Symmetric minimally entangled typical thermal states

We extend White's minimally entangled typically thermal states approach (METTS) to allow Abelian and non-Ablian symmetries to be exploited when computing finite-temperature response functions in one-dimensional (1D) quantum systems. Our approach, called SYMETTS, starts from a METTS sample of states that are not symmetry eigenstates, and generates from each a symmetry eigenstate. These symmetry states are then used to calculate dynamic response functions. SYMETTS is ideally suited to determine the low-temperature spectra of 1D quantum systems with high resolution. We employ this method to study a generalized diamond chain model for the natural mineral azurite $\mathrm{Cu}{}_3\left(\mathrm{CO}{}_3{)}_2\right({\mathrm{OH})}_2$, which features a plateau at $\frac{1}{3}$ in the magnetization curve at low temperatures. Our calculations provide new insight into the effects of temperature on magnetization and excitation spectra in the plateau phase, which can be fully understood in terms of the microscopic model.

Figure 1

Schematic illustration of the METTS algorithm. For details on how to explicitly evaluate response functions of the type ${\langle \widehat{B}\left(t\right)\widehat{C}\rangle }_{\beta }$, see Sec. 4b.

Figure 2

Energy of the spin-$\frac{1}{2}$ Heisenberg chain with $N=100$ spins for $\beta =4,8,20$. Details on the setup of the imaginary-time evolution can be found in Sec. 4. Starting from an ensemble of 100 randomly generated CPS, we conduct 10 thermal steps with each state and measure the ensemble average of the total energy in each step. The different basis choices for the CPS collapse become apparent in the autocorrelation times. Whereas measuring along the $z$ axis only (squares) leads to strong autocorrelations that prevent the energy to converge towards the exact value (black lines), randomly chosen measurement bases (circles) result in short autocorrelation times of a few thermal steps [31].

Figure 3

Schematic illustration of the SPS (a) in Eq. (6) and (b) in Eq. (7).

Figure 4

Schematic illustration of the SYMETTS sampling algorithm. For details on how to explicitly evaluate response functions of the type ${\langle \widehat{B}\left(t\right)\widehat{C}\rangle }_{\beta }$, see Sec. 4b.

Figure 5

U(1)-SYMETTS sampling of the thermal energy for the isotropic Heisenberg chain with $h=1$. Panels (a)–(d) in the upper row display the thermal energy of each symmetry sector entering the SYMETTS sample for $\beta$ ranging between 4 and 20; dashed lines indicate the overall ensemble average ${\langle E\rangle }_{\beta }$ determined by SYMETTS. Moreover, a comparison to benchmark calculation based on METTS (dotted lines) and purification (solid lines) is provided. Panels (e)–(h) in the second row show $\beta (E-{\langle E\rangle }_{\beta })$, in order to zoom into an energy window of order of the temperature around ${\langle E\rangle }_{\beta }$. Panels (i)–(l) in the last row illustrate the subsample size $M^{S_{\mathrm{tot}}^z}$ of different symmetry sectors for a fixed total sampling size $M=500$.

Figure 6

SU(2)-SYMETTS sampling of the thermal energy for the isotropic Heisenberg chain with $h=0$, using the same layout as Fig. 5.

Figure 7

Schematic illustration of the dynamic calculation with SYMETTS [34].

Figure 8

tDMRG calculation for the dynamic spin structure factor of the XX model using $M=300$ ensemble states. Panel (a) illustrates the time evolution of individual SYMETTS (thin lines) at $\beta =4$ used to calculate $S^{zz}(\pi /4,t)$ by taking the ensemble average (thick red line). (b) Displays the SYMETTS ensemble average for various inverse temperatures, which is then used to compute the dynamic spin structure factor in frequency space. Panels (c) and (d) show the frequency data obtained from Fourier transform for $k\approx \pi /4$ and $\pi /2$. For all considered inverse temperatures, we find excellent agreement with the exact result (dashed lines).

Figure 9

SU(2)-SYMETTS calculation (solid lines) for (a) $\widehat{\mathit{S}}(3\pi /4,t)$ and (b) $\widehat{\mathit{S}}(3\pi /4,\omega )$ of the isotropic Heisenberg chain using $M=300$ ensemble states. For all considered inverse temperatures, we find very good agreement with data obtained from matrix-product purification (dashed lines).

Figure 10

(a) Average multiplet bond dimension ${\overline{D}}^{*}$ and (b) the corresponding bond dimension $\overline{D}$ of SU(2) SYMETTS during the calculation of $\widehat{\mathit{S}}(3\pi /4,t)$ (solid lines) in comparison to the purified density matrix (dashed lines). In both cases, we keep all singular values $>{10}^{-4}$ during real-time evolution. For $\beta =4,8$ we include a backwards time evolution of the auxiliary bonds of the purified density matrix, as this leads to a reduction of the entanglement growth [24, 53]. (c) Average bond dimension $\overline{D}$ of U(1)-SYMETTS sample during the calculation of $S^{zz}(3\pi /4,t)$ for comparison.

Figure 11

(a) Illustration of the generalized diamond chain model with the antiferromagnetic exchange couplings $J_1,J_2,J_3,$ and $J_m$. One unit cell of the system is highlighted by the gray area. (b) Dependence of the magnetization on an external magnetic field $H$, which we calculated by employing DMRG and SYMETTS at zero and finite temperature, respectively. In all calculations, we keep every singular value larger than the truncation threshold $s^{\mathrm{tol}}={10}^{-5}$ and use a sample size of $M=1000$. For a system of $N=90$ spins in total, the emergence of the $\frac{1}{3}$ plateau can be observed for fields in the range of $H_{l,c}\le H\le H_{u,c}$. At finite temperatures, the plateau is washed out and the magnetization curve becomes a linear function of $H$. The vertical dashed lines indicate the parameter choices for our dynamical SYMETTS calculations in Sec. 5b.

Figure 12

Transverse dynamic spin structure factor $S^{xx}(k,\omega )$ of a generalized diamond chain model for azurite. The intensity is displayed in arbitrary units. Each column indicates a different magnetic field strength, $H=14,20,27$ T, corresponding to distinct points in the $\frac{1}{3}$ plateau phase. The ground state spectra are displayed in panels (a)–(c) in the first row, whereas the panels (d)–(i) show finite-temperature results obtained using tDMRG in combination with U(1) SYMETTS (for details see text).

Figure 13

Transverse dynamic spin structure factor $S^{xx}(k,\omega )$ at $k\approx 4\pi /5$ of a generalized diamond chain model for azurite for three different field strengths in the plateau phase. At the edges of the plateau phase [(a) and (c)], the large peaks indicating the monomer and dimer branches are already washed out at intermediate temperatures. Thermal fluctuations redistribute spectral weight in-between the two excitation peaks in these cases. In the middle of the plateau phase (b) thermal broadening is almost not present at $T=4.125$ K. Only at high temperatures is the height of the combined peak of monomer and dimer excitations significantly reduced.

Figure 14

CheMETTS calculation for spin structure factor of the XX model with $N=50,s_{\mathrm{tol}}^{\mathrm{dyn}}=5\times {10}^{-4},N_{\mathrm{Che}}=300,\eta =0.1$, and $M=300$. For all considered inverse temperatures, we find excellent agreement with the exact result. However, note that the required numerical resources clearly exceed those used in Fig. 8, where tDMRG was employed for a system with twice as many spins!

Figure 15

(a), (b) Display the numerical performance of SYMETTS and METTS in terms of the average cumulative CPU time ${\overline{t}}_{\mathrm{CPU}}$ as a function of $t{J}_2$, when carrying out the real-time evolution to determine the dynamic spin structure factor $S^{xx}(k,\omega )$ of azurite in Eq. (19) using the parameters $N=90,H=14$ T, and $k=4/5\pi$. (c), (d) Show the average computation time of a single time step $\tau$ as a function of $\overline{D}$. We note that the ensemble states have a $\overline{D}\approx 20$ at both temperatures after the initial imaginary-time evolution. The dashed lines in Fig. 15 are guide to the eye illustrating the ${\overline{t}}_{\mathrm{CPU}}\sim D^3$ scaling of the CPU time for larger bond dimensions. The employed tDMRG setup is described in Sec. 5b.