tenax-tn 0.5.1.dev20260425

JAX-based tensor network library with symmetry-aware block-sparse tensors

Tenax

Website | Docs | PyPI

A JAX-based tensor network library with symmetry-aware block-sparse tensors and label-based contraction.

The name Tenax combines Tensor network + Jax, and is also Latin for "holding fast" — reflecting how tensor networks bind indices together through contraction.

Experimental project — This library is under active development and largely written with the assistance of Claude Code (AI). While we test extensively, AI-generated code can contain subtle bugs. Please verify results against known benchmarks before using them in research. Bug reports and contributions are welcome.

Features

• Block-sparse symmetric tensors — only symmetry-allowed charge sectors stored (U(1), Z_n)
• Label-based contraction — legs are identified by string/integer labels; shared labels are automatically contracted (Cytnx-style)
• opt_einsum integration — optimal contraction path finding for multi-tensor contractions
• Network class — graph-based tensor network container with contraction caching
• .net file support — cytnx-style declarative network topology; parse once, load tensors, contract repeatedly (template pattern)
• Algorithms — DMRG, iDMRG (1D chain & infinite cylinder), TRG, HOTRG, iPEPS (simple update with 1-site or 2-site unit cell & AD optimization), fermionic iPEPS (fPEPS), quasiparticle excitations
• GPU/TPU-accelerated DMRG — JIT-compiled sweeps via jax.lax.scan for dense tensors and per-operation JIT for block-sparse symmetric tensors; automatic warmup-to-JIT transition when bond dimensions are growing; multi-GPU sharding via GSPMD for large bond dimensions (DMRGConfig(accelerator="jit"|"sharded"))
• AutoMPO — build Hamiltonian MPOs from symbolic operator descriptions (custom couplings, NNN, arbitrary spin); supports symmetric=True for U(1) block-sparse MPOs
• AD-based iPEPS optimization — gradient optimization via implicit differentiation through CTM fixed point, supporting 1-site and 2-site unit cells (Francuz et al. PRR 7, 013237); L-BFGS with Hager-Zhang line search and metric preconditioning (Rader et al.), Adam (with cosine lr decay), and conjugate gradient optimizers; implicit AD via iterative VJP (default) and optional GMRES route; explicit AD through unrolled CTM iterations for 1-site C4v path; 2-site shared-tensor C4v path (unit_cell="2site" + gs_c4v=True) where a single C4v tensor is optimized and the second sublattice is derived by spin-π rotation, stable across χ=8–24 for spin-1/2 AFMs; opt-in reference-mode dense C4v Appendix C-F mode (ctm_ad_mode="c4v_reference") with Krylov implicit backward (bicgstab + gmres fallback); sigma gauge fixing (forward_gauge="sigma") for stable elementwise CTM convergence; C4v symmetry enforcement via explicit basis parameterization; chi-ramping schedule (optimize_gs_ad_chi_schedule) for progressive refinement- SVD and QR CTMRG projectors — SVD (Fishman) projectors (projector_method="svd") and QR projectors for faster CTM convergence alongside the default eigh
• Split-CTMRG — ket/bra-separated CTM environment tensors for O(χ³D³) projector cost instead of O(χ³D⁶); works with both DenseTensor and SymmetricTensor via the Tensor protocol (Naumann et al., arXiv:2502.10298)
• Quasiparticle excitations — iPEPS excitation spectra at arbitrary Brillouin-zone momenta (Ponsioen et al. 2022)
• Polymorphic tensor arithmetic — +, -, *, -T, max_abs, inner(), conj(), dagger(), bar() work identically on DenseTensor and SymmetricTensor, enabling algorithm code that is agnostic to the underlying storage
• Block-sparse SVD, QR, and eigh — native symmetry-aware decompositions in tenax.linalg for SymmetricTensor
• Sector-based TensorIndex — legs store sorted charge sectors and multiplicities for O(n_sectors) lookups; FuseInfo tracks parent legs so split_index can reverse fuse_indices
• Cython BLAS fast path — fused Cython Lanczos solver and block-sparse contractions via direct BLAS calls with zero Python reentry for high-performance CPU DMRG
• iDMRG transfer matrix environments — fixed-point environment computation for self-consistent infinite boundary conditions
• Extensible symmetry system — non-Abelian symmetry interface for future SU(2) support
• Benchmark suite — CLI-driven performance benchmarks for all algorithms across CPU, CUDA, TPU, and Metal backends

Installation

Note: The PyPI package (tenax-tn) is not yet available. Install from source using the instructions below.

git clone https://github.com/tenax-lab/tenax.git
cd tenax

# With uv (recommended)
uv sync --all-extras --dev

# Or with pip
pip install -e .

Hardware acceleration

Tenax uses JAX as its backend. To enable GPU or TPU acceleration, install the appropriate JAX variant before installing Tenax:

# NVIDIA GPU (CUDA 13, recommended)
pip install -U "jax[cuda13]"

# NVIDIA GPU (CUDA 12)
pip install -U "jax[cuda12]"

# Google Cloud TPU
pip install -U "jax[tpu]"

# Apple Silicon GPU (macOS only, experimental)
pip install jax-metal

See the JAX installation guide for the latest accelerator options.

Quick Start

import jax
import jax.numpy as jnp
import numpy as np
from tenax import (
    U1Symmetry,
    TensorIndex,
    FlowDirection,
    SymmetricTensor,
    TensorNetwork,
    contract,
)

# Define U(1) symmetric tensor indices with named legs
u1 = U1Symmetry()
phys_charges = np.array([-1, 1], dtype=np.int32)
bond_charges = np.array([-1, 0, 1], dtype=np.int32)
key = jax.random.PRNGKey(0)

A = SymmetricTensor.random_normal(
    indices=(
        TensorIndex(u1, phys_charges, FlowDirection.IN, label="p0"),
        TensorIndex(u1, bond_charges, FlowDirection.IN, label="left"),
        TensorIndex(u1, bond_charges, FlowDirection.OUT, label="bond"),
    ),
    key=key,
)
B = SymmetricTensor.random_normal(
    indices=(
        TensorIndex(u1, phys_charges, FlowDirection.IN, label="p1"),
        TensorIndex(u1, bond_charges, FlowDirection.IN, label="bond"),  # shared label
        TensorIndex(u1, bond_charges, FlowDirection.OUT, label="right"),
    ),
    key=jax.random.PRNGKey(1),
)

# Contract by matching shared labels — "bond" is summed over automatically
result = contract(A, B)
print(result.labels())  # ('p0', 'left', 'p1', 'right')

# Build a tensor network and contract
tn = TensorNetwork()
tn.add_node("A", A)
tn.add_node("B", B)
tn.connect_by_shared_label("A", "B")
result = tn.contract()

Network Blueprint (.net file) Example

from tenax import NetworkBlueprint

# Define network topology as a string (or read from a .net file)
bp = NetworkBlueprint("""
L: a, b, c
M: a, p, q, d
A: b, p, s, e
M2: e, q, t, f
R: d, f, g
TOUT: c, s, t, g
""")

# Load tensors (can be DenseTensor or SymmetricTensor)
bp.put_tensors({"L": L, "M": M, "A": A, "M2": M2, "R": R})
result = bp.launch()  # contracts the full network

# Reuse with different tensors (e.g. in a DMRG sweep)
bp.put_tensor("A", new_A)
result2 = bp.launch()

DMRG Example

Performance note: Tenax's DMRG uses a fused Cython BLAS pipeline on CPU for high-throughput block-sparse contractions. GPU/TPU acceleration is available via DMRGConfig(accelerator="jit") for dense tensors and accelerator="sharded" for multi-GPU runs.

from tenax.algorithms.dmrg import dmrg, build_mpo_heisenberg, DMRGConfig
from tenax.network.network import build_mps

L = 10  # chain length
mpo = build_mpo_heisenberg(L, Jz=1.0, Jxy=1.0)

# Build random initial MPS
# ...

config = DMRGConfig(max_bond_dim=50, num_sweeps=10)
result = dmrg(mpo, initial_mps, config)
print(f"Ground state energy: {result.energy:.8f}")

2D Cylinder DMRG Example

from tenax import AutoMPO, DMRGConfig, build_random_mps, dmrg

# Build Heisenberg Hamiltonian on a 6x3 cylinder via AutoMPO
Lx, Ly, N = 6, 3, 18
auto = AutoMPO(L=N, d=2)
for x in range(Lx):
    for y in range(Ly):
        # Within-ring bond (periodic y)
        i, j = x * Ly + y, x * Ly + (y + 1) % Ly
        auto += (1.0, "Sz", min(i, j), "Sz", max(i, j))
        auto += (0.5, "Sp", min(i, j), "Sm", max(i, j))
        auto += (0.5, "Sm", min(i, j), "Sp", max(i, j))
        # Between-ring bond (open x)
        if x < Lx - 1:
            i, j = x * Ly + y, (x + 1) * Ly + y
            auto += (1.0, "Sz", i, "Sz", j)
            auto += (0.5, "Sp", i, "Sm", j)
            auto += (0.5, "Sm", i, "Sp", j)

mpo = auto.to_mpo(compress=True)
mps = build_random_mps(N, physical_dim=2, bond_dim=16)
config = DMRGConfig(max_bond_dim=100, num_sweeps=10, verbose=True)
result = dmrg(mpo, mps, config)
print(f"E/N = {result.energy / N:.8f}")  # converges in a few sweeps

See examples/heisenberg_cylinder.py for a full working example with 4x2, 6x3, and 8x4 cylinders.

iDMRG Example

from tenax import idmrg, build_bulk_mpo_heisenberg, iDMRGConfig

W = build_bulk_mpo_heisenberg(Jz=1.0, Jxy=1.0)
config = iDMRGConfig(max_bond_dim=32, max_iterations=100, convergence_tol=1e-8)
result = idmrg(W, config)
print(f"Energy per site: {result.energy_per_site:.6f}")  # ~ -0.4431
print(f"Converged: {result.converged}")

Infinite Cylinder iDMRG Example

from tenax import build_bulk_mpo_heisenberg_cylinder, iDMRGConfig, idmrg

# Ly=4 cylinder: each super-site is a ring of 4 spins (d=16, D_w=14)
# Only even Ly is supported (odd Ly frustrates AFM order).
W = build_bulk_mpo_heisenberg_cylinder(Ly=4)
config = iDMRGConfig(max_bond_dim=200, max_iterations=200, convergence_tol=1e-4)
result = idmrg(W, config, d=16)
e_per_spin = result.energy_per_site / 4
print(f"Energy per spin: {e_per_spin:.6f}")

See examples/heisenberg_infinite_cylinder.py for Ly=2 and Ly=4 cylinders with ED cross-checks.

TRG Example

from tenax import TRGConfig, trg, compute_ising_tensor, ising_free_energy_exact

beta = 0.44  # near critical temperature
T = compute_ising_tensor(beta)

config = TRGConfig(max_bond_dim=16, num_steps=20)
log_z_per_n = trg(T, config)
f_trg = float(-log_z_per_n / beta)
f_exact = ising_free_energy_exact(beta)
print(f"TRG:   {f_trg:.8f}")
print(f"Exact: {f_exact:.8f}")

See examples/ising_trg.py and examples/ising_hotrg.py for full TRG and HOTRG examples at multiple temperatures compared against the Onsager exact solution.

AutoMPO Example

from tenax import AutoMPO, build_auto_mpo

# Class-based interface: build a Heisenberg chain
L = 10
auto = AutoMPO(L)
for i in range(L - 1):
    auto += (1.0, "Sz", i, "Sz", i + 1)
    auto += (0.5, "Sp", i, "Sm", i + 1)
    auto += (0.5, "Sm", i, "Sp", i + 1)
mpo = auto.to_mpo()

# Or use the functional interface with custom operators
import numpy as np

custom_ops = {
    "X": np.array([[0.0, 1.0], [1.0, 0.0]]),
    "Z": np.array([[1.0, 0.0], [0.0, -1.0]]),
    "Id": np.eye(2),
}
terms = [(1.0, "Z", i, "Z", i + 1) for i in range(L - 1)]
terms += [(0.5, "X", i) for i in range(L)]
mpo = build_auto_mpo(terms, L=L, site_ops=custom_ops)

# Build a symmetric (U(1) block-sparse) MPO
mpo_sym = auto.to_mpo(symmetric=True)

iPEPS Simple Update (2-site unit cell)

import jax.numpy as jnp
from tenax import iPEPSConfig, CTMConfig, ipeps

# Build a 2-site Heisenberg gate
Sz = 0.5 * jnp.array([[1.0, 0.0], [0.0, -1.0]])
Sp = jnp.array([[0.0, 1.0], [0.0, 0.0]])
Sm = jnp.array([[0.0, 0.0], [1.0, 0.0]])
gate = jnp.einsum("ij,kl->ikjl", Sz, Sz) + 0.5 * (
    jnp.einsum("ij,kl->ikjl", Sp, Sm) + jnp.einsum("ij,kl->ikjl", Sm, Sp)
)

# 2-site checkerboard iPEPS — captures Neel order
config = iPEPSConfig(
    max_bond_dim=2,
    num_imaginary_steps=200,
    dt=0.3,
    ctm=CTMConfig(chi=10, max_iter=40),
    unit_cell="2site",
)
energy, peps, (env_A, env_B) = ipeps(gate, None, config)
print(f"Energy per site: {energy:.6f}")  # ~ -0.65

See examples/heisenberg_ipeps_su.py for 1-site and 2-site unit cell examples.

iPEPS AD Optimization and Excitations

import jax.numpy as jnp
from tenax import (
    iPEPSConfig,
    CTMConfig,
    optimize_gs_ad,
    optimize_gs_ad_chi_schedule,
    ExcitationConfig,
    compute_excitations,
    make_momentum_path,
)

# Build a 2-site Heisenberg gate
Sz = 0.5 * jnp.array([[1.0, 0.0], [0.0, -1.0]])
Sp = jnp.array([[0.0, 1.0], [0.0, 0.0]])
Sm = jnp.array([[0.0, 0.0], [1.0, 0.0]])
gate = jnp.einsum("ij,kl->ikjl", Sz, Sz) + 0.5 * (
    jnp.einsum("ij,kl->ikjl", Sp, Sm) + jnp.einsum("ij,kl->ikjl", Sm, Sp)
)

# Recommended AD configuration: L-BFGS + explicit AD + QR projectors.
# forward_gauge defaults to "phase" (variPEPS-style Frobenius + phase
# fix), correct for both implicit and explicit AD. Reaches E=-0.6628
# at D=2, chi=16 (literature: -0.6548 at D=2).
config = iPEPSConfig(
    max_bond_dim=2,
    ctm=CTMConfig(
        chi=16,
        max_iter=80,
        projector_method="qr",  # recommended projector for explicit AD
    ),
    # gs_implicit_ad=False is the default (explicit AD)
    gs_projector_method="qr",
    gs_optimizer="lbfgs",  # L-BFGS with Hager-Zhang line search
    gs_line_search_method="hager_zhang",
    gs_metric_precond=True,  # metric preconditioning (Rader et al.)
    gs_c4v=True,  # C4v basis parameterization
    su_init=True,
)
A_opt, env, E_gs = optimize_gs_ad(gate, None, config)
print(f"Ground-state energy: {E_gs:.6f}")

# Chi-ramping schedule: progressively increase chi for faster convergence
chi_schedule = [4, 8, 16]
A_opt, env, E_gs = optimize_gs_ad_chi_schedule(gate, None, config, chi_schedule)

# 2-site shared-tensor C4v AD for antiferromagnets (Neel order)
# A single C4v-parameterized tensor is optimized; B is derived from A via
# sublattice rotation B = e^{i pi sigma^y/2} on the physical leg.  This
# ties the two sublattices together and avoids the A/B drift that makes
# the unconstrained 2-site AD path unstable.  Spin-1/2 (d=2) only.
config_2site = iPEPSConfig(
    max_bond_dim=2,
    ctm=CTMConfig(chi=16, max_iter=100, min_iter=50),
    gs_optimizer="lbfgs",
    gs_explicit_ad_steps=10,
    gs_explicit_ad_warmup=2,
    gs_num_steps=50,
    gs_line_search=True,
    unit_cell="2site",
    gs_c4v=True,
    su_init=True,
    num_imaginary_steps=100,
    dt=0.3,
)
(A_opt, B_opt), (env_A, env_B), E_gs = optimize_gs_ad(gate, None, config_2site)

# SVD (Fishman) projectors — alternative to eigh and QR
config_svd = iPEPSConfig(
    max_bond_dim=2,
    ctm=CTMConfig(chi=16, max_iter=50, projector_method="svd"),
    gs_num_steps=200,
    gs_optimizer="lbfgs",
    gs_line_search_method="hager_zhang",
)
A_opt, env, E_gs = optimize_gs_ad(gate, None, config_svd)

# Opt-in reference-mode dense C4v mode (Francuz et al., App. C-F)
config_reference = iPEPSConfig(
    max_bond_dim=2,
    ctm=CTMConfig(
        chi=16,
        max_iter=80,
        projector_method="eigh",
        ctm_ad_mode="c4v_reference",
        adjoint_solver="bicgstab",
        adjoint_maxiter=50,
        adjoint_tol=1e-8,
    ),
    gs_implicit_ad=True,
    gs_c4v=True,
    unit_cell="1x1",
    gs_num_steps=100,
    gs_optimizer="adam",
)
A_opt, env, E_gs = optimize_gs_ad(gate, None, config_reference)

# Quasiparticle excitations (Ponsioen et al. 2022)
momenta = make_momentum_path("brillouin", num_points=20)
exc_config = ExcitationConfig(num_excitations=3)
result = compute_excitations(A_opt, env, gate, E_gs, momenta, exc_config)
print(result.energies.shape)  # (20, 3)

See examples/heisenberg_ipeps_ad.py for AD optimization with random vs simple update initialization, and examples/heisenberg_ipeps_excitations.py for the full excitation spectrum along Gamma-X-M-Gamma.

Split-CTMRG

from tenax import CTMConfig, ctm_split, compute_energy_split_ctm

# Split-CTMRG keeps ket/bra layers separate for O(χ³D³) projector cost
# instead of O(χ³D⁶) — significant speedup at large bond dimension D
config = CTMConfig(chi=20, max_iter=100, chi_I=10)
env = ctm_split(A, config)
E = compute_energy_split_ctm(A, env, gate, d=2)

Examples

Runnable example scripts are in the examples/ directory:

Script	Algorithm	Model
heisenberg_cylinder.py	DMRG	Heisenberg on 4x2, 6x3, 8x4 cylinders
heisenberg_infinite_cylinder.py	iDMRG	Heisenberg on infinite Ly=2, Ly=4 cylinders
heisenberg_ipeps_su.py	iPEPS simple update	Heisenberg (1x1 and 2-site unit cells)
heisenberg_ipeps_ad.py	iPEPS AD optimization	Heisenberg (random vs SU init)
heisenberg_ipeps_excitations.py	iPEPS excitations	Heisenberg dispersion along Γ-X-M-Γ
spinless_fermion_fpeps.py	fPEPS simple update	Spinless fermions (free and interacting)
ising_trg.py	TRG	2D Ising vs Onsager exact
ising_hotrg.py	HOTRG	2D Ising vs Onsager exact

Run any example with:

uv run python examples/<script>.py

Symmetry System

from tenax import U1Symmetry, ZnSymmetry, ProductSymmetry, FermionParity
import numpy as np

# U(1): integer charges, fusion by addition
u1 = U1Symmetry()
charges = np.array([-1, 0, 1], dtype=np.int32)
print(u1.fuse(charges, charges))  # [-2, 0, 2]
print(u1.dual(charges))  # [1, 0, -1]

# Z_3: charges mod 3
z3 = ZnSymmetry(3)
print(
    z3.fuse(np.array([1, 2], dtype=np.int32), np.array([2, 2], dtype=np.int32))
)  # [0, 1]

# Product symmetry: combine two symmetries (e.g., charge × S_z)
sym = ProductSymmetry(U1Symmetry(), U1Symmetry())
packed = ProductSymmetry.encode_charges(
    np.array([0, 1, -1], dtype=np.int32),  # charge
    np.array([1, 0, -1], dtype=np.int32),  # S_z
)
q1, q2 = ProductSymmetry.decode_charges(packed)

Limitations: ProductSymmetry combines exactly two factors by bit-packing two int16 charges into one int32. Nesting is not supported, so three-factor groups (e.g., U(1)×U(1)×Z₂) require a future MultiProductSymmetry. Each factor charge must fit in the int16 range [-32768, 32767].

Gotchas

Float64 precision and JAX_ENABLE_X64

Tenax defaults to float64 for all tensors and algorithms. Importing tenax automatically calls jax.config.update("jax_enable_x64", True), so 64-bit arithmetic is enabled out of the box.

If you import JAX before tenax and create arrays in that window, they will still be float32. To avoid surprises, either import tenax first or enable x64 manually:

import jax

jax.config.update("jax_enable_x64", True)

import tenax

MPO index convention

The MPO W-tensor uses the convention W[w_l, ket, bra, w_r] — the two middle indices are physical (ket on top, bra on bottom) and the outer indices are bond dimensions.

NumPy >= 2.0 casting

Adding a Python complex scalar (even 1+0j) into a float64 array raises UFuncOutputCastingError under NumPy >= 2.0. Use .real or an explicit complex128 dtype instead.

Local test failures on macOS x86_64

uv run pytest may fail on macOS x86_64 if jaxlib has no wheel for that platform.

Benchmarks

A CLI-driven benchmark suite measures wall-clock performance of every algorithm across hardware backends.

# Quick smoke test (TRG, small size, 1 trial)
python -m benchmarks.run --backend cpu --algorithm trg --size small --trials 1

# Full CPU baseline
python -m benchmarks.run --backend cpu -o benchmarks/results/cpu_baseline.json

# GPU comparison
python -m benchmarks.run --backend cuda -o benchmarks/results/cuda.json

# Specific algorithms and sizes
python -m benchmarks.run -b cpu -a dmrg idmrg -s small medium -n 5

# CSV output for analysis
python -m benchmarks.run -b cpu -a all -s all --csv results.csv

# Show available backends
python -m benchmarks.run --list-backends

Each run prints a summary table and saves full results (timings, parameters, device info) to JSON. See docs/guide/benchmarks.md for the complete guide.

Development

# Clone and install with dev dependencies
git clone https://github.com/tenax-lab/tenax.git
cd tenax
uv sync --all-extras --dev

# Install pre-commit hooks (ruff lint + format on every commit)
uv run pre-commit install

# Run tests
uv run pytest -m core          # fast core tests only
uv run pytest -m algorithm     # algorithm tests (DMRG, TRG, iPEPS, integration)
uv run pytest -m "not slow"    # skip expensive tests
uv run pytest                  # full suite

# Lint
uv run ruff check src/ tests/

Work-in-progress design documents live in design/.

Documentation

Full API documentation is built with Sphinx:

cd docs && make html

The generated HTML is in docs/_build/html/.

References

• H.-J. Liao, J.-G. Liu, L. Wang, T. Xiang, Phys. Rev. X 9, 031041 (2019) — AD-based iPEPS ground-state optimization
• A. Francuz, N. Schuch, B. Vanhecke, PRR 7, 013237 (2025) — Stable AD through CTM (SVD regularization, truncation correction, implicit differentiation)
• M. Rader, L. Gresista, C. Hubig, S. Montangero, A. Weichselbaum, J. von Delft, arXiv:2511.09546 (2025) — Metric preconditioning and Hager-Zhang line search for iPEPS optimization
• L. Ponsioen, F. F. Assaad, P. Corboz, SciPost Phys. 12, 006 (2022) — Quasiparticle excitations for iPEPS
• J. Naumann, E. L. Weerda, J. Eisert, M. Rizzi, P. Schmoll, arXiv:2502.10298 (2025) — Split-CTMRG with factored projectors for efficient iPEPS environments

License

Apache 2.0