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 infinite pyrochlore and cubic lattices with unprecedented accuracy. Our technique is therefore an ideal tool to benchmark realistic and interesting optical-lattice experiments.
We present a 2D bosonization duality using the language of tensor networks. Specifically, we construct a tensor network operator (TNO) that implements an exact 2D bosonization duality. The primary benefit of the TNO is that it allows for bosonization at the level of quantum states. Thus, we use the TNO to provide an explicit algorithm for bosonizing fermionic projected entangled pair states (fPEPs). A key step in the algorithm is to account for a choice of spin-structure, encoded in a set of bonds of the bosonized fPEPS. This enables our tensor network approach to bosonization to be applied to…
The tensor network representation of many-body quantum states, given by local tensors, provides a promising numerical tool for the study of strongly correlated topological phases in two dimension. However, tensor network representations may be vulnerable to instabilities caused by small perturbations of the local tensor, especially when the local tensor is not injective. For example, the topological order in tensor network representations of the toric code ground state has been shown to be unstable under certain small variations of the local tensor, if these small variations do not obey a loca…
We investigate the thermal physics of a Bose-Hubbard model with Rashba spin-orbit coupling starting from a strong coupling mean-field ground state. The essential role of the spin-orbit coupling $\left(γ\right)$ is to promote condensation of the bosons at a finite wavevector ${k}_{0}$. We find that the bosons display either homogeneous or phase-twisted or orbital ordered superfluid phases, depending on $γ$ and the inter-species interaction strength $λ$. We show that an increase of $γ$ leads to suppression of the critical interaction $U_c$ for the superfluid to Mott insulator transition in the g…
We present a method that generalises the standard mean field theory of correlated lattice bosons to include amplitude and phase fluctuations of the $U(1)$ field that induces onsite particle number mixing. This arises formally from an auxiliary field decomposition of the kinetic term in a Bose Hubbard model. We solve the resulting problem, initially, by using a classical approximation for the particle number mixing field and a Monte Carlo treatment of the resulting bosonic model. In two dimensions we obtain $T_c$ scales that dramatically improve on mean field theory and are within about 20% of …
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…
Tensor networks states allow to find the low energy states of local lattice Hamiltonians through variational optimization. Recently, a construction of such states in the continuum was put forward, providing a first step towards the goal of solving quantum field theories (QFTs) variationally. However, the proposed manifold of continuous tensor network states (CTNSs) is difficult to study in full generality, because the expectation values of local observables cannot be computed analytically. In this paper, we study a tractable subclass of CTNSs, the Gaussian CTNSs (GCTNSs), and benchmark them on…
Motivated by recent transport measurements in high-$T_c$ cuprate superconductors in a magnetic field, we study the thermal Hall conductivity in materials with topological order, focusing on the contribution from neutral spinons. Specifically, different Schwinger boson mean-field ansätze for the Heisenberg antiferromagnet on the square lattice are analyzed. We allow for both Dzyaloshinskii-Moriya interactions, and additional terms associated with scalar spin chiralities that break time-reversal and reflection symmetries, but preserve their product. It is shown that these scalar spin chiralities…
Recently, uncovering the sources of thermal Hall effect in insulators has become an important issue. In the case of ferromagnetic insulators, it is well known that Dzyaloshinskii-Moriya (DM) interaction can induce magnon thermal Hall effect. Specifically, the DM vector parallel to the magnetization direction induces complex magnon hopping amplitudes, so that magnons act as if they feel Lorentz force. However, the DM vector which is orthogonal to the magnetization direction has hitherto been neglected as a possible source of magnon thermal Hall effect. This is because they play no role in the l…
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…
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 …
