3,091 research outputs found
Structure of the thermodynamic arrow of time in classical and quantum theories
In this work we analyse the structure of the thermodynamic arrow of time,
defined by transformations that leave the thermal equilibrium state unchanged,
in classical (incoherent) and quantum (coherent) regimes. We note that in the
infinite-temperature limit the thermodynamic ordering of states in both regimes
exhibits a lattice structure. This means that when energy does not matter and
the only thermodynamic resource is given by information, the thermodynamic
arrow of time has a very specific structure. Namely, for any two states at
present there exists a unique state in the past consistent with them and with
all possible joint pasts. Similarly, there also exists a unique state in the
future consistent with those states and with all possible joint futures. We
also show that the lattice structure in the classical regime is broken at
finite temperatures, i.e., when energy is a relevant thermodynamic resource.
Surprisingly, however, we prove that in the simplest quantum scenario of a
two-dimensional system, this structure is preserved at finite temperatures. We
provide the physical interpretation of these results by introducing and
analysing the history erasure process, and point out that quantum coherence may
be a necessary resource for the existence of an optimal erasure process.Comment: 14 pages, 10 figures. Published version. Expanded discussion and a
new section on history erasure process adde
Tensor product representation of topological ordered phase: necessary symmetry conditions
The tensor product representation of quantum states leads to a promising
variational approach to study quantum phase and quantum phase transitions,
especially topological ordered phases which are impossible to handle with
conventional methods due to their long range entanglement. However, an
important issue arises when we use tensor product states (TPS) as variational
states to find the ground state of a Hamiltonian: can arbitrary variations in
the tensors that represent ground state of a Hamiltonian be induced by local
perturbations to the Hamiltonian? Starting from a tensor product state which is
the exact ground state of a Hamiltonian with topological order,
we show that, surprisingly, not all variations of the tensors correspond to the
variation of the ground state caused by local perturbations of the Hamiltonian.
Even in the absence of any symmetry requirement of the perturbed Hamiltonian,
one necessary condition for the variations of the tensors to be physical is
that they respect certain symmetry. We support this claim by
calculating explicitly the change in topological entanglement entropy with
different variations in the tensors. This finding will provide important
guidance to numerical variational study of topological phase and phase
transitions. It is also a crucial step in using TPS to study universal
properties of a quantum phase and its topological order.Comment: 10 pages, 6 figure
The parameterised complexity of counting connected subgraphs and graph motifs
We introduce a family of parameterised counting problems on graphs, p-#Induced Subgraph With Property(Φ), which generalises a number of problems which have previously been studied. This paper focuses on the case in which Φ defines a family of graphs whose edge-minimal elements all have bounded treewidth; this includes the special case in which Φ describes the property of being connected. We show that exactly counting the number of connected induced k-vertex subgraphs in an n-vertex graph is #W[1]-hard, but on the other hand there exists an FPTRAS for the problem; more generally, we show that there exists an FPTRAS for p-#Induced Subgraph With Property(Φ) whenever Φ is monotone and all the minimal graphs satisfying Φ have bounded treewidth. We then apply these results to a counting version of the Graph Motif problem
Noncommutative geometry for three-dimensional topological insulators
We generalize the noncommutative relations obeyed by the guiding centers in
the two-dimensional quantum Hall effect to those obeyed by the projected
position operators in three-dimensional (3D) topological band insulators. The
noncommutativity in 3D space is tied to the integral over the 3D Brillouin zone
of a Chern-Simons invariant in momentum-space. We provide an example of a model
on the cubic lattice for which the chiral symmetry guarantees a macroscopic
number of zero-energy modes that form a perfectly flat band. This lattice model
realizes a chiral 3D noncommutative geometry. Finally, we find conditions on
the density-density structure factors that lead to a gapped 3D fractional
chiral topological insulator within Feynman's single-mode approximation.Comment: 41 pages, 3 figure
Locality Estimates for Quantum Spin Systems
We review some recent results that express or rely on the locality properties
of the dynamics of quantum spin systems. In particular, we present a slightly
sharper version of the recently obtained Lieb-Robinson bound on the group
velocity for such systems on a large class of metric graphs. Using this bound
we provide expressions of the quasi-locality of the dynamics in various forms,
present a proof of the Exponential Clustering Theorem, and discuss a
multi-dimensional Lieb-Schultz-Mattis Theorem.Comment: Contribution for the proceedings of ICMP XV, Rio de Janeiro, 200
Weak embeddings of posets to the Boolean lattice
The goal of this paper is to prove that several variants of deciding whether
a poset can be (weakly) embedded into a small Boolean lattice, or to a few
consecutive levels of a Boolean lattice, are NP-complete, answering a question
of Griggs and of Patkos. As an equivalent reformulation of one of these
problems, we also derive that it is NP-complete to decide whether a given graph
can be embedded to the two middle levels of some hypercube
Anomaly Manifestation of Lieb-Schultz-Mattis Theorem and Topological Phases
The Lieb-Schultz-Mattis (LSM) theorem dictates that emergent low-energy
states from a lattice model cannot be a trivial symmetric insulator if the
filling per unit cell is not integral and if the lattice translation symmetry
and particle number conservation are strictly imposed. In this paper, we
compare the one-dimensional gapless states enforced by the LSM theorem and the
boundaries of one-higher dimensional strong symmetry-protected topological
(SPT) phases from the perspective of quantum anomalies. We first note that,
they can be both described by the same low-energy effective field theory with
the same effective symmetry realizations on low-energy modes, wherein
non-on-site lattice translation symmetry is encoded as if it is a local
symmetry. In spite of the identical form of the low-energy effective field
theories, we show that the quantum anomalies of the theories play different
roles in the two systems. In particular, We find that the chiral anomaly is
equivalent to the LSM theorem, whereas there is another anomaly, which is not
related to the LSM theorem but is intrinsic to the SPT states. As an
application, we extend the conventional LSM theorem to multiple-charge
multiple-species problems and construct several exotic symmetric insulators. We
also find that the (3+1)d chiral anomaly provides only the perturbative
stability of the gapless-ness local in the parameter space.Comment: 14 + 3 pages, 1 figure. (The first two authors contributed equally to
the work.
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