Symmetric minimally entangled typical thermal states for canonical and grand-canonical ensembles

Based on the density matrix renormalization group (DMRG), strongly correlated quantum many-body systems at finite temperatures can be simulated by sampling over a certain class of pure matrix product states (MPS) called minimally entangled typical thermal states (METTS). When a system features symmetries, these can be utilized to substantially reduce MPS computation costs. It is conceptually straightforward to simulate canonical ensembles using symmetric METTS. In practice, it is important to alternate between different symmetric collapse bases to decrease autocorrelations in the Markov chain of METTS. To this purpose, we introduce symmetric Fourier and Haar-random block bases that are efficiently mixing. We also show how grand-canonical ensembles can be simulated efficiently with symmetric METTS. We demonstrate these approaches for spin-$1/2$ $XXZ$ chains and discuss how the choice of the collapse bases influences autocorrelations as well as the distribution of measurement values and, hence, convergence speeds.

Figure 1

Illustration of collapse bases. The top depicts the original collapse basis using the $\left\{{\widehat{S}}_i^z\right\}$ eigenbasis on each site (Sz). Below, we alternate between the Sz basis and its global symmetric Fourier transform (8) (SF-Sz). These are maximally mixing. In practice, we approximate the SF-Sz scheme by restricting the basis construction to blocks of $b$ sites, which are shifted by $b/2$ lattice sites for every second block collapse $\left(\mathrm{SF}b\text{−}\mathrm{Sz}\right)$. Alternatively, a Haar-random symmetric basis can be chosen on these blocks $\left(\mathrm{SR}b\text{−}\mathrm{Sz}\right)$. One can also omit the Sz collapse in every second step and only use the Fourier basis, again shifting blocks in every second collapse $\left(\mathrm{SF}b\right)$. The swap collapse randomly partitions the lattice into pairs of sites, where, on each pair, the eigenstates of the swap operator (9) are chosen. The overall collapse basis is the tensor product of these. Again, we approximate the ideal swap collapse by restricting the pairing to blocks of $b$ sites $\left(\mathrm{SWAP}b\right)$, where we shift by $b/2$ sites in every second collapse.

Figure 2

METTS sampling for the canonical ensemble of an isotropic spin-$1/2$ Heisenberg antiferromagnet at inverse temperature $\beta =1$, zero magnetization, and with system size $L=64$. The figure illustrates the effect of different symmetric collapse bases: Sz collapses only (left), and alternating Sz and symmetric Haar-random collapses $\left(\mathrm{SR}b\text{−}\mathrm{Sz}\right)$ with block sizes $b=2$ (center) and $b=8$ (right). The panels show the $\left\{{\widehat{S}}_i^z\right\}$ eigenstates after Sz collapses. Thermalization of the Markov chains was ensured by discarding the first 1000 samples.

Figure 3

Convergence of METTS for different symmetry-conserving collapse bases in the CE. We study spin-$1/2$ $XXZ$ chains with $L=64$ sites and zero magnetization at the isotropic point [left two columns, anisotropy parameter $\mathrm{\Delta }=1$ in Eq. (10)] and in the gapped phase $\left(\mathrm{\Delta }=3\right)$ for different inverse temperatures $\beta$. The considered observables are the energy per site $\langle \widehat{H}/L\rangle$ (top), the correlator $\langle {\widehat{S}}_0^{+}{\widehat{S}}_1^{-}\rangle$ (center), and the correlator $\langle {\widehat{S}}_0^{+}{\widehat{S}}_3^{-}\rangle$ (bottom), where site $i=0$ refers to the center of the lattice. For the collapse bases, we compare the $\left\{{\widehat{S}}_i^z\right\}$ eigenbasis on each site (Sz), the symmetric Fourier $\left(\mathrm{SF}b\right)$, and the symmetric random $\left(\mathrm{SR}b\right)$ bases on blocks of $b$ sites. As a reference, we also show the convergence for alternating $\left\{{\widehat{S}}_i^x\right\}$ and $\left\{{\widehat{S}}_i^z\right\}$ eigenbases (Sx-Sz) in a simulation of the GCE. Numbers in the lower left corners of the panels state the quasiexact values of the observables in the CE.

Figure 4

Influence of collapse bases on the convergence in the GCE without symmetries for the antiferromagnetic spin-$1/2$ Heisenberg chain $\mathrm{[}\mathrm{\Delta }=1$ in Eq. (10)] at inverse temperature $\beta =1$. We show METTS errors for the energy per site $\langle \widehat{H}/L\rangle$ as a function of the number of samples $N$. The curves compare nonsymmetric Haar-random collapse bases $\left(\mathrm{R}b\right)$ on blocks of $b=1$, 2, 4, and 8 sites.

Figure 5

Illustration of our symmetric METTS algorithm for the GCE using small sets of symmetry eigenstates. An initial symmetric basis state $|\mathit{n}\rangle$ is evolved in imaginary time to obtain the METTS $|{\varphi }_{\mathit{n}}\rangle$. After the evaluation of observables, a suitable symmetry-breaking collapse results in a nonsymmetric basis state $|{\mathit{n}}^{\prime }\rangle$. It is split into its symmetric components $\left\{\right|{\mathit{n}}_j^{\prime }\rangle \}$, which are separately evolved to obtain a small collection of symmetric states $\left\{\right|{\varphi }_{{\mathit{n}}_j^{\prime }}\rangle \}$. Observables are evaluated with respect to this collection according to Eq. (13), and a collapse using a symmetric basis yields a new symmetric basis state $|{\mathit{n}}^{\prime \prime }\rangle$. We keep alternating between symmetry-breaking and symmetric collapse bases to explore different symmetry sectors according to their weights in the GCE.

Figure 6

Simulations of grand-canonical ensembles for the antiferromagnetic spin-$1/2$ Heisenberg chain $\mathrm{[}\mathrm{\Delta }=1$ in Eq. (10)] with $\langle {\widehat{S}}_{\mathrm{tot}}^z\rangle =0$ and system size $L=64$. The density plots show the distribution $\left\{p_M\right\}$ of the total magnetization $M$, computed with MPDOs as described in Ref. [15]. Lines show exemplary trajectories of $M\left(\nu \right)$ in Markov chains of symmetric METTS simulations using the SF4-Sz/Sx collapse bases as discussed in Sec. 8. They are characterized by $n_x$, the number of sites on which the symmetry-breaking ${\widehat{S}}_i^x$ eigenbasis is used. Note that for even iteration steps $\nu$, the state is an eigenstate of ${\widehat{S}}_{\mathrm{tot}}^z$ with quantum number $M\left(\nu \right)$ while, for odd $\nu$, we use the expectation value $M\left(\nu \right)=\langle {\varphi }_{{\mathit{n}}^{\nu }}\left|{\widehat{S}}_{\mathrm{tot}}^z\right|{\varphi }_{{\mathit{n}}^{\nu }}\rangle$.

Figure 7

Convergence of the symmetric METTS algorithm for grand-canonical ensembles, applied to the antiferromagnetic spin-$1/2$ Heisenberg chain with $L=64$ and $\langle {\widehat{S}}_{\mathrm{tot}}^z\rangle =0$. We show the METTS error as a function of the number of samples $N$ at inverse temperatures $\beta =1$ (left), 4 (center), and 16 (right), for the energy density $\langle \widehat{H}/L\rangle$ (top), the correlator $\langle {\widehat{S}}_0^{+}{\widehat{S}}_1^{-}\rangle$ (center), and the correlator $\langle {\widehat{S}}_0^{+}{\widehat{S}}_3^{-}\rangle$ (bottom). For collapses, we use the SF4-Sz/Sx bases as described in Sec. 8, and compare their performance to the nonsymmetric Sx-Sz collapse. Numbers in the lower left corners of the panels state the quasiexact values of the observables.