Infinite MPS

An infinite MPS describes a bulk state through a repeating finite unit cell. It removes finite-chain boundaries while retaining a bond-dimension approximation and a choice of unit-cell period. Computing observables requires transfer-operator fixed points; multiplying infinitely many tensors is neither necessary nor numerically meaningful.

Required background. MPS canonicalization fixes the gauge language, and entanglement and bond dimension explains finite-rank limitations.

The transfer operator and its fixed points

For a one-site tensor $A^s$, define the completely positive map

$$\mathcal E(X)=\sum_s A^sX(A^s)^\dagger.$$

Find its dominant eigenvalue $\lambda_0$ and positive right and left eigenmatrices $r,l$, satisfying $\mathcal E(r)=\lambda_0r$ and $\mathcal E^\dagger(l)=\lambda_0l$. Rescale $A^s$ by $\lambda_0^{-1/2}$, and normalize $\operatorname{tr}(lr)=1$.

For an injective normalized MPS, the dominant eigenvalue is nondegenerate and the remaining eigenvalues have modulus below one. Degeneracy or additional unit-modulus eigenvalues can indicate symmetry breaking, a noninjective representation, or an inadequately chosen unit cell. Do not impose uniqueness without testing it.

For a $q$-site unit cell, compose the $q$ transfer maps in physical order. The cell transfer eigenvalues then advance distances by $q$ lattice spacings.

Bulk observables and correlation lengths

Insert a local operator by defining

$$\mathcal E_O(X)=\sum_{s,t}\langle t|O|s\rangle A^sX(A^t)^\dagger.$$

The expectation is $\operatorname{tr}[l\mathcal E_O(r)]$. Two insertions separated by $n$ sites have intervening ordinary transfer maps. For a diagonalizable one-site transfer operator, subtracting disconnected contributions leaves terms proportional to $\lambda_\alpha^{n-1}$. Jordan blocks can also supply polynomial factors in the separation.

If the leading coupled subdominant eigenvalue has modulus $|\lambda_1|<1$, the corresponding correlation length is

$$\xi=-\frac{q}{\log|\lambda_1|}.$$

The numerator is $q$ for cell eigenvalues, and one for a one-site transfer map. A complex eigenvalue also contributes an oscillation wave number through its argument. An eigenmode contributes to a particular correlator only if both operator insertions overlap it.

A numerical fixed-point example

Suppose a two-site cell has normalized leading eigenvalues $1$ and $0.64$. The correlation length in lattice-site units is $-2/\log0.64\approx4.48$. Reporting $-1/\log0.64\approx2.24$ silently changes the length unit to cells.

Check the fixed-point residuals and positivity of the Hermitianized $l,r$. Small negative eigenvalues at roundoff may be numerical noise; substantial negativity indicates a faulty contraction, nonconvergence, or an unsuitable eigenvector from a degenerate eigenspace. Never cure a large residual by clipping eigenvalues.

A product-state MPS with $D=1$ provides the simplest test: every connected correlation between disjoint local observables is zero. An independently constructed noninjective GHZ tensor tests the code’s handling of degenerate fixed points and boundary-sector choices.

An observable-selective contraction test

A four-dimensional physical site gives a small exact test of both fixed points and operator insertions. Label its basis $|0\rangle,|x\rangle,|y\rangle,|z\rangle$, and choose bond dimension two with

$$A^s=\sqrt{p_s}\,\sigma_s, \qquad (p_0,p_x,p_y,p_z)=(0.6,0.25,0.1,0.05),$$

where $\sigma_0=I$ and $\sigma_x,\sigma_y,\sigma_z$ are Pauli matrices acting on the virtual two-dimensional space. The physical site has four states; these Pauli matrices are tensor entries, not its local observables. Both canonical sums equal $I$, so choose $r=I$ and $l=I/2$. Direct Pauli conjugation gives transfer eigenvalues $(1,0.7,0.4,0.3)$ on $(I,X,Y,Z)$.

Now use the physical observables $O_s=|0\rangle\langle s|+|s\rangle\langle0|$, for $s=x,z$. Their means vanish, while $\mathcal E_{O_s}(I)=2\sqrt{p_0p_s}\,\sigma_s$ and $\mathcal E_{O_s}(\sigma_s)=2\sqrt{p_0p_s}\,I$. At separation $n\geq1$ sites this gives

$$\langle O_x(0)O_x(n)\rangle_c=0.6(0.7)^{n-1}, \qquad \langle O_z(0)O_z(n)\rangle_c=0.12(0.3)^{n-1}.$$

An $O_z$ measurement therefore misses the slowest transfer mode. Test these contractions as well as the eigenvalues; a spectrum alone cannot verify the physical insertion indices. Blocking two identical sites squares the transfer eigenvalues, and using $q=2$ must leave both lengths in site units unchanged. For a physical spin-chain example with staggered correlations, see the AKLT calculation.

Optimizing and interpreting the infinite state

Infinite TEBD applies a repeated gate pattern and converges projection time, time step, and bond dimension. Infinite DMRG and uniform variational methods optimize energy density using self-consistent environments. In all cases, check energy density, local reduced states, fixed-point residuals, and correlation length under increasing $D$.

Repeat with a larger unit cell when translation symmetry may break. Enforcing one-site translation invariance of local expectations excludes a selected period-two ordered state even at enormous bond dimension; a translation-invariant superposition of ordered states has different local expectations. Noninjective tensors also require explicit boundary-sector choices. At criticality, the finite correlation length of an injective finite-$D$ approximation should grow as the approximation improves; it need not indicate a physical gap.

Exercises

A one-site transfer operator has $\lambda_1=-0.8$. What qualitative connected-correlation behavior does that mode produce?

Solution

It produces an alternating sign with separation and an envelope $0.8^n$. Its correlation length is $-1/\log0.8\approx4.48$ sites, and its oscillation wave number is $\pi$ modulo $2\pi$. The result applies only to observables that couple to this eigenmode.

References

• Schollwöck, Ulrich, “The density-matrix renormalization group in the age of matrix product states,” Annals of Physics 326, 96–192 (2011), arXiv:1008.3477 — canonical gauges, contractions, and finite-chain algorithms.
• Vanderstraeten, Laurens, Jutho Haegeman, and Frank Verstraete, “Tangent-space methods for uniform matrix product states,” SciPost Physics Lecture Notes 7 (2019), arXiv:1810.07006 — fixed points, observables, and uniform-state optimization.