Ground-State Properties of the Hydrogen Chain: Dimerization, Insulator-to-Metal Transition, and Magnetic Phases

Accurate and predictive computations of the quantum-mechanical behavior of many interacting electrons in realistic atomic environments are critical for the theoretical design of materials with desired properties, and they require solving the grand-challenge problem of the many-electron Schrödinger equation. An infinite chain of equispaced hydrogen atoms is perhaps the simplest realistic model for a bulk material, embodying several central themes of modern condensed-matter physics and chemistry while retaining a connection to the paradigmatic Hubbard model. Here, we report a combined application of cutting-edge computational methods to determine the properties of the hydrogen chain in its quantum-mechanical ground state. Varying the separation between the nuclei leads to a rich phase diagram, including a Mott phase with quasi-long-range antiferromagnetic order, electron density dimerization with power-law correlations, an insulator-to-metal transition, and an intricate set of intertwined magnetic orders.

Popular Summary

Materials in which electrons strongly interact with one another exhibit a fascinating variety of structural, electronic, and magnetic properties. Capturing the many underlying effects responsible for these properties is essential for understanding and predicting material behavior but requires a reliable treatment of the grand-challenge problem of solving the many-electron Schrödinger equation. We take a step toward this goal by undertaking an in-depth study of the ground state and physical properties of what is perhaps the simplest realistic model for a bulk material: an infinite chain of equally spaced hydrogen atoms.

We combine different cutting-edge computational methods to show that varying the separation between nuclei leads to a rich phase diagram, including a Mott phase (an insulating state that conventional theory predicts should be conducting) with quasi-long-range antiferromagnetic order, an insulator-to-metal transition, and an intricate set of intertwined magnetic orders. The insulator-to-metal transition is found to be driven by a self-doping mechanism, in which the traditional picture breaks down and the interplay of correlation physics with realistic electronic structure is important.

Our work establishes the hydrogen chain as a key benchmark for numerical methods and an important model system for the field of correlated electrons. The results will motivate experimental realizations, stimulate further efforts to characterize phase diagrams of low-dimensional materials, provide new insights into the metal-insulator transitions in three-dimensional materials, and encourage further methodological improvements.

Figure 1

Insulating phase—Antiferromagnetic correlations. Main panel: Correlation function $C_i$ at $R=3.6$, 2.8, and $2.0{\mathrm{a}}_{\mathrm{B}}$ (red circles, blue crosses, and orange triangles) computed with DMRG for a chain of $N=50$ with OBC. The oscillations, with wavelength $\lambda =2R$, show AFM correlations. Inset: Oscillation amplitudes $C_{2i+1}-C_{2i}$ versus $i$ (log-log scale). Lines show the results of a linear fit.

Figure 2

Insulating phase—Density dimerization. (a) Top diagram: Electronic density $n(z)$ for $R=2.0{\mathrm{a}}_B$, with OBC. Bottom diagram: Definition of the dimerization measure ${\mathrm{\Delta }}_N$ for a system under OBC. The gray vertical lines indicate how $n(z)$ is integrated along thin (width $\delta z=0.1{\mathrm{a}}_{\mathrm{B}}$) slabs perpendicular to the chain. (b) Dependence of the dimerization measure ${\mathrm{\Delta }}_N$ on the number of atoms in the open chain, $N$, from AFQMC, sb-DMRG, VMC, and DMC. Here, ${\mathrm{\Delta }}_N$ decays as a power law, ${\mathrm{\Delta }}_N\propto N^{-d}$, with exponent $d\simeq 0.5$, from correlated methods. Results from HF are also shown for reference. (c) Expectation value $h(z)$ of the kinetic energy operator (top) and von Neumann entropy $S_{vN}(z)$ (bottom) from sb-DMRG, for a 10-atom chain with OBC, as a function of position $z$ along the chain axis for $R=3.6$ and $2.0{\mathrm{a}}_B$, respectively.

Figure 3

Metal-insulator transition. (a) Complex polarization measure as a function of $R$ from multiple methods, identifying a MIT separating a region at smaller $R$ with $|Z_N|=0$ from $|Z_N|>0$ at larger $R$. Here, $|Z_N|$ is plotted for $N=40$. (b) Magnetic phases from DFT-PBE, as indicated by $M_{\mathrm{tot}}=\int m(r)d^{3}r/N$ and $M_{\mathrm{abs}}=\int |m(r)|d^{3}r/N$, where $m(r)$ is the magnetization of the simulation cell. (c) Spin-density computed from DFT-PBE, shown along a plane containing the chain, illustrating long-wavelength ferromagnetic domains at $R=0.9{\mathrm{a}}_B$.

Figure 4

Mechanism of the insulator-to-metal transition. (a) Schematic illustration of the self-doping mechanism inducing a MIT in the H chain. Dark green line: Large $R$, no band overlap. Light green lines: Small $R$, multiple bands crossing ${\epsilon }_F$. (b) Structure factor of the spin-spin correlation function at $R=0.9{\mathrm{a}}_B$ and at $R=2.5{\mathrm{a}}_B$, computed from AFQMC, DMC, and DMRG. Wave vectors $q$ are along the chain. Vertical lines, as an aid to the eye, mark the kinks in $S(q)$ associated with the Fermi surfaces.

Figure 5

Insight into the many-body metallic state. (a) Occupancy of NOs, obtained from the one-body density matrix of the many-body ground state at $R=0.9{\mathrm{a}}_B$, from AFQMC (supercell with $N=24$ atoms, twist-averaged) and sb-DMRG ($N=10$ atoms with hard-wall boundary). The orbital characteristics are indicated, in each data set, by the marker type. (b,c) Contour plots illustrating the characters of representative occupied NOs (ordered high to low in $n_i$), projected in a transverse plane slicing through an atom, at $R=0.9{\mathrm{a}}_{\mathrm{B}}$ for a ${\mathrm{H}}_{10}$ with OBC and hard-wall boundary, by sb-DMRG (b); and a ${\mathrm{H}}_{24}$ supercell from DMRG, at twists $k=0$, 0.1, and 0.4, respectively (c1)–(c3).