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arXiv:2301.02939 · Physical review. B./Physical review. B

Mean-field treatment of the long-range transverse field Ising model with fermionic Gaussian states

Michael Kaicher, Davide Vodola, Simon B. Jäger

Published Apr 25, 2023• 11 citations
arXivHTMLPDF

Abstract

We numerically study the one-dimensional long-range transverse field Ising model (TFIM) in the antiferromagnetic (AFM) regime at zero temperature using generalized Hartree-Fock (GHF) theory. The spin-spin interaction extends to all spins in the lattice and decays as $1/{r}^{\ensuremath{\alpha}}$, where $r$ denotes the distance between two spins and $\ensuremath{\alpha}$ is a tunable exponent. We map the spin operators to Majorana operators and approximate the ground state of the Hamiltonian with a fermionic Gaussian state (FGS). Using this approximation, we calculate the ground-state energy and the entanglement entropy, which allows us to map the phase diagram for different values of $\ensuremath{\alpha}$. In addition, we compute the scaling behavior of the entanglement entropy with the system size to determine the central charge at criticality for the case of $\ensuremath{\alpha}>1$. For $\ensuremath{\alpha}<1$ we find a logarithmic divergence of the entanglement entropy even far away from the critical point, a feature of systems with long-range interactions. We provide a detailed comparison of our results to outcomes of density matrix renormalization group (DMRG) and the linked cluster expansion (LCE) methods. In particular, we find excellent agreement of GHF with DMRG and LCE in the weak long-range regime $\ensuremath{\alpha}\ensuremath{\ge}1$, and qualitative agreement with DMRG in the strong-long range regime $\ensuremath{\alpha}\ensuremath{\le}1$. Our results highlight the power of the computationally efficient GHF method in simulating interacting quantum systems.

DOI
10.1103/physrevb.107.165144

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We numerically study the one-dimensional long-range transverse field Ising model (TFIM) in the antiferromagnetic (AFM) regime at zero temperature using generalized Hartree-Fock (GHF) theory. The spin-spin interaction extends to all spins in the lattice and decays as 1/r\ensuremathα1/{r}^{\ensuremath{\alpha}}1/r\ensuremathα, where rrr denotes the distance between two spins and \ensuremathα\ensuremath{\alpha}\ensuremathα is a tunable exponent. We map the spin operators to Majorana operators and approximate the ground state of the Hamiltonian with a fermionic Gaussian state (FGS). Using this approximation, we calculate the ground-state energy and the entanglement entropy, which allows us to map the phase diagram for different values of \ensuremathα\ensuremath{\alpha}\ensuremathα. In addition, we compute the scaling behavior of the entanglement entropy with the system size to determine the central charge at criticality for the case of \ensuremathα>1\ensuremath{\alpha}>1\ensuremathα>1. For \ensuremathα<1\ensuremath{\alpha}<1\ensuremathα<1 we find a logarithmic divergence of the entanglement entropy even far away from the critical point, a feature of systems with long-range interactions. We provide a detailed comparison of our results to outcomes of density matrix renormalization group (DMRG) and the linked cluster expansion (LCE) methods. In particular, we find excellent agreement of GHF with DMRG and LCE in the weak long-range regime \ensuremathα\ensuremath≥1\ensuremath{\alpha}\ensuremath{\ge}1\ensuremathα\ensuremath≥1, and qualitative agreement with DMRG in the strong-long range regime \ensuremathα\ensuremath≤1\ensuremath{\alpha}\ensuremath{\le}1\ensuremathα\ensuremath≤1. Our results highlight the power of the computationally efficient GHF method in simulating interacting quantum systems.
We investigate a generalized antiferromagnetic cluster XY\mathit{XY}XY model in a transverse magnetic field, where long-range interactions decay algebraically with distance. This model can be exactly solvable within a free fermion framework. By analyzing the gap, we explicitly derive the critical exponents $\ensuremath{\nu}$ and zzz, finding that the relationship $\ensuremath{\nu}z=1$ still holds. However, the values of $\ensuremath{\nu}$ and zzz depend on the decaying exponent $\ensuremath{\alpha}$, in contrast to those for the quantum long-range antiferromagnetic Ising chain. To optimize scaling behavior, we verify these critical exponents using correlation functions and fidelity susceptibility, achieving excellent data collapse across various system sizes by adjusting fitting parameters. Finally, we compute the entanglement entropy at the critical point to determine the central charge ccc, and find it also varies with $\ensuremath{\alpha}$. This study provides insights into the unique effect of long-range cluster interactions on the critical properties of quantum spin systems.
We consider the Ising model in a transverse field with long-range antiferromagnetic interactions that decay as a power law with their distance. We study both the phase diagram and the entanglement properties as a function of the exponent of the interaction. The phase diagram can be used as a guide for future experiments with trapped ions. We find two gapped phases, one dominated by the transverse field, exhibiting quasi-long-range order, and one dominated by the long-range interaction, with long-range Néel ordered ground states. We determine the location of the quantum critical points separating those two phases. We determine their critical exponents and central charges. In the phase with quasi-long-range order the ground states exhibit exotic corrections to the area law for the entanglement entropy coexisting with gapped entanglement spectra.
We study the low-energy properties of the long-range random transverse-field Ising chain with ferromagnetic interactions decaying as a power of the distance. Using variants of the strong-disorder renormalization group method, the critical behavior is found to be controlled by a strong-disorder fixed point with a finite dynamical exponent . Approaching the critical point, the correlation length diverges exponentially. In the critical point, the magnetization shows an -independent logarithmic finite-size scaling and the entanglement entropy satisfies the area law. These observations are argued to hold for other systems with long-range interactions, even in higher dimensions.
The quantum-critical properties of the transverse-field Ising model with algebraically decaying interactions are investigated by means of stochastic series expansion quantum Monte Carlo, on both the one-dimensional linear chain and the two-dimensional square lattice. We extract the critical exponents $\ensuremath{\nu}$ and $\ensuremath{\beta}$ as a function of the decay exponent of the long-range interactions. For ferromagnetic Ising interactions, we resolve the limiting regimes known from field theory, ranging from the nearest-neighbor Ising to the long-range Gaussian universality classes, as well as the intermediate regime with continuously varying critical exponents. In the long-range Gaussian regime, we treat the effect of dangerous irrelevant variables on finite-size scaling forms. For antiferromagnetic and therefore competing Ising interactions, the stochastic series expansion algorithm displays growing autocorrelation times leading to a reduced performance. Nevertheless, our results are consistent with the nearest-neighbor Ising universality for all investigated interaction ranges both on the linear chain and the square lattice.
The criticality of the ferromagnetic long-range transverse-field Ising chain (LRTFIC) with periodic boundary conditions and power-law decaying Ising coupling is studied using the Density Matrix Renormalization Group (DMRG). The ground states of the systems at different values of the transverse field hhh are obtained using DMRG. From these ground states, a quantity called crosscap overlap can be extracted efficiently. This thesis suggests that the crosscap overlap can serve as an indicator to identify the critical points of the LRTFIC. The estimated critical points are further verified by data collapse. For the decay exponent α=3.5\alpha=3.5α=3.5, the LRTFIC belongs to the short-range regime. DMRG is performed on systems with different sizes up to N=280N=280N=280. The critical point is estimated to be hc≈1.25h_c \approx 1.25hc​≈1.25, at which the crosscap overlaps approach the exact value of the short-range Ising chain as the system size grows.
We study numerically the entanglement entropy and spatial correlations of the one-dimensional transverse-field Ising model with three different perturbations. First, we focus on the out-of-equilibrium steady state with an energy current passing through the system. By employing a variety of matrix-product state based methods, we confirm the phase diagram and compute the entanglement entropy. Second, we consider a small perturbation that takes the system away from integrability and calculate the correlations, the central charge, and the entanglement entropy. Third, we consider periodically weakened bonds, exploring the phase diagram and entanglement properties first in the situation when the weak and strong bonds alternate (period two bonds) and then the general situation of a period of nnn bonds. In the latter case we find a critical weak bond that scales with the transverse field as ${J}_{c}^{\ensuremath{'}}/J={(h/J)}^{n}$, where JJJ is the strength of the strong bond, ${J}^{\ensuremath{'}}$ is that of the weak bond, and hhh is the transverse field. We explicitly show that the energy current is not a conserved quantity in this case.