Density Matrix Renormalization Group for Continuous Quantum Systems

We introduce a versatile and practical framework for applying matrix product state techniques to continuous quantum systems. We divide space into multiple segments and generate continuous basis functions for the many-body state in each segment. By combining this mapping with existing numerical density matrix renormalization group routines, we show how one can accurately obtain the ground-state wave function, spatial correlations, and spatial entanglement entropy directly in the continuum. For a prototypical mesoscopic system of strongly interacting bosons we demonstrate faster convergence than standard grid-based discretization. We illustrate the power of our approach by studying a superfluid-insulator transition in an external potential. We outline how one can directly apply or generalize this technique to a wide variety of experimentally relevant problems across condensed matter physics and quantum field theory.

Figure 1

Examples of (a) one-body and (b) two-body basis functions in a segment between $X_{j-1}$ and $X_j$. The cusp at $x_1=x_2$ in (b) encodes the physics of contact interactions.

Figure 2

Ground-state calculation for 5 bosons with strong interactions ($\gamma =50$) in a uniform box, divided into 8 segments. (a),(b) Starting from a localized state, as the penalty $\mathrm{\Lambda }$ is increased in powers of 10, the discontinuity falls as $1/{\mathrm{\Lambda }}^2$ and the energy approaches the asymptotic value $E_{*}$, which is within $5\times {10}^{-6}$ of the exact Bethe-ansatz result $E_{\mathrm{BA}}$. (The remaining discrepancy is due to the finite basis set used.) Here, $E_c={\sum }_j⟨{\widehat{K}}_j+{\widehat{U}}_j⟩$ and $E_{\mathrm{disc}}≔(N/L){\sum }_j⟨{\widehat{\mathrm{\Upsilon }}}_{j,j+1}⟩$ [see Eqs. (4)–(5)]. The inset shows $E_c$ approaching $E_{*}$ as $1/\mathrm{\Lambda }$. (c),(d) Density and correlations in the ground state, showing Friedel oscillations similar to exact results for the Tonks gas ($\gamma \to \infty$) and far from a mean-field Gross-Pitaevskii theory. The mismatch between the cDMRG and Tonks curves is due to the different values of $\gamma$. See Supplemental Material [45] for a full description of the basis states and the DMRG parameters.

Figure 3

(a) Relative error vs number of segments (grid points) and (b) CPU time vs relative error in finding ground states for $N=10$ using cDMRG (solid lines) and discretization on a grid (dotted lines). For cDMRG, $E$ is the extrapolated energy $E_{*}>E_{\mathrm{BA}}$ (see Fig. 2). We retained up to cubic basis functions in each segment, causing the error to fall off as $M^{-6}$, instead of $M^{-2}$ for discretization. Empirically, $t_{\mathrm{CPU}}\sim |1-E/E_{\mathrm{BA}}{|}^{-0.2}$ for cDMRG at small errors, whereas for discretization this exponent is $-0.75$ for $\gamma =0.1$ and $-1$ for $\gamma =10$. Note, $t_{\mathrm{CPU}}$ was measured in seconds from the number of clock cycles during all DMRG sweeps on single quad-core CPUs. The saturation at large errors in (b) is due to larger bond dimensions [45].

Figure 4

(a) Ground-state energy and (b) condensate fraction for 10 bosons in 10 potential wells of depth $V_0$ using cDMRG with 20 segments and quartic basis functions (solid lines) and using a tight-binding approximation (dash-dotted lines). Here, $N_0$ is the occupation of the single-particle ground state. Dotted lines in (a),(b) and crosses in (a) show exact solutions for the Tonks gas and from Bethe ansatz, respectively; the condensate fractions are found using Monte Carlo integration [45]. The arrow in (b) marks the $N_0$ for $\gamma \approx 4.22$, when the ground state for $V_0\to 0$ becomes Mott insulating. In our finite-size system this transition is a crossover. (c),(d) Single-particle correlations, $⟨{\widehat{\psi }}^{†}(x)\widehat{\psi }(x^{\prime })⟩L/N$, in a superfluid and a Mott state, corresponding to circled points in (b) [45].

Figure 5

(a) Schematic of how one can decompose a tensor $T_j$ of the MPS by a basis splitting ${\varphi }^{(j)}\to {\varphi }^{+}\otimes {\varphi }^{-}$ to calculate the entanglement between arbitrary bipartitions $[0,X]$ and $[X,L]$. (b) Ground-state entanglement entropy for 10 bosons in a shallow lattice ($V_0=E_r$) with interactions $\gamma =0.2$ (red), 2 (green), 20 (blue), using cDMRG with the same numerical parameters as in Fig. 4. We have split each segment at 20 intermediate points [45]. Dashed and dotted curves show exact results for $\gamma =0$ [50] and $\gamma \to \infty$ [48], respectively.