In a unified fashion, we establish Lieb-Schultz-Mattis (LSM) theorems and their generalizations i... more In a unified fashion, we establish Lieb-Schultz-Mattis (LSM) theorems and their generalizations in systems with long-range interactions. We show that, for a quantum spin chain, if the interactions decay fast enough as their ranges increase and the Hamiltonian has an anomalous symmetry, the Hamiltonian cannot have a unique gapped symmetric ground state. If the Hamiltonian contains only 2-spin interactions, these theorems hold when the interactions decay faster than 1/r 2 , with r the distance between the two interacting spins. Moreover, any pure state with an anomalous symmetry, which may not be a ground state of any natural Hamiltonian, must be long-range entangled. The symmetries we consider include on-site internal symmetries combined with lattice translation symmetries, and they can also extend to purely internal but non-on-site symmetries. Moreover, these internal symmetries can be discrete or continuous. We explore the applications of the theorems through various examples.
One of the fundamental facts of condensed matter physics is that sufficient amount of disorder al... more One of the fundamental facts of condensed matter physics is that sufficient amount of disorder always turns a Fermi liquid metal into an Anderson insulator: a compressible, but non-conducting phase of matter. Recently, topological semimetals have emerged as another way a metallic phase may be realized. In this paper we point out that, unlike ordinary metals, at least some topological semimetals are immune to localization, provided certain conditions are satisfied. We present several physical arguments, based on diagrammatic perturbation theory and Keldysh field theory, as well as decorated domain wall construction, to back up this claim.
We establish rigorous connections between quantum circuit complexity and approximate quantum erro... more We establish rigorous connections between quantum circuit complexity and approximate quantum error correction (AQEC) capability, two properties of fundamental importance to the physics and practical use of quantum many-body systems, covering both all-to-all and geometric scenarios like lattice systems in finite spatial dimensions. To this end, we introduce a type of code parameter that we call subsystem variance, which is closely related to the optimal AQEC precision. Our key finding is that if the subsystem variance is below an O(k/n) threshold then any state in the code subspace must obey certain circuit complexity lower bounds, which identify nontrivial "phases" of codes. Based on our results, we propose O(k/n) as a boundary between subspaces that should and should not count as AQEC codes. This theory of AQEC provides a versatile framework for understanding the quantum complexity and order of many-body quantum systems, offering new insights for wide-ranging physical scenarios, in particular topological order and critical quantum systems which are of outstanding importance in many-body and high energy physics. We observe from various different perspectives that roughly O(1/n) represents a common, physically significant "scaling threshold" of subsystem variance for features associated with nontrivial quantum order.
It has recently been demonstrated that it is possible to open a gap in a magnetic Weyl semimetal,... more It has recently been demonstrated that it is possible to open a gap in a magnetic Weyl semimetal, while preserving the chiral anomaly along with the charge conservation and translational symmetries, which all protect the gapless nodes in a weakly interacting semimetal. The resulting state was shown to be a nontrivial generalization of a nonabelian fractional quantum Hall liquid to three dimensions. Here we point out that a second fractional quantum Hall state exists in this case. This state has exactly the same electrical and thermal Hall responses as the first, but a distinct (fracton) topological order. Moreover, the existence of this second fractional quantum Hall state necessarily implies a gapless phase, which has identical topological response to a noninteracting Weyl semimetal, but is distinct from it. This may be viewed as a generalization (in a weaker form) of the known duality between a noninteracting two-dimensional Dirac fermion and QED3 to 3 + 1 dimensions. In addition we discuss a (3 + 1)-dimensional topologically ordered state, obtained by gapping a nodal line semimetal without breaking symmetries.
Ning Sun, ∗ Jinmin Yi, 2, ∗ Pengfei Zhang, Huitao Shen, and Hui Zhai 4 Institute for Advanced Stu... more Ning Sun, ∗ Jinmin Yi, 2, ∗ Pengfei Zhang, Huitao Shen, and Hui Zhai 4 Institute for Advanced Study, Tsinghua University, Beijing, 100084, China Department of Physics, Peking University, Beijing, 100871, China Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA Collaborative Innovation Center of Quantum Matter, Beijing, 100084, China (Dated: May 29, 2018)
In this work we design and train deep neural networks to predict topological invariants for one-d... more In this work we design and train deep neural networks to predict topological invariants for one-dimensional four-band insulators in AIII class whose topological invariant is the winding number, and two-dimensional two-band insulators in A class whose topological invariant is the Chern number. Given Hamiltonians in the momentum space as the input, neural networks can predict topological invariants for both classes with accuracy close to or higher than 90%, even for Hamiltonians whose invariants are beyond the training data set. Despite the complexity of the neural network, we find that the output of certain intermediate hidden layers resembles either the winding angle for models in AIII class or the solid angle (Berry curvature) for models in A class, indicating that neural networks essentially capture the mathematical formula of topological invariants. Our work demonstrates the ability of neural networks to predict topological invariants for complicated models with local Hamiltonians as the only input, and offers an example that even a deep neural network is understandable.
It has recently been demonstrated that it is possible to open a gap in a magnetic Weyl semimetal,... more It has recently been demonstrated that it is possible to open a gap in a magnetic Weyl semimetal, while preserving the chiral anomaly along with the charge conservation and translational symmetries, which all protect the gapless nodes in a weakly interacting semimetal. The resulting state was shown to be a nontrivial generalization of a non-Abelian fractional quantum Hall liquid to three dimensions. Here we point out that a second fractional quantum Hall state exists in this case. This state has exactly the same electrical and thermal Hall responses as the first, but a distinct (fracton) topological order. Moreover, the existence of this second fractional quantum Hall state necessarily implies a gapless phase, which has identical topological response to a noninteracting Weyl semimetal, but is distinct from it. This may be viewed as a generalization (in a weaker form) of the known duality between a noninteracting two-dimensional Dirac fermion and QED 3 to 3 + 1 dimensions. In addition we discuss a (3 + 1)-dimensional topologically ordered state, obtained by gapping a nodal line semimetal without breaking symmetries.
In a unified fashion, we establish Lieb-Schultz-Mattis (LSM) theorems and their generalizations i... more In a unified fashion, we establish Lieb-Schultz-Mattis (LSM) theorems and their generalizations in systems with long-range interactions. We show that, for a quantum spin chain, if the interactions decay fast enough as their ranges increase and the Hamiltonian has an anomalous symmetry, the Hamiltonian cannot have a unique gapped symmetric ground state. If the Hamiltonian contains only 2-spin interactions, these theorems hold when the interactions decay faster than 1/r 2 , with r the distance between the two interacting spins. Moreover, any pure state with an anomalous symmetry, which may not be a ground state of any natural Hamiltonian, must be long-range entangled. The symmetries we consider include on-site internal symmetries combined with lattice translation symmetries, and they can also extend to purely internal but non-on-site symmetries. Moreover, these internal symmetries can be discrete or continuous. We explore the applications of the theorems through various examples.
One of the fundamental facts of condensed matter physics is that sufficient amount of disorder al... more One of the fundamental facts of condensed matter physics is that sufficient amount of disorder always turns a Fermi liquid metal into an Anderson insulator: a compressible, but non-conducting phase of matter. Recently, topological semimetals have emerged as another way a metallic phase may be realized. In this paper we point out that, unlike ordinary metals, at least some topological semimetals are immune to localization, provided certain conditions are satisfied. We present several physical arguments, based on diagrammatic perturbation theory and Keldysh field theory, as well as decorated domain wall construction, to back up this claim.
We establish rigorous connections between quantum circuit complexity and approximate quantum erro... more We establish rigorous connections between quantum circuit complexity and approximate quantum error correction (AQEC) capability, two properties of fundamental importance to the physics and practical use of quantum many-body systems, covering both all-to-all and geometric scenarios like lattice systems in finite spatial dimensions. To this end, we introduce a type of code parameter that we call subsystem variance, which is closely related to the optimal AQEC precision. Our key finding is that if the subsystem variance is below an O(k/n) threshold then any state in the code subspace must obey certain circuit complexity lower bounds, which identify nontrivial "phases" of codes. Based on our results, we propose O(k/n) as a boundary between subspaces that should and should not count as AQEC codes. This theory of AQEC provides a versatile framework for understanding the quantum complexity and order of many-body quantum systems, offering new insights for wide-ranging physical scenarios, in particular topological order and critical quantum systems which are of outstanding importance in many-body and high energy physics. We observe from various different perspectives that roughly O(1/n) represents a common, physically significant "scaling threshold" of subsystem variance for features associated with nontrivial quantum order.
It has recently been demonstrated that it is possible to open a gap in a magnetic Weyl semimetal,... more It has recently been demonstrated that it is possible to open a gap in a magnetic Weyl semimetal, while preserving the chiral anomaly along with the charge conservation and translational symmetries, which all protect the gapless nodes in a weakly interacting semimetal. The resulting state was shown to be a nontrivial generalization of a nonabelian fractional quantum Hall liquid to three dimensions. Here we point out that a second fractional quantum Hall state exists in this case. This state has exactly the same electrical and thermal Hall responses as the first, but a distinct (fracton) topological order. Moreover, the existence of this second fractional quantum Hall state necessarily implies a gapless phase, which has identical topological response to a noninteracting Weyl semimetal, but is distinct from it. This may be viewed as a generalization (in a weaker form) of the known duality between a noninteracting two-dimensional Dirac fermion and QED3 to 3 + 1 dimensions. In addition we discuss a (3 + 1)-dimensional topologically ordered state, obtained by gapping a nodal line semimetal without breaking symmetries.
Ning Sun, ∗ Jinmin Yi, 2, ∗ Pengfei Zhang, Huitao Shen, and Hui Zhai 4 Institute for Advanced Stu... more Ning Sun, ∗ Jinmin Yi, 2, ∗ Pengfei Zhang, Huitao Shen, and Hui Zhai 4 Institute for Advanced Study, Tsinghua University, Beijing, 100084, China Department of Physics, Peking University, Beijing, 100871, China Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA Collaborative Innovation Center of Quantum Matter, Beijing, 100084, China (Dated: May 29, 2018)
In this work we design and train deep neural networks to predict topological invariants for one-d... more In this work we design and train deep neural networks to predict topological invariants for one-dimensional four-band insulators in AIII class whose topological invariant is the winding number, and two-dimensional two-band insulators in A class whose topological invariant is the Chern number. Given Hamiltonians in the momentum space as the input, neural networks can predict topological invariants for both classes with accuracy close to or higher than 90%, even for Hamiltonians whose invariants are beyond the training data set. Despite the complexity of the neural network, we find that the output of certain intermediate hidden layers resembles either the winding angle for models in AIII class or the solid angle (Berry curvature) for models in A class, indicating that neural networks essentially capture the mathematical formula of topological invariants. Our work demonstrates the ability of neural networks to predict topological invariants for complicated models with local Hamiltonians as the only input, and offers an example that even a deep neural network is understandable.
It has recently been demonstrated that it is possible to open a gap in a magnetic Weyl semimetal,... more It has recently been demonstrated that it is possible to open a gap in a magnetic Weyl semimetal, while preserving the chiral anomaly along with the charge conservation and translational symmetries, which all protect the gapless nodes in a weakly interacting semimetal. The resulting state was shown to be a nontrivial generalization of a non-Abelian fractional quantum Hall liquid to three dimensions. Here we point out that a second fractional quantum Hall state exists in this case. This state has exactly the same electrical and thermal Hall responses as the first, but a distinct (fracton) topological order. Moreover, the existence of this second fractional quantum Hall state necessarily implies a gapless phase, which has identical topological response to a noninteracting Weyl semimetal, but is distinct from it. This may be viewed as a generalization (in a weaker form) of the known duality between a noninteracting two-dimensional Dirac fermion and QED 3 to 3 + 1 dimensions. In addition we discuss a (3 + 1)-dimensional topologically ordered state, obtained by gapping a nodal line semimetal without breaking symmetries.
Uploads
Papers by Jinmin Yi