Lieb-Robinson bounds

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In the study of quantum systems such as quantum optics, quantum information theory, atomic physics and condensed-matter physics it is important to know that there is a finite speed with which information can propagate. The existence of such a finite speed was discovered mathematically by Lieb and Robinson, (1972). It turns the locality properties of physical systems into the existence of, and upper bound for this speed. The bound is now known as the Lieb-Robinson bound and the speed is known as the Lieb-Robinson velocity.

Set up

Lieb-Robinson bounds are considered on the \( \nu \) dimensional lattice \(\Gamma = \mathbb{Z}^\nu\).

A Hilbert space of states \( \mathcal{H}_x\) is associated with each vertex \(x\in\Gamma\). The dimension of this space is finite, but this was generalized in 2008 (see below). Physicists call this a quantum spin system.

For every finite subset \(X \subset\Gamma\) the associated Hilbert space is \[\mathcal{H}_X=\otimes_{x\in X} \mathcal{H}_x. \] Obviously, \(\mathcal{H}_X\) can be viewed as a subspace of \(\mathcal{H}_Y\) if \(X \subset Y\).

An observable \(A\) supported on a finite set \(X\subset \Gamma\) is a bounded linear operator on the Hilbert space \(\mathcal{H}_X\).

When \(\mathcal{H}_x\) is finite dimensional we may choose a finite basis of operators that span the set of bounded linear operators on \(\mathcal{H}_x\). Then any observable on \(\mathcal{H}_x\) can be written as a sum of basis operators on \(\mathcal{H}_x\).

The Hamiltonian of the system is described by an interaction \(\Phi(\cdot)\). The interaction is a function from finite sets \(X\subset\Gamma\) to self-adjoint observables \(\Phi(X)\) supported in \(X\). The interaction is assumed to be of finite range and translation invariant. These requirements were lifted later by Hastings and Koma, (2006) and Nachtergaele, Ogata and Sims, (2006).

Although translation invariance is usually assumed, it is not necessary to do so. It is enough to assume that the interaction is bounded above and below on its domain. A finite range is essential, however. An interaction is said to be of finite range if there is a finite number \(R\) such that for any set \(X\) with diameter greater than \(R\) the interaction is zero, i.e., \(\Phi(X)=0\).

The Hamiltonian of a system with interaction \(\Phi\) is defined formally by: \[H_\Phi=\sum_{X\subset\Gamma}\Phi(X).\]

For every observable \(A\) with a finite support the Hamiltonian defines a strongly continuous one-parameter group of automorphisms \(\tau_t\): \[\tau_t(A)=e^{itH_\Phi}Ae^{-itH_\Phi}.\] The time evolution is defined as a norm-convergent series \(\tau_t(A)=A+it[H,A]+\frac{(it)^2}{2!}[H, [H, A]]+\cdots \), see Ruelle, 1969, Theorem 7.6.2, which is an adaptation from (Robinson, 1968).

See (Lieb, Robinson, 1972) for more rigorous details.

Lieb-Robinson bounds

Lieb and Robinson, (1972) proved the following bound: for any observables \(A\) and \( B\) with finite supports \(X\subset\Gamma\) and \(Y\subset\Gamma\), respectively, and for any time \(t\in\mathbb{R}\) the following holds \[\tag{1}\|[\tau_t(A), B]\|\leq c\, \exp\{-a(d(X,Y)-v|t|)\},\] where \(d(X,Y)\) denotes the distance between the sets \(X\) and \(Y\). A positive constant \(c\) depends on the norms of the observables \(A\) and \(B\), the sizes of the supports \(X\) and \(Y\), the interaction, the lattice structure and the dimension of the Hilbert space \(\mathcal{H}_x\). A positive constant \(v\) depends on the interaction and the lattice structure only. The number \(a>0\) can be chosen at will provided \(d(X,Y)/v|t|\) is chosen sufficiently large. In other words, the further out one goes on the light cone, \(d(X,Y)-v|t|\), the sharper the exponential decay rate is. (In later works authors tended to regard \(a\) as a fixed constant.) The constant \(v\) is called the group velocity or Lieb-Robinson velocity.

The Lieb-Robinson bound (1) is presented differently from the equation in (Lieb, Robinson, 1972). This more explicit form (1) can be seen from the proof of the bound in (Lieb, Robinson, 1972).

From the Lieb-Robinson bound we see that for times \(|t| < d(X, Y )/v\) the norm on the right-hand side is exponentially small.

The reason for considering the commutator on the left-hand side of the Lieb-Robinson bounds is the following:

The commutator between observables \(A\) and \( B\) is zero if their supports are disjoint.

The converse is also true: if observable \(A\) is such that its commutator with any observable \(B\) supported outside some set \(X\) is zero, then \(A\) has a support inside set \(X\).

This statement is also approximately true in the following sense: suppose that there exists some \(\epsilon > 0\) such that \(\|[A, B]\| \leq \epsilon \|B\|\) for some observable \(A\) and any observable \(B\) that is supported outside the set \(X\). Then there exists an observable \(A(\epsilon)\) with support inside set \(X\) that approximates an observable \(A\), i.e. \( \|A - A(\epsilon)\| \leq \epsilon\). See (Bachmann et al., 2011).

Thus, Lieb-Robinson bounds say that the time evolution of an observable \(A\) with support in a set \(X\) is mainly supported in a \(\delta\)-neighborhood of set \(X\), where \(\delta > v|t|\) with \(v\) being the Lieb-Robinson velocity.

In other words, this bound asserts that the speed of propagation of perturbations in quantum spin systems is bounded.

Improvements of the Lieb-Robinson bounds

In 1976 Robinson generalized the bound (1) by considering exponentially decaying interactions (that need not be translation invariant), i.e., for which the strength of the interaction exponentially decays with the diameter of the set. This result is discussed in detail in (Bratteli, Robinson, 1981), (Bratteli, Robinson, 1997) Chapter 6. No great interest was shown in the Lieb-Robinson bounds until 2004 when Hastings, (2004) applied them to the Lieb–Schultz–Mattis theorem. Subsequently Nachtergaele and Sims, (2006) extended the results of Robinson (1976) to include models on vertices with a metric and to derive exponential decay of correlations.

From 2005–2006 interest in Lieb–Robinson bounds strengthened with additional applications to exponential decay of correlations (see (Hastings and Koma, 2006) (Nachtergaele, Ogata and Sims, 2006) (Hastings, 2004) and the sections below). New proofs of the bounds were developed and, in particular, the constant in (1) was improved making it independent of the dimension of the Hilbert space.

Several further improvements of the constant \(c\) in (1) were made by Nachtergaele and Sims, (2009). In 2008 the Lieb-Robinson bound was extended to the case in which each \(H_x\) is infinite dimensional by Nachtergaele et al., (2009).

It was shown by Nachtergaele et al., (2009) that the on-site unbounded perturbations do not change the Lieb-Robinson bound, i.e. the Hamiltonian of the following form is considered on a finite subset \(\Lambda\subset\Gamma\): \[H_\Lambda=\sum_{x\in\Lambda}H_x+\sum_{X\subset\Lambda}\Phi(X),\] where \(H_x\) is a self-adjoint operator over \(\mathcal{H}_x\), which needs not to be bounded.

Harmonic and Anharmonic Hamiltonians

In (Nachtergaele et al., 2009) the Lieb-Robinson bounds were extended to a harmonic Hamiltonian, which in a finite volume \(\Gamma_L=(-L, L]^\nu\cap\mathbb{Z}^\nu,\) where \(L, \nu\) are positive integers, takes the form: \[\sum_{x\in\Gamma_L}p_x^2+\omega^2q_x^2+\sum_{x\in\Gamma_L}\sum_{j=1}^\nu\lambda_j(q_x-q_{x+e_j})^2,\] where the periodic boundary conditions are imposed and \(\lambda_j\geq 0\), \(\omega>0\). Here \(\{e_j\}\) are canonical basis vectors in \(\mathbb{Z}^\nu\).

Anharmonic Hamiltonians with on-site and multiple-site perturbations were considered and the Lieb-Robinson bounds were derived for them in (Nachtergaele et al., 2009) and (Nachtergaele et al., 2010).

Further generalizations of the harmonic lattice were discussed in (Cramer, Serafini, Eisert, 2008) and (Juenemann et al., 2013).

Irreversible dynamics

Another generalization of the Lieb-Robinson bounds is to irreversible dynamics, in which case the dynamics has a Hamiltonian part and also a dissipative part. The dissipative part is described by terms of Lindblad form, so that the dynamics \(\tau_t\) satisfies the Lindblad-Kossakowski master equation.

Lieb-Robinson bounds for the irreversible dynamics were considered by Hastings, (2004) in the classical context and by Poulin, (2010) for a class of quantum lattice systems with finite-range interactions. Lieb-Robinson bounds for lattice models with a dynamics generated by both Hamiltonian and dissipative interactions with suitably fast decay in space, and that may depend on time, were proved by Nachtergaele, Vershynina and Zagrebnov, (2011), where they also proved the existence of the infinite dynamics as a strongly continuous cocycle of unit preserving completely positive maps.

Some applications

Lieb-Robinson bounds are used in many areas of mathematical physics. Among the main applications of the bound there is the existence of the thermodynamic limit, the exponential decay of correlations and the Lieb-Schultz-Mattis theorem.

Thermodynamic limit of the dynamics

One of the important properties of any model meant to describe properties of bulk matter is the existence of the thermodynamic limit of the dynamics. Any of these properties should be essentially independent of the size of the system which, in any experimental setup, is ﬁnite.

Thermodynamic limit from equilibrium point of view was settled much before the Lieb-Robinson bound was proved, see (Ruelle, 1969) for example. In certain cases one can use the Lieb-Robinson bound to establish the existence of a dynamics \(\tau_t^{\Gamma}\) for an infinite lattice \(\Gamma\) as the limit of finite lattice dynamics. The limit is usually considered over an increasing, exhausting sequence of finite subsets \(\Lambda_n\subset\Gamma\), i.e. such that for \(n<m\), there is an inclusion \( \Lambda_n\subset\Lambda_m\). In order to prove the existence of the infinite dynamics \(\tau_t^\Gamma\) as a strongly continuous, one-parameter group of automorphisms, the papers mentioned above proceeded by proving that \( \{\tau_t^{\Lambda_n} \}_n\) is a Cauchy sequence and consequently converges. By elementary properties the existence of the thermodynamic limit follows. See (Bratteli, Robinson, 1997) section 6.2 for more detailed discussion of the thermodynamic limit.

Robinson, (1976) was the first to show the existence of the thermodynamic limit for exponentially decaying interactions. Later, Nachtergaele et al. (Nachtergaele, Ogata, Sims, 2006), (Nachtergaele et al., 2010), (Nachtergaele, Vershynina, Zagrebnov, 2011) showed the existence of the infinite volume dynamics for almost every type of the interaction described in the section Improvements of Lieb-Robinson bounds above.

Exponential decay of correlations

Let \(<A>_\Omega \) denote the expectation value of the observable \(A\) with respect to a state \(\Omega\). The correlation function between two observables \(A\) and \(B\) is defined as \(<AB>_\Omega-<A>_\Omega< B >_\Omega.\)

Lieb-Robinson bounds are used to show that Hamiltonians with a non-vanishing spectral gap have ground states with exponentially decaying correlations. Let \(\Omega\) be a non-degenerate ground state for a Hamiltonian with a non-vanishing spectral gap above it, see (Hastings, Koma, 2004), Nachtergaele, Sims, (2006). Then the inequality \[|<AB>_\Omega-<A>_\Omega< B >_\Omega|\leq K\|A\|\|B\|\min(|X|,|Y|) e^{-a\, d(X,Y)}, \] holds for observables \(A\) and \(B\) with support in sets \(X\) and \(Y\) respectively. Here \(K\) and \(a\) are some constants.

One can also start with a product state \(\Omega\) and show the exponential decay of correlations as it is done by Nachtergaele, Ogata and Sims, (2006).

Such a decay was long known for relativistic dynamics, but only guessed for Newtonian dynamics. The noteworthy point here is that the Lieb-Robinson bounds succeed in replacing the relativistic symmetry by local estimates.

Lieb-Schultz-Mattis theorem

The theorem by Lieb and Mattis, (1962) implies that the ground state of the Heisenberg antiferromagnet on a bipartite lattice with isomorphic sublattices, is non-degenerate, i.e., unique, but the gap can be very small. For one-dimensional and quasi-one-dimensional systems of even length and with half-integral spin Aﬄeck and Lieb, (1986), generalizing the original result by Lieb, Schultz, and Mattis, (1961), proved that the gap \(\gamma_L \) in the spectrum above the ground state is bounded above by \[\gamma_L\leq c/L,\] where \( L \) is the size of the lattice and \( c\) is a constant.

The Lieb-Robinson bound was utilized in a proof of the Lieb-Schultz-Mattis Theorem for higher-dimensional cases by Hastings, (2004) and by Nachtergaele and Sims, (2007). In these papers the following bound on the gap was obtained: \[\gamma_L\leq c\log (L)/L. \]

Experiments

The first experimental observation of the Lieb-Robinson velocity was done by Cheneau et.al., (2012).

References

Aﬄeck(1986). A proof of part of Haldane’s conjecture on quantum spin chains. Lett. Math. Phys. 12: 57–69. doi:10.1007/bf00400304.

Bachmann, S; Michalakis, S; Nachtergaele, B and Sims, R (2012). Automorphic Equivalence within Gapped Phases of Quantum Lattice Systems. Commun. Math. Phys. 309: 835-871. doi:10.1007/s00220-011-1380-0.

Cheneau, M, et al. (2012). Light-cone-like spreading of correlations in a quantum many-body system. Nature 481: 484-487. doi:10.1038/nature10748.

Cramer, M; Serafini, A and Eisert, J (2008). Locality of dynamics in general harmonic quantum systems. arXiv:0803.0890 : .

Hastings, M B (2004). Locality in quantum and Markov dynamics on lattices and networks. Phys. Rev. Lett. 93: 140402-140406. doi:10.1103/physrevlett.93.140402.

Hastings, M B (2004). Lieb-Schultz-Mattis in higher dimensions. Phys. Rev. Lett. B 69: 1104431–10444. doi:10.1103/physrevb.69.104431.

Hastings(2006). Spectral gap and exponential decay of correlations. Commun. Math. Phys. 256: 781. doi:10.1007/s00220-006-0030-4.

Juenemann, J; Cadarso, A; Perez-Garcia, D; Bermudez, A and Garcia-Ripoll, J.J (2013). Lieb-Robinson bounds for spin-boson lattice models and trapped ions. arXiv:1307.1992 : .

Lieb(1962). Ordering energy levels in interacting spin chains. J. Math. Phys. 3: 749–751. doi:10.1063/1.1724276.

Lieb(1972). The ﬁnite group velocity of quantum spin systems. Commun. Math. Phys. 28: 251-257. doi:10.1007/bf01645779.

Lieb, E H; Schultz, T and Mattis, D (1961). Two soluble models of an antiferromagnetic chain. Ann. Phys. (N.Y.) 16: 407–466. doi:10.1016/0003-4916(61)90115-4.

Nachtergaele, B; Ogata, Y and Sims, R (2006). Propagation of correlations in quantum lattice systems. J. Stat. Phys. 124: 1-13. doi:10.1007/s10955-006-9143-6.

Nachtergaele, B.; Raz, H; Schlein, B and Sims, R (2009). Lieb-Robinson bounds for harmonic and anharmonic lattice systems. Commun. Math. Phys. 286: 1073-1098. doi:10.1007/s00220-008-0630-2.

Nachtergaele, B; Schlein, B; Sims, R; Starr, S and Zagrebnov, V (2010). On the existence of the dynamics for anharmonic quantum oscillator systems. Rev. Math. Phys. 22: 207-231. doi:10.1142/s0129055x1000393x.

Nachtergaele(2006). Lieb-Robinson bounds and the exponential clustering theorem. Commun. Math. Phys. 265: 119-130. doi:10.1007/s00220-006-1556-1.

Nachtergaele(2007). A multi-dimensional Lieb-Schultz-Mattis theorem. Commun. Math. Phys. 276: 437-472. doi:10.1007/s00220-007-0342-z.

Nachtergaele(2009). Locality estimates for quantum spin systems. Sidoravicius, Vladas (Ed.), New Trends in Mathematical Physics. Selected contributions of the XVth International Congress on Mathematical Physics, Springer Verlag : 591-614.

Nachtergaele, B; Vershynina, A and Zagrebnov, V (2011). Lieb-Robinson bounds and existence of the thermodynamic limit for a class of irreversible quantum dynamics. AMS Contemporary Mathematics 552: 161-175. doi:10.1090/conm/552/10916.

Poulin, D (2010). Lieb-Robinson bound and locality for general Markovian quantum dynamics. Phys. Rev. Lett. 104: 190401. doi:10.1103/physrevlett.104.190401.

Robinson, D W (1976). Properties of propagation of quantum spin systems. J. Austral. Math. Soc. 19 (Series B): 387-399. doi:10.1017/s0334270000001260.

Robinson, D W (1968). Statistical mechanics of quantum spin systems II. Comm. Math. Phys. 7: 337-348. doi:10.1007/bf01646665.

Bratteli, O and Robinson, D W (1981). Operator algebras and quantum statistical mechanics. Springer, 1 ed., vol. 2.

Bratteli, O and Robinson, D W (1997). Operator algebras and quantum statistical mechanics. Springer, 2 ed., vol. 2.

Ruelle, D (1969). Statistical mechanics. Rigorous results. Benjamin, New York.

Lieb-Robinson bounds

Contents

Set up

Lieb-Robinson bounds

Improvements of the Lieb-Robinson bounds

Harmonic and Anharmonic Hamiltonians

Irreversible dynamics

Some applications

Thermodynamic limit of the dynamics

Exponential decay of correlations

Lieb-Schultz-Mattis theorem

Experiments

References

Recommended reading

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