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Many-Body Multi-Valuedness of Particle-Current Variance in Closed and Open Cold-Atom Systems

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Many-Body Multi-Valuedness of Particle-Current Variance in Closed and Open Cold-Atom Systems Many-body multi-valuedness of particle-current variance in closed and open cold-atom systems Mekena Metcalf,1 Chen-Yen Lai,2, 3 Massimiliano Di Ventra,4 and Chih-Chun Chien1, ∗ 1School of Natural Sciences, University of California, Merced, Merced, CA 95343, USA. 2Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA 3Center for Integrated Nanotechnologies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA 4Department of Physics, University of California, San Diego, La Jolla, CA 92093, USA. (Dated: November 6, 2018) The quantum variance of an observable is a fundamental quantity in quantum mechanics, and the variance provides additional information other than the average itself. By examining the relation between the particle- current variance (δJ)2 and the average current J in both closed and open interacting fermionic systems, we show the emergence of a multi-valued Lissajous curve between δJ and J due to interactions. As a closed system we considered the persistent current in a benzene-like lattice enclosing an effective magnetic flux and solved it by exact diagonalization. For the open system, the steady-state current flowing through a few lattice sites coupled to two particle reservoirs was investigated using a Lindblad equation. In both cases, interactions open a loop and change the topology of the corresponding δJ-J Lissajous curve, showing that this effect is model-independent. We finally discuss how the predicted phenomena can be observed in ultracold atoms, thus offering an alternative way of probing the dynamics of many-body systems. Quantum fluctuations of the current flowing in a system reservoir couplings. Exact diagonalization [15, 16] is used to provide more information than the average current itself [1– find the closed system many-body ground state, and a Lind- 3]. This fact has been demonstrated in several experimen- blad equation [17–19] is implemented to simulate the time- tal and theoretical studies ranging from quantum dots [4] to evolved density matrix for the open system. nanoscale systems [5, 6], to name a few. In all those studies, In both cases we plot the square root of the quantum vari- two-time correlations of current are measured (or calculated) ance, or the standard deviation δJ, versus average current J away from the average current, and from their spectrum one as the interaction strength varies. We find two distinct classes can infer the type of physical processes at play [2]. On the of the δJ-J Lissajous curve [20–22]: (1) A one-to-one cor- other hand, equal-time density fluctuations at different spatial respondence between δJ and J in the absence of interactions locations have been measured in ultracold atoms [7], revealing except possible isolated points due to energy degeneracy and spatial correlations in quantum gases. (2) a loop or more complicated patterns in the presence of in- However, one could also study quantum variance of the teractions. The Lissajous loop thus establishes a multi-valued current, a property of fundamental importance in quantum relation between δJ and J for interacting many-body systems. mechanics because of the uncertainty principle [8]. This The similarity of results in both closed and open systems sug- equal-time, equal-space quantity has been less explored, pre- gests our findings are model-independent. However, as the sumably because of experimental difficulty in measuring it in interaction vanishes, the Lissajous curve closes abruptly in a current-carrying system. Emergence of cold atoms [9–11] as the closed system case but smoothly in the open system one. new model systems to study a host of phenomena otherwise The difference is due to quantum degeneracy of noninteract- difficult to probe using traditional solid-state materials, makes ing fermions in the closed case versus the open one. Irre- this transport property readily accessible experimentally [12]. spective, these results show that the presence of interactions It is then natural to ask what information the variance would between the particles can be detected as the degree of multi- reveal, and how that information might be useful in character- valuedness of the corresponding δJ-J Lissajous curve. We izing the many-body dynamics. finally discuss how to verify this many-body effect in cold- atom experiments. In this paper, having in mind cold-atom systems as possi- Closed system – As a model closed system amenable to ex- ble experimental verification of our predictions, we study the act diagonalization, we consider a benzene-like ring lattice quantum variance, (δJ)2, of current flowing in a fermionic with two-component fermions (labeled by the spin ) many-body system, and relate this quantity to the average cur- σ = ; hopping between the six sites. The lattice is half-filled with" # rent, J. We consider two experimentally realizable situations: same number of up-spin and down-spin fermions, N" = N#. arXiv:1805.06443v1 [cond-mat.quant-gas] 16 May 2018 the persistent current of a periodic system and the steady-state When a perpendicular, static magnetic field threads the ring, limit of the current in an open system. The latter case is more as illustrated in Fig. 1(b), a persistent current in the ring amenable to an easier experimental realization in ultracold emerges [23–26]. The effect of the magnetic flux is included atoms [13, 14]. by the Peierls substitution [27, 28] in the tight-binding ap- In order to solve the many-body problem exactly (hence proximation, and we model the system by the Fermi-Hubbard beyond mean field), we have considered, as a closed system, model with the Hamiltonian a benzene-like ring lattice with a static magnetic flux to sus- tain a persistent current. The open system is a triple-site lat- ^ X y X H = hijc^iσc^jσ + U n^i"n^i#: (1) tice connected to two particle reservoirs with tunable system- i6=j,σ i 2 Therefore, the current variance reflects the density-density correlations between the sites across which the current flows. Since the system is translationally invariant on a lattice, we will drop the subscript ij. The total current is the sum over the spin components J^ = J^" + J^# ; (5) h i h i h i ^ ^ ^2 ^2 and since we work at half-filling, J" = J# and J" = J# . The total current variance is thenh i h i h i h i 2 2 2 δJ = 2 J^ + 2 J^"J^# 4 J^" : (6) h " i h i − h i The cross-component current correlation is ^ ^ 2 y y 2 y y J"J# = ( hijc^i"c^j"c^i#c^j# hij c^i"c^j"c^j#c^i# h 2i y y− h ∗ 2 y iy − hj j i − hij c^j"c^i"c^i#c^j# + hij c^j"c^i"c^j#c^i# ). As expected, in thehj presencej of interactions,i h the total currenti variance is not generally a simple sum of the current variance from each spin 2 P 2 FIG. 1. A benzene-like lattice with N" =N# =3 fermions threaded component, i.e., δJ = σ δJij,σ. Only in the noninteracting by a magnetic field perpendicular to the plane (illustrated in the in- case with an equal6 population of both species, the Wick set). The hopping coefficient from site i to site j is hij . The onsite decomposition leads to an equality between the total current coupling constant is U and the tunneling coefficient is t¯. Ground- variance and the sum of the current variance from the two state persistent currents (top row) and its standard deviation (bottom spins. row) as functions of the Peierls phase φ for (a) noninteracting and (b) The persistent current J and its standard deviation δJ of U=t¯=1. The time unit T0 = ~=t¯where ~ = 1. non-interacting fermions in the benzene-like lattice are shown in Fig. 1(a). They exhibit periodic structure as the Peierls phase φ increases. We remark that each value of φ corre- y Here c^i (c^i) denotes the fermion creation (annihilation) oper- sponds to a static magnetic flux and the persistent current is y an equilibrium property [24]. Interestingly, there are discon- ator at site i, and n^i =c ^i c^i. The hopping coefficient between ∗ tinuities in both J and δJ, as shown by the isolated points site i and site j is hij and hji = hij. Onsite contact inter- actions between fermions of opposite spins has the coupling in Fig. 1(a). These discontinuities are due to level crossings constant U. in the energy spectrum, which are known in the study of per- We first focus on a system with only nearest-neighbor hop- sistent currents in a ring [23–25]. (See the Supplemental In- iφ formation (SI) for details of the level crossing.) At a level- ping, hi;i+1 = te¯ . Here, t¯is the tunneling coefficient and φ is the Peierls phase. For a uniform magnetic field, φ is pro- crossing point, the values of J and δJ are determined by as- portional to the magnetic field strength [24, 27]. The energy signing each degenerate state equal statistical weight, so the result is consistent with the zero-temperature limit [24]. Sim- unit is t¯ and the time unit is T0 = ~=t¯. We consider the zero- temperature case, where the persistent current is a property ilar discontinuities have also been observed in the superfluid of the many-body ground state [24, 25]. For a moderate lat- velocity and its square for clean superconductors in the Little- tice size, we use the exact diagonalization technique [15, 16] Parks experiment (see, e.g., Ref. [29]). to obtain the ground state and excited states along with their In the presence of the onsite interaction, the persistent cur- energies.
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