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Continuum Limits of Quantum Lattice Systems

Tobias J. Osborne

arXiv:1901.06124v1quant-phcond-mat.str-elhep-th

TL;DR

The paper constructs quantum continuum limits for lattice states, including tensor-network families, and extends them to dynamics. The resulting continuum theories recover continuous tensor-network counterparts, allow non-Gaussian correlations, and satisfy strict causality under stated conditions.

Abstract

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We describe a general procedure to give effective continuous descriptions of quantum lattice systems in terms of quantum fields. There are two key novelties of our method: firstly, it is framed in the hamiltonian setting and applies equally to distinguishable quantum spins, bosons, and fermions and, secondly, it works for arbitrary variational tensor network states and can easily produce computable non-gaussian quantum field states. Our construction extends the mean-field fluctuation formalism of Hepp and Lieb (developed later by Verbeure and coworkers) to identify emergent continuous large-scale degrees of freedom - the continuous degrees of freedom are not identified beforehand. We apply the construction to to tensor network states, including, matrix product states and projected entangled-pair states, where we recover their recently introduced continuous counterparts, and also for tree tensor networks and the multi-scale entanglement renormalisation ansatz. Finally, extending the continuum limit to include dynamics we obtain a strict light cone for the propagation of information.

The spacetime continuum limit

The spacetime continuum limit extends the construction to dynamics by treating time and space together, with rapidly decaying test functions on spacetime. Its existence is required to define the resulting quantum spacetime limit and can also accommodate discrete-time dynamics.

The spacetime continuum limit

  • The dynamical extension studies quantum lattice evolution using a spacetime continuum limit rather than kinematics alone.The construction is extended from Hilbert-space structure to dynamics generated by lattice Hamiltonians.Source: The spacetime continuum limit · p. 6
  • Spacetime field operators are defined using rapidly decaying functions on R × R^D.Both classical and quantum spacetime fluctuation operators are constructed from such test functions.Source: The spacetime continuum limit · p. 7
  • The spacetime quantum continuum limit is defined through limits of expectation values of products of quantum fluctuation operators as a →0.The corresponding classical limit uses expectation values of the discretised spacetime fields.Source: The spacetime continuum limit · p. 7
  • The framework also allows continuum limits for dynamics generated by discrete groups, such as quantum cellular automata, by replacing time integrals with sums.This provides a discrete-time counterpart to the spacetime construction.Source: The spacetime continuum limit · p. 7

Strict causality for the quantum continuum limit

For tensor-network or equilibrium states with local parent Hamiltonians, the continuum dynamics obey a strict causality condition: sufficiently spacelike-separated observables commute in the limit. The result follows from Lieb–Robinson bounds and requires a spacetime continuum limit, whose existence is assured generally for equilibrium states.

Strict causality for the quantum continuum limit

  • A constant velocity c defines a light cone outside which spacelike-separated continuum observables satisfy the strict causality condition.The result applies to continuum limits of tensor networks and equilibrium states with local parent Hamiltonians.Source: Strict causality for the quantum… · p. 7
  • The causality proof exploits Lieb–Robinson bounds for local Hamiltonians with bounded interaction norms and finite lattice degree.The bound controls propagation according to lattice graph distance and constants determined by the interaction strengths and geometry.Source: Strict causality for the quantum… · p. 7
  • Spacelike separation places the supports outside the lattice-derived light cone, whose geometry is diamond-shaped and may have direction-dependent speed.The effective velocity is c = ℓ/τ.Source: Strict causality for the quantum… · p. 8
  • The existence of a spacetime continuum limit is a demanding restriction, but it is fulfilled generally when ρa is an equilibrium state, such as a ground or thermal state.The argument is also expected to apply to finite-density excitations above equilibrium.Source: Strict causality for the quantum… · p. 8

Product state case

For product states, the classical continuum limit is determined by single-site expectation values, while the quantum fluctuations form canonical bosonic fields. Generically, a D-dimensional quantum spin yields D −1 conjugate bosonic field pairs alongside classical and identity fields.

Product state case

  • The classical product-state fields are ϕ(x) = π(x) = 0 and ξ(x) = 1 after choosing a basis adapted to the single-site state.The identity component factorises, while the remaining field is constant in the classical limit.Source: Product state case · pp. 8–9
  • The fluctuation operators obey canonical commutation relations and identify the continuum limit with a single bosonic field in the qubit example.The associated limiting vector is annihilated by the bosonic annihilation operator for every test function.Source: Product state case · pp. 9–10
  • For nonzero single-site Bloch vectors, changing the state rotates the symplectic structure and correspondingly rotates the canonical field identification without changing the continuum description.The local fluctuation neighbourhood is generically the same for every such state.Source: Product state case · p. 10
  • The product-state construction generically yields bosonic quantum fields, D −1 classical fields, and the identity field.The fluctuation content follows from the rank and kernel structure of the antisymmetric Θ matrix.Source: Product state case · p. 10

Matrix product states

Applying the construction to matrix product states recovers continuous matrix product states as continuum limits. Their fluctuations form a bosonic field, but unlike product states the limiting state has nontrivial nongaussian correlations.

Matrix product states

  • The continuum-limit construction for matrix product states produces the continuous matrix product state class.The analysis considers a two-level physical system for simplicity, with a generalisation indicated afterward.Source: Matrix product states · p. 10
  • The MPS classical continuum limit is a pure factorised state with fields φx(x) = 0, φy(x) = 0, and φz(x) = 1.Higher-order classical correlation functions factorise in the same way.Source: Matrix product states · p. 11
  • The quantum fluctuations again contain a canonical bosonic field, with the fluctuation operators satisfying the standard commutation relations.The limit factorises with respect to the commuting ξ fluctuation.Source: Matrix product states · pp. 11–12
  • Unlike the product-state case, the limiting MPS vector is not annihilated by the bosonic annihilation operator and exhibits nontrivial nongaussian correlations.These correlations are computed through continuum n-point functions built from the MPS transfer structure.Source: Matrix product states · pp. 12–13

Projected entangled-pair states

The PEPS construction maps a tractable two-dimensional tensor network to classical Ising-model data, then uses critical scaling and fluctuation constraints to obtain quantum continuum limits. Different approaches to the critical point can produce distinct limits, including two limits for the Ising PEPS.

Scope

  • The worked PEPS example is physically motivated but not fully rigorous because several mathematical properties of the underlying Ising model remain open.The authors explicitly characterize this subsection as being at a physical level of rigour.Source: Projected entangled-pair states · p. 13

PEPS construction

  • The PEPS example maps all n-point correlation functions to classical Ising-model expectation values and uses the Ising correlation length to choose the lattice scale a(β).The scale a(β) tends to zero as the critical inverse temperature is approached.Source: Projected entangled-pair states · pp. 13–14

Critical scaling

  • The continuum limit depends on how the Ising critical point is approached: approaching T_c from above or below yields different quantum continuum limits.The passage attributes this to the number of relevant operators involved in approaching the critical point.Source: Projected entangled-pair states · p. 14

Fluctuation fields

  • The fluctuation scaling is chosen to make correlation-function limits exist while also enforcing canonical commutation relations, with the latter taking priority when the requirements conflict.The construction may therefore retain only field-strength n-point functions when the full set of limits is incompatible with canonical commutation relations.Source: Projected entangled-pair states · p. 15

Results

  • For the Ising PEPS, the construction produces two possible quantum continuum limits, denoted |Ω_+⟩ and |Ω_-⟩.Their existence follows after selecting scaling functions that support the field correlations and canonical commutation relations.Source: Projected entangled-pair states · p. 15

Tree tensor network states

For TTN states, the continuum construction uses ascending-operator eigenoperators and fusion maps to derive correlation functions and scaling behavior. The same approach extends essentially without modification to MERA, while discrete scale invariance creates oscillatory limits that require a subsequence choice.

Scope

  • The construction applies to tree tensor networks and extends essentially without modification to the multiscale entanglement renormalisation ansatz (MERA).The analysis is presented for one-dimensional chains of d-dimensional spins, with higher-dimensional generalization stated to be straightforward.Source: Tree tensor network states · p. 15

Operator structure

  • Primary operators are obtained from eigenvectors of the ascending operator, and simple primary operators provide a basis for analyzing TTN observables.The ascending operator is a completely positive map whose eigenvalue equation defines the relevant operators.Source: Tree tensor network states · p. 16

Correlation functions

  • The TTN state has correlation length ξ_ℓ = 2^ℓ+1, and sufficiently separated n-point correlators factor exactly into products.Exact factorization occurs when neighboring operator separations satisfy |j_k+1 − j_k| ≥ 2^ℓ+1.Source: Tree tensor network states · p. 17
  • Fusion maps F_e and F_o reduce lattice two-point functions to calculations in the primary-operator basis.The construction exploits the TTN isometry and translation invariance to evaluate correlators at separations below the correlation length.Source: Tree tensor network states · p. 17

Continuum limit

  • TTN discrete scale covariance makes correlation functions oscillate rather than converge continuously, so the continuum limit is restricted to the subsequence a_j = 2^-j.The oscillations arise from scale-dependent functions such as M_a(x) and N_a(x), which exhibit discrete scale invariance.Source: Tree tensor network states · p. 18
  • Scaling eigenvalues generate power-law behavior multiplied by bounded, discretely scale-invariant oscillatory factors.The factors M_a(x) and N_a(x) oscillate within ranges determined by the corresponding eigenvalues.Source: Tree tensor network states · p. 18

The classical continuum limit

The classical TTN continuum limit is obtained by taking the lattice spacing along a dyadic subsequence, because discrete scale invariance prevents ordinary convergence. In the generic case, all classical n-point functions factorise.

Classical continuum limit

  • Because TTNs are covariant under discrete position rescaling, their correlation functions oscillate around a value instead of approaching a continuous limit.The oscillations are caused by the discretely scale-invariant functions M_a(x) and N_a(x).Source: The classical continuum limit · p. 19
  • The continuum limit is therefore defined along the convergent subsequence a_j = 2^-j.This subsequence aligns the lattice spacing with the TTN's dyadic scaling structure.Source: The classical continuum limit · p. 19

Alternative prescription

  • An alternative random-spacing prescription averages over lattice spacings in [a/2, a], producing a limit but making the correlation functions awkward.The prescription is equivalent to smearing field operators over randomly selected lattice spacings.Source: The classical continuum limit · p. 19

Classical continuum limit

The quantum continuum limit

The quantum continuum construction derives canonical discretised fields from primary operators and their correlation structure. In the simplified treatment, the canonical fields are assumed to correspond directly to simple primary operators.

Quantum continuum limit

Canonical fields

Assumption

  • For simplicity, the analysis assumes that canonical field operators are directly associated with simple primary operators, a condition that holds in many examples.The general case is described as only slightly more involved.Source: The quantum continuum limit · p. 20

A simple example

The example constructs quantum fluctuation fields for a tensor-network state, choosing normalization and commutation relations so their correlation functions and canonical continuum limit exist.

A simple example

  • The state contains one excitation type, with isometry terms describing pair creation and scale transformation of that excitation.The construction interprets |ψp⟩⟨0| as nucleating magnon pairs and |φ+⟩⟨1| as changing their scale behavior.Source: A simple example · p. 20
  • The simple scaling operators, their eigenvalues, and the right eigenvector of the ascending operator determine the classical fields and their continuum expectation values.The theta matrix is then specified for these scaling operators.Source: A simple example · p. 21
  • The example introduces quantum fluctuation fields whose commutation relations are chosen to recover canonical commutation relations as a→0.The normalization is selected to ensure that the limiting n-point correlation functions exist.Source: A simple example · pp. 21–22
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