions between topological and non-topological superconductors (TRITOPS-S). We focus on long junctions for which several channels intervene in the tunneling coupling through the junction. We present a description of the topological edge modes for different TRITOPS models including $p$-wave pairing and the combination of $s$-wave pairing with spin-orbit coupling. We derive effective low-energy Hamiltonians to describe the Josephson junction, which can be solved analytically to explain the contribution of the edge states to the Josephson current as a function of the phase bias. We find that edge-modes yield singular corrections to the CPR for both junction types. The primary effects occur for the response of the Majorana zero-modes at half-flux quantum phase $\phi\approx \pi$ in TRITOPS-TRITOPS junctions and for integer flux quantum phase $\phi \approx 0$ for TRITOPS-S junctions, respectively. The former effect is particularly strong two-component nematic superconductors. The latter effect leads to a spontaneously broken time-reversal symmetry in the TRITOPS-S junction and to a breakdown of the bulk-boundary correspondence.

Spin-valley polarized transport in a field-controllable bilayer silicene superlattice

We have theoretically investigated the spin-valley asymmetric transport of massive Dirac fermions in the field-controllable bilayer silicene superlattices. The spin-valley dependent ballistic transmission, conductance, and polarization have been systematically calculated by formulating the scattering matrix method for the completed four-band low-energy effective Hamiltonian. Our results uncover that for a single valley transport, near-perfect spin polarization and its perfect switching could be efficiently modulated by the gate field engineering. Under the one-dimensional periodic field modulation, two types of flat bands with less dispersion and, importantly, the perfect contrast in the spin-dependent subbands are observed for the bilayer silicene superlattice. Together with its larger spin-orbit coupling and better stability, these spin-valley asymmetric characteristics engineered by the gate field indicate that the field-controllable bilayer silicene could be a potential component candidate to achieve a fully spin-valley polarized beam for quantum logic applications.

Variational Adiabatic Gauge Transformation on Real Quantum Hardware for Effective Low-Energy Hamiltonians and Accurate Diagonalization

Effective low-energy theories represent powerful theoretical tools to reduce the complexity in modeling interacting quantum many-particle systems. However, common theoretical methods rely on perturbation theory, which limits their applicability to weak interactions. Here we introduce the Variational Adiabatic Gauge Transformation (VAGT), a nonperturbative hybrid quantum algorithm that can use nowadays quantum computers to learn the variational parameters of the unitary circuit that brings the Hamiltonian to either its block-diagonal or full-diagonal form. If a Hamiltonian can be diagonalized via a shallow quantum circuit, then VAGT can learn the optimal parameters using a polynomial number of runs. The accuracy of VAGT is tested through numerical simulations, as well as simulations on Rigetti and IonQ quantum computers.