Optimally Localized Conserved Quantities

• Optimally localized conserved quantities are operators engineered to be spatially confined (via techniques such as time averaging and Q-matrix optimization) while exactly or approximately commuting with the Hamiltonian.
• They are constructed in various frameworks—ranging from many-body localization using LIOMs to integrable lattice models and entanglement Hamiltonian methods—each emphasizing trade-offs between exact conservation and localized support.
• In general relativity, similar optimization principles define quasi-local energy-momentum and angular momentum on closed 2-surfaces through optimal isometric embedding, illustrating the broad applicability of the concept.

Searching arXiv for the cited works and closely related literature on optimally localized conserved quantities across MBL, integrable systems, entanglement-Hamiltonian methods, and quasi-local GR charges. arXiv search query: "(Goihl et al., 2017) many-body localization conserved quantities effective models exact constants of motion" Optimally localized conserved quantities are conserved operators or charges constructed to be as confined as possible to a prescribed site, finite support window, subregion, or closed boundary while retaining exact conservation, controlled approximate conservation, or a canonical quasi-local meaning. In disordered quantum matter, the canonical examples are quasi-local $\tau_i^z$ operators or LIOMs in the many-body-localized phase; in integrable and nearly integrable lattices they appear as local or quasilocal charges extracted by time averaging, support-restricted diagonalization, or perturbative minimization; in entanglement-based constructions they arise as EHSM eigen-operators with small relative commutation errors; and in general relativity they denote quasi-local energy-momentum, angular momentum, and center-of-mass assigned to a spacelike $2$-surface through optimal isometric embedding or canonical boundary terms (Goihl et al., 2017, Craps et al., 10 Dec 2025, Lian, 2021, Chen et al., 2013).

1. Terminological scope and mathematical meaning

The phrase does not have a single domain-independent definition. In the many-body-localization literature, “optimal localization” refers to exact constants of motion that remain exponentially localized in real space while commuting exactly with the Hamiltonian. In integrable and nearly integrable lattice systems, the same phrase is used for charges that maximize overlap with a chosen local operator space or minimize a residual non-conservation norm under a support constraint. In subregion constructions, it refers to operators supported in a subsystem that diagonalize the covariance structure of entanglement Hamiltonians. In general relativity, it refers to quasi-local charges attached to a closed spacelike $2$-surface by solving an optimal matching problem to a flat reference geometry or by fixing a unique boundary correction term under natural axioms (Goihl et al., 2017, Mierzejewski et al., 2014, Ulčakar et al., 2023, Bart, 2019).

Domain	Conserved object	Localization principle
Many-body localization	$\tau_i^z$, LIOMs, $l$-bits	Exact commutation with $H$ plus quasi-local support or maximal local overlap
Integrable or nearly integrable lattices	Local, quasilocal, or perturbative charges	Time averaging, finite-support projection, or residual minimization
Subregions of eigenstates	EHSM eigen-operators $\overline Q_A^{(n)}$	Diagonalization of entanglement-Hamiltonian covariance with $O(l_{AB}/L_A)$ errors
General relativity	Quasi-local energy, angular momentum, center of mass	Optimal isometric embedding or canonical boundary correction on a closed $2$-surface

A recurring misconception is that optimal localization must mean strict on-site support. The literature distinguishes strictly local, quasi-local, quasilocal, subregionally quasilocal, perturbatively conserved, and quasi-local $2$-surface charges. Another recurrent point is that localization and conservation are often competing constraints: exact diagonalization of a Hamiltonian makes conservation manifest but can delocalize operators, whereas localization-preserving constructions generally require additional optimization, support truncation, or geometric matching.

2. Exact quasi-local constants of motion in many-body localization

For a spin-$2$0 chain of $2$1 sites with Hamiltonian $2$2, a defining MBL structure is a complete family of independent, mutually commuting, quasi-local operators $2$3 such that

$2$4

and each $2$5 has eigenvalues $2$6. In the construction of Goihl, Gluza, Krumnow, and Eisert, the unitary is built by exact diagonalization followed by an iterative reordering of eigenvectors so that the diagonal representatives of local magnetization operators become as close as possible to ideal Pauli-$2$7 operators in energy space (Goihl et al., 2017).

The starting point is the infinite-time average

$2$8

which is diagonal in the energy basis but does not have exactly $2$9 eigenvalues. The eigenvectors are then permuted iteratively: first one sorts the diagonal entries of $2$0, then sorts $2$1 within the blocks induced by the first step, and so on up to site $2$2. If $2$3 diagonalizes $2$4, the reordered diagonalizing unitary is

$2$5

and the exact conserved operators are

$2$6

with $2$7 the ideal Pauli-$2$8 operator acting on site $2$9 in the reordered energy-space tensor structure. By construction, $\tau_i^z$0 remains diagonal, so $\tau_i^z$1 commute exactly with $\tau_i^z$2, commute mutually, and carry the exact Pauli spectrum.

The significance of this construction is that it avoids the common compromise in numerical LIOM schemes between exact conservation and locality. The infinite-time-averaged operators $\tau_i^z$3 are exactly conserved but generally fail to realize a Pauli algebra, while exact diagonal operators with prescribed $\tau_i^z$4 spectra need not be local. The reordering scheme is designed to retain the locality profile of the dephased magnetization operators while restoring the Pauli-$\tau_i^z$5 spectrum exactly.

3. Effective Hamiltonians and locality diagnostics in the MBL phase

Once the exact $\tau_i^z$6 are available, the Hamiltonian admits an expansion in the orthonormal basis of diagonal operators

$\tau_i^z$7

Grouping terms by Hamming weight yields the familiar truncated effective Hamiltonian

$\tau_i^z$8

The associated phenomenology is encoded in two numerical regularities: inter-order suppression, meaning that if $\tau_i^z$9 then typically $l$0, and spatial decay with the span

$l$1

for which the numerics show

$l$2

A fit of $l$3 versus $l$4 shows clear exponential decay well inside the localized regime (Goihl et al., 2017).

Localization is quantified by the truncation error

$l$5

where $l$6 is a contiguous buffer region of odd cardinality around site $l$7. Optimal localization means that for each $l$8 and sufficiently large disorder $l$9,

$H$0

In the exact-diagonalization study, system sizes reached $H$1 up to $H$2 with periodic boundary conditions, disorder $H$3, $H$4, and $H$5–$H$6 realizations per point. In the MBL phase, identified numerically as $H$7, $H$8 decays exponentially with $H$9; in the ergodic phase this decay is lost and the operators quickly become non-local. A local quench from the Néel state further shows that $\overline Q_A^{(n)}$0 reproduces the slow dephasing and saturation of the imbalance over intermediate-to-long times at large disorder (Goihl et al., 2017).

These observations provide the operator-level explanation of standard MBL phenomenology. Exact conservation of the $\overline Q_A^{(n)}$1 implies absence of DC transport, exponentially decaying couplings in $\overline Q_A^{(n)}$2 imply logarithmic entanglement growth through dephasing, and product structure in the $\overline Q_A^{(n)}$3-basis implies area-law eigenstates up to exponentially small real-space tails.

4. Variational optimization and the Q-matrix formulation of LIOMs

A later unifying formulation treats the construction of exact LIOMs as an optimization problem with locality weights specified a priori rather than emerging a posteriori. In this framework one chooses an orthonormal Hermitian operator basis $\overline Q_A^{(n)}$4, assigns locality weights $\overline Q_A^{(n)}$5, and defines for any candidate operator $\overline Q_A^{(n)}$6 the local overlap

$\overline Q_A^{(n)}$7

and the non-locality cost

$\overline Q_A^{(n)}$8

subject to

$\overline Q_A^{(n)}$9

optionally together with a prescribed spectrum. Since any operator commuting with a nondegenerate $O(l_{AB}/L_A)$0 is diagonal in the energy basis, the problem reduces to maximizing a quadratic form

$O(l_{AB}/L_A)$1

over the diagonal coefficients $O(l_{AB}/L_A)$2. The unconstrained optimum is the top eigenvector of $O(l_{AB}/L_A)$3, while the binary-spectrum requirement $O(l_{AB}/L_A)$4 maps the problem to the QUBO

$O(l_{AB}/L_A)$5

equivalently the ground-state problem of a classical Ising spin glass (Craps et al., 10 Dec 2025).

This formulation reproduces several earlier constructions as special cases. The time-averaging method of Chandran–Huse–Vidal–Abanin appears as the unconstrained single-site choice; the binary-spectrum $O(l_{AB}/L_A)$6-bit construction appears as the QUBO version; bounded-support schemes correspond to locality weights restricted to Pauli strings up to a chosen length. The framework also emphasizes “designer locality”: the favored support criterion is imposed at the outset, and enlarging the favored block can trade a small increase in core support for exponentially reduced tails.

Concrete examples in the same formulation include a Heisenberg chain of length $O(l_{AB}/L_A)$7 at disorder $O(l_{AB}/L_A)$8, where one-site unconstrained LIOMs yield support probabilities $O(l_{AB}/L_A)$9 with best-fit localization length $2$0, while a three-site favored block yields a shorter apparent decay length $2$1. In an Anderson chain of length $2$2 at disorder $2$3, analogous constructions give $2$4 with $2$5 (Craps et al., 10 Dec 2025). This suggests that optimal localization can be cast uniformly as a constrained quadratic optimization problem whose complexity depends on whether the operator spectrum is left free or fixed to a binary $2$6-bit form.

5. Local, quasilocal, and perturbatively conserved operators in integrable and nearly integrable lattices

In integrable spin chains, one systematic approach starts from the operator Hilbert space $2$7 of translationally invariant, traceless operators supported on at most $2$8 consecutive sites. For $2$9,

$2$0

For each seed operator $2$1, one forms the infinite-time average

$2$2

which is the orthogonal projection of $2$3 onto the commuting subspace. Diagonalizing the stiffness matrix

$2$4

classifies the resulting conserved operators: $2$5 corresponds to strictly local charges, finite $2$6 signals quasilocal conserved quantities, and $2$7 corresponds to effectively non-local projections. In the XXZ chain, the total number of independent local and quasilocal conserved operators with support up to $2$8 grows linearly with $2$9, numerically

$2$00

for $2$01, and novel quasilocal charges appear in both parity sectors (Mierzejewski et al., 2014).

For weakly perturbed nearly integrable systems, the conserved object is instead a perturbative series

$2$02

for $2$03, with recursion

$2$04

Because exact solution would generally require non-local $2$05, one truncates $2$06 to support range $2$07 and minimizes the non-conserved remainder in the Hilbert–Schmidt norm or a weighted locality norm: $2$08 In the XXZ chain with a weak transverse field, the first correction obtained in this way is

$2$09

and the coefficients of higher-order local strings decay exponentially with their spatial range, $2$10 (Ulčakar et al., 2023).

A complementary exact-locality result is available for the one-dimensional Hubbard model. There, the full tower $2$11 of local charges can be written in a diagrammatic operator basis with recursively determined coefficients $2$12. Each $2$13 is strictly $2$14-local, meaning that it acts nontrivially only on $2$15 consecutive sites and no diagram of range $2$16 appears. The recursion implies that the $2$17 are the unique local combinations with the commutativity property $2$18, and any local operator with support $2$19 that commutes with the Hamiltonian is a linear combination of $2$20 (Fukai, 2023). In this usage, “optimal” refers not to exponential tails but to exact minimal support and uniqueness within the local operator algebra.

6. Subregional conserved quantities from entanglement Hamiltonians

For a bipartition $2$21 and an energy eigenstate $2$22, the reduced density matrix $2$23 defines a subregion entanglement Hamiltonian through

$2$24

Given an ensemble of eigenstates $2$25 with weights $2$26, one introduces the entanglement-Hamiltonian superdensity matrix

$2$27

whose eigen-operators $2$28 are the optimally supported approximate conserved quantities on $2$29. Their defining property is not exact commutation with the full Hamiltonian, but a controlled small relative commutator in the subregion. If $2$30 is subregionally quasilocal in $2$31, then

$2$32

and similarly for mutual commutators of the $2$33. Under these conditions, each entanglement Hamiltonian admits an approximate expansion in subregional conserved quantities with corrections of order $2$34 (Lian, 2021).

The free-fermion benchmark exhibits a sharp counting structure. In one dimension, $2$35 has exactly $2$36 nonzero eigenvalues with $2$37; in two dimensions, $2$38 depending on subregion aspect ratio. Some of the corresponding eigen-operators are nonlocal on $2$39, but still subregionally quasilocal in the sense of the commutator bounds. In the strongly localized regime, the cutoff shrinks to $2$40, and the modes become the $2$41 on-site occupations. In the interacting XYZ chain, exact diagonalization for $2$42 and $2$43 finds a power-law decay

$2$44

with $2$45 in extended integrable phases and $2$46 deep in the MBL phase. In fully chaotic systems, only two EHSM eigenvalues are significantly nonzero, and their eigen-operators correspond to the identity and the subregion Hamiltonian (Lian, 2021).

This framework sharpens the distinction between exact LIOMs and approximate subregional integrals. The latter are optimized with respect to an ensemble of entanglement Hamiltonians, not directly with respect to $2$47, and their locality is controlled by the boundary-to-volume ratio rather than by exponential tails alone.

7. Quasi-local conserved charges in general relativity

In general relativity, “optimally localized conserved quantities” refers to quasi-local charges attached to a closed, spacelike $2$48-surface $2$49, not to bulk local operator algebras. In the Chen–Wang–Yau construction, one starts from the intrinsic metric $2$50, the norm $2$51 of the mean-curvature vector, and the normal-bundle connection $2$52 in the mean-curvature gauge. One then seeks an isometric embedding $2$53 together with a constant future-timelike unit Killing field $2$54 that solves the optimal isometric embedding equations. For an optimal pair $2$55, any Minkowski Killing field $2$56 defines a quasi-local conserved quantity

$2$57

with $2$58 and $2$59 extracted from the Hamilton–Jacobi boundary term. Rotation generators yield quasi-local angular momentum, and boost generators yield quasi-local center of mass. For harmonic asymptotic initial data, the embedding equations can be solved term by term on coordinate spheres, leading to explicit asymptotic formulas for total center of mass and total angular momentum in terms of asymptotic coefficients $2$60. These charges are coordinate-invariant, finite without extra parity assumptions, vanish in Minkowski space, and under vacuum Einstein evolution satisfy

$2$61

or, equivalently in the nonlinear asymptotically flat setting,

$2$62

(Chen et al., 2014, Chen et al., 2013).

A related Hamiltonian formulation constructs quasi-local charges for an arbitrary vector field $2$63 through

$2$64

Imposing consistency, linearity in $2$65, orthogonality, and a Minkowski zero-point condition fixes the boundary correction term uniquely under canonical Dirichlet data. The resulting quasi-local charge is valid for any vector field on any closed spacelike $2$66-surface $2$67, and in Newman–Unti gauge its specialization to BMS generators reproduces the Bondi mass and angular-momentum aspects at null infinity, the Vaidya mass in spherical symmetry, and the irreducible mass $2$68 at the Reissner–Nordström horizon. The $2$69 mode $2$70 is argued to be a promising definition of quasi-local energy because it vanishes in flat space and reproduces the expected asymptotic and symmetric limits (Bart, 2019).

In this relativistic setting, “optimally localized” therefore does not mean microscopic locality. It means that the conserved quantity is determined intrinsically by the chosen $2$71-surface through an exact minimization or uniqueness principle, with no dependence on arbitrary asymptotic coordinates or ad hoc reference Killing fields. This usage is conceptually distinct from LIOM constructions, but both traditions treat localization as a constrained optimization of conserved structure against a prescribed notion of spatial support or boundary attachment.