Low-temperature Gibbs states with tensor networks

TL;DR

The paper tackles the challenge of obtaining thermal states of quantum many-body systems at low temperatures by introducing a tensor-network approach that starts from the ground state. By projecting the Hamiltonian onto the subspace spanned by the ground-state Schmidt vectors, constructing an effective Hamiltonian $\tilde{H}$ of dimension $D^2$, and using its spectrum $\{\tilde{E}_k\}$ to form a low-rank Gibbs ensemble, the method maps back to the full space to obtain $\rho_\beta$ with reduced computational cost. The TTN/MPS implementation enables efficient computation of thermodynamic quantities and entanglement measures (e.g., logarithmic negativity), with numerical benchmarks in 1D and 2D transverse-field Ising models showing good performance near criticality and controlled degradation away from criticality. The approach provides a complementary, low-temperature alternative to imaginary-time evolution and can be extended to other tensor-network structures (e.g., MERA, PEPS) to study very-low-$T$ physics and finite-size effects in higher dimensions. Overall, the method advances practical access to low-temperature quantum thermodynamics and entanglement in critical and near-critical systems.

Abstract

We introduce a tensor network method for approximating thermal equilibrium states of quantum many-body systems at low temperatures. Whereas the usual approach starts from infinite temperature and evolves the state in imaginary time (toward lower temperature), our ansatz is constructed from the zero-temperature limit, the ground state, which can be found with a standard tensor network approach. Motivated by properties of the ground state for conformal field theories, our ansatz is especially well suited near criticality. Moreover, it allows an efficient computation of thermodynamic quantities and entanglement properties. We demonstrate the performance of our approach with a tree tensor network ansatz, although it can be extended to other tensor networks, and present results illustrating its effectiveness in capturing the finite-temperature properties in one- and two-dimensional scenarios. In particular, in the critical one-dimensional case we show how the ansatz reproduces the finite temperature scaling of entanglement in a conformal field theory.

Figures (7)

• Figure 1: Graphical representation of the proposed TN method. (a) Description of the method. 1. The TN for the ground state in canonical form with respect to a bond defines the isometries $\Phi_{A(B)}$; $d$ is the physical dimension and $D$ is the bond dimension. 2. Projecting the full Hamiltonian with the Schmidt vectors of the ground state $\Phi_{A(B)}$ yields the corresponding effective Hamiltonian $\tilde{H}$. The dimension of the isometry $\phi_A \otimes \phi_B$ is $D^2 \times 2^N$, yielding an effective Hamiltonian $\tilde{H}$ dimension of $D^2 \times D^2$. 3. Diagonalizing $\tilde{H}$, through e.g., exact diagonalization (ED), provides the set of eigenvalues $\tilde{E}$ and eigenvectors $\ket{\tilde{E}}$. 4. Thermal state representation via TN of the reduced Hamiltonian $\tilde{H}$ and its decomposition in terms of energy eigenvalues and eigenvectors. The circle in the vertical bond represents the Boltzmann weights from $\tilde{E}$. (b) Basic elements for the computation of entanglement quantities. On the left, a TN representation of the reduced density matrix when subsystem B is traced out, $\tilde{\rho_A}$. This would allow computing the Von Neumann entropy of the thermal state. On the right, a TN representation of the partial transpose $\tilde{\rho^{T_A}}$, necessary when computing logarithmic negativity, a valid measure of entanglement for mixed states plenio2005logarithmic. (c) Basic elements for the computations of observables and expectation values. First, $\tilde{\rho}$ needs to be mapped back into the physical space, using the isometries from step 1(a), yielding an approximation for the thermal state $\rho_\beta$. Then, any operator $O$ can be applied to the TTO. In picture, a single site operator, acting on site 2, $O_2$. To compute $\Tr(\rho_\beta O)$, we need to perform standard contractions of the physical legs of the resulting TTO, as graphically indicated by the dashed lines.
• Figure 2: Numerical results for the $1\mathrm{D}$ TFI. (a) Free entropy (main plot) and relative error in energy spectrum $\delta E_k^R$ (inset) at $g=1$, $L=128$ with open boundary conditions. Error bars (obtained comparing $D'=70$ and $D=100$) are smaller than the size of the markers. (b) Dependence on the field $g$ of the maximal temperature that can be simulated while keeping the relative error in free energy below a predefine threshold $\varepsilon_{th}=10^{-2}$. Two system sizes $L=64,\ 128$ are shown (filled symbols), along with the exact energy gap $\Delta$ computed for $L=128$ (dashed line). (c) Entanglement negativity for the critical case with periodic boundary conditions. The inset shows a fit of our data to the low-temperature prediction of $\varepsilon_s(T)$. The bond dimension used for all system sizes is $D=10^2$ and the number of excited states included in the thermal ensemble is $\chi=10^4$.
• Figure 3: Numerical results for the 2D TFI model on a $L\times L$ square lattice with periodic boundary conditions. (a) Free energy (main plot) and relative error in energy spectrum $\delta E_k^R$ (inset) at $g=1.8$, $L=4$, for bond dimensions $D=40$ (light circles) and $100$ (dark circles), compared to results from exact diagonalization in full (black solid line), or truncated to a fixed number of excitations (blue solid line). (b) Absolute value of the average magnetization $\abs{\braket{Z}}$ as a function of temperature. Results are presented at $g=2.5$ for $L=8$ (purple diamonds) and $12$ (green circles), and at $g=3.04$ for $L=12$ (blue circles). $L=8$ is shown for $D=70$ and $L=12$ is shown for both $D=100$ (darker) and $D=70$ (lighter symbols), to illustrate convergence in $D$. The red line indicates the expected critical temperature for $g=2.5$czarnik2019finite, and the inset shows schematically the phase diagram, with the probed values of $g$ marked by vertical lines.
• Figure 4: Performance of our method and its one-excitation extension in the gapped regime. The figure shows the free energy as a function of temperature for the $1\mathrm{D}$ TFI at $g=1.5$, for $L=96$, using $D=100$, for our original ansatz (blue circles) and the improved one in \ref{['eq:improved_E1']} (squares). Solid lines indicate the result from a fixed number $k$ of exactly computed excitations, from $k=2$ (lightest grey, less accurate than our ansatz), to $k=5$ (comparable to the improved ansatz) and the full exact result (black line, for reference).
• Figure 5: Convergence of the results with bond dimension as a function of temperature, for the $1\mathrm{D}$ TFI at $g=1,$$L=128$. The upper panel shows entanglement negativity in PBC, while the main figure shows the free energy in OBC, for the same set of bond dimensions.