Single-hole spectral functions in 1D quantum magnets with different ground states

TL;DR

This work uses constrained stochastic analytic continuation (SAC) applied to imaginary-time Green's functions from stochastic series expansion (SSE) quantum Monte Carlo to extract the zero-temperature single-hole spectral function $A(k,\omega)$ in 1D $S=1/2$ spin systems. It systematically studies the uniform $t$-$J$ chain and the $t$-$J$-$Q$ chain, examining spin-charge separation and edge structures, and then investigates how dimerization toward a valence-bond solid (VBS) or bond alternation modifies the spectrum. In the uniform and critical regimes, SAC captures holon/spinon edge features consistent with spin-charge separation, though with momentum-dependent deviations such as gaps between holon bands at $k=0$ and $k=\pi$ in some parameter ranges. In the VBS phase, evidence for spinon-holon binding emerges as a spin polaron, including isolated lower-edge states and higher parity bands, while bond-alternating chains reveal multiple narrowly separated spin-polaron bands with even/odd internal parity. Overall, the study demonstrates the power of constrained SAC to resolve sharp spectral features inaccessible to other analytic continuation methods and highlights rich spinon-holon physics and bound-state phenomena in 1D quantum magnets.

Abstract

Thanks to improved methods for numerical analytic continuation with constraints, spectral functions with sharp features can now be extracted from imaginary-time correlation functions computed by quantum Monte Carlo (QMC) simulations. Here we test these new approaches on various one-dimensional $S=1/2$ spin systems with a single ejected fermion, i.e., extracting the single-hole spectral function $A(k,ω)$. We compute the Green's function $G(r,τ)$ via a canonical transformation of the fermionic Hamiltonian, implementing it for stochastic series expansion QMC simulations. Our calculations of $A(k,ω)$ focus on the different characteristics of systems with spin-charge separation and those in which a spin polaron forms instead due to effectively attractive interactions between the spin and the charge. Spin-charge separation is well established in the conventional $t$-$J$ chain, which we confirm here as a demonstration of the method. Turning on a multi-spin interaction $Q$ that eventually drives the system into a spontaneously dimerized (valence-bond solid, VBS) state, we can observe the features of spin-charge separation until the VBS transition takes place. While generally good agreement is found with the conventional analytical spin-charge separation ansatz, we point out the formation of a gap between two holon bands that in the ansatz are degenerate at $k=0$ and $k=π$. Inside the VBS phase, effectively attractive interactions may lead to the binding of the spinon and holon, of which we find evidence at large $Q/J$. In the statically dimerized $t$-$J$ chain, we find equally spaced spin polaron bands corresponding to increasingly large bound states with two internal spin polaron modes -- even and odd with respect to parton permutation. Our results overall demonstrate the power of modern analytic continuation tools in combination with large-scale QMC simulations.

Figures (17)

• Figure 1: Illustration of an SSE configuration of a system with eight spins and time slices including $M=8$ operators in Eq. (\ref{['ssez']}). Spins $\uparrow$ and $\downarrow$ are represented by open and solid circles, respectively, and the operators $K_{l,s}$ shown as bars are drawn from the sets defined in Eqs. (\ref{['hops']}), with the operator types corresponding to colors as follows: white for $a=1$, black for $a=2$, and yellow for $a=3$. No fill-in unit operators are included here but will appear in actual SSE configurations. The red dot represents an injected hole replacing the $\uparrow$ spin at site $1$, and the red line illustrates the different paths this hole can take upon evolution in imaginary time. Note that the hole in this representation can propagate only on sites with $\uparrow$ spins, reflecting the definition of the three-state basis, Eq. (\ref{['etabasis']}).
• Figure 2: Comparisons of the exact $k=\pi/2$ Green's function (blue curve, from full diagonalization of $H$) of $N=8$ chains and corresponding SSE results (red points with error bars) for (a) the standard $t$-$J$ chain at $t=1$, $J=4$ and (b) The $t$-$J$-$Q$ chain with $t=1$, $J=2$, and $Q=0.5$. The left and right insets show, respectively, the absolute and relative difference between the SSE data and the exact results. In the analytic continuation we use only data points for which the statistical error (one standard deviation of the mean) is less than $0.1$, which corresponds to a similar relative error with respect to the exact value, indicated here with dashed lines. Note that the log scale in the insets showing $|\Delta G|/G_{\rm exact}$ does not visually reflect well the size of the lower part of the error bar (one standard deviation of the mean). In (a) the deviation from the exact value at $\tau=4$ is $1.0$ standard deviation, while in (b) it is $2.1$ standard deviations. The deviations overall are statistically likely.
• Figure 3: Two parameterizations of the spectral function $A(\omega)$ used in unconstrained SAC with a large number $N_\omega$ (typically thousands) of $\delta$-functions in the frequency continuum (in practice on a grid with very small spacing; $\Delta_\omega=10^{-5}$ or smaller). In (a), each $\delta$-function carries the same fixed weight $N^{-1}_\omega$ of the normalized spectrum and only the locations $\omega_i$ are sampled, while in (b) also the amplitudes are sampled, with the total spectral weight conserved.
• Figure 4: Spectral function $A(k,\omega)$ of the $L=64$$t$-$J$ chain with $t=1$ and $J=0.4$ obtained by SAC with one unconstrained and two constrained $\delta$-function parametrizations. (a) with unconstrained variable amplitudes as in Fig. \ref{['Fig.sacpara1']}(b). (b) with a double edge representing the spectral bounds and unconstrained contributions between the edges, combining Figs. \ref{['Fig.edgecontpara']}(a), \ref{['Fig.edgecontpara']}(b), and \ref{['Fig.edgecontpara']}(c). In (c), the single-edge constraint illustrated in Fig. \ref{['Fig.edgecontpara']}(a) has been combined with unconstrained $\delta$-functions above the edge as in Fig. \ref{['Fig.edgecontpara']}(b). The black dashed lines show the upper and lower holon branch (on which the spinon has momentum $k=\pi/2$ and energy $0$) according to the spin-charge separation ansatz. The blue solid lines show the lowest and highest energies of the combined spinon and holon excitations when not coinciding with the holon branches. These predictions are based on the dispersion relations in Eqs. (\ref{['disp']}) with $J_s=J$ and $t_h=2t$. The circles in (b) show the locations of local maximums inside the continuum that we identify with the holon branches.
• Figure 5: Spectral profiles for fixed $k=\pi/8$ (red curves) and $\pi/2$ (blue curves) for three different parameter sets of the $t$-$J$ ($Q=0$) and $t$-$J$-$Q$ models, as indicated on top of the respective columns. In each case, three different SAC parametrizations were used; unconstrained in (a),(d),(g), double-edge in (b),(e),(h) and lower edge only in (c),(f),(i). Parameter sets correspond to the heat-map representation of the entire spectral function in Figs. \ref{['Fig.t1J04logcombine']}, \ref{['Fig.t1J2combine']}, and \ref{['Fig.t1J1Q01645logmorecombine']}.