Undecidability of the Spectral Gap

Toby Cubitt,1 David Perez-Garcia,2 and Michael M. Wolf3

1DAMTP, University of Cambridge, Centre for Mathematical Sciences,Wilberforce Road, Cambridge CB3 0WA, United Kingdom

2Departamento de Analisis Matematico, Facultad de CC Matematicas,Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040 Madrid, Spain

3Department of Mathematics, Technische Universitat Munchen, 85748 Garching, Germany

The spectral gap—the difference in energy between the ground state and the first excited state—is one of the most important prop-erties of a quantum many-body system. Quantum phase transitions occur when the spectral gap vanishes and the system becomescritical. Much of physicsis concerned with understanding the phase diagrams of quantum systems, and some of the most challengingand long-standing open problems in theoretical physics concern the spectral gap,1–3 such as the Haldane conjecture4 that the Heisen-berg chain is gapped for integer spin, proving existence of a gapped topological spin liquid phase,2,3 or the Yang-Mills gap conjecture5

(one of the Millennium Prize problems). These problems are all particular cases of the general spectral gap problem: Given a quan-tum many-body Hamiltonian, is the system it describes gapped or gapless? Here we show that this problem is undecidable, in thesame sense as the Halting Problem was proven to be undecidable by Turing.6 A consequence of this is that the spectral gap of certainquantum many-body Hamiltonians is not determined by the axioms of mathematics, much as Godels incompleteness theorem impliesthat certain theorems are mathematically unprovable. We extend these results to prove undecidability of other low temperature prop-erties, such as correlation functions. The proof hinges on simple quantum many-body models that exhibit highly unusual physics inthe thermodynamic limit.

Many seminal results in condensed matter theory prove thatspecific systems are gapped or gapless. For example, Lieb,Schultz and Mattis’ proof that the Heisenberg chain is gaplessfor half-integer spin7 (later extended to higher dimensions byHastings8), or Affleck et al.’s proof that the 1D AKLT model isgapped.1 Similarly, many famous and long-standing open prob-lems in theoretical physics concern the presence or absence of aspectral gap. A paradigmatic example is the antiferromagneticHeisenberg model. A 1D chain of quantum spins, each inter-acting with its neighbours by a rotationally invariant spin-spincoupling, is arguably the simplest of all interacting quantummany-body models. Yet even in 1D, the case of integer spinremains open: the “Haldane conjecture” that the Heisenbergchain is gapped for integer spin, first formulated in 1983,4 hasso far resisted rigorous proof despite strong supporting nu-merical evidence (see e.g.9 and references therein). The samequestion in the case of 2D non-bipartite lattices (such as theKagome lattice) was posed by Anderson in 1973.2 The latestnumerical evidence10 strongly indicates that these systems maybe topological spin liquids. This problem has attracted signifi-cant attention recently3 as materials such as herbertsmithite11

have emerged whose interactions are well-approximated by theHeisenberg coupling. The presence of a spectral gap in thesemodels remains one of the main unsolved questions concerningthe long-sought topological spin liquid phase. In the relatedsetting of quantum field theory, proving that Yang-Mills theoryis gapped would resolve one of the most important questionsin fundamental physics: a full explanation of the phenomenonof quark confinement.5 Again, whilst there is strong numericalevidence from numerical lattice QCD calculations,12 the Yang-Mills gap conjecture remains open. Indeed, proving this is oneof the Millennium Prize problems.5

All of these problems are specific instances of the generalspectral gap problem: Given a quantum many-body Hamilto-nian, is the system it describes gapped or gapless? Our mainresult is to prove that the spectral gap problem is undecidable.This says something much stronger than merely showing that

the spectral gap is difficult to compute. Though one may beable to solve the spectral gap problem in specific cases, ourresult implies that it is in general logically impossible to saywhether a many-body quantum system is gapped or gapless.This has two subtly distinct meanings, and we prove both ofthese:

1. The spectral gap problem is algorithmically undecidable:there cannot exist any procedure which, given the ma-trix elements of the local interactions of the Hamiltonian,determines whether the resulting model is gapped or gap-less. This is precisely the same sense in which the HaltingProblem is undecidable.6

2. The spectral gap problem is axiomatically independent:given any consistent axiomatisation of mathematics, thereexist quantum many-body Hamiltonians for which thepresence or absence of the spectral gap is not determinedby the axioms of mathematics. This is the form of undecid-ability encountered in Godel’s incompleteness theorem.13

Precise statement of resultsIt is important to be absolutely precise in what we mean by thespectral gap problem, to avoid any possibility of “cheating” insome way by answering a subtly different question. To this end,we must first specify carefully the systems we are considering.Since we are proving undecidability, the simpler the system, thestronger the result. We therefore restrict ourselves to one of thesimplest types of quantum many-body system, and a workhorseof condensed matter theory: nearest-neighbour, translationallyinvariant spin lattice models on a 2D square lattice of sizeL × L, with local Hilbert space dimension d (i.e. spin- d−1

2particles). Any such Hamiltonian HL is completely specified byat most three finite-dimensional Hermitian matrices describingthe local interactions of the system: two d2 × d2 matrices hrowand hcol giving the interactions along the rows and columns,and a d×d matrix h1 giving the on-site interaction (if any). Wemust be careful to avoid allowing the Hamiltonian itself to be

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FIG. 1. (a) A gapped system has a unique ground state λ0(H) anda constant lower-bound γ on the spectral gap ∆(H) in the thermody-namic limit. (b) A gapless system has continuous spectrum above theground state in the thermodynamic limit.

uncomputable, as this would render the result meaningless. Sowe will restrict the matrix elements to be algebraic numbers(i.e. simple roots of polynomial equations).

We must also be precise in what we mean by “gapped” and“gapless” (see Figure 1). Since quantum phase transitions occurin the thermodynamic limit of arbitrarily large system size, weare interested in the spectral gap ∆(HL) = λ1(HL) − λ0(HL)as the system size L → ∞ (where λ0, λ1 are the lowest andsecond-lowest eigenvalues). We take “gapped” to mean thesystem has a unique ground state and a constant lower boundon the spectral gap: ∆(HL) ≥ γ for all sufficiently large L. Wetake “gapless” to mean the system has continuous spectrumabove the ground state in the thermodynamic limit. (Formally,there is a constant c > 0 such that for all ε > 0 there existsan L0 such that, for all lattice sizes L ≥ L0, any point in theinterval [λ0(HL), λ0(HL) + c] is within ε of some eigenvalue ofHL.)

Note that gapped is not the negation of gapless here; thereare systems that fall into neither category. The reason foradopting such strong definitions is to deliberately exclude am-biguous cases, such as systems with degenerate ground states.A Hamiltonian that is gapped or gapless according to these def-initions is recognised as such throughout the literature. Sincewe are proving undecidability, this only strengthens the result.We prove that the spectral gap problem is undecidable evenwhen ambiguous cases are excluded, and the Hamiltonians arerestricted to translationally-invariant 2D spin lattice models.

From spectral gap to ground state energy densityWe prove our result by showing that the spectral gap problemis equivalent to the Halting Problem. Recall that a TuringMachine is a simple, abstract model of computation in which ahead reads and writes symbols from some finite alphabet onan infinite tape and moves left or right, according a finite setof update rules. The Halting Problem asks: given an initialinput written on the tape, does the Turing Machine halt on thatinput? (Informally, this is asking: given the source code of

a computer program, will that program halt or run forever?)Turing famously proved that this problem is undecidable.6 Ourapproach is to first show that undecidability of the spectralgap follows from undecidability of another important physicalquantity: the ground state energy density. We then relate thelatter to the Halting Problem.

To transform the ground state energy density problem intoan equivalent spectral gap problem, we require two ingredients:

(i). A translationally-invariant Hamiltonian Hu(ϕ) on a2D square lattice with local interactions hu(ϕ), whoseground state energy density is either strictly positiveor tends to 0 from below in the thermodynamic limit,depending on the value of an external parameter ϕ. How-ever, determining which case holds should be undecid-able. Constructing such a Hamiltonian is challenging,and constitutes the main technical work of our result.

(ii). A gapless Hamiltonian Hd with translationally-invariantlocal interactions hd and ground state energy 0. Thereare many well-known examples of such Hamiltonians,e.g. the critical XY-model on a chain:7 hd = σx ⊗ σx +

σy ⊗ σy + σz ⊗ 1 + 1 ⊗ σz, where σx,y,z are the Paulimatrices.

From these two Hamiltonians, we construct a new HamiltonianH(ϕ) which is gapped or gapless, depending on the value of ϕ.Given hu and hd, constructing such an h is not difficult. Thelocal Hilbert space is the tensor product of those of hu and hdtogether with one additional energy level: H = |0〉 ⊕ Hu ⊗Hd.The interaction h(i, j) between nearest-neighbour sites i and j isthen given by the following formula:

h(ϕ)(i, j) = |0〉〈0|(i)⊗(1−|0〉〈0|)( j) +h(i, j)u (ϕ)⊗1(i, j)

d +1(i, j)u ⊗h(i, j)

d .(1)

To see that this Hamiltonian has the desired properties, as-sign a signature σ ∈ {0, 1}L

2to every product basis state, by

assigning a 0 to the |0〉 state, and a 1 to any state in theHu⊗Hdsubspace. The overall Hilbert space then decomposes intosubspaces of given signature:

⊗iH

(i) '⊕

σHσ. Since(1) is manifestly block-diagonal with respect to this directsum decomposition, the overall Hamiltonian decomposes asH =

∑〈i, j〉 h(ϕ)(i, j) '

⊕Hσ where Hσ acts on Hσ. Thus its

spectrum is given by spec H(ϕ) =⋃σ spec Hσ. and it is suffi-

cient to analyse the spectra for each signature separately. Wedistinguish three cases:

1. σ = (0, . . . , 0): In this sector Hσ = 0.

2. σ = (1, . . . , 1): In this sector, the first term in (1) is 0 andHσ = Hu ⊗1+1⊗Hd. Thus the spectrum stemming fromthis subspace is spec Hu + spec Hd.

3. σ , (0, . . . , 0) but σi = 0 for some site i. In this case theremust be at least two14 locations in the lattice where a |0〉state is adjacent to a state from the Hu ⊗ Hd subspace.These pick up an energy contribution of 2 from the firstterm in (1). One can show that even if the ground stateenergy density of Hu is negative, since it tends to zero inthe thermodynamic limit in this case (see (i), above), the

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negative energy contribution from the hu term is at most−1 (see ref. 15 for a detailed proof). Thus the spectrum Sfrom the entire mixed-signature sector is lower-boundedby S ≥ 1.

The spectrum of the overall Hamiltonian is therefore

spec H(ϕ) = {0} ∪{

spec Hu(ϕ) + spec Hd

}∪ S . (2)

Recalling that we chose Hd to be gapless, we see immediatelyfrom (2) that if the ground state energy density of Hu tendsto zero from below (so that λ0(Hu) < 0) then H(ϕ) is gapless,but if Hu has strictly positive ground state energy density (sothat λ0(Hu) diverges to +∞) then it has spectral gap ≥ 1, asrequired. (Illustrated in Figure 2.)

Much more difficult is to construct hu(ϕ). We explain thisin detail below. However, before we do so, observe that thisconstruction is rather general. It is readily extended to proveundecidability of many other low-energy properties. In fact,by choosing different hd, we obtain undecidability of any phys-ical property that distinguishes a Hamiltonian from a gappedsystem with unique product ground state. Correlation func-tions provide an important example. Determining whether theground state has long-range (algebraically decaying) correla-tions, or no ground state correlations whatsoever (strictly 0connected correlation functions) is undecidable by this argu-ment.

Undecidable ground state energy densityThe more challenging part remains: relating the ground stateenergy density problem to the Halting Problem. The lat-ter concerns the dynamics of a classical system—a TuringMachine. How can this be related to the spectral gap orground state energy density—both static properties of a quan-tum system? The idea of relating ground states of quantumsystems to computation, which dates back to Feynman,16 isto construct a Hamiltonian whose ground state encodes theentire history of the computation in superposition. I.e. ifthe state of the computation at time t is represented by thestate vector |ψt〉, the ground state is the so-called computa-tional history state 1

√T

∑T−1t=0 |t〉 |ψt〉. If there are no other con-

straints, writing down such a Hamiltonian is straightforward:H = |0〉〈0| ⊗ (1− |ψ0〉〈ψ0|) +

∑t(|t〉〈t| ⊗ |ψt〉〈ψt |+ |t + 1〉〈t + 1| ⊗

|ψt+1〉〈ψt+1|− |t + 1〉〈t|⊗ |ψt+1〉〈ψt |− |t〉〈t + 1|⊗ |ψt〉〈ψt+1|). How-ever, this Hamiltonian is completely non-local, so bears littleresemblance to that of a many-body system. This idea was sig-nificantly developed by Kitaev,17 who showed that it is possibleto construct a quantum many-body Hamiltonian with a historystate ground state using 5-body interactions. A long sequenceof results18–20 improved this to ever-simpler local Hamiltoni-ans, culminating in Gottesman and Irani’s construction for 1Dspin chains with translationally-invariant nearest-neighbourinteractions.21

At first sight, undecidability might now appear to be a triv-ial consequence: simply encode the evolution of a universalTuring Machine using the Gottesman-Irani construction, andadd an additional term |⊥〉〈⊥| to the resulting Hamiltonianthat increases the energy of the state |⊥〉 that represents thehalting state of the Turing Machine. In this way, the ground

FIG. 2. To relate ground state energy density and spectral gap, weneed: (i) a Hamiltonian Hu(ϕ) whose ground state energy densityis either strictly positive or tends to 0 from below in the thermody-namic limit, but determining which is undecidable, and (ii) a gaplessHamiltonian Hd with ground state energy 0. We combine Hu(ϕ) andHd to form a new local interaction, h(ϕ), in such a way that h(ϕ)has (iii) an additional non-degenerate 0-energy eigenstate, and thecontinuous spectrum of Hd is shifted immediately above the groundstate energy of Hu. (The formula for h(ϕ) is given in the main text.)(a) If the ground state energy density of Hu(ϕ) is strictly positive, itsground state energy in the thermodynamic limit must diverge to +∞,and h(ϕ) is gapped. (b) Whereas if the ground state energy densityof Hu(ϕ) tends to 0 from below, then its ground state energy in thethermodynamic limit must be ≤ 0, and h(ϕ) is gapless.

state energy depends on whether or not the computation halts.However, this argument fails for two crucial and fundamentalreasons:

1. The local Hilbert space dimension is fixed, so the Hamilto-nian is completely specified by the finite number of matrixelements defining its local interactions. It is not at all clearhow to encode the countably infinite number of HaltingProblem instances into this finite number of parameters.

2. The ground state energy of history state Hamiltoniansscales inverse-polynomially in the system size. Thus theground state energy in the thermodynamic limit—andhence the ground state energy density—is always zero,independent of the output of the computation encoded inthe ground state.

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Obstacle 1: Finite local dimensionIf the Halting Problem—or indeed any problem—is restrictedto finitely many possible instances, it becomes trivially algo-rithmically decidable: the answers can simply be enumerated.The translationally-invariant Hamiltonians we are consideringare completely specified by a finite number of matrix elements.However, each matrix element can be an arbitrarily precisecomplex (algebraic) number. The only way we can hope toparametrise infinitely many problem instances and overcomeobstacle 1 is to encode them in the decimal digits of one ofthe matrix elements. However, this entails constructing theHamiltonian in such a way that whether it is gapped or gaplessdepends sensitively on the precise value of this matrix element,no matter how minute the change in that value.

It is easy to see that this is impossible if we try to encodeclassical computation in the Hamiltonian, unless we allowthe local dimension of the Hamiltonian to grow arbitrarilylarge. The same issue arises if we instead try to use undecid-ability of other classical problems. For example, the Wangtiling problem22 asks whether a given set of square tiles withcoloured edges can tile the infinite plane such that adjacentcolours match. This was famously proven to be undecidable byBerger,23 and can easily be recast as a translationally-invariant(classical) Hamiltonian on a square lattice. However, the tilingproblem is only undecidable if there is no limit on the numberof colours, which again leads to an unbounded local Hilbertspace dimension.

It is necessary, therefore, to exploit the fact that the groundstate can encode quantum computation. There is a well-knownquantum algorithm that does something close to what we need:the quantum phase estimation algorithm,24 which extracts thedecimal digits of a unitary phase rotation. The evolution ofthis algorithm can be encoded in the Hamiltonian in exactlythe same way as above, using the Gottesman-Irani history stateconstruction. The phase to be estimated then becomes oneof the matrix elements of the Hamiltonian, which can dependon an external parameter ϕ. In this way, we obtain a localHamiltonian whose ground state is a computational historystate that encodes (in superposition) the evolution of: (1) thequantum phase estimation algorithm, which extracts a stringof digits determined by an external parameter ϕ, and feeds thisstring as input to (2) the universal Turing Machine.

Formally, to make this argument rigorous15 we need to con-struct, for each natural number n, a Quantum Turing Machine(QTM) of fixed size (independent of n) that writes the binaryexpansion of n on its tape and then halts deterministically.Without these very precise properties, the matrix elements ofthe Hamiltonian ultimately end up being uncomputable, whichwould render the result meaningless. We obtain this by feedingthe Turing machine as input a number N written in unary, anddemand that n is written on the tape as output and the machinehalts deterministically only if N is larger than the number ofbits of n. This is sufficient for our purposes, as we can use theoverall system size to provide a suitable N. More precisely, bya careful explicit construction,15 we show: There is a familyof Quantum Turing Machines Pn indexed by integers n, allwith the same set of internal states and tape symbols (only thetransition rules differ), such that on unary input N > n, Pn

halts after poly(N)2N time using only N + 3 space, leaving thebinary expansion of n written on the tape.

For technical reasons, when it comes to encoding the time-evolution |ψt〉 of Pn in a 1D history state, the Gottesman-Irani21

construction is insufficient. It cannot encode computationslonger than poly(N) in the length N of the chain, becausethe “clock” |t〉 part of their history state

∑t |t〉 |ψt〉 counts in

unary. Instead, we give a new construction – heavily inspiredby Gottesman-Irani – where the clock counts in binary, whichallows exponentially-long computations to be encoded. (Thisalso requires addressing a subtle unitarity issue; see ref. 15for technical details.) In fact, we prove a more general re-sult: For any QTM M, we construct a translationally-invariantHamiltonian H(N) on a 1D spin chain of length N with nearest-neighbour interactions hq, whose local Hilbert space dimen-sion depends only on the number of internal states and tapesymbols of M, such that the unique ground state of H(N) is thehistory state encoding the evolution of M on input N − 3 repre-sented in unary, where M is restricted to a finite tape segmentof length N and O(2cN) time-steps (for any given constant c).

Obstacle 2: The thermodynamic limitThe thermodynamic limit presents a more serious obstacle.There is an inherent trade-off between run-time and energyin the Feynman-Kitaev Hamiltonian construction, with theground state energy scaling (at best) inverse-polynomially withthe run-time of the encoded computation. On a lattice this trans-lates into an inverse-polynomial scaling with lattice size.21 Butthe spectral gap and ground state energy density problems con-cern the thermodynamic limit of arbitrarily large lattice size.An inverse-polynomial scaling is of no use in our case, as itvanishes in the thermodynamic limit. Even the ground stateenergy problem becomes trivial in this setting, as it requiresat least a constant difference in ground state energy betweenthe halting and non-halting cases, independent of system size.This cannot be achieved by any known variant of the Feynman-Kitaev Hamiltonian construction, and may well be impossible.Undecidability of the ground state energy density is more de-manding still: it requires an energy difference that grows atleast quadratically with the lattice size.

To overcome this obstacle, we return to Wang tilings. Butinstead of blindly representing tiling problems as Hamiltonians,we prove and then exploit very particular properties of anaperiodic tiling due to Robinson,25 and combine this with theFeynman-Kitaev construction. Although the pattern of tiles inthe Robinson tiling extends infinitely in all directions, it neverrepeats. More precisely, it contains periodically repeating sub-patterns forming squares with sizes equal to every power of 4(see Figure 3). This will allow us to encode in the ground statethe parallel execution of the same universal Turing Machinerunning on the same input (extracted from the value ϕ of amatrix element of the Hamiltonian by the quantum phase-estimation procedure described above). But now, thanks to theaperiodic tiling pattern, the encoded Turing Machines run ontapes of all possible finite lengths and for every possible finiterun-time (see Figure 3 for details).

We do this by sandwiching the 1D quantum history-stateHamiltonian hq “on top of” the (classical) Robinson tiling

5

Hamiltonian hc to form two “layers”, so that the local Hilbertspace at each site is H = Hc ⊗ (He ⊕ Hq) (where He = |0〉is an additional energy level). If |L〉 , |R〉 are the tiles at thetop-left and top-right corners of the squares in the Robinsontiling, and | 〉 , | 〉 are states that occur only at the left andright ends of the 1D history state, then the local interactionsalong the columns are just the classical tiling interactions hcol

c ,but the local interactions along the rows couple the tiling andquantum layers of neighbouring sites i and j = i + 1:

h(i, j)row = hrow

c ⊗ 1(i, j)eq + 1

(i, j)c ⊗ hq (3a)

+ |L〉〈L|(i)c ⊗ (1eq − | 〉〈 |)(i) ⊗ 1( j)ceq (3b)

+ (1c − |L〉〈L|c)(i) ⊗ | 〉〈 |(i)⊗ 1

( j)ceq (3c)

+ 1(i)ceq ⊗ |R〉〈R|

( j)c ⊗ (1eq − | 〉〈 |)( j) (3d)

+ 1(i)ceq ⊗ (1c − |R〉〈R|c)( j) ⊗ | 〉〈 |

( j)(3e)

+ 1(i)c ⊗ |0〉〈0|

(i)e ⊗ |R〉〈R|

( j)c ⊗ 1

( j)eq (3f)

+ |L〉〈L|(i)c ⊗ 1(i)eq ⊗ 1

( j)c ⊗ |0〉〈0|

( j)e (3g)

+ 1(i)c ⊗ |0〉〈0|

(i)e ⊗ (1c − |L〉〈L|c)( j) ⊗ (1eq − |0〉〈0|e)( j) (3h)

+ (1c − |R〉〈R|c)(i) ⊗ (1eq − |0〉〈0|e)(i) ⊗ 1( j)c ⊗ |0〉〈0|

( j)e . (3i)

This interaction looks complicated, but it simply forces thedesired structure in the ground state: terms (3b)–(3e) force a| 〉 (| 〉) state to appear in the quantum layer above every |L〉(|R〉) in the tiling layer, and the hq term then forces a historystate between | 〉 and | 〉; terms (3f)–(3i) force |0〉’s every-where else in the quantum layer. By assigning signatures in asimilar way to before (cf. (1)), one can show that this forcesthe ground state to have 1D history states along the top edge ofeach square in the Robinson tiling. (See ref. 15 for a detailedproof.)

The inspiration for encoding the parallel evolution of thesame Turing Machine on tapes of all possible finite lengths,instead of encoding the evolution of a single Turing Machineon an infinite tape, dates back to Berger’s original proof23

of undecidability of tiling. However, in our case it servesa different—and purely quantum—purpose: it allows us todecouple the energy dependence of the computational historyground state from the overall system size, thereby overcomingobstacle 2.

If the universal Turing Machine eventually halts on inputϕ, then all Turing Machines running on sufficiently long tapeswill halt. If we add to the Hamiltonian an on-site interactionh1 = |⊥〉〈⊥| that gives an additional energy to the state |⊥〉representing the halting state of the Turing Machine in thehistory state, then the ground state will pick up additionalenergy from all encoded Turing Machines that halt. This energystill decreases inverse-polynomially with the effective size ofthe system that the Feynman-Kitaev part of the Hamiltonianacts on. But, crucially, this effective size is now the size of thecorresponding sub-pattern in the Robinson tiling (see Figure 3),not the overall system size. The energy now depends onlyon the length of tape used by the universal Turing Machinebefore it halts (which is some finite number that depends onthe external parameter ϕ in the halting case, and can be infinite

FIG. 3. The Robinson tiles enforce a recursive pattern of interlockingsquares, of sizes given by all integer powers of 4 (middle figure).As with any Wang tiling, we can readily represent this as a classicalHamiltonian whose ground state has the same quasi-periodic structure.Since the set of tiles is fixed, the local dimension of this Hamilto-nian is constant. By adding a “quantum layer” on top of this tilingHamiltonian and choosing a suitable translationally-invariant cou-pling between the layers (see (3) in the main text the precise formula),we can effectively place copies of the history state Hamiltonian (topfigure) along the top edge of all the squares. The ground state of thisHamiltonian consists of the Robinson tiling configuration in the tilinglayer, with computational history states in the quantum layer alongthe top edge of all of the squares in the tiling (bottom figure). Eachof these encodes the evolution of the same quantum phase estimationalgorithm and universal Turing Machine. The effective tape lengthavailable to the Turing Machine is determined by the size of the squareit “runs” on.

in the non-halting case). Thus we have decoupled the groundstate energy from the overall system size.

However, this is still not sufficient. We have shown thatthe difference in energy between the halting and non-haltingcases now depends inverse-polynomially on the amount of tapeused, which depends on the input ϕ. Not only does this fail toprovide a constant bound on the energy difference, indepen-dent of the Hamiltonian parameter ϕ, this energy difference isuncomputable!

In fact, this is a limitation of the above analysis, not the truesituation. Instead of being uncomputably small, the true dif-ference in ground state energy between the two cases divergesto infinity. To show this requires a more careful analysis ofthe low-energy excitations. It is easy to see that the eigenstateconsisting of a valid Robinson tiling, together with compu-tational history states along the top edges of all the squares,has energy that diverges with lattice size in the halting case.Once the lattice is large enough, the number of borders that aresufficiently large for the encoded Turing Machine to halt growsquadratically, and each of them contributes a small but non-zero energy. The difficulty is that, since its energy diverges,this eigenstate may not be the ground state in the halting case.We may be able to lower the energy by introducing defects

6

in the tiling layer, that effectively “break” some of the TuringMachines so that they do not halt.

However, we prove that the Robinson tiling is robust againstsuch defects: a tile mismatch can only affect the pattern ofsquares in a finite region around the defect. More precisely:15

There exists a modification of Robinson’s tiles leading to thesame pattern of interlocking squares (see Figure 3) with theproperty that in any tiling of an L×H rectangle (width L, heightH) with d defects, the total number of complete top edges ofsquare borders of size 4n appearing in the resulting patternis at least bH/22n+1c

(bL/22n+1c − 1

)− 2d (where b·c denotes

rounding down to the nearest integer).Therefore, destroying n squares (thus “breaking” n Turing

Machines) requires O(n) defects, and each defect contributesO(1) energy. Meanwhile, Turing Machines on the remain-ing intact squares each contribute O(1) energy in the haltingcase. Thus, no matter how many defects we introduce intothe configuration of the tiling layer, the overall energy of bothlayers together will still grow quadratically with lattice size.The energy difference between the halting and non-haltingcases therefore diverges quadratically in the thermodynamiclimit. Since our system is on a 2D square lattice, this givesthe required difference in ground state energy density in thethermodynamic limit. Undecidability of the ground state en-ergy density follows immediately from undecidability of theHalting Problem. Finally, since the ground state energy densitytends to 0 in the thermodynamic limit, we can add an addi-tional identity term h1 = −α1 to the Hamiltonian (effectivelyan energy shift of −α), and choose the value of α to ensure theground state energy density is negative (and tending to 0 frombelow) in the non-halting case, but still strictly positive in thehalting case. Undecidability of the spectral gap now followsfrom our previous arguments, and establishes our main result:

Theorem 1 We can construct explicitly a dimension d, d2 × d2

matrices A, B,C,D and a rational number β so that

(i). A is Hermitian and with coefficients in Z + βZ +β√

2Z,

(ii). B,C have integer coefficients,

(iii). D is Hermitian and with coefficients in {0, 1, β}.

For each natural number n, define the local interactions of atranslationally-invariant nearest-neighbour Hamiltonian H(n)on a 2D square lattice as:

h1(n) = α(n)1 (α2(n) an algebraic computable number)hrow(n) = D (independently of n)

hcol(n) = A + β(eiπϕB + e−iπϕB† + eiπ2−|ϕ|C + e−iπ2−|ϕ|C†

).

Then:

(i). The local interaction strength is ≤ 1 (i.e.‖h1(n)‖, ‖hrow(n)‖, ‖hcol(n)‖ ≤ 1).

(ii). If the universal Turing Machine halts on input n, thenthe Hamiltonian H(n) is gapped (in the strong sensedefine earlier: it has a unique ground state and constantspectral gap). Moreover, the spectral gap γ ≥ 1.

(iii). If the universal Turing Machine does not halt on inputn, then the Hamiltonian H(n) is gapless (in the strongsense defined earlier: it has continuous spectrum).

(For a full technical account containing complete proof details,which justify the specific form of the Hamiltonian stated above,see ref. 15.) From the classic result of Turing6 that the HaltingProblem undecidable, this immediately means that the spectralgap problem is also undecidable.

DiscussionWe have proven that determining whether a quantum many-body system is gapped or gapless is an undecidable problem.Indeed, it remains undecidable even if we restrict to transla-tionally invariant, nearest-neighbour spin Hamiltonians on a2D square lattice, insist on the strongest definitions of gappedand gapless, and promise that it has a spectral gap larger atleast as large as the interaction strength in the gapped case.This result extends to other low-temperature properties, suchas ground state correlation functions. In fact, our results proveundecidability of any physical property that distinguishes aHamiltonian from a gapped system with unique product groundstate.

But what does it mean for a physical property to be undecid-able? After all, if it is physical, surely we could in principleconstruct the system in the laboratory, and measure it. A realquantum many-body system might exhibit gapped physicsor gapless critical physics, but it must exhibit some kind ofphysics!

The key to reconciling this with our results is to realise thatthe thermodynamic limit is an idealisation. A real physicalsystem necessarily has a finite size (albeit very large in the caseof a typical many-body system consisting of O(1026) atoms).Nonetheless, signatures of undecidability appear in the veryunusual finite-size behaviour of these models.

In reality, we usually probe the idealised infinite thermo-dynamic limit by studying how the system behaves as wetake larger and larger finite systems. In experimental quan-tum many-body physics, one often assumes that the systems,though finite, are so large that we are already seeing the asymp-totic behaviour. In numerical simulations of condensed mattersystems, one typically simulates finite systems of increasingsize and extrapolates the asymptotic behaviour from finite sizescaling.26 Similarly, lattice QCD calculations simulate finitelattice spacings, and extrapolate the results to the continuum.12

Renormalisation group techniques accomplish much the samething mathematically.27

However, the undecidable quantum many-body models con-structed in this work exhibit behaviour that defeats all of theseapproaches. As the system size increases, the Hamiltonian willinitially look exactly like a gapless system, with the low-energyspectrum appearing to converge to a continuum. But at somethreshold lattice size, a constant spectral gap will suddenlyappear. (Our construction can also be used to produce theopposite behaviour, with the system having a constant spectralgap up to some threshold lattice size, beyond which it abruptlyswitches to gapless physics.15) Not only can the lattice sizeat which the system switches from gapless to gapped be ar-

7

bitrarily large, the threshold at which this transition occurs isuncomputable.

This implies that we can never know whether the systemis truly gapless, or whether increasing the lattice size—evenby just one more lattice site—would reveal it to be gapped.The analogous implication also holds for all other undecid-able low-temperature properties. Any method of extrapolatingthe asymptotic behaviour from finite system sizes must failin general, or it would give a contradiction with undecidabil-ity. Could it be that the numerical evidence for the Haldaneconjecture from numerical studies, or for a Yang-Mills massgap from lattice QCD calculations, would evaporate at largersystem sizes? Could it be that, no matter how large the systemswe succeed in simulating numerically, this question can neverbe answered? Though this seems unlikely, our results showthat it is in principle possible; whilst our current techniquesare not strong enough to show this for the Haldane or Yang-Mills models, they do imply exactly this for certain artificiallyconstructed models.

Signatures of undecidability also appear in the very unusualphysics exhibited by these models. Systems with infinitelymany phases are known in connection with the quantum Halleffect, where fractal phase diagrams such as the Hoftstadterbutterfly can be obtained.28,29 The phase diagrams of the mod-els appearing in our result are more complex still. Critical andnon-critical phases are so intertwined that arbitrarily close toany non-critical point is a critical point, and vice versa. Indeed,the phase diagrams are so complex they are not just fractal,they are uncomputable. The physics of these models thereforeexhibits extreme sensitivity to perturbations. A change in theexternal parameters of the system, however small, can drive itthrough infinitely many phase transitions.

Our undecidability results imply (by a standard argument30)that there exist quantum many-body model whose basicproperties—such whether they are gapped or gapless, or

whether they have long-range ground state correlations—arenot determined by the axioms of mathematics. This raises anintriguing possibility: could some of the long-standing openproblems in theoretical physics concerning the spectral gapbe provably unsolvable? Our techniques are a long way fromshowing this for the specific models relevant to questions suchas the Haldane conjecture or the Yang-Mills gap conjecture.But they do already imply this for certain simple, if artificiallyconstructed, quantum many-body models.

AcknowledgementsTSC would like to thank IBM T. J. Watson Laboratory fortheir hospitality during many visits, and Charlie Bennett inparticular for insightful discussions about aperiodic tilings andreversible Turing Machines.

TSC, DPG and MMW thank the Isaac Newton Institutefor Mathematical Sciences, Cambridge for their hospitalityduring the programme “Mathematical Challenges in QuantumInformation”, where part of this work was carried out. TSCand DPG are both very grateful to MMW and the TechnischeUniversitat Munchen for their hospitality over summer 2014,when another part of the work was completed.

TSC is supported by the Royal Society. DPG acknowledgessupport from MINECO (grant MTM2011-26912), Comunidadde Madrid (grant QUITEMAD+-CM, ref. S2013/ICE-2801),and financial support from the John von Neumann Professor-ship program of the Technische Universtiat Munchen. DPGand MMW acknowledge support from the European CHIST-ERA project CQC (funded partially by MINECO grant PRI-PIMCHI-2011-1071).

This work was made possible through the support of grant#48322 from the John Templeton Foundation. The opinionsexpressed in this publication are those of the authors and do notnecessarily reflect the views of the John Templeton Foundation.

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