Generalized Reduced-Density-Matrix Quantum Monte Carlo Gives Access to More

Abstract

For a long time, people have been focusing on how to extract more information, such as off-diagonal observables, from the quantum Monte Carlo (QMC) simulation of the partition function, but there have been numerous difficulties, and many of them are insurmountable. In this article, we point out that all the difficulties stem from the starting point of the simulation: calculating a partition function. We introduce a paradigm shift: when we transform the simulated object from a partition function to a generalized reduced density matrix (GRDM), the difficult problem of measurement can be readily solved. By designing the GRDM, both equal-time and nonequal-time off-diagonal observables have been measured easily in QMC with a polynomial computation complexity. As a demonstration, the GRDM enables direct access to nonequal-time correlators for dynamical spectra as well as Rényi-1 correlators that reveal strong-to-weak symmetry breaking in the mixed state, capabilities that lie beyond the reach of prior methods. This establishes a unified framework for holographic characterization within QMC.

Figures (7)

• Figure 1: Schematic illustration of the reduced density matrix (RDM) and operator-inserted generalized RDM on the A-OBC/B-PBC manifold. (a) The RDM of subsystem $A$ is obtained by tracing out subsystem $B$, yielding $\rho_A \propto {\rm Tr}_B(e^{-\beta H})$. Region $A$ (colored) is open in imaginary time, with boundary states $\alpha_A',\alpha_A$ at $\tau=0,\beta$, while region $B$ (gray) remains periodic and is traced over through the states $\xi_B$. (b) Generalized RDM with an operator insertion $\hat{O}_A$ at imaginary time $\tau$ inside subsystem $A$, $\rho_A^{\hat{O}} \propto {\rm Tr}_B \left(e^{-(\beta-\tau)H}\hat{O}_A e^{-\tau H}\right)$. The two panels are connected (yellow arrow) by the switch between the inserted operator $\hat{O}_A$ and the identity operator.
• Figure 2: Benchmark results for the $L=4$ XXZ chain with $\beta=10$. The hollow markers (SSE) are consistent with the ED results (dashed lines) within error bars. (a) Imaginary-time correlation function $\langle S^x_i(\tau) S^x_{i+1}(0) \rangle$ as a function of $\tau$ for various anisotropy values $\Delta$. (b) Equal-time correlation function $\langle S^x_i(0) S^x_{i+1}(0) \rangle$ calculated using $\tilde{\rho}^I_A$ from the GRDM framework.
• Figure 3: Spectral function $S^{xx}(k,\omega)$ for the spin-1/2 XXZ chain at $\Delta=0.2$.
• Figure 5: Comparison between the two-string construction and the boundary-hole construction for directed-loop updates on the A-OBC/B-PBC manifold. The gray bond operators denote local Hamiltonian terms distributed along both the spatial (horizontal) and imaginary-time (vertical) directions in the SSE operator-string configuration. (a) In the two-string scheme, the loop head starts from a bulk leg and propagates by standard directed-loop scatterings. Upon reaching the open imaginary-time boundary, the first string (red) must terminate because no boundary scattering rule is available. Since the loop is not yet closed, one must restart from the original starting point and generate a second string (blue), which propagates until it also terminates at the boundary. (b) In the boundary-hole construction, the open boundary endpoints are promoted to boundary holes $h_{i0},h_{j0},h_{i\beta},h_{j\beta}$, which serve as explicit entrance and exit channels for the loop head. The update is then completed through a hole-to-hole jump (red arrow) between boundary holes.
• Figure 6: Diagnosing and resolving the directed-loop drift on the A-OBC/B-PBC manifold. (a) Benchmark of $\langle S_1^z S_2^z\rangle$ versus anisotropy $\Delta$ comparing ED (all solid black lines) with directed-loop measurements under imaginary-time PBC and the two-holes OBC (open imaginary-time boundaries at sites $i$ and $j$), demonstrating that the discrepancy originates from the OBC sector. (b) For $\langle S_1^x S_2^x\rangle$, we compare ED with three directed-loop implementations: a fixed $\varepsilon=0.25$ choice (blue) and the lower-bound $\varepsilon$ (red) taken from the bounce-free construction of Refs. DL1_OlavF_2002, both used directed-loop string update, and compare the correct results of directed-loop update using boundary-hole trick (green). (c)-(d) The remaining drift comes from DL string method is exposed by longer-range correlators $\langle S_1^x S_3^x\rangle$ and $\langle S_1^x S_4^x\rangle$. In contrast, the boundary-hole trick (green) removes the drift in (b)-(d) and restores agreement with ED across $\Delta$.