MPS¶

This section contains the API documentation for the mps module.

States¶

class MpsTensor(tensor, alegs)¶

Bases: home.docs.checkouts.readthedocs.org.user_builds.tensortrack.checkouts.latest.src.tensors.AbstractTensor

Generic MPS tensor objects that have a notion of virtual, physical and auxiliary legs.

Properties

var (Tensor or SparseTensor) – tensor data of the MPS tensor.
plegs (int) – number of physical legs.
alegs (int) – number of auxiliary legs.

Todo

Document all methods.

insert_onespace(tsrc, varargin)¶: Insert a trivial space at position i, corresponding to an additional auxiliary leg.

See also

Tensor.insert_onespace()

pspace(A)¶: The physical space of an MpsTensor.

leftvspace(A)¶: The left virtual space of an MpsTensor.

rightvspace(A)¶: The right virtual space of an MpsTensor.

ctranspose(t)¶

Compute the adjoint of a tensor. This is defined as swapping the codomain and domain, while computing the adjoint of the matrix blocks.

Usage

t = ctranspose(t)

t = t'

Arguments: t (MpsTensor) – input tensor.
Returns: t (MpsTensor) – adjoint tensor.

initializeC(A)¶: Initialize a set of gauge tensors for the given mpstensors.

expand(A, leftvspace, rightvspace, noisefactor)¶: Expand a tensor to the given virtual spaces.

multiplyleft(A, C)¶: Multiply a gauge matrix from the left.

multiplyright(A, C)¶: Multiply a gauge matrix from the right.

vectorize(t, type)¶

Collect all parameters in a vector, weighted to reproduce the correct inproduct.

Arguments

t (MpsTensor) – input tensor.
type (‘real’ or ‘complex’) – optionally specify if complex entries should be seen as 1 or 2 parameters. Defaults to ‘complex’, with complex parameters.

Returns

v (numeric) – real or complex vector containing the parameters of the tensor.

devectorize(v, t, type)¶

Collect all parameters from a vector, and insert into a tensor.

Arguments

v (numeric) – real or complex vector containing the parameters of the tensor.
t (MpsTensor) – input tensor.
type (char, ‘real’ or ‘complex’) – optionally specify if complex entries should be seen as 1 or 2 parameters. Defaults to ‘complex’, with complex parameters.

Returns

t (MpsTensor) – output tensor, filled with the parameters.

static decompose_local_state(psi, kwargs)¶

convert a tensor into a product of local operators.

Usage

local_operators = MpoTensor.decompose_local_operator(H, kwargs).

Arguments: psi (AbstractTensor) – tensor representing a local state on N sites.
Keyword Arguments: ‘Trunc’ (cell) – optional truncation method for the decomposition. See also Tensor.tsvd()

class FiniteMps(varargin)¶

Finite matrix product state.

Properties

A (cell of MpsTensor) – set of tensors that define a finite MPS.
center (int) – location of center gauge, such that every tensor to the left (right) of center is in left (right) gauge.

Todo

Document.

class UniformMps(varargin)¶

Implementation of infinite translation invariant MPS

The center gauge is defined to have:

\[AL_w \cdot C_w = AC_w = C_{w-1} \cdot AR_w\]

Properties

AL (MpsTensor) – left-gauged mps tensors.
AR (MpsTensor) – right-gauged mps tensors.
C (Tensor) – center gauge transform.
AC (MpsTensor) – center-gauged mps tensors.

Todo

Document all methods.

UniformMps(varargin)¶

Usage

mps = UniformMps(A)

mps = UniformMps(AL, AR, C [, AC])

Arguments

A (cell of MpsTensor) – set of tensors per site that define an MPS to be gauged.
AL, AR, AC (cell of MpsTensor) – set of gauged MpsTensors.
C (cell of Tensor) – gauge tensor.

Returns

mps (UniformMps) – gauged uniform MPS.

static new(fun, pspaces, vspaces)¶

Create a uniform matrix product state with data using a function handle.

Usage

UniformMps.new(fun, pspaces, vspaces)

Arguments

fun (function_handle) – function to initialize the tensor.

Repeating Arguments

pspaces (AbstractSpace) – physical spaces for each site.
vspaces (AbstractSpace) – virtual spaces between each site. (entry i corresponds to left of site i.)

static randnc(pspaces, vspaces)¶: Create a uniform matrix product state with random entries.

See also

UniformMps.new()

period(mps)¶: Period over which the mps is translation invariant.

depth(mps)¶: Number of lines in a multi-line mps.

leftvspace(mps, w)¶: Return the virtual space to the left of site w.

pspace(mps, w)¶: Return the physical space at site w.

rightvspace(mps, w)¶: Return the virtual space to the right of site w.

canonicalize(mps, kwargs)¶

Compute the center-gauged form of an mps.

Usage

[mps, lambda] = canonicalize(mps, kwargs)

Arguments

mps (UniformMps) – input mps, from which AL or AR is used as the state, and optionally C as an initial guess for the gauge.

Keyword Arguments

Tol (numeric) – tolerance for the algorithm.
MaxIter (integer) – maximum number of iterations.
Method (char) – algorithm used for decomposition. Must be ‘polar’, ‘qr’ or ‘qrpos’.
Verbosity (Verbosity) – level of output.
DiagC (logical) – flag to indicate if C needs to be diagonalized.
ComputeAC (logical) – flag to indicate if AC needs to be computed.
Order (char, ‘lr’ or ‘rl’) – order of gauge fixing:
- ‘lr’ uses AL as input tensors, first leftorth, then rightorth.
- ‘rl’ uses AR as input tensors, first rightorth, then leftorth.

diagonalizeC(mps)¶: Gauge transform an mps such that C is diagonal.

normalize(mps)¶: Normalize an mps state.

transfermatrix(mps1, mps2, sites, kwargs)¶

A finite matrix product operator that represents the transfer matrix of an mps.

Usage

T = transfermatrix(mps1, mps2, sites, kwargs)

Arguments

mps1 (UniformMps) – input mps for top layer.
mps2 (UniformMps) – input mps for bottom layer, by default equal to the top.
sites (int) – optionally slice the unit cell of the mps and only define the transfer matrix for this slice.

Keyword Arguments

Type (char) – ‘LL’, ‘LR’, ‘RL’, ‘RR’ to determine if the top or bottom respectively are AL or AR.

Returns

T (FiniteMpo) – transfer matrix of an mps, acting to the left.

fixedpoint(mps, type, w)¶

compute the fixed point of the transfer matrix of an mps.

Usage

rho = fixedpoint(mps, type, w)

Arguments

mps (UniformMps) – input state.
type (char) – specification of the type of transfer matrix: general format: sprintf(%c_%c%c, side, top, bot) where side is ‘l’ or ‘r’ to determine which fixedpoint, and top and bot are ‘L’ or ‘R’ to specify whether to use AL or AR in the transfer matrix.
w (int) – position within the mps unitcell of the fixed point.

transfereigs(mps1, mps2, howmany, which, eigopts, kwargs)¶

Compute the eigenvalues of the transfer matrix of an mps.

Usage

[V, D] = transfereigs(mps1, mps2, howmany, which, eigopts, kwargs)

Arguments

mps1 (UniformMps) – input mps for top layer.
mps2 (UniformMps) – input mps for bottom layer. Default value equal to mps1.
howmany (int) – number of eigenvectors and eigenvalues to compute.
which (char) – type of eigenvectors to target.

Keyword Arguments

eigopts – see keyword arguments for eigsolve().
Verbosity (int) – detail level for output.
Type (char) – type of transfer matrix to construct, see UniformMps.transfermatrix().
Charge (AbstractCharge) – charge of eigenvectors to target.

fidelity(mps1, mps2, kwargs)¶: Compute the fidelity between two uniform MPSs.

expectation_value(mps1, O, mps2)¶: Compute the expectation value of an operator.

local_expectation_value(mps, O, offset)¶: Compute the expectation value of a local operator.

schmidt_values(mps, w)¶

Compute the Schmidt values and corresponding charges for an entanglement cut to the right of site w.

Usage

[svals, charges] = schmidt_values(mps, w)

plot_entanglementspectrum(mps, d, w, ax, kwargs)¶: Plot the entanglement spectrum of a uniform MPS.

correlation_length(mps, charge)¶

Compute the correlation length of an MPS in a given charge sector.

Usage

[xi, theta] = correlation_length(mps, charge)

Arguments

mps (UniformMps) – input mps.
charge (AbstractCharge) – charge sector for correlation length to target.

Returns

xi (numeric) – correlation length in the given charge sector.
theta (numeric) – angle of the corresponding oscillation period.

marek_gap(mps, charge, kwargs)¶

Compute the Marek gap of an MPS in a given charge sector.

Usage

[epsilon, delta, spectrum] = marek_gap(mps, charge, kwargs)

Arguments

mps (UniformMps) – input mps.
charge (AbstractCharge) – charge sector for correlation length to target.

Keyword Arguments

HowMany (int) – amount of transfer matrix eigenvalues to compute.
Angle (numeric) – angle in radians around which the gap should be computed.
AngleTol (numeric) – tolerance in radians for angles to be considered equal.

Returns

epsilon (numeric) – inverse correlation length in the given charge sector.
delta (numeric) – refinement parameter.
spectrum (numeric) – computed partial transfer matrix spectrum.

entanglement_entropy(mps, w)¶: Compute the entanglement entropy of a uniform MPS for a cut to the right of site w

renyi_entropy(mps, n, w)¶: Compute the n-th Renyi entropy of a uniform MPS for a cut to the right of site w

truncate(mps, trunc)¶: Truncate a uniform MPS according to the options specified in trunc (see Tensor.tsvd() for details on the truncation options).

class InfQP(varargin)¶

Infinite Quasi-Particle states

Todo

Document.

fixedpoint(qp, type, w)¶

compute the fixed point of the transfer matrix of a quasi-particle state.

Usage

rho = fixedpoint(qp, type, w)

Arguments

mps (InfQP) – input quasi-particle state.
type (char) – specification of the type of transfer matrix: general format: sprintf(%c_%c%c, side, top, bot) where side is ‘l’ or ‘r’ to determine which fixedpoint, and top and bot are ‘L’ or ‘R’ to specify whether to use AL or AR in the transfer matrix.
w (int) – position within the mps unitcell of the fixed point.

vectorize(qp, type)¶

Collect all parameters in a vector, weighted to reproduce the correct inner product.

Arguments

qp (InfQP) – input quasi-particle state.
type (char, ‘real’ or ‘complex’) – optionally specify if complex entries should be seen as 1 or 2 parameters. Defaults to ‘complex’, with complex parameters.

Returns

v (numeric) – real or complex vector containing the parameters of the quasi-particle state.

devectorize(v, qp, type)¶

Collect all parameters from a vector, and insert into a quasi-particle state.

Arguments

v (numeric) – real or complex vector containing the parameters of the quasi-particle state.
qp (InfQP) – input quasi-particle state.
type (char, ‘real’ or ‘complex’) – optionally specify if complex entries should be seen as 1 or 2 parameters. Defaults to ‘complex’, with complex parameters.

Returns

qp (InfQP) – output quasi-particle state, filled with the parameters.

Operators¶

class MpoTensor(varargin)¶

Bases: home.docs.checkouts.readthedocs.org.user_builds.tensortrack.checkouts.latest.src.tensors.AbstractTensor

Matrix product operator building block.

This object represents the MPO tensor at a single site as the sum of rank (2,2) (sparse) tensors and some scalars, which will be implicitly used as unit tensors.

       4
       v
       |
1 -<-- O --<- 3
       |
       v
       2

Todo

Document.

static decompose_local_operator(H, kwargs)¶

Convert a tensor into a product of local operators.

Usage

local_operators = MpoTensor.decompose_local_operator(H, kwargs).

Arguments: H (AbstractTensor) – tensor representing a local operator on N sites.
Keyword Arguments: ‘Trunc’ (cell) – optional truncation method for the decomposition. See also Tensor.tsvd()

class FiniteMpo(L, O, R)¶

Finite matrix product operator.

Properties

L (MpsTensor) – left end tensor.
O (cell of MpoTensor or PepsSandwich) – bulk MPO tensors.
R (MpsTensor) – right end tensor.

Todo

Document.

initialize_fixedpoint(mpo)¶: Initialize a dense tensor for the fixedpoint of a FiniteMPO.

class InfMpo(varargin)¶

Infinite translation invariant matrix product operator.

Properties: O (cell of MpoTensor or PepsSandwich) – cell of MPO tensors in translation invariant unit cell.

Todo

Document

class InfJMpo(varargin)¶

Bases: src.mps.InfMpo

Infinite translation invariant matrix product operator with a Jordan block structure.

Todo

Document.

class PepsTensor(tensor)¶

Generic PEPS tensor object that has a notion of virtual and physical legs.

This object represents the PEPS tensor at a single site as a rank (1, 4) tensor, where the physical index lies in the codomain and the virtual indices lie in the domain.

       5   1
       |  /
       v ^
       |/
2 ->-- O --<- 4
       |
       ^
       |
       3

Properties: var (Tensor) – PEPS tensor data.

Todo

Document.

class PepsSandwich(top, bot)¶

Data structure representing a pair of PEPS tensors in an overlap, which behave as an MPO tensor.

Properties

top (PepsTensor) – top-layer PEPS tensor, usually interpreted as the ‘ket’ in the overlap.
bot (PepsTensor) – bottom-layer PEPS tensor, usually interpreted as the ‘bra’ in the overlap.

Todo

Document.

class UniformPeps(varargin)¶

Implementation of infinite translation invariant PEPS

Properties: A (cell of PepsTensor) – cell array of PEPS tensors in 2D unit cell.

Todo

Document.

UniformPeps(varargin)¶

Usage

peps = UniformPeps(A)

Arguments: A (cell of PepsTensor) – cell array of PEPS tensors in 2D unit cell.
Returns: peps (UniformPeps) – infinite translation-invariant PEPS.