Tenax

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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), iTEBD (numerically stable infinite TEBD, incl. inversion-free Hastings update), 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
• In-CTM χ-bump (variPEPS §2.8.2) — recommended reactive growth of the CTM bond dimension inside CTM convergence (CTMConfig.ctmrg_heuristic_increase_chi=True with chi_max set); the env is always converged at the new χ before the optimizer sees it, avoiding the zero-padded-env cliff-edge artifact that the legacy end-of-outer-step chi_auto_bump and scheduled chi_ramp introduce between L-BFGS steps. Both legacy knobs still work but emit DeprecationWarning (see issue #512) and will be removed in a future release. References: Naumann et al., SciPost Phys. Lect. Notes 86, 2024
• SVD and QR CTMRG projectors — SVD (Fishman) projectors (projector_method="svd", default) and eigh projectors, plus a reduced-corner QR-CTMRG projector (projector_method="qr", arXiv:2505.00494) on the dense single-site path, usable both forward-only and under AD ground-state optimization via gs_recipe="1x1" + gs_projector_method="qr" (Phase 2, dense; block-sparse is a later phase)
• 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)
• Split-CTM energy entry points — compute_energy_split_ctm_tensor_2site and compute_energy_split_ctm_tensor_multisite for 2-site checkerboard and multisite unit cells (kagome PESS, etc.) at large D
• Split-CTM AD ground-state optimization — optimize_gs_ad with CTMConfig(fuse_virtual_legs=False) drives the whole single-site optimizer (unit_cell="1x1", gs_recipe="1x1") through the split χ²·D⁴ forward instead of the fused χ²·D⁶ double layer: implicit AD via a Γ-gauge-fixed fixed-point custom_vjp (Neumann backward), with the line-search probe, warm-start, and final environment all routed through the same split forward (returns a SplitCTMTensorEnv). Implicit gradient matches the trusted explicit-AD gradient to ~1e-12. DenseTensor only (SymmetricTensor/fermionic split AD is a later phase); fixed χ (the χ-changing knobs are rejected on this path); the memory win over fused is a large-D effect (D≳16). References: Naumann et al., arXiv:2502.10298
• Honeycomb iPEPS CTM (native) — rank-4, 6-corner, 3-direction, 2-sublattice CTMRG for honeycomb iPEPS (replaces the dummy-bond brick-wall workaround). Public entry honeycomb_ctm_energy_implicit provides jax.custom_vjp with a JIT-fused GMRES backward; default Corboz biorthogonal projector + per-column phase fix; configurable energy_fn hook for kagome iPESS triangle energies. References: Lukin & Sotnikov, PRB 107, 054424 (2023) for the 6-corner CTMRG and the bipartite extension in PRE 109, 045305 (2024) §II.C.
• Quasiparticle excitations — iPEPS excitation spectra at arbitrary Brillouin-zone momenta (Ponsioen et al. 2022)
• Model gate helpers — pre-built 2-site Hamiltonian tensors: heisenberg_gate (dense DenseTensor with trivial charges), heisenberg_gate_u1sz (U(1)-Sz block-sparse SymmetricTensor with charges [+1, −1] for spin-↑/↓), xxz_gate (XXZ anisotropy), spinless_fermion_gate (fPEPS hopping + interaction with FermionParity symmetry)
• 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)
)

# Explicit-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,  # opt into explicit AD (the default is implicit)
    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.
# Each entry is (chi, num_steps) — run `num_steps` AD steps at logical χ=chi.
# Internally the schedule runs as a single `optimize_gs_ad` call with envs
# padded to max(chi) from step 1, so the JIT-compiled CTM / energy / backward
# kernels never see a shape change (issue #453).
chi_schedule = [(4, 30), (8, 30), (16, 20)]
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)

Honeycomb iPEPS CTM (native rank-4)

Native rank-4 CTMRG for honeycomb iPEPS — six corners, three edge directions, two sublattices — without the dummy-bond brick-wall hack. Custom jax.custom_vjp forward with a JIT-fused GMRES backward.

import jax
import jax.numpy as jnp
import numpy as np
from tenax import (
    HONEYCOMB_DIRECTIONS,
    honeycomb_ctm_energy_implicit,
    honeycomb_ctm_run,
)
from tenax.core.index import FlowDirection, TensorIndex
from tenax.core.symmetry import U1Symmetry
from tenax.core.tensor import DenseTensor


def _make_site(D=2, d=2, key=jax.random.PRNGKey(0)):
    sym = U1Symmetry()
    virt = np.zeros(D, dtype=np.int32)
    phys = np.zeros(d, dtype=np.int32)
    indices = (
        TensorIndex.from_charges(sym, virt.copy(), FlowDirection.OUT, label="e0"),
        TensorIndex.from_charges(sym, virt.copy(), FlowDirection.OUT, label="e1"),
        TensorIndex.from_charges(sym, virt.copy(), FlowDirection.OUT, label="e2"),
        TensorIndex.from_charges(sym, phys.copy(), FlowDirection.IN, label="phys"),
    )
    re = jax.random.normal(key, (D, D, D, d))
    im = jax.random.normal(jax.random.fold_in(key, 1), (D, D, D, d))
    return DenseTensor((re + 1j * im).astype(jnp.complex128), indices)


# Spin-1/2 Heisenberg bond operator (4×4)
sx = 0.5 * np.array([[0, 1], [1, 0]], dtype=np.complex128)
sy = 0.5 * np.array([[0, -1j], [1j, 0]], dtype=np.complex128)
sz = 0.5 * np.array([[1, 0], [0, -1]], dtype=np.complex128)
H_bond = jnp.asarray(np.kron(sx, sx) + np.kron(sy, sy) + np.kron(sz, sz))

# Honeycomb iPEPS uses two rank-4 sites at coords (0,0) and (1,0); legs
# (e0, e1, e2, phys). All virtuals OUT, phys IN.
A = _make_site(D=2, d=2, key=jax.random.PRNGKey(0))
B = _make_site(D=2, d=2, key=jax.random.PRNGKey(1))
sites = {(0, 0): A, (1, 0): B}

# Forward only: returns the converged per-sublattice env dict + info.
envs, info = honeycomb_ctm_run(
    sites, chi=8, max_iter=80, conv_tol=1e-8,
    projector_method="biorthogonal",  # default; eigh/svd are A=B opt-ins
    forward_gauge="phase",            # default; sigma reserved for A=B opt-in
)

# Implicit-AD energy: takes jax.grad through the CTM fixed point via
# JIT-fused GMRES on (I - dF/denv) lambda = dE/denv.
energy = honeycomb_ctm_energy_implicit(
    sites, H_bond, chi=8, max_iter=80, conv_tol=1e-8,
)
grad_fn = jax.grad(
    lambda Ad: honeycomb_ctm_energy_implicit(
        {(0, 0): DenseTensor(Ad, A.indices), (1, 0): B},
        H_bond, chi=8, max_iter=40,
    )
)
gA = grad_fn(A.todense())

The default energy is the 3-edge nearest-neighbor bond sum Σ_α Tr(ρ_α · H_bond). Pass energy_fn=compute_honeycomb_triangle_energy for the kagome iPESS use case where each site is a 3-spin triangle and the Hamiltonian is the intra-triangle 3-spin operator.

Kagome iPESS with AD

Differentiable iPESS pipeline for kagome XXZ ground states (Liao et al., PRX 9, 031041, 2019). Two simplex tensors T_u, T_d and three site tensors R_a, R_b, R_c define the variational state; triangle simple update gives the SU warm start, then L-BFGS through the square-coarse-grained CTM (Convention C) refines (R_a, R_b, R_c, T_u, lambdas). T_d is held frozen during AD — its variational role is absorbed by the down-bond gauges.

import jax
from tenax import (
    CTMConfig,
    IPESSState,
    kagome_triangle_xxz_hamiltonian,
    kagome_xxz_pess_cg_gates,
    pess_simple_update,
    optimize_pess_ad,
)

D, d = 2, 3  # spin-1
H = kagome_triangle_xxz_hamiltonian(delta=1.0, d=d)
cg_gates = kagome_xxz_pess_cg_gates(delta=1.0, d=d)

state = IPESSState.random(D=D, d=d, key=jax.random.PRNGKey(0))
state = pess_simple_update(state, H,
                           dt_schedule=[(0.1, 200), (0.01, 200), (0.001, 100)],
                           D_max=D)

config = CTMConfig(chi=8, max_iter=30, conv_tol=1e-7,
                   projector_method="svd", forward_gauge="phase",
                   ctm_conv_method="elementwise")
state, e_per_site = optimize_pess_ad(state, cg_gates, config, max_iter=30)
print(f"E/site = {e_per_site:.6f}")  # spin-1 D=2 lands around -1.0

The full kagome Hamiltonian (3 up-triangle bonds + 3 down-triangle bonds per unit cell) is reconstructed via compute_energy_cg's intra-cell + horizontal/vertical/diagonal inter-cell 2-site RDMs; see examples/kagome_spin12_pess_ad_benchmark.py and examples/kagome_spin1_pess_ad_benchmark.py for full sweeps.

Multisite path (3-site kagome on a square unit cell)

For the multisite encoding pess_to_kagome_3site_multisite, where the kagome unit cell maps to three sites (u, v, w) on a square lattice and the energy uses 4 NN bonds + 2 marginalised-3-site contributions, use build_pess_loss_3site_multisite and optimize_pess_3site_multisite_ad:

from tenax import (
    build_pess_loss_3site_multisite,
    optimize_pess_3site_multisite_ad,
    pess_to_kagome_3site_multisite,
)
from tenax.algorithms._pess_multisite_energy import kagome_3site_bond_gates

bond_gates = kagome_3site_bond_gates(delta=1.0, d=d)
state, e_per_site = optimize_pess_3site_multisite_ad(
    state, bond_gates, config, max_iter=30,
)

The optimizer warm-starts CTM envs across L-BFGS steps via an internal env_cache, returns the best-seen energy across the trajectory, and gates CTMConfig at entry on the implicit-AD invariants (projector_method='svd', forward_gauge='phase', ctm_conv_method='elementwise').

Examples

Runnable example scripts are in the examples/ directory:

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