The spectral gap the energy difference between the ground state and first excited stateis central to quantum manybody physics. This research area is a focal topic that transcends traditional discipline boundaries, unifying the otherwise disparate fields of condensed matter physics. Undecidability of the spectral gap full version authors. This is true even with the promise that each hamiltonian is either. The spectral gap the energy difference between the ground state and first excited state is central to quantum manybody physics. We consider a square distance matrix which arises from a preconditioned jacobian matrix for the numerical computation of the cahnhilliard problem. The difference between the largest eigenvalue and the nextlargest of a graph laplacian equivalently, of the random walk markov chain on the graph is the spectral gap, related to the cheeger cons. A spectral gap precludes lowdimensional embeddings assaf naor abstract. A small spectral gap the energy needed to transfer an electron from a lowenergy state to an excited state is the central property of semiconductors. Lvzhou chen university of chicago spectral gap of stable commutator length ams sectional meeting 7 11 a new proof of spectral gap theorem proof continued. Rayleing quotient and rayleingritz theorem as a particular case. Specifically, we construct families of translationallyinvariant, nearestneighbour hamiltonians on a 2d square lattice of dlevel quantum systems d constant, for which determining whether the system is gapped or gapless is an undecidable problem. The spectral gap the difference in energy between the ground state and the first excited state is of central importance to quantum manybody physics.
The spectral gap problemwhether the hamiltonian of a quantum manybody problem is gapped or gaplessis rigorously proved to be undecidable. Experimental observation of spectral gap in microwave ndisk systems s. Paradox at the heart of mathematics makes physics problem. The spectral gap the energy difference between the ground state and first excited state of a systemis central to quantum manybody physics. Wolf the spectral gap the difference in energy between the ground state and the first excited stateis one of the most important prop erties of a quantum manybody system. On the spectral gap of a square distance matrix springerlink.
We will see a connection between spectral gaps and signrank. In addition, devices such as photonic crystals, wave guides, solar cells and detectors require a preknowledge of the refractive index and energy gap. Add to your lists download to your calendar using vcal. Spectral gap and coercivity estimates for linearized. In the simplest case of nearestneighbour frustrationfree qubit interactions, there is a complete classification. Spectral analysis of signalspetre stoica and randolph moses p. The spectral gap is one of the most important physical properties of a quantum manybody system, determining much of its lowenergy physics. Undecidability of the spectral gap toby cubitt, david perezgarcia, and michael m. Spectral gap and logarithmic sobolev inequality for kawasaki. The spectral gap problem determining whether the energy spectrum of a system has an energy gap above ground state, or if there is a continuous range of lowenergy excitations pervades quantum manybody physics. If you have a question about this talk, please contact katarzyna macieszczak. There is a vast literature on poincar e inequalities, with many examples and. The closer those two get, the slower you have to go. This is true even with the promise that each hamiltonian is.
Apr 20, 2004 we say that a real function on the real line has a spectral gap if its fourier transform is zero in a neighborhood of the origin. A quantum spin system is said to be gapped if the spectral gap can be uniformly bounded from below by a quantity. In each section, we introduce the results, explain the ideas and present a large number of examples which should be considered as a critical part of the paper in illustration of the results. Ask an unbounded question, get an uncomputable answer. This is a recent result of bourgain and gamburd building on earlier work of gamburd,jakobson,and sarnak 1,6. Just to be clear, the title to our nature article is undecidability of the spectral gap. Given a unitary representation \\pi,\mathcalh\ of g, we study spectral properties of the operator. Experimental observation of spectral gap in microwave disk. It is known that functions with a spectral gap oscillate. Entropy jumps in the presence of a spectral gap keith ball.
A number of famous longstanding open problems in mathematical physics concern spectral gaps of particular manybody models, such as the haldane conjecture for the 1d heisenberg chain, or. Undecidability of the spectral gap full version core. A walk will mix quickly if theres a lot connected to everything else. History of computing theory, quantum computation theory. Many challenging open problems, such as the haldane. This conjecture is referred to as the spectral gap conjecture and is a question in 14,15. A spectral gap property for random walks under unitary. On the other extreme, for two or higher dimensional.
Namely, one cannot construct an algorithm to determine whether a translationally invariant spin lattice system is. Energy gaprefractive index relations in semiconductors an. An article published in nature yesterday proves that finding the spectral gap of a material based on a complete quantum level description of the material is undecidable in the turing sense one of the authors is quoted from a more philosophical perspective, they also challenge the reductionists point of view, as the insurmountable difficulty lies precisely in the derivation of. In quantum mechanics, the spectral gap of a system is the energy difference between its ground state and its first excited state. Many challenging open problems, such as the haldane conjecture, the question of the existence of gapped topological spin liquid phases, and the yangmills gap conjecture, concern spectral. Many challenging open problems, such as the haldane conjecture, existence of gapped topological spin liquid phases, or the yangmills gap conjecture, concern spectral gaps. Wolfz5 1department of computer science, university college london, gower street, london wc1e 6bt, united kingdom 2damtp, university of cambridge, centre for mathematical sciences, wilberforce road, cambridge cb3 0wa, united kingdom. Johannes bausch, toby cubitt, angelo lucia, david perezgarcia download pdf. In mathematics, the spectral gap is the difference between the moduli of the two largest eigenvalues of a matrix or operator. Specifically, we construct families of translationallyinvariant. Once we have this, it is relatively easy to lift it to undecidability of the spectral gap. The conjecture is only known for ntuples which have,up to conjugacy in su2, all matrix entries of all. The mass gap is the spectral gap between the vacuum and the lightest particle. Assistant vice president of production and manufacturing.
We discuss lower and upper estimates for the spectral gap of the laplace operator on a finite compact connected metric graph. Full text also available in the acm digital library as pdf html digital edition. More precisely, we give a reduction from the ground state energy density problem to the spectral gap problem. Markov chain monte carlo, mixing, and the spectral gap. Paradox at the heart of mathematics makes physics problem unanswerable this ought. Undecidability of the spectral gap school of physics and.
In communication engineering, such functions are called highpass signals. The signrank of a boolean matrix is the minimum dimension of real space in which the matrix can be realized as a pointhalfspace incidence matrix. In particular, it determines the phase diagram of the system, with quantum phase transitions occurring at critical points, where the spectral gap vanishes. Various theorems relate this difference to other properties of the system. These include the original version for approximating the permanent, estimating volume of a convex body, counting total orderings of a partially ordered set, etc. Many challenging open problems, such as the haldane conjecture, the question of the existence of gapped topological spin liquid phases, and the yangmills gap conjecture, concern spectral gaps. Undecidability of the spectral gap in one dimension. Lecture 10 conductance and the spectral gap wednesday, september 8 many of the early results in studying mixing times were derived by geometric methods. Ill assume you refer to the results reported in undecidability of the spectral gap by t.
The center for theory of quantum matter ctqm conducts theoretical physics research focused on macroscopic quantum matter. Conceptually, the process x t may be described as follows. This is the same sense in which the halting problem is undecidable. Undecidability of the spectral gap higgs centre for. Undecidability of the spectral gap quantum field theory. A recent example is the proof of the undecidability of the spectral gap 30. A short version of this paper was published in nature cpw15a. Nature just published a paper by cubitt, perezgarcia and wolf titled undecidability of the spectral gap, there is an extended version on arxiv which is 146 pages long. Wolfz5 1department of computer science, university college london, gower street, london wc1e 6ea, united kingdom 2damtp, university of cambridge, centre for mathematical sciences, wilberforce road, cambridge cb3 0wa, united kingdom. The spectral gap problem is axiomatically independent. Pdf on the spectral gap and the automorphism group of. The main novelty is an idea based on electric network reduction.
It determines the phase diagram at low temperature, with quantum phase transitions and critical phenomena occurring when the gap vanishes. The spectral gap problem is algorithmically undecidable. The spectral gap problem has been proven to be undecidable. Aug 30, 2010 a well known conjecture asserts that on any graph the random walk and the interchange process have the same spectral gap. A measure satisfying a poincar e inequality is also said to have spectral gap. The spectral gap the energy difference between the ground state and first excited. We show that the spectral gap problem is undecidable. We now state a result bounding the pmixing times in terms of the spectral gap. On the spectral gap for laplacians on metric graphs. The title a paradox at the heart of mathematics makes a physics problem unanswerable is that of the nature news article, written by davide castelvecci though i dont know if he wrote the title.
Dr johannes bausch, damtp, university of cambridge. A spectral gap theorem in sud article in journal of the european mathematical society 145 august 2011 with 24 reads how we measure reads. Undecidability of the spectral gap short version arxiv. If the graph is not regular then, in general, there is no simple or useful relation between the eigenvalue of the adjacency matrix and the eigenvalues of the laplacian. Undecidability of the spectral gap eprints complutense. Undecidability of the spectral gap toby cubitt,1 david perezgarcia,2 and michael m. But the undecidability at infinity means that even if the spectral gap is known for a certain finitesize lattice, it could change abruptly from gapless to gapped or vice versa when. Proof of aldous spectral gap conjecture microsoft research. Why some physicists are excited about the undecidability. The spectral gapthe energy difference between the ground state and first excited state of a systemis central to quantum manybody physics. Existence of stein kernels under a spectral gap, and. They cannot be restricted by axiomatic systems subject to godel incompleteness. The behaviour of the spectral gap is intimately related to the phase diagram of a quantum manybody system, with quantum phase transitions occurring at critical points where the gap vanishes.
Related properties, like ergodicity roughly, the equivalence between averages over time and averages over the state space in a markov chain, also fall under the umbrella of mixing but we will not address them. We then show how to transform the halting problem into a question about ground state energy densities. In particular, it determines the phase diagram of the system, with quantum phase transitions occurring where the spectral gap vanishes. The spectral gap of a random subgraph of a graph fan chung. Spectral gaps and geometric representations simons. Zworski 4 1 fachbereich physik, philippsuniversit at marburg, renthof 5, 35032 marburg, germany.
Supplementary discussion toby cubitt 1, david perezgarciay3,4, and michael m. If the gaps get closer like e n where n is the size of some problem, then you need exponentially more time to solve the problem and youre no better off than a classical computer. Matrix product operators and states nphardness and undecidability m. Does the undecidability of spectral gaps imply that we.
Estimation of spectral gap for elliptic operators 1241 then go to the full line section 3 and nally studying the general case section 4. We prove strict negativity of all but one associated eigenvalues. The spectral gap b ound will b e used in sections 4 and 5 to achiev e motion lower bounds through lemma 1. Determining whether a finite set of upper triangular. Undecidability of the spectral gap in one dimension in the simplest case of nearestneighbour frustrationfree qubit interactions, there is a complete classification. Domingues and others published evidence of nonexistence of a spectralgap in turbulent data measured above rondonia, brazil. That depends on the spectral gap between the ground state and the first excited state a nonsolution.
The minimum possible energy level is called the ground state energy, and the associated states are called ground states. International weekly journal of science, issn 00280836, vol. Wolfz3 1damtp, university of cambridge, centre for mathematical sciences, wilberforce road. Spectral gap for the interchange exclusion process on a. Introduction the perronfrobenius theorem per07, fro08 asserts that a real square matrix with strictly positive entries has a spectral gap, i.
The spectral gap of a quantum manybody hamiltonian plays a crucial role in determining its physical properties. The spectral gap the energy difference between the ground state and first excited state of a system is central to quantum manybody physics. A notion of standard vertex conditions for beam operara tors the fourth derivative on metric graphs is presented, and the spectral gap the. Undecidability of the spectral gap toby cubitt 1,2, david perezgarciay3,4, and michael m. A hamiltonian with a spectral gap is called a gapped hamiltonian, and those that do not are called gapless in solidstate physics, the most important spectral gap is for the. Spectral gap and coercivity estimates for linearized boltzmann collision operators without angular cutoff clement mouhota. On the other extreme, for two or higher dimensional models with nearestneighbour interactions this problem can be reduced to the halting problem, and it is therefore undecidable. This little code is intended to compute numerically the lowest eigenvalue of the discrete laplacian of a finite unoriented graph this is quantity which controls, in particular, the rate of convergence of the simple random walk. The spectral gap problem illuminates the metamathematical nature of quantum systems. A spectral gap or eigengap of a selfadjoint operator is a closed real interval to which no eigenvalues belongs. On the spectral gap for networks of beams diva portal. Chapter 1 introduction in a 2015 paper by cubitt et al.
These and other problems are particular cases of the general spectral gap. Lloyd, on the uncomputability of the spectral gap, arxiv. Spectral gaps of matrices are related to many basic properties, like mixing times, expansion, isoperimetry and more. The aim of this short note is to clarify some of the claims made in the comparison made in s. The most important step in proving undecidability of the spectral gap is to prove undecidability of another relevant quantity. Richthammer, in which we prove the conjecture using a recursive strategy. It determines the phase diagram at low temperature, with quantum phase transitions and critical phenomena occurring when the gap. We establish the spectral gap property for dense subgroups generated by algebraic elements in any compact simple lie group, generalizing earlier results of. Evidence and interpretation of a spectral gap in the turbulent. Undecidability of the spectral gap problem isaac newton. We explain the spectral gap problem, its importance for physics and possible consequences of this exciting new result. The spectral gap problem consist in deciding, given a local interaction, whether the corresponding translationally invariant hamiltonian on a lattice has a spectral gap independent of the system size or not.