Tree Tensor Network Impurity Solver

• Tree Tensor Network (TTN) impurity solver is an advanced computational method that represents impurity-plus-bath wavefunctions in a loop-free tree structure.
• It computes spectral and dynamical correlation functions directly in real time and frequency, bypassing the challenges of analytic continuation.
• Its scalable, symmetry-preserving design enables accurate simulations of multi-orbital, highly correlated, and open quantum systems.

A Tree Tensor Network (TTN) impurity solver is an advanced computational method for simulating quantum impurity models, particularly in the context of nonequilibrium and dynamical mean-field theory (DMFT) calculations. The approach is grounded in the representation of the full impurity-plus-bath many-body wavefunction as a tree-structured tensor network, enabling efficient encoding of the entanglement structure characteristic of impurity-bath systems. TTN impurity solvers operate in real time or frequency and are capable of directly computing spectral and dynamical correlation functions with high and uniform energy resolution across the entire spectrum, bypassing the numerical instabilities linked to analytic continuation. Their efficiency and scalability make them highly suitable for tackling multi-orbital, highly correlated, or open quantum systems, including those involving spin-orbit coupling or complex bath geometries (Cao et al., 2021, Zhan et al., 25 Jan 2026, Ke, 2023, Lindoy et al., 19 Mar 2025).

1. Tensor Network Architecture and Bath Mapping

The TTN impurity solver expresses the full Fock space as a product of local tensors arranged in a loop-free tree. Each node (vertex) corresponds to either an auxiliary node (carrying only bond indices) or a physical site (impurity orbital or bath site, with an attached physical index). For a general multiorbital Anderson impurity model, a binary tree is constructed to depth $\Delta = \lceil \log_2 N_o \rceil$ (where $N_o$ is the number of orbitals). Impurity sites are attached at an intermediate layer (layer 4 in the canonical construction), from which bath subtrees (chains or branches) emanate downward.

Key features include:

• Each tensor $M^{[v]}$ (site $v$) has up to three virtual legs and, for a leaf node, a single physical index of dimension $d_i$.
• Tensors are block-sparse by symmetry: global conservation laws (such as $U(1)_{\mathrm{charge}}$, $U(1)_{S_z}$, or $U(1)_{J_z}$) are imposed so that only compatible quantum number sectors are present.
• Implementation of the bath is geometry-dependent: options include star, chain, and natural-orbital (NO) configurations. The NO geometry, which interleaves reservoir sites with impurity orbitals, further mitigates long-range entanglement and supports faster convergence.

The noninteracting bath Hamiltonian is fit using a discretized hybridization (Weiss) function, yielding parameters $\{ \alpha^b_i, \beta^b_i \}$ for $N_b$ bath sites (Cao et al., 2021, Zhan et al., 25 Jan 2026).

2. Impurity Hamiltonians and Supported Models

The TTN impurity solver is capable of treating general multiorbital Anderson impurity models of the form

$H_{\rm imp} = H_{\rm loc} + H_{\rm bath} \,,$

with $H_{\rm loc}$ containing single-particle energy and rotationally invariant Kanamori interaction terms, and the bath coupled in star, chain, or Cayley-tree form. Spin-orbit coupling (SOC) terms are handled trivially at the Hamiltonian level and preserved as block quantum numbers throughout the network.

Explicitly, in the presence of SOC: $$H_{\rm loc} = \sum_{\tau_1,\tau_2}\epsilon_{\tau_1\tau_2} c^\dag_{\tau_1} c_{\tau_2} + \sum_{\tau_1 \dots \tau_4} U_{\tau_1\tau_2\tau_3\tau_4} c^\dag_{\tau_1} c^\dag_{\tau_2} c_{\tau_4} c_{\tau_3} + H_{\rm soc}\,.$$ For the single-impurity Anderson model (SIAM): $$H_{\rm SIAM} = \sum_\sigma \varepsilon_d n_{d\sigma} + U n_{d\uparrow} n_{d\downarrow} + \sum_{k\sigma} \varepsilon_k c^\dag_{k\sigma} c_{k\sigma} + \sum_{k\sigma} ( V_k d^\dag_\sigma c_{k\sigma} + h.c. )$$ The solver is also extensible to open quantum system settings through integration with the hierarchical equations of motion (HEOM): TTN is used to represent the extended system–bath wavefunction or auxiliary density operator space (Ke, 2023, Lindoy et al., 19 Mar 2025).

3. Real-Time and Frequency Algorithms

Real-time dynamics and spectral functions are accessed via TDVP-based time evolution of the TTN. The core procedure is as follows:

1. Ground State Preparation: The zero-temperature ground state is found via single-site DMRG augmented by subspace expansion.
2. Time Propagation: Real-time evolution is implemented by the time-dependent variational principle (TDVP) on the TTN manifold. For each tensor $M^{[v]}$:

$$M^v(t+\Delta t) = e^{-i\Delta t\,\hat H^{v}_{(1)}} M^v(t),$$

and similarly for bond (center matrix) updates. A second-order Trotter splitting of the projected Schrödinger equation is used.

1. Dynamic Correlation Functions: One-particle Green's functions are computed as:

$$G_{\tau_i\tau_j}(t) = -i \Theta(t) \langle \{ c_{\tau_i}(t), c^\dag_{\tau_j}(0) \} \rangle_0 \,,$$

by evolving perturbed states $c_{\tau_j}^\dag |0\rangle$ and $c_{\tau_i} e^{-i(H-E_0)t}|0\rangle$.

2. Frequency Domain Transformation: After real-time evolution up to $t_\mathrm{max}$, linear prediction is applied to the time series, which is then Fourier transformed to obtain $G(\omega)$ with high, energy-uniform resolution.

The methodology is directly applicable to two-particle observables by evolving an appropriate set of excited states (e.g., for spin or orbital susceptibilities) (Cao et al., 2021, Zhan et al., 25 Jan 2026).

4. Integration with DMFT and Open Quantum System Methods

Within DMFT, TTN is embedded into the real-frequency self-consistency loop:

1. Construct the impurity Hamiltonian from bath discretization.
2. Solve for the impurity Green's function using the TTN impurity solver.
3. Extract the self-energy and update the lattice-local Green's function.
4. Recompute the Weiss function and iterate until convergence.

Bath mapping is revisited at each iteration, and the Cayley-tree TTN construction is updated according to the new hybridization (Cao et al., 2021, Zhan et al., 25 Jan 2026).

For open quantum systems, TTN is combined with HEOM. Bath correlation functions are expanded in sums of damped exponentials, mapped to effective-bath-mode Fock states. The associated non-Hermitian super-Hamiltonians are encoded as tree tensor network operators, and time evolution proceeds via single-site TDVP sweeps (Ke, 2023). This HEOM+TTN method yields a computational scaling only linear in the number of bath poles and truncation tiers, a dramatic improvement over pure HEOM. Accurate simulations of the spin-boson model demonstrate speed-ups of $240\times$ in time and $760\times$ in memory relative to standard HEOM, and a $4\times$ time advantage over MPS-based schemes at similar accuracy.

5. Computational Complexity, Scaling, and Practical Implementation

The computational cost per step is determined by the network structure and bond dimension. For the canonical binary TTN:

• Each auxiliary tensor update (rank-3) involves SVDs with cost $\mathcal O(m^4)$, where $m$ is the bond dimension.
• Subspace-expansion combined with single-site TDVP maintains the per-step auxiliary cost at $\mathcal O(m^4)$.
• Physical-site updates (if present) can scale as $\mathcal O(m^5)$, yet tree geometries such as natural orbitals or Cayley trees minimize the number of highly entangled bonds, accelerating convergence.
• Entanglement is peaked near the impurity and decays with distance from the root; thus, outer bath layers are handled with lower $m$.

Benchmarks for three-orbital Hund metals show that NO tree geometries outperform star and chain, achieving ground-state variance $\sim10^{-12} D^2$ at $m\approx200$ and exponential accuracy in Green's functions with increasing $m$ (Cao et al., 2021). In SIAM studies, TTN achieves accurate results with $D=30$ compared to $D=90$ for MPS (Zhan et al., 25 Jan 2026). The overall walltime and scaling are linear in bath sites for the hSOP representation (Lindoy et al., 19 Mar 2025).

A representative workflow (as implemented in pyTTN) involves:

1. Defining the symbolic Hamiltonian and discretizing the bath.
2. Constructing the tree topology, assigning physical indices to each node.
3. Assembling the TTN state and operator.
4. Configuring and running the TDVP projector-splitting evolution with adaptive bond dimension control.
5. Accumulating desired observables across time slices.

6. Applications and Benchmark Results

The TTN impurity solver has enabled unprecedented real-frequency resolution in multi-orbital and realistic model systems. For Sr$_2$RuO$_4$, real-axis DFT+DMFT using TTN+NO geometry accurately resolves van Hove singularities, high-energy Hubbard bands, and multiplet splittings, in both the presence and absence of SOC. Quasiparticle weights extracted from self-energies are in excellent agreement with ARPES measurements. TTN captures the fine features of two-particle susceptibilities, including the manifestation of spin-orbital separation and the effects of SOC-induced channel mixing at energies above the Fermi-liquid scale (Cao et al., 2021).

In SIAM benchmarks, TTN realizes ground-state energies and spectral functions at lower bond dimension than MPS, and accurately reproduces entanglement entropy scaling and Green's function evolution. The real-time Green's function, as well as the full spectral density, are extracted directly in real frequency, making analytic continuation unnecessary (Zhan et al., 25 Jan 2026, Lindoy et al., 19 Mar 2025).

Open quantum system calculations with HEOM+TTN demonstrate efficient treatment of dissipative spin models, complex exciton-phonon interfaces, and large-scale molecular systems (Ke, 2023, Lindoy et al., 19 Mar 2025). The pyTTN package provides a unified Python interface for constructing and running TTN impurity solvers, with full support for adaptive bond expansion, different mapping schemes, and direct observable extraction.

7. Advantages, Limitations, and Outlook

TTN impurity solvers offer several clear advantages:

• Alignment of the network structure with the physical entanglement graph yields much reduced maximum bond dimension compared to chain-based (MPS/DMRG) methods, which translates into lower computational cost for comparable accuracy.
• Real-time and real-frequency access allows uniform high-resolution spectroscopy, especially at low and high energies, without analytic continuation artifacts.
• The block-sparse nature of tensors under symmetry constraints further optimizes memory and runtime.
• The method generalizes to multi-impurity, multi-orbital, bosonic, and open quantum systems.
• The tree topology can be tuned (balanced, unbalanced, optimized) to match the problem-specific entanglement and physical bath structure.

Limitations include:

• The loop-free nature of the TTN restricts the direct treatment of 2D or higher-dimensional baths; highly nonlocal hybridization structures may require sophisticated tree design.
• For extremely long evolution times or in the absence of significant entanglement decay with “distance” from the impurity (e.g., star geometry), bond-dimension demands can still become large.

Ongoing advances such as Krylov-based subspace expansion and hierarchical compression continue to extend the reach of TTN impurity solvers, as evidenced by recent open-source developments and expanding application domains (Lindoy et al., 19 Mar 2025).