Matrix product purifications for canonical ensembles and quantum number distributions

Matrix product purifications (MPPs) are a very efficient tool for the simulation of strongly correlated quantum many-body systems at finite temperatures. When a system features symmetries, these can be used to reduce computation costs substantially. It is straightforward to compute an MPP of a grand-canonical ensemble, also when symmetries are exploited. This paper provides and demonstrates methods for the efficient computation of MPPs of canonical ensembles under utilization of symmetries. Furthermore, we present a scheme for the evaluation of global quantum number distributions using matrix product density operators (MPDOs). We provide exact matrix product representations for canonical infinite-temperature states, and discuss how they can be constructed alternatively by applying matrix product operators to vacuum-type states or by using entangler Hamiltonians. A demonstration of the techniques for Heisenberg spin-$1/2$ chains explains why the difference in the energy densities of canonical and grand-canonical ensembles decays as $1/L$.

Figure 1

Ranges (16) of possible left-block quantum numbers $N_i={\sum }_{j=1}^i{n}_i$ for a bosonic or fermionic system with $N_{\mathrm{tot}}$ particles or a spin-$s$ system with $N_{\mathrm{tot}}:=S_{\mathrm{tot}}^z+sL$. The system has $L=60$ sites and the diagram displays quantum number ranges for fillings $N_{\mathrm{tot}}=30\tilde{d}$ and $N_{\mathrm{tot}}=15\tilde{d}$, where $\tilde{d}=d-1$ is the maximum possible or considered number of particles per site—in particular, $\tilde{d}=2s$ for spin $s$. In matrix product representations (15) of the infinite-temperature canonical ensembles, every quantum number sector has dimension one such that bond dimensions are $D_i=N_i^{\text{max}}-N_i^{\text{min}}+1$. For fixed filling $N_{\mathrm{tot}}/L$, the maximum bond dimension is hence proportional to $L$.

Figure 2

Variational approach to construct MPPs of canonical infinite-temperature states (10) in a DMRG ground state computation with a so-called entangler Hamiltonian (17). The figure shows the convergence to the closely related equal-weight superposition $|{\tilde{\varrho }}_{0,Q}^{\mathrm{c}}\rangle$ in Eq. (18), which is the zero-energy ground state of ${\widehat{H}}^{\mathrm{c}}$ [Eq. (19)]. Top: energy expectation value $\langle {\widehat{H}}^{\mathrm{c}}\rangle$. Bottom: fidelity measure $1-|\langle \psi |{\tilde{\varrho }}_{0,Q}^{\mathrm{c}}\rangle |$. Left: spin-$1/2$ chains of different lengths with magnetization $S_{\mathrm{tot}}^z=0$. Right: bosons on 1D lattices with a maximum of $\tilde{d}=4$ bosons per site and unit filling $N/L=1$. Different curves for the same system size $L$ refer to different initial random MPSs for the variational computation. Bond dimensions have been fixed to those of the corresponding exact MPP representations given in Sec. 6c.

Figure 3

MPP simulations of the canonical and grand-canonical ensembles for zero magnetization, which are equivalent in the thermodynamic limit. For the antiferromagnetic spin-$1/2$ Heisenberg chain (21), the upper panel shows the difference of the energy densities for the two ensembles that decays as $1/L$ with the system size $L$. The lower panel shows the standard deviation of the energy density in the grand-canonical ensemble. As is known from thermodynamics, it decays as $1/\sqrt{L}$.

Figure 4

Probability distribution $p_M$ [Eq. (20)] of the total magnetization $M$ in the grand-canonical ensemble $e^{-\beta \widehat{H}}/Z={\sum }_M{p}_M{e}^{-\beta {\widehat{H}}_M}/Z_M$ for antiferromagnetic spin-$1/2$ Heisenberg chains (21) without magnetic field, computed with MPDO simulations as described in Sec. 7. The properly rescaled distribution $(M,p_M)\to (M/\sqrt{L},p_M/\sqrt{L})$ converges in the thermodynamic limit $L\to \infty$.

Figure 5

Energy densities $E_M/L$ of canonical ensembles $e^{-\beta {\widehat{H}}_M}/Z_M$ for antiferromagnetic spin-$1/2$ Heisenberg chains (21), computed with MPP simulations. The energy density $E_M/L$ as a function of the magnetization $M/L$ converges in the thermodynamic limit $L\to \infty$. The energy densities for the different temperatures have minima at $M=0$ as this is the ground-state sector. Due to the equivalency of ensembles, ${lim}_{L\to \infty }{E}_{M=0}/L$ coincides with the energy density ${lim}_{L\to \infty }{E}_{\mathrm{gc}}/L$ of the grand-canonical ensemble $e^{-\beta \widehat{H}}/Z$ for zero magnetization ${\langle {\widehat{S}}_{\mathrm{tot}}^z\rangle }_{\mathrm{gc}}=0$.