QTen

A torch-based tensor library for quantum-related computations.

Installation

QTen requires Python 3.11 or newer.

Install PyTorch first. For CPU or CUDA-specific builds, use the official PyTorch installation command for your platform.

Then install QTen:

pip install qten

If you want plotting support:

pip install qten-plots

If you use uv:

uv add torch
uv add qten
uv add qten-plots

For development from this repository:

uv sync --extra cpu --group dev

If you want a CUDA-specific PyTorch build in this repository, choose one of the defined extras:

uv sync --extra cu126 --group dev
uv sync --extra cu128 --group dev
uv sync --extra cu129 --group dev
uv sync --extra cu130 --group dev

Quick Start

The package root exports the main tensor helpers: Tensor, eye, zeros, ones, and set_precision.

For simple examples, the easiest axis type is IndexSpace.

Tensors

Create a tensor with named dimensions:

import torch

import qten
from qten.symbolics import IndexSpace

rows = IndexSpace.linear(2)
cols = IndexSpace.linear(3)

x = qten.Tensor(
    data=torch.arange(6, dtype=torch.float64).reshape(2, 3),
    dims=(rows, cols),
)

print(x.rank())      # 2
print(x.data.shape)  # torch.Size([2, 3])
print(x.dims)        # (IndexSpace(size=2), IndexSpace(size=3))

Create standard tensors on the same space:

import torch

import qten
from qten.symbolics import IndexSpace

space = IndexSpace.linear(2)

a = qten.Tensor(
    data=torch.tensor([[1.0, 2.0], [3.0, 4.0]]),
    dims=(space, space),
)

identity = qten.eye((space, space))
filled = qten.ones((space, space))
empty = qten.zeros((space, space))
product = a @ identity

print(product.data)
print(filled.data.shape)  # torch.Size([2, 2])
print(empty.data.shape)   # torch.Size([2, 2])

If you want QTen to use single precision instead of double precision:

import qten

qten.set_precision("32")

Lattices

Build a square lattice:

import sympy as sy
import torch

from qten.geometries import Lattice

lattice = Lattice(
    basis=sy.ImmutableDenseMatrix([[1, 0], [0, 1]]),
    shape=(3, 3),
)

print(lattice.shape)                 # (3, 3)
print(len(lattice.cartes()))         # 9
print(lattice.cartes(torch.Tensor))  # Cartesian coordinates

Boundary Conditions

Finite Lattice objects use periodic boundary conditions. When you pass shape=(Lx, Ly, ...), QTen treats lattice indices modulo those extents, so a site at (Lx, y) is identified with (0, y).

That means:

• shape=(3, 3) creates a 3 x 3 torus, not an open patch.
• Distances are measured using the nearest periodic image.
• shape is shorthand for a diagonal PeriodicBoundary.

If you want to specify the periodic identifications explicitly, pass boundaries=PeriodicBoundary(...):

import sympy as sy

from qten.geometries import Lattice, PeriodicBoundary

lattice = Lattice(
    basis=sy.ImmutableDenseMatrix([[1, 0], [0, 1]]),
    boundaries=PeriodicBoundary(sy.ImmutableDenseMatrix([[3, 0], [0, 3]])),
)

print(lattice.shape)  # (3, 3)

For a unit cell with more than one site:

import sympy as sy

from qten.geometries import Lattice

lattice = Lattice(
    basis=sy.ImmutableDenseMatrix([[1, 0], [0, 1]]),
    shape=(2, 2),
    unit_cell={
        "A": sy.ImmutableDenseMatrix([0, 0]),
        "B": sy.ImmutableDenseMatrix([sy.Rational(1, 2), sy.Rational(1, 2)]),
    },
)

print(lattice.shape)         # (2, 2)
print(len(lattice.cartes())) # 8

Plotting

Plotting requires qten-plots.

import torch

import qten
from qten.symbolics import IndexSpace

rows = IndexSpace.linear(2)
cols = IndexSpace.linear(3)

x = qten.Tensor(
    data=torch.arange(6, dtype=torch.float64).reshape(2, 3),
    dims=(rows, cols),
)

fig = x.plot("heatmap", show=False)
fig.show()

You can also plot a lattice:

import sympy as sy

from qten.geometries import Lattice

lattice = Lattice(
    basis=sy.ImmutableDenseMatrix([[1, 0], [0, 1]]),
    shape=(3, 3),
)

fig = lattice.plot("structure", show=False)
fig.show()

IO

QTen includes a small versioned pickle-based storage helper at qten.io.

import qten

qten.io.iodir(".data")
qten.io.env("demo")

qten.io.save({"loss": 0.12}, "metrics")
qten.io.save({"loss": 0.08}, "metrics")

print(qten.io.load("metrics", -1))
print(qten.io.list_saved("metrics"))

Worked Example: Dirac Semi-Metal On The Honeycomb Lattice

The example does six things:

1. Builds the real-space Dirac semi-metal Hamiltonian on a honeycomb lattice.
2. Fourier transforms it into Bloch form.
3. Folds the bands into a larger real-space cell.
4. Transforms the blocked Hamiltonian by C6 and compares the result.
5. Builds the ground-state correlation matrix and restricts it to a local region.
6. Extracts a local C6-symmetric basis from the restricted eigenvectors.

1. Imports And Lattice Setup

Start by defining the honeycomb lattice, the C6 symmetry, and the two primitive translations.

import sympy as sy
import qten
import qten.ops as Q

from qten.bands import bandfillings, bandfold, bandtransform
from qten.geometries import BasisTransform, Lattice, Offset
from qten.phys import FFObservable
from qten.pointgroups import AbelianOpr, pointgroup
from qten.symbolics import FuncOpr, HilbertSpace, U1Basis, brillouin_zone
from qten_plots.plottables import PointCloud


triangular = sy.ImmutableMatrix(
    [
        [sy.sqrt(3) / 2, 0],
        [-sy.Rational(1, 2), 1],
    ]
)

honeycomb = Lattice(
    basis=triangular,
    unit_cell={
        "a": sy.ImmutableMatrix([sy.Rational(1, 3), sy.Rational(2, 3)]),
        "b": sy.ImmutableMatrix([sy.Rational(2, 3), sy.Rational(1, 3)]),
    },
    shape=(96, 96),
)

c6 = pointgroup("c6-xy:xy")
T_c6 = AbelianOpr(c6)

a1, a2 = honeycomb.basis_vectors()
R_a1 = FuncOpr(Offset, lambda r: r + a1)
R_a2 = FuncOpr(Offset, lambda r: r + a2)

What this does:

• triangular is the triangular Bravais lattice underlying the honeycomb lattice.
• unit_cell adds the two sublattice sites a and b.
• c6 is the sixfold point-group generator.
• T_c6 = AbelianOpr(c6) is the affine version of that rotation. At this stage it only contains the linear C6 action, centered at the canonical origin.
• R_a1 and R_a2 are translation operators that shift an orbital by one primitive lattice vector.

2. Define The Two Bloch Basis Orbitals

Use one orbital on each sublattice site, both transforming in the same C6 basis channel.

psi_a = U1Basis.new(honeycomb.at("a"), c6.basis_table[1])
psi_b = U1Basis.new(honeycomb.at("b"), c6.basis_table[1])

Here:

• honeycomb.at("a") and honeycomb.at("b") pick the two sites in the reference unit cell.
• c6.basis_table is a lookup table from C6 eigenvalue to an AbelianBasis.
• c6.basis_table[1] picks the eigen-basis with eigenvalue 1, which is the trivial C6 character. In this case it is the constant basis function e.
• Using that basis means the internal orbital itself is C6 invariant; the nontrivial symmetry action in this example comes from how the rotation moves the lattice position, not from an extra internal orbital phase.

3. Build The Dirac Semi-Metal Hamiltonian In Real Space

Now add the nearest-neighbor hoppings that define the honeycomb Dirac Hamiltonian.

t = sy.Number(1)

dirac = FFObservable()
dirac.add_bond(-t, psi_a, psi_b)         # A(R) <-> B(R)
dirac.add_bond(-t, psi_a, R_a2 @ psi_b)  # A(R) <-> B(R + a2)
dirac.add_bond(-t, psi_b, R_a1 @ psi_a)  # B(R) <-> A(R + a1)

harmonic = dirac.to_tensor()

Interpretation:

• FFObservable() collects quadratic fermion terms.
• Each add_bond adds one hopping channel.
• harmonic is the real-space tensor form of the tight-binding Hamiltonian.

The three bonds are enough because the remaining nearest-neighbor hoppings are generated automatically by translation and Hermitian completion in the observable-to-tensor construction. In other words, you specify one representative set of bonds in the unit cell, not every bond in the full finite lattice.

This is the standard nearest-neighbor honeycomb model, so its Bloch spectrum contains Dirac cones.

4. Build The Bloch Hamiltonian

Create the Bloch basis, the Brillouin-zone momentum space, and the Fourier transform.

bloch_space = HilbertSpace.new([psi_a, psi_b])
k_space = brillouin_zone(honeycomb.dual)

F = Q.fourier_transform(k_space, bloch_space, harmonic.dims[0])
bloch = F @ harmonic @ F.h(-2, -1)

What happens here:

• bloch_space is the two-orbital basis inside one unit cell.
• k_space is the sampled momentum space of the finite lattice.
• F maps the real-space basis into the Bloch basis.
• bloch is the momentum-space Hamiltonian with dims (MomentumSpace, HilbertSpace, HilbertSpace).

This is the first place where the model becomes a band Hamiltonian in the usual condensed-matter sense: for each crystal momentum k, bloch[k] is a finite matrix acting on the orbital content of one unit cell.

5. Block The Hamiltonian Into A Larger Cell

Here the real-space cell is doubled in both primitive directions.

blocking = BasisTransform(sy.ImmutableMatrix.eye(2) * 2)
blocked_bloch = bandfold(blocking, bloch, opt="both")

This does a band folding operation:

• the real-space unit cell becomes larger,
• the Brillouin zone becomes smaller,
• the number of Bloch bands increases accordingly.

Using opt="both" means the basis change is applied consistently on the bra and ket sides of the operator, so the result is still a Hamiltonian in the blocked basis rather than a mixed-basis object.

6. Transform The Blocked Hamiltonian By C6

Apply the sixfold symmetry operator to the blocked Bloch Hamiltonian.

c6_bloch = bandtransform(T_c6, blocked_bloch)

To compare the transformed Hamiltonian with the original one:

_ = (blocked_bloch - c6_bloch).plot("bandstructure")

If the construction is symmetry-compatible, this difference should be small or vanish up to the expected numerical and representation conventions.

This comparison is worth doing after blocking because folding changes the band labeling and basis organization. Checking bandtransform(T_c6, blocked_bloch) against blocked_bloch is a direct sanity check that the blocked model still carries the intended C6 symmetry.

7. Plot The Blocked Band Structure

Plot the blocked Hamiltonian itself:

_ = blocked_bloch.plot("bandstructure")

This gives the folded band structure after enlarging the unit cell.

8. Find The Ground-State Occupied Subspace

Fill half the bands to form the free-fermion ground state.

gs = bandfillings(blocked_bloch, 0.5)
P_gs = gs @ gs.h(-2, -1)
C_gs = 1.0 * qten.eye(P_gs.dims) - P_gs

Meaning:

• gs contains the occupied Bloch eigenvectors.
• P_gs is the projector onto the occupied subspace.
• C_gs is the single-particle correlation matrix for the ground state.

For a free-fermion ground state, this is the main object you need for entanglement and local symmetry analysis.

The filling fraction 0.5 means half filling of the total one-particle spectrum. bandfillings diagonalizes the Hamiltonian across all sampled momenta, finds the global filling threshold, and keeps the states below that threshold. For this blocked Dirac model, that gives the half-filled ground state. The relation

$$ C_\mathrm{gs} = I - P_\mathrm{gs} $$

is the correlation matrix convention used in this workflow: it stores the complementary one-particle projector associated with the chosen filling.

9. Choose A Local Region In Real Space

Pick a cluster of sites near a chosen center.

region = Q.nearest_sites(honeycomb, 2 * a1 + 2 * a2, 3)

The third argument is n_nearest, the number of distinct distance shells to include around the chosen center.

• 1 means only the nearest-distance shell,
• 2 means the first two distinct shells,
• 3 means the first three distinct shells.

So this call does not mean “take exactly 3 sites” or “use radius 3”. It means “collect every lattice site whose distance from 2 * a1 + 2 * a2 lies in one of the first three distinct distance shells”.

You can inspect that region on the lattice:

_ = honeycomb.plot("structure", highlights=[PointCloud(region)])

This is useful to confirm that the restricted subsystem is centered where you expect.

The center 2 * a1 + 2 * a2 matters later when constructing the local symmetry action. A rotation is not determined only by “rotate by 60 degrees”; it also needs a center. The local symmetry operator must be centered on the same point as the local patch if you want the patch and its restricted modes to transform into themselves.

10. Build The Restricted Correlation Matrix

Construct the restriction operator and then compress the global correlation matrix into the chosen region.

R = Q.region_restrict(C_gs, region)
C_R = (R.h(-2, -1) @ C_gs @ R).mean(dim=0)

Here:

• R maps the full lattice Hilbert space down to the selected region.
• C_R is the restricted correlation matrix seen by that region.
• .mean(dim=0) combines the contributions from all crystal momenta to reconstruct the real-space correlation matrix of the chosen subsystem. Physically, the local reduced state is built from the full occupied Fermi sea, not from any single momentum sector, so the restricted correlator is obtained only after summing or averaging over all k.

The mean over momentum is what turns the translationally organized band description into the correlation matrix of one finite real-space patch. Without that step you would still have a momentum-resolved family of partial contributions rather than the physical local correlator of the region.

You can inspect C_R directly:

C_R

And plot it:

_ = C_R.plot("heatmap")
_ = C_R.plot("spectrum")

The heatmap shows the local correlation structure, and the spectrum is the entanglement-spectrum-style eigenvalue distribution of the restricted free-fermion state.

11. Diagonalize The Restricted Correlation Matrix

Get eigenvalues and eigenvectors of the restricted matrix.

v, u = qten.linalg.eigh(C_R)

At this point:

• v contains the eigenvalues of C_R,
• u contains the corresponding local eigenmodes as columns.

These local modes are the starting point for symmetry-adapted basis construction.

12. Build The Local C6 Action On The Restricted Hilbert Space

Center the symmetry at the same point used for the local region and build its matrix representation on the local basis.

g_c6 = Q.hilbert_opr_repr(T_c6.fixpoint_at(2 * a1 + 2 * a2), u.dims[0])

This gives the local unitary representation of the sixfold rotation acting on the restricted Hilbert space.

The .fixpoint_at(2 * a1 + 2 * a2) part is essential. T_c6 by itself is only the affine rotation with its default center. fixpoint_at(r) changes the affine translation part so that the point r is invariant under the operation. In formula form, if the affine action is

x -> R x + t

then fixing the point r means choosing t so that

R r + t = r

or equivalently

t = (I - R) r

That is exactly what fixpoint_at(...) does internally.

Why this matters here:

• the local region was chosen around 2 * a1 + 2 * a2,
• the restricted Hilbert space u.dims[0] is the Hilbert space of that local patch,
• the local C6 operator should rotate around the same center as the patch.

If you skipped fixpoint_at(...), you would generally be representing a rotation around the wrong point, so the local subspace would not transform the way you intend.

13. Get A Local C6-Symmetric Basis Directly

Now symmetry-adapt the first few local modes.

u_sym = Q.abelian_column_symmetrize(
    T_c6.fixpoint_at(2 * a1 + 2 * a2),
    u[:, :3],
)

This is the simplest high-level way to get a C6-adapted basis from the local unitary structure.

Conceptually, this takes a set of columns spanning a small local subspace and rotates them into columns with definite abelian symmetry character under C6. That is usually the cleanest entry point if your goal is “give me local modes with good symmetry labels”.

You can visualize the resulting basis vectors:

_ = u_sym.plot("column_scatter", default_size=15)

And confirm the transformed column space:

T_c6 @ u_sym.dims[1]

14. Get The Local C6-Symmetric Basis Manually From The Unitary

If you want to see the linear algebra explicitly, you can also do the symmetry adaptation by hand inside the selected subspace.

First choose a working subspace:

w = u[:, :3]

Project the local C6 operator into that subspace:

g_c6_w = w.h(-2, -1) @ g_c6 @ w

Now diagonalize the symmetry operator inside the subspace:

v_c6, u_c6 = qten.linalg.eig(g_c6_w)
print(v_c6.data)

Finally rotate the original basis w by the eigenvectors of the projected symmetry operator:

w_sym = w @ u_c6

This w_sym is a local C6 eigenbasis extracted from the unitary itself.

The reason this works is standard subspace linear algebra:

• w is a basis for the subspace you care about,
• g_c6_w = w^† g_c6 w is the symmetry operator written in that subspace basis,
• diagonalizing g_c6_w finds the combinations of columns of w that carry definite C6 eigenvalues,
• multiplying back by w lifts those symmetry eigenvectors to the full local Hilbert space.

So w_sym and u_sym are solving the same problem in two different ways:

• u_sym uses the high-level helper,
• w_sym constructs the symmetry-adapted basis explicitly from the projected unitary.

To compare before and after:

_ = w.plot("column_scatter")
_ = w_sym.plot("column_scatter")

This manual route is useful when you want to inspect the symmetry representation explicitly instead of using the higher-level helper.

Notes

• QTen tensors always carry axis metadata in dims.
• IndexSpace is the simplest place to start.
• Geometry, symbolic, point-group, and band tools are available from submodules when you need them.