In this work we propose a series-expansion thermal tensor network (SETTN) approach for efficient simulations of quantum lattice models. This continuous-time SETTN method is based on the numerically exact Taylor series expansion of equilibrium density operator $e^{-βH}$ (with $H$ the total Hamiltonian and $β$ the imaginary time), and is thus Trotter-error free. We discover, through simulating XXZ spin chain and square-lattice quantum Ising models, that not only the Hamiltonian $H$, but also its powers $H^n$, can be efficiently expressed as matrix product operators, which enables us to calculate with high precision the equilibrium and dynamical properties of quantum lattice models at finite temperatures. Our SETTN method provides an alternative to conventional Trotter-Suzuki renormalization group (RG) approaches, and achieves an unprecedented standard of thermal RG simulations in terms of accuracy and flexibility.
We provide an efficient approximation for the exponential of a local operator in quantum spin systems using tensor-network representations of a cluster expansion. We benchmark this cluster tensor network operator (cluster TNO) for one-dimensional systems, and show that the approximation works well for large real- or imaginary-time steps. We use this formalism for representing the thermal density operator of a two-dimensional quantum spin system at a certain temperature as a single cluster TNO, which we can then contract by standard contraction methods for two-dimensional tensor networks. We ap…
We demonstrate that perturbative expansions for quantum many-body systems can be rephrased in terms of tensor networks, thereby providing a natural framework for interpolating perturbative expansions across a quantum phase transition. This approach leads to classes of tensor-network states parametrized by few parameters with a clear physical meaning, while still providing excellent variational energies. We also demonstrate how to construct perturbative expansions of the entanglement Hamiltonian, whose eigenvalues form the entanglement spectrum, and how the tensor-network approach gives rise to…
Tensor network methods are routinely used in approximating various equilibrium and non-equilibrium scenarios, with the algorithms requiring a small bond dimension at low enough time or inverse temperature. These approaches so far lacked a rigorous mathematical justification, since existing approximations to thermal states and time evolution demand a bond dimension growing with system size. To address this problem, we construct PEPOs that approximate, for all local observables, $i)$ their thermal expectation values and $ii)$ their Heisenberg time evolution. The bond dimension required does not …
Cluster expansions for the exponential of local operators are constructed using tensor networks. In contrast to other approaches, the cluster expansion does not break any spatial or internal symmetries and exhibits a very favourable prefactor to the error scaling versus bond dimension. This is illustrated by time evolving a matrix product state using very large time steps, and by constructing a novel robust algorithm for finding ground states of 2-dimensional Hamiltonians using projected entangled pair states as fixed points of 2-dimensional transfer matrices.
Angular dependence of resonant inelastic x-ray scattering: a spherical tensor expansion
球张量展开进行表示的共振非弹性散射截面的总和的产品的基本光谱与张量涉及波矢量和偏振矢量的入射和散射光子。本文提出的表达式不同于Carra等人的有影响力的文章（Phys. Rev. Lett. 74，3700，1995），因为当将每个光子耦合到其自身时，它没有省略电偶极子和四极贡献之间的干涉项。讨论了球面张量展开的一些特殊情况。例如，各向同性样品的情况下被认为是和横截面表示为只有三个基本光谱的组合的情况下，电偶极子或电四极跃迁的吸收过程中，其次是电偶极子跃迁的发射。这种情况包括无纹理粉末样品的情况，这对应于实验中遇到的最常见的情况。最后，它是预测，一些圆二色性可以观察到各向同性的样品提供的散射光束的圆偏振可以检测。
Ultracold atoms in optical lattices are one of the most promising experimental setups to simulate strongly correlated systems. However, efficient numerical algorithms able to benchmark experiments at low-temperatures in interesting 3d lattices are lacking. To this aim, here we introduce an efficient tensor network algorithm to accurately simulate thermal states of local Hamiltonians in any infinite lattice, and in any dimension. We apply the method to simulate thermal bosons in optical lattices. In particular, we study the physics of the (soft-core and hard-core) Bose-Hubbard model on the infi…
In this paper, we perform a comprehensive study of the renormalization group (RG) method on thermal tensor networks (TTN). By Trotter-Suzuki decomposition, one obtains the 1+1D TTN representing the partition function of 1D quantum lattice models, and then employs efficient RG contractions to obtain the thermodynamic properties with high precision. The linearized tensor renormalization group (LTRG) method, which can be used to contract TTN in an efficient and accurate way, is briefly reviewed. In addition, the single-layer LTRG can be generalized to a bilayer form, dubbed as LTRG++, in both fin…
We investigated the temperature evolution of crystal structure of orthorhombic CoZr3, which is a superconductor with a transition temperature of 4.3 K, by synchrotron and laboratory (CuKα) X-ray diffraction. Uniaxial negative thermal expansion along the c-axis, which is similar to that observed in tetragonal CoZr2, has been observed at a wide temperature range of T = 90-800 K in CoZr3, while a-and b-axis exhibit positive thermal expansion.
We present a method for measuring thermal expansion under tunable uniaxial stresses, and show measurements of the thermal expansion of Mn$_3$Sn, a room temperature antiferromagnet that exhibits a spontaneous Hall effect, under uniaxial stresses of up to 1.51 GPa compression. Measurement of thermal expansion provides thermodynamic data about the nature of phase transitions, and uniaxial stress provides a powerful tuning method that does not introduce disorder. Mn$_3$Sn exhibits an anomaly in its thermal expansion near $\sim$270 K, associated with a first-order change in its magnetic structure. …
