Constructing quantum Hamiltonians for simulating and controlling the exotic physics of many-body systems belongs to the most important topics of condensed matter physics and quantum technologies. The main challenge that hinders the future investigations is the extremely high complexity for either their numerical simulations or experimental realizations. In this work, we propose the temperature-adaptive entanglement simulator (TAES) that mimics and tunes the thermodynamics of the one-dimensional (1D) many-body system by embedding a small-size model in an entanglement bath. The entanglement bath is described by the interactions located at the boundaries of the small-size model, whose coupling constants are optimized by means of differentiable tensor network at target temperatures. With the benchmark on 1D spin chains, TAES surpasses the state-of-the-art accuracy compared with the existing finite-temperature approaches such as linearized and differential tensor renormalization group algorithms. By tuning the couplings of the entanglement bath with the temperature fixed, the bulk entropy exhibits similar behavior compared to that obtained by tuning the temperature. Our work provides novel opportunities of engineering the distribution of fluctuations and mimicking the non-equilibrium phenomena in a uniform temperature within the canonical ensemble framework using the optimized boundary interactions.
Constructing quantum Hamiltonians for simulating and controlling the exotic physics of many-body systems belongs to the most important topics of condensed matter physics and quantum technologies. The main challenge that hinders the future investigations is the extremely high complexity for either their numerical simulations or experimental realizations. In this work, we propose the temperature-adaptive entanglement simulator (TAES) that mimics and tunes the thermodynamics of the one-dimensional (1D) many-body system by embedding a small-size model in an entanglement bath. The entanglement bath is described by the interactions located at the boundaries of the small-size model, whose coupling constants are optimized by means of differentiable tensor network at target temperatures. With the benchmark on 1D spin chains, TAES surpasses the state-of-the-art accuracy compared with the existing finite-temperature approaches such as linearized and differential tensor renormalization group algorithms. By tuning the couplings of the entanglement bath with the temperature fixed, the bulk entropy exhibits similar behavior compared to that obtained by tuning the temperature. Our work provides novel opportunities of engineering the distribution of fluctuations and mimicking the non-equilibrium phenomena in a uniform temperature within the canonical ensemble framework using the optimized boundary interactions.
In this work we propose to simulate many-body thermodynamics of infinite-size quantum lattice models in one, two, and three dimensions, in terms of few-body models of only O(10) sites, which we coin as quantum entanglement simulators (QES's). The QES is described by a temperature-independent Hamiltonian, with the boundary interactions optimized by the tensor network methods to mimic the entanglement between the bulk and environment in a finite-size canonical ensemble. The reduced density matrix of the physical bulk then gives that of the infinite-size canonical ensemble under interest. We show that the QES can, for instance, accurately simulate varieties of many-body phenomena, including finite-temperature crossover and algebraic excitations of the one-dimensional spin liquid, the phase transitions and low-temperature physics of the two- and three-dimensional antiferromagnets, and the crossovers of the two-dimensional topological system. Our work provides an efficient way to explore the thermodynamics of intractable quantum many-body systems with easily accessible systems.
Due to the presence of strong correlations, theoretical or experimental investigations of quantum many-body systems belong to the most challenging tasks in modern physics. Stimulated by tensor networks, we propose a scheme of constructing the few-body models that can be easily accessed by theoretical or experimental means, to accurately capture the ground-state properties of infinite many-body systems in higher dimensions. The general idea is to embed a small bulk of the infinite model in an ``entanglement bath'' so that the many-body effects can be faithfully mimicked. The approach we propose is efficient, simple, flexible, sign-problem free, and it directly accesses the thermodynamic limit. The numerical results of the spin models on honeycomb and simple cubic lattices show that the ground-state properties including quantum phase transitions and the critical behaviors are accurately captured by only O(10) physical and bath sites. Moreover, since the few-body Hamiltonian only contains local interactions among a handful of sites, our work provides different ways of studying the many-body phenomena in the infinite strongly correlated systems by mimicking them in the few-body experiments using cold atoms/ions, or developing quantum devices by utilizing the many-body features.
Various successful approaches using numerical annealing and renormalization groups have been conceived to study the thermodynamics of strongly correlated systems where perturbation or expansion theories fail to work. As the process of lowering the temperatures is usually involved in different manners, these approaches in general become much less efficient or accurate at the low temperatures. In this work, we propose to access the finite-temperature properties from the tensor network (TN) representing the zero-temperature partition function. We propose to ``scissor'' a finite part from such an infinite-size TN and ``stitch'' it to possess the periodic boundary condition along the imaginary-time direction. We dub this approach ``TN tailoring.'' Exceptional accuracy is achieved with a fine-tuning process, surpassing the previous methods including the linearized tensor renormalization group [W. Li, S.-J. Ran, S.-S. Gong, Y. Zhao, B. Xi, F. Ye, and G. Su, Phys. Rev. Lett. 106, 127202 (2011)], the continuous matrix product operator [W. Tang, H.-H. Tu, and L. Wang, Phys. Rev. Lett. 125, 170604 (2020)], etc. High efficiency is demonstrated, where the time cost is nearly independent of the target temperature, including the extremely low temperatures. The proposed idea can be extended to higher-dimensional systems of bosons and fermions.
We speed up thermal simulations of quantum many-body systems in both one-(1D) and twodimensional (2D) models in an exponential way by iteratively projecting the thermal density matrix ρ ¼ e -β Ĥ onto itself.We refer to this scheme of doubling β in each step of the imaginary time evolution as the exponential tensor renormalization group (XTRG).This approach is in stark contrast to conventional Trotter-Suzuki-type methods which evolve ρ on a linear quasicontinuous grid in inverse temperature β ≡ 1=T.As an aside, the large steps in XTRG allow one to swiftly jump across finite-temperature phase transitions, i.e., without the need to resolve each singularly expensive phase-transition point right away, e.g., when interested in low-energy behavior.A fine temperature resolution can be obtained, nevertheless, by using interleaved temperature grids.In general, XTRG can reach low temperatures exponentially fast and, thus, not only saves computational time but also merits better accuracy due to significantly fewer truncation steps.For similar reasons, we also find that the series expansion thermal tensor network approach benefits in both efficiency and precision, from the logarithmic temperature scale setup.We work in an (effective) 1D setting exploiting matrix product operators (MPOs), which allows us to fully and uniquely implement non-Abelian and Abelian symmetries to greatly enhance numerical performance.We use our XTRG machinery to explore the thermal properties of Heisenberg models on 1D chains and 2D square and triangular lattices down to low temperatures approaching ground-state properties.The entanglement properties, as well as the renormalization-group flow of entanglement spectra in MPOs, are discussed, where logarithmic entropies (approximately ln β) are shown in both spin chains and squarelattice models with gapless towers of states.We also reveal that XTRG can be employed to accurately simulate the Heisenberg XXZ model on the square lattice which undergoes a thermal phase transition.We determine its critical temperature based on thermal physical observables, as well as entanglement measures.Overall, we demonstrate that XTRG provides an elegant, versatile, and highly competitive approach to explore thermal properties, including finite-temperature thermal phase transitions as well as the different ordering tendencies at various temperature scales for frustrated systems.
Numerical simulations of strongly correlated fermions at finite temperature are essential for studying high-temperature superconductivity and other quantum many-body phenomena. The recently developed tangent-space tensor renormalization group (tanTRG) provides an efficient and accurate framework by representing thermal density operators as matrix product operators. However, the particle number generally varies during the cooling process. The conventional strategy of fine-tuning chemical potentials to reach a target filling is computationally demanding. Here we propose a fixed-N tanTRG algorithm that stabilizes the average particle number by adaptively tuning the chemical potential within the imaginary-time evolution. We benchmark its accuracy on exactly solvable free fermions, and further apply it to the square-lattice Hubbard model. For hole-doped cases, we study the temperature evolution of charge and spin correlations, identifying several characteristic temperature scales for stripe formation. Our results establish fixed-N tanTRG as an efficient and reliable tool for finite-temperature studies of correlated fermion systems.
This chapter is focused on the quantum entanglement simulation approach (Ran et al., Phys Rev B 96:155120, 2017). The idea is to use few-body models embedded in the “entanglement bath” to mimic the properties of large and infinite-size quantum systems. The few-body models are dubbed as quantum entanglement simulators. Generally speaking, the QES Quantum entanglement simulation/simulator (QES) approach consists of three stages: first, determine the effective interactions that give the infinite boundary condition (Ran et al., Phys Rev B 96:155120, 2017; Phien et al., Phys Rev B 86:245107, 2012) by the MPS/PEPS Matrix product state (MPS) Projected entangled pair state (PEPS) methods, such as iDMRG Infinite density matrix renormalization group (iDMRG) in one dimension or zero-loop scheme in two and three dimensions; second, construct the simulators by surrounding a finite-size cluster of the targeted model with the effective interactions; third, simulate the properties of the quantum entanglement simulator by the finite-size algorithms or quantum hardware, and extract the properties of the targeted model within the bulk.