Unique Ground State Research Articles

Rigorous RG Algorithms and Area Laws for Low Energy Eigenstates in 1D

One of the central challenges in the study of quantum many-body systems is the complexity of simulating them on a classical computer. A recent advance (Landau et al. in Nat Phys, 2015) gave a polynomial time algorithm to compute a succinct classical description for unique ground states of gapped 1D quantum systems. Despite this progress many questions remained unsolved, including whether there exist efficient algorithms when the ground space is degenerate (and of polynomial dimension in the system size), or for the polynomially many lowest energy states, or even whether such states admit succinct classical descriptions or area laws. In this paper we give a new algorithm, based on a rigorously justified RG type transformation, for finding low energy states for 1D Hamiltonians acting on a chain of n particles. In the process we resolve some of the aforementioned open questions, including giving a polynomial time algorithm for poly(n) degenerate ground spaces and an nO(logn) algorithm for the poly(n) lowest energy states (under a mild density condition). For these classes of systems the existence of a succinct classical description and area laws were not rigorously proved before this work. The algorithms are natural and efficient, and for the case of finding unique ground states for frustration-free Hamiltonians the running time is {tilde{O}(nM(n))} , where M(n) is the time required to multiply two n × n matrices.

The gap of the area-weighted Motzkin spin chain is exponentially small

We prove that the energy gap of the model proposed by Zhang et al (2016 arXiv:1606.07795) is exponentially small in the square of the system size. In Movassagh and Shor (2016 Proc. Natl Acad. Sci. USA) a class of exactly solvable quantum spin chain models was proposed that have integer spins (s), with a nearest neighbors Hamiltonian, and a unique ground state. The ground state can be seen as a uniform superposition of all s-colored Motzkin walks. The half-chain entanglement entropy provably violates the area law by a square root factor in the system’s size (∼) for s > 1. For s = 1, the violation is logarithmic (Bravyi et al 2012 Phys. Rev. Lett. 109 207202). Moreover in Movassagh and Shor (2016 Proc. Natl Acad. Sci. USA) it was proved that the gap vanishes polynomially and is O(n−c) with .Recently, a deformation of Movassagh and Shor (2016 Proc. Natl Acad. Sci. USA), which we call ‘weighted Motzkin quantum spin chain’ was proposed Zhang et al (2016 arXiv:1606.07795). This model has a unique ground state that is a superposition of the s-colored Motzkin walks weighted by with t > 1. The most surprising feature of this model is that it violates the area law by a factor of n. Here we prove that the gap of this model is upper bounded by for t > 1 and s > 1.

Ground states and zero-temperature measures at the boundary of rotation sets

We consider a continuous dynamical system$f:X\rightarrow X$on a compact metric space$X$equipped with an$m$-dimensional continuous potential$\unicode[STIX]{x1D6F7}=(\unicode[STIX]{x1D719}_{1},\ldots ,\unicode[STIX]{x1D719}_{m}):X\rightarrow \mathbb{R}^{m}$. We study the set of ground states$GS(\unicode[STIX]{x1D6FC})$of the potential$\unicode[STIX]{x1D6FC}\cdot \unicode[STIX]{x1D6F7}$as a function of the direction vector$\unicode[STIX]{x1D6FC}\in S^{m-1}$. We show that the structure of the ground state sets is naturally related to the geometry of the generalized rotation set of$\unicode[STIX]{x1D6F7}$. In particular, for each$\unicode[STIX]{x1D6FC}$the set of rotation vectors of$GS(\unicode[STIX]{x1D6FC})$forms a non-empty, compact and connected subset of a face$F_{\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D6F7})$of the rotation set associated with$\unicode[STIX]{x1D6FC}$. Moreover, every ground state maximizes entropy among all invariant measures with rotation vectors in$F_{\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D6F7})$. We further establish the occurrence of several quite unexpected phenomena. Namely, we construct for any$m\in \mathbb{N}$examples with an exposed boundary point (that is,$F_{\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D6F7})$being a singleton) without a unique ground state. Further, we establish the possibility of a line segment face$F_{\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D6F7})$with a unique but non-ergodic ground state. Finally, we establish the possibility that the set of rotation vectors of$GS(\unicode[STIX]{x1D6FC})$is a non-trivial line segment.

Interaction-driven distinctive electronic states of artificial atoms at the ZnO interface

We have investigated the electronic states of planar quantum dots at the ZnO interface containing a few interacting electrons in an externally applied magnetic field. The electron–electron interaction effects are expected to be much stronger in this case than in traditional semiconductor quantum systems, such as in GaAs or InAs quantum dots. In order to highlight that stronger Coulomb effects in the ZnO quantum dots, we have compared the energy spectra and the magnetization in this system to those of the InAs quantum dots. We have found that in the ZnO quantum dots the signatures of stronger Coulomb interaction manifests in an unique ground state that has very different properties than the corresponding ones in the InAs dot. Our results for the magnetization also exhibits behaviors never before observed in a quantum dot for a realistic set of parameters. We have found a stronger temperature dependence and other unexpected features, such as paramagnetic-like behavior at high temperatures for a quantum-dot helium.

Of Mice and Men: In Vivo and In Vitro Studies of Primordial Germ Cell Specification.

Specification of mouse primordial germ cells (PGCs), the precursors of sperm and eggs, involves three major molecular events: repression of the somatic program, reacquisition of pluripotency, and reprogramming to a unique epigenetic ground state. Gene knockout studies in mouse models, along with global transcriptome analyses, have revealed the key signaling pathways and transcription factors essential for PGC specification. Knowledge obtained from these studies has been utilized extensively to develop robust in vitro PGC induction models not only in mice but also in humans. These models have, in turn, formed the basis for a detailed understanding of the signaling pathways and epigenetic dynamics during in vivo PGC specification and development. Recapitulation of human PGC specification in culture is of tremendous significance for understanding the mechanisms of human germ cell development in normal and disease states and has implications for addressing germ-cell-based causes of infertility.

Quantum chaos and holographic tensor models

A class of tensor models were recently outlined as potentially calculable examples of holography: their perturbative large-N behavior is similar to the Sachdev-Ye-Kitaev (SYK) model, but they are fully quantum mechanical (in the sense that there is no quenched disorder averaging). These facts make them intriguing tentative models for quantum black holes. In this note, we explicitly diagonalize the simplest non-trivial Gurau-Witten tensor model and study its spectral and late-time properties. We find parallels to (a single sample of) SYK where some of these features were recently attributed to random matrix behavior and quantum chaos. In particular, the spectral form factor exhibits a dip-ramp-plateau structure after a running time average, in qualitative agreement with SYK. But we also observe that even though the spectrum has a unique ground state, it has a huge (quasi-?)degeneracy of intermediate energy states, not seen in SYK. If one ignores the delta function due to the degeneracies however, there is level repulsion in the unfolded spacing distribution hinting chaos. Furthermore, there are gaps in the spectrum. The system also has a spectral mirror symmetry which we trace back to the presence of a unitary operator with which the Hamiltonian anticommutes. We use it to argue that to the extent that the model exhibits random matrix behavior, it is controlled not by the Dyson ensembles, but by the BDI (chiral orthogonal) class in the Altland-Zirnbauer classification.

Superconductivity due to soft super-symmetry breaking

Revisiting the super-symmetric dyons in N = 2 super-symmetric theory and analyzing the possible soft breaking of N = 2 super-symmetric Yang Mills theory to N = 0 by making the dynamically generated mass scale ∧ a function of dilation spurion, it has been demonstrated that the scalar and auxiliary components of pre-potential, constructed in terms of dilation, are frozen to be constant to generate soft breaking of N = 2 theory and it has been shown that, as soon as these soft breaking terms are turned on, monopole condensation appears and we get a unique ground state and the superconducting phase. It is also shown that in this soft breaking of N = 2 super-symmetry, the superconductivity phase occurs due to condensation of monopoles only and the dyons do not condensate near the real u -plane.

Entanglement and correlation functions of the quantum Motzkin spin-chain

We present exact results on the exactly solvable spin chain of Bravyi et al. [Phys. Rev. Lett. 109, 207202 (2012)]. This model is a spin one chain and has a Hamiltonian that is local and translationally invariant in the bulk. It has a unique (frustration free) ground state with an energy gap that is polynomially small in the system’s size (2n). The half-chain entanglement entropy of the ground state is 12logn+const.[Bravyi et al., Phys. Rev. Lett. 109, 207202 (2012)]. Here we first write the Hamiltonian in the standard spin-basis representation. We prove that at zero temperature, the magnetization is along the z-direction, i.e., ⟨sx⟩=⟨sy⟩=0 (everywhere on the chain). We then analytically calculate ⟨sz⟩ and the two-point correlation functions of sz. By analytically diagonalizing the reduced density matrices, we calculate the Schmidt rank, von Neumann, and Rényi entanglement entropies for the following: 1. Any partition of the chain into two pieces (not necessarily in the middle) and 2. L consecutive spins centered in the middle. Further, we identify entanglement Hamiltonians (Eqs. (49) and (59)). We prove a small lemma (Lemma (1)) on the combinatorics of lattice paths using the reflection principle to relate and calculate the Motzkin walk “height” to spin expected values. We also calculate the, closely related (scaled), correlation functions of Brownian excursions. The known features of this model are summarized in a table in Sec. I.

Neutron Scattering Study in Breathing Pyrochlore Antiferromagnet Ba3Yb2Zn5O11

Inelastic neutron scattering (INS) study on breathing pyrochlore antiferromagnet Ba3Yb2Zn5O11 is presented. Observed crystalline electric field (CEF) excitations are explained by a Hamiltonian of Kramers ion Yb3+ of which the local symmetry exhibits C3v point group symmetry. The magnetic susceptibility is consistently reproduced by the energy scheme of the CEF excitations. The INS spectra in the low-energy range are quantitatively explained by spin-1/2 single-tetrahedron model having XXZ anisotropy and Dzyaloshinskii-Moriya interaction. This model has a two-fold degeneracy of the lowest-energy state per tetrahedron and well reproduces the bulk properties at T ≥ 0.5K. At lower temperatures, however, we observe a broad maximum in the heat capacity around 63 mK, demonstrating that a unique quantum ground state is selected due to extra perturbations.

Local reversibility and entanglement structure of many-body ground states

The low-temperature physics of quantum many-body systems is largely governed by the structure of their ground states. Minimizing the energy of local interactions, ground states often reflect strong properties of locality such as the area law for entanglement entropy and the exponential decay of correlations between spatially separated observables. Here, we present a novel characterization of quantum states, which we call ‘local reversibility’. It characterizes the type of operations that are needed to reverse the action of a general disturbance on the state. We prove that unique ground states of gapped local Hamiltonian are locally reversible. This way, we identify new universal features of many-body ground states, which cannot be derived from the aforementioned properties. We use local reversibility to distinguish between states enjoying microscopic and macroscopic quantum phenomena. To demonstrate the potential of our approach, we prove specific properties of ground states, which are relevant both to critical and non-critical theories.

A Class of Asymmetric Gapped Hamiltonians on Quantum Spin Chains and its Characterization III

In this paper, we consider classification problem of asymmetric gapped Hamiltonians, which are given as the non-degenerate part of the Hamiltonians introduced in [O1]. We consider the $C^1$-classification, which takes into account the effect of boundaries. We show that the left and right degeneracies of edge ground states are the complete invariant. As a corollary, we consider the bulk-classification problem. We study Hamiltonians that1.are given by translation invariant finite range interactions, 2.are gapped in the bulk, 3.are frustration-free, 4.have uniformly bounded ground state degeneracy on finite intervals, and 5.have a unique bulk ground state. We show that for the bulk-classification, any such Hamiltonians are equivalent.

Sharp asymptotics for the minimal mass blow up solution of the critical gKdV equation

Let S be a minimal mass blow up solution of the critical generalized KdV equation as constructed in [25]. We prove both time and space sharp asymptotics for S close to the blow up time. Let Q be the unique ground state of (gKdV), satisfying Q″+Q5=Q.First, we show that there exist universal smooth profiles Qk∈S(R) (with Q0=Q) and a constant c0∈R such that, fixing the blow up time at t=0 and appropriate scaling and translation parameters, S satisfies, for any m⩾0,∂xmS(t)−∑k=0[m/2]1t12+m−2kQk(m−k)(⋅+1tt+c0)→0 in L2as t↓0. Second, we prove that, for 0<t≪1, x⩽−1t−1,S(t,x)∼−12‖Q‖L1|x|−3/2, and related bounds for the derivatives of S(t) of any order. We also prove ∫RS(t,x)dx=0.

Weak Interactions in a Background of a Uniform Magnetic Field. A Mathematical Model for the Inverse β Decay. I.

In this paper we consider a mathematical model for the inverse β decay in a uniform magnetic field. With this model we associate a Hamiltonian with cutoffs in an appropriate Fock space. No infrared regularization is assumed. The Hamiltonian is self-adjoint and has a unique ground state. We study the essential spectrum and determine the spectrum. The coupling constant is supposed sufficiently small.

{Fe(NO)(2)(9) Dinitrosyl Iron Complex Acting as a Vehicle for the NO Radical.

To carry and deliver nitric oxide with a controlled redox state and rate is crucial for its pharmaceutical/medicinal applications. In this study, the capability of cationic {Fe(NO)2}9 dinitrosyl iron complexes (DNICs) [(RDDB)Fe(NO)2]+ (R = Me, Et, Iso; RDDB = N,N'-bis(2,6-dialkylphenyl)-1,4-diaza-2,3-dimethyl-1,3-butadiene) carrying nearly unperturbed nitric oxide radical to form [(RDDB)Fe(NO)2(•NO)]+ was demonstrated and characterized by IR, UV-vis, EPR, NMR, and single-crystal X-ray diffractions. The unique triplet ground state of [(RDDB)Fe(NO)2(•NO)]+ results from the ferromagnetic coupling between two strictly orthogonal orbitals, one from Fe dz2 and the other a π*op orbital of a unique bent axial NO ligand, which is responsible for the growth of a half-field transition (ΔMS = 2) from 70 to 4 K in variable-temperature EPR measurements. Consistent with the NO radical character of coordinated axial NO ligand in complex [(MeDDB)Fe(NO)2(•NO)]+, the simple addition of MeCN/H2O into CH2Cl2 solution of complexes [(RDDB)Fe(NO)2(•NO)]+ at 25 °C released NO as a neutral radical, as demonstrated by the formation of [S5Fe(NO)2]- from [S5Fe(μ-S)2FeS5]2-.

Emergence of string valence-bond-solid state in the frustrated J1−J2 transverse field Ising model on the square lattice

We investigate the ground state nature of the transverse field Ising model on the $J_1-J_2$ square lattice at the highly frustrated point $J_2/J_1=0.5$. At zero field, the model has an exponentially large degenerate classical ground state, which can be affected by quantum fluctuations for non-zero field toward a unique quantum ground state. We consider two types of quantum fluctuations, harmonic ones by using linear spin wave theory (LSWT) with single-spin flip excitations above a long range magnetically ordered background and anharmonic fluctuations, by employing a cluster-operator approach (COA) with multi-spin cluster type fluctuations above a non-magnetic cluster ordered background. Our findings reveal that the harmonic fluctuations of LSWT fail to lift the extensive degeneracy as well as signaling a violation of the Hellmann-Feynman theorem. However, the string-type anharmonic fluctuations of COA are able to lift the degeneracy toward a string-valence bond solid (VBS) state, which is obtained from an effective theory consistent with the Hellmann-Feynman theorem as well. Our results are further confirmed by implementing numerical tree tensor network simulation. The emergent non-magnetic string-VBS phase is gapped and breaks lattice rotational symmetry with only two-fold degeneracy, which bears a continuous quantum phase transition at $\Gamma/J_1 \cong 0.50$ to the quantum paramagnet phase of high fields. The critical behavior is characterized by $\nu \cong 1.0$ and $\gamma \cong 0.33$ exponents.

Lattice effects on Laughlin wave functions and parent Hamiltonians

We investigate lattice effects on wave functions that are lattice analogues of bosonic and fermionic Laughlin wave functions with number of particles per flux $\nu=1/q$ in the Landau levels. These wave functions are defined analytically on lattices with $\mu$ particles per lattice site, where $\mu$ may be different than $\nu$. We give numerical evidence that these states have the same topological properties as the corresponding continuum Laughlin states for different values of $q$ and for different fillings $\mu$. These states define, in particular, particle-hole symmetric lattice Fractional Quantum Hall states when the lattice is half-filled. On the square lattice it is observed that for $q\leq 4$ this particle-hole symmetric state displays the topological properties of the continuum Laughlin state at filling fraction $\nu=1/q$, while for larger $q$ there is a transition towards long-range ordered anti-ferromagnets. This effect does not persist if the lattice is deformed from a square to a triangular lattice, or on the Kagome lattice, in which case the topological properties of the state are recovered. We then show that changing the number of particles while keeping the expression of these wave functions identical gives rise to edge states that have the same correlations in the bulk as the reference lattice Laughlin states but a different density at the edge. We derive an exact parent Hamiltonian for which all these edge states are ground states with different number of particles. In addition this Hamiltonian admits the reference lattice Laughlin state as its unique ground state of filling factor $1/q$. Parent Hamiltonians are also derived for the lattice Laughlin states at other fillings of the lattice, when $\mu\leq 1/q$ or $\mu\geq 1-1/q$ and when $q=4$ also at half-filling.

Construction of a minimal mass blow up solution of the modified Benjamin–Ono equation

We construct a minimal mass blow up solution of the modified Benjamin–Ono equation (mBO) mBO $$\begin{aligned} u_{t}+(u^3-D^1 u)_{x}=0, \end{aligned}$$ which is a standard mass critical dispersive model. Let $$Q\in H^{\frac{1}{2}}$$ , $$Q>0$$ , be the unique ground state solution of $$D^1 Q +Q=Q^3$$ , constructed using variational arguments by Weinstein (Commun. Part. Differ. Equations 12:1133–1173, 1987a; J. Differ. Equations 69:192–203, 1987b) and Albert et al. (Proc. R. Soc. Lond. A 453:1233–1260, 1997), and whose uniqueness was recently proved by Frank and Lenzmann (Acta Math. 210:261–318, 2013). We show the existence of a solution S of (mBO) satisfying $$\Vert S \Vert _{L^2}=\Vert Q\Vert _{L^2}$$ and $$\begin{aligned} S(t)-\frac{1}{\lambda ^{\frac{1}{2}}(t)} Q\left( \frac{\cdot - x(t)}{\lambda (t)}\right) \rightarrow 0\quad \text{ in } H^{\frac{1}{2}}(\mathbb R) \text{ as } t\downarrow 0, \end{aligned}$$ where $$\begin{aligned} \lambda (t)\sim t,\quad x(t) \sim -|\ln t| \quad \hbox {and}\quad \Vert S(t)\Vert _{\dot{H}^{\frac{1}{2}}} \sim t^{-\frac{1}{2}}\Vert Q\Vert _{\dot{H}^{\frac{1}{2}}} \quad \hbox {as} t\downarrow 0. \end{aligned}$$ This existence result is analogous to the one obtained by Martel et al. (J. Eur. Math. Soc. 17:1855–1925, 2015) for the mass critical generalized Korteweg-de Vries equation. However, in contrast with the (gKdV) equation, for which the blow up problem is now well-understood in a neighborhood of the ground state, S is the first example of blow up solution for (mBO). The proof involves the construction of a blow up profile, energy estimates as well as refined localization arguments, developed in the context of Benjamin–Ono type equations by Kenig et al. (Ann. Inst. H. Poincare Anal. Non Lin. 28:853–887, 2011). Due to the lack of information on the (mBO) flow around the ground state, the energy estimates have to be considerably sharpened in the present paper.

Preparing topologically ordered states by Hamiltonian interpolation

We study the preparation of topologically ordered states by interpolating between an initial Hamiltonian with a unique product ground state and a Hamiltonian with a topologically degenerate ground state space. By simulating the dynamics for small systems, we numerically observe a certain stability of the prepared state as a function of the initial Hamiltonian. For small systems or long interpolation times, we argue that the resulting state can be identified by computing suitable effective Hamiltonians. For effective anyon models, this analysis singles out the relevant physical processes and extends the study of the splitting of the topological degeneracy by Bonderson (2009 Phys. Rev. Lett. 103 110403). We illustrate our findings using Kitaev’s Majorana chain, effective anyon chains, the toric code and Levin–Wen string-net models.

Quantum rotor theory of systems of spin-2 bosons

We consider quantum phases of tightly-confined spin-2 bosons in an external field under the presence of rotationally-invariant interactions. Generalizing previous treatments, we show how this system can be mapped onto a quantum rotor model. Within the rotor framework, low-energy excitations about fragmented states, which cannot be accessed within standard Bogoliubov theory, can be obtained. In the spatially extended system in the thermodynamic limit there exists a mean-field ground state degeneracy between a family of nematic states for appropriate interaction parameters. It has been established that quantum fluctuations lift this degeneracy through the mechanism of order-by-disorder and select either a uniaxial or square-biaxial ground state. On the other hand, in the full quantum treatment of the analogous single-spatial mode problem with finite particle number it is known that, due to symmetry restoring fluctuations, there is a unique ground state across the entire nematic region of the phase diagram. Within the established rotor framework we investigate the possible quantum phases under the presence of a quadratic Zeeman field, a problem which has previously received little attention. By investigating wave function overlaps we do not find any signatures of the order-by-disorder phenomenon which is present in the continuum case. Motivated by this we consider an alternative external potential which breaks less symmetry than the quadratic Zeeman field. For this case we do find the phenomenon of order-by-disorder in the fully quantum system. This is established within the rotor framework and with exact diagonalization.

Low-energy excitations and ground-state selection in the quantum breathing pyrochlore antiferromagnetBa3Yb2Zn5O11

We study low energy excitations in the quantum breathing pyrochlore antiferromagnet Ba$_3$Yb$_2$Zn$_5$O$_{11}$ by combination of inelastic neutron scattering (INS) and thermodynamical properties measurements. The INS spectra are quantitatively explained by spin-1/2 single-tetrahedron model having $XXZ$ anisotropy and Dzyaloshinskii-Moriya interaction. This model has a two-fold degeneracy of the lowest-energy state per tetrahedron and well reproduces the magnetization curve at 0.5 K and heat capacity above 1.5 K. At lower temperatures, however, we observe a broad maximum in the heat capacity around 63 mK, demonstrating that a unique quantum ground state is selected due to extra perturbations with energy scale smaller than the instrumental resolution of INS.